On Invariants and the Theory of Numbers
-_
-. _______l_-
-.__-.__x.y-“-____-.
--
[THE
MADISON
COLLOQUIUM,
1913, PART
I]
On Invariants and the Theory of Numbers b Leonard Eugene Dickson
DOVER
-. .-__--_~
PUBLICATIONS,
INC.,
NEW
YORK
,---.-..----
___-
Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd., 16 Orange Street, .London WC 2.
This Dover edition, first published in 1966, is an unabridged and unaltered republication of the work originally published by the American Mathematical Society in 1914 as Part I of The Madison Colloquium (1913), Volume IV of the Colloquium Lectures. Part II of The Madison Colloquium is reprinted separately by Dover Publications under the title Topics in the Theory of Several Complex Variables, by William Fogg Osgood. This edition is published by special arrangement with the American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02964.
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of
Congress
Manufactured
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Card Number:
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in the United States of America Dover Publications, Inc. 180 Varick Street New York, N. Y. 16614
CONTENTS
INTRODUCTION
.. ... ..................................
LECTURE A
!hiEORY
OF
1
I
s APPLICABLE MODTJLAR FORMS
h’hRIlLNT
TO ALGEBRAIC
AND
l-3.
Introduction to the algebraic side of the theory by means of the example of an algebraic quadratic form in m variables. . . . . . . . . . . . . . . . . . . . . . . . . 4 47. Introduction to the number theory side of the theory of invariants by means of the example of a modular quadratic form.. . . . . . . . . . . . . . . . . . . 6 8-Q. Modular invariants are rational and integral. . . . . . 12 10. Characteristic modular invariants. . . . . . . . . . . . . . . 13 11. Number of linearly independent modular invariants 13 12. Fundamental system of modular invariants. . . . . . . 14 13. Minor r81e of modular covariants . . . . . . . . . . . . . . . 15 14. References to further developments. . . . . . . . . . . . . 15 LECTURE SEMINV -s l-6.
7-10. 11.
OF ALGEBRAIC
II
AND MODULAR
BINARY
FORMS
Introductory example of the binary quartic form. . 16 Fundamental system of modular seminvariants of a binary n-ic derived by induction from n - 1 to n 21 Explicit fundamental system when the modulus p exceeds n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
.. . Vlll
CONTENTS.
12. Anothermethodforthecasep> It . . . . . . . . . . . . . . 27 13. Number of linearly independent seminvariants. . . . 28 14-15. Derivation of modular invariants from seminvariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
LECTURE
III
INVARIANTS OF A MODULAR GROUP. FORMAL INVARIANTS AND COVARIANTS OF MODULAR FORMS. APPLICATIONS
l-4. 5-11. 12. 13. 14. 15.
Invariants of certain modular groups; problem of Hurwitz.................................... Formal invariants and seminvariants of binary modular forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem of Miss Sanderson. . . . . . . . . . . . . . . . . . . . Fundamental systems of modular covariants. . . . . . Form problem for the total binary modular group Invariantive classification of forms. . . . . . . . . . . . . .
LECTURE MODULAR
1-2. 3.
33 40 54 55 58 61
IV
GEOMETRY AND COVARIANTIVE RATIC FORM IN m VARIABLES
THEORY OF A QUADMODULO 2
Introduction. The polar locus. . . . . . . . . . . . . . . . . Odd number of variables; apex; linear tangential equation................................... 4. Covariant line of a conic. , . . . . . . . . . . . . . . . . . . . . . 5. Even number of variables. . . . . . . . . . . . . . . . . . . . . . 6. Covariant plane of a degenerate quadric surface . . 7. A configuration defined by the quinary surface. . . . 8. Certain formal and modular covariants of a conic 9-32. Fundamental system of covariants of a conic. . . . . 33. References on modular geometry.. , . . . . . . . . . . . . .
65 66 69 70 71 72 73 76 98
CONTENTS. LECTURE
ix
V
A THEORY OF PIANE CUBIC CURVES NITH A REAL INFLEXION POINT VALID IN ORDINARY AND IN MODULAR GEOMETRY 1. Normal form of a ternary cubic. . . . . . . . . . . . . . . . . 99 2. The invariants s and t. . . . . . . . . . . . . . . . . . . . . . . . 99 3. The four inflexion triangles. . . . . . . . . . . . . . . . . . . . . 100 4. The parameter 6 in the normal form. . . . . . . . . . . . . 101 5-9. Criteria for 9,3 or 1 real inflexion points; sub-cases 101
__I______ -__
-___-...
_-
INTRODUCTION A simple theory of invariants for the modular forms and linear transformations employed in the theory of numbers should be of an importance commensurate with that of the theory of invariants in modern algebra and analytic projective geometry, and should have the advantage of introducing into the theory of numbers methods uniform with those of algebra and geometry. In considering the invariants of a modular form (a homogeneous polynomial with integral coefficients taken modulo p, where p is a prime), we see at once that the rational integral invariants of the corresponding algebraic form with arbitrary variables as coefficients give rise to as many modular invariants of the modular form, and that there are numerous additional invariants peculiar to the case of the theory of numbers. Moreover, nearly all of the processes of the theory of algebraic invariants, whether symbolic or not, either fail for modular invariants or else become so complicated as to be useless. For instance, the annihilators are no longer linear differential operators. The attempt to construct a simple theory of modular invariants from the standpoints in vogue in the algebraic theory was a failure, although useful special results were obtained in this laborious way. Later I discovered a new standpoint which led to a remarkably simple theory of modular invariants. This standpoint is of function-theoretic character, employing the 1
2
THE
MADISON
COLLOQUIUM.
values of the invariant, and using linear transformations only in the preliminary problem of separating into classes the particular forms obtained by assigning special values to the coefficients of the ground form. As to the practical value of the new theory as a working tool, it may be observed that the problem to find a fundamental system of modular seminvariants of a binary form is from the new standpoint a much simpler problem than the corresponding one in the algebraic case; indeed, we shall exhibit explicitly the fundamental system of modular seminvariants for a binary form of general degree. It will now be clear why these Lectures make no use of the technical theories of algebraic invariants. On the contrary, they afford an introduction to that subject from a new standpoint and, in particular, throw considerable new light on the relations between the subjects of rational integral invariants and transcendental invariants of algebraic forms and the corresponding questions for seminvariants. Again, I shall make no use of technical theory of numbers, presupposing merely the concepts of congruence and primitive roots, Fermat’s theorem, and (in Lectures III and V) the concept of quadratic residues. The developments given in these Lectures are new, with exceptions in the case of Lecture I, which presents an introduction to the theory, and in the case of the earlier and final sections of Lecture III. But in these cases the exposition is considerably simpler and more elementary than that in my published papers on the same topics. The contacts with the work of other writers will be indicated at the appropriate places. Much light is thrown upon the unsolved problem of Hurwitz concerning formal invariants. In many parts of these Lectures, I have not aimed at complete generality and exhaustiveness, but rather at an illumination of typical and suggestive topics, treated by that particular method which I have found to be the best of various possible methods. Surely in a new subject in which most of the possible methods are very complex, it is desirable to put on record an account of the
INVARLWTS
AND
NUMBER
THEORY.
3
simple successful methods. Finally, it may be remarked that the present theory is equally simple when the coefficients of the forms and linear transformations are not integers, but are elements of any Snite field. I am much indebted to Dr. Sanderson and Professors Cole and Glenn for reading the proof sheets.
LECTURE A THEORY
I
OF INVARIANTS APPLICABLE MODULAR FORMS
TO ALGEBRAIC
AND
INTRODUCTION TO THE ALGEBRAIC SIDE OF THE THEORY BY MEANS OF THE EXAMPLE OF AN ALGEBRAIC QUADRATIC FORM IN m VARIABLES, $0 l-3 1. Classes of Algebraic Quadratic Forma.-Let the coefficients of
be ordinary real or complex numbers.
Let the determinant (i,j
D = l&l
(2)
= 1, s-s, m)
of a particular form qm be of rank r (r > 0); then every minor of order exceeding t is zero, while at least one minor of order r is not zero. There exists a linear transformation of determinant unity which replaces this qm by a form* (3)
W12
+
***+cY$.x:r2
(a1$:0,“‘,a,*0).
Indeed, if 811 =l=0, we obtain a form lacking 21x2, . . ., ~1% by substituting Xl
-
811~v312s
+
* * * +
PlA)
xc for x1 and - 21 XI. If BII = 0, Bi6 * 0, we substitute for xi; while, if every /3kk = 0, and 812 =I=0, we substitute x2 + x1 for x2; in either case we obtain a form in which the coefficient of xl2 is not zero. We now have ~1x12+ 4, where crl =j=0 and Cpinvolves only x2, . . . , G. Proceeding similarly with 4, we ultimately obtain a form (3). Now (3) is replaced by a similar form having al = 1 by the for
* Note for later use that each ar and each coeflicient of the transformation ia a rational function of the B’s with integral coefficients. 4
IN-VARIANTS
AND
NUMBER
THEORY.
5
.*
transformation 51 = cQ-*xl’,
xm = a,+&‘,
xj = Xi’
(i=2,
**.,m-1)
of determinant unity. Hence there exists a linear transformation with complex coefficients of determinant unity which replaces qni by (4)
Xl2
+
’ - * +
xi-1
+
Da;n2,
Xl2
+
* * * +
x,2,
according as r = m or r < m. In the first case, the final coefficient is D since the determinant (2) of a form q,,, equals that of the form derived from q,,, by any linear transformation of determinant unity. Hence all quadratic forms (1) may be separated into the classes (5)
C=,D, C,
(DPO,r=O,L--.,m-l),
where, for a particular number D =j=0, the class C,,,,D is composed of all forms qn, of determinant D, each being transformable into (41); while, for 0 < r < m, the class C, is composed of all forms of rank r, each being transformable into (42) ; and, finally, the class Co is composed of the single form with every coefficient zero. In the last case, the determinant D is said to be of rank zero. Using also the fact that the rank of the determinant of a quadratic form is not altered by linear transformation, we conclude that two quadratic forms are transformable into each other by linear transformations of akterminant unity if and only if they belong to the same &a.? (5). 2. Single-palued InvaGz& of q,,,.-Using the term function in Dirichlet’s sense of correspondence, we shall say that a singlevalued function I$ of the undetermined coefficients /9ij of the general quadratic form qfii is an invariunt of q,,, if C#Ihas the same value for all sets pi,, /3$, * * of coefficients of forms q:, qz, - belonging to the same class.* The values v,,,,=, v, of I$ for the various classes (5) are in general different. For example, the determinant D is an invariant; likewise the single-valued funcl
* Briefly,
if + has the same value for all fonm
l
in any class.
6
THE
MADISON
COLLOQUIUM.
tion Tof the undetermined coefficients p. so that
unity Gn = (2 + PYxm, Choose* integers cy, /3
Pw+P2) = 1 (7) (mod PI. Hence the sum of two terms of (6) with the coefficient p can be transformed into a sum of two squares. Thus by means of a linear transformation, with integral coefficients of determinant unity, q,,, can be reduced to one of the forms
(8) a2+ - - - + XT-, + x,2, xl2 + . - - +xE-,+pxr2 Next, let r = m. We obtain initially
(O=
+(a0,
82,
SS,
841
aoj
,
where + is a rational integral function of its arguments. We therefore seek such functions 4 as are divisible by a power of ao, and hence by (11) in which the terms involving only aI cancel. The function of lowest degree is evidently (12)
S4
+ 3~92~= ao21, I = aoa4 - 4ala8 + 3e2.
The next lowest degree is 6 and the function is d&i&
+ e8S2+ (3d + 4e)&3s
The coefficient of d is ao21Ss,that of e is S2 -I- 4st8 = ao2D (13)
(D z a&Q - 6aoamaa + 4aoa2 + 4aPar - 3a12az2).
Henceford= (14)
I&-D=aoJ,
l,e=
- 1, the function is the product of ao2and J=aaoa2a4-a,,aa2+2ala2a~-a~2a4--3.
We do not retain D since it is expressible in terms of the other functions. Eliminating D between (13) and (14), we get (15)
8a2 + 482’ - ao21S2+ ao3J = 0.
Now I and J are aeminvariank Indeed, if a0 + 0, they are expressible in terms of the parameters ao, SC in (3) and hence each has the same value for any form in a class (3) ; while (16)
I = - 513, J = -
514
(if a0 = O),
INVARIANTS
AND
NUMBER
THEORY.
19
so that each has the same value for any form in a class (4); finally, (if a0 = al = 0), I = 3at2, J = - a33 (17) so that each has the same value for any form in a class (5)-(7). From 4 we eliminate S* by means of (12) and then the second and higher powers of S3 by means of (15). Thus S equals N/aok, where N is a rational integral function of (18)
ao, &,
83,
I,
J,
of degree 0 or 1 in S3. If k > 0, we may evidently assume that not every term of the polynomial N in the arguments (18) has the factor a~. Let P(&, 6’3, I, J) denote the aggregate of the terms of N not involving a0 explicitly. We shall prove that, if k > 0, N/aok isthen not a rational integral function of a~, * * a, ~24. For, if it be, P vanishes when a0 = 0. By (11) and (16), the terms independent of a0 in J involve a4, while those in I, SH, S’s do not. Hence J does not occur in P. Then, by (11) and the term 3as2in I, we conclude that I does not occur in P. Thus P is a polynomial in & and S3 of degree 0 or 1 in SS and is not identically zero. By (ll), it cannot vanish for a0 = 0. Under the initial assumption that a0 =j=0, we have now proved that any rational integral seminvariant S equals a polynomial in the functions (18). The resulting equality is therefore an identity. Th-e seminvariank, (18) form a ficndamental system of rational integral seminvariants of the algebraic quurtic form.* They are connected by the relation, or syzygy, (15). 4. Invariantive Characterization of the Classes.-By 0 3, the classes (3) are completely characterized by the seminvariants a~, S’s, S3, I. These with J characterize the classes (4) having = 0, al + 0. For, by (ll), AS’Sand S3 determine al; while, a0 by (16), I and J determine the remaining parameters in (4). * The above proof differs from that by Cayley in minor details and in the obtaining the functions (18) and the vetication that they are seminvariants (the present method being based upon the classes).
method of
20
THE
MADISON
COLLOQUIUM.
The parameter ~42(a2 =I=0) in (5) is determined by I and J, in view of (17). We have now gone as far as is possible in the characterization of the classes by means of rational integral seminvariants S, since the parameters S24,u3, a4 in (5)-(7) cannot be determined Indeed,* for a0 = al = 0, we have by such seminvariants. I!32 = ss = 0 by (1 l), while I and J reduce to powers of a2 by (17). 5. Single+&& Semin~uriunts.-We may, however, construct a single-valued seminvariant which shall determine these outstanding parameters 6’24, a3, ~4. To this end consider the singlevalued function V defined as follows by its values in the sense of Dir&let. We take V = 0 if a0 =l=0 or if al $: 0, and v = S24, us, u4 in the respective cases (5), (6), (7). Since V has the same value for all forms in any class, it is a seminvariant. The seminvariants (18) and V completely characterize the classes (3)-(7) and hence, by 0 2 of Lecture I, form a fundamental system of single-valued seminvariants of the algebraic binary quartic form. 6. Seminvuriants of a Modular Qua&c Form.-Passing to the number theory case, let the coefficients of the quartic form f be integers taken modulo p, where p is a prime exceeding 3. The denominator in (2) is then not divisible by p, so that the classes are again (3)-(7). By the general theory in Lecture I, it is possible to characterize all of the classes by means of rational integral seminvariants, and the latter will then form a fundamental system. In particular, we do not now require the use of such a bizarre function as that used in 0 5. * A proof of this fact, not bssed upon the f&l theorem of 0 3, would afford a better insight into the nature of the last steps in $3 and explain, in particular, why we stopped with I and J and did not consider combinations of the Si of higher than the sixth degree in the u’s. To this end, let S be a seminvariant homogeneous of total degree i, in the a’s, and isobaric, of constant weight W. As well known, 4i 2 220. Thus S cannot have a term u8f or a& and cannot reduce, when a0 = al = 0, to U&S’LP (m > 0), of degree 2 + 2m and weight 21+&n.
INVARIANTS
AND
NUMBER
21
THEORY.
We shall make frequent use of the abbreviation (19)
P, = (1 - a$-‘)(1
- UP--‘) * ** (1 - UP-‘).
Then PI&, Psa, and Ptal are seminvariants* since each takes the same value for all forms in any class. For the classes (5), (6), (7), their values are S24, a3 and a4, respectively. Hence the jive seminvarianti (18) together with P&J, Ptaa and Psar completely ch4zracterize the classes and therefore form a fundamental system of rational integral seminvariank, of the qua& form f with integral coefients taken module p, p > 3.
SEMINVARIANTS OF 7. Fundamental by Induction from
A
MODULAR BINARY [FORM OF ORDER n,
System of Moclulur Seminvariants Derived n - 1 to n.-It is necessary to distinguish the
case in which the modulus p is prime to n from the case in which p divides n. Binomial coefficients for the form are not permissible in the second case and often not in the first case (for example, if n = 4, p = 3, since (3 is then divisible by p). Denote the form by F, = AG”
(20)
+ A#-ly
+ . . . + A,y”.
First, let p be prime to n. For A0 =+ 0, we can transform F,, into a form lacking the second term and having as coefficients the quotients of = nAoA2 - *(n - l)A?, g2 (21)
a3=n2A>A3-(n-2)n&4A++(n-l)(n-2)Ar3,
These may also be obtained from (8) by
by powers of nA0. identifying F, with (22)
fn = aoxn + nalxn-ly
* The first is one-half PIfly4
+
I
._
= Pl(6w9
_...
I.
_-
n(n 2
the discriminant
.__-_-_
-_
1) azxn-2y~+ . . . .
of the semicovariant
+ da&q/ + a@)
and the last two are the seminvariants
-...
***
-.-..
‘.-I-.--
(mod P), of PzT11/s = P2(4asl: + a&
-
i._.._.
.-
.”
-
__..“~..
-_-..,
(mod p).
-
..-
1
22
THE
MADISON
COLLOQUIUM.
For p prime to n, a fundamental system of seminvariants of F, is given by Ao, ~2, . * . , u, together with a fundamental system of the particular form of order n - 1 FL (23)
= PoF& E P&x”-‘+
P~A~x”-~Y+
m- a+PoA,y-’
(mod p>,
where PO = 1 - A#‘+. Indeed, Ao, ~2, . . . , u,, completely characterize the classes of forms F, with A0 + 0. Since yF,+< = F, identically, when A0 = 0, the classes of forms F, with A0 = 0 are completely
characterized by the seminvariants of the fundamental system for F-1’. For example, A0 and PoAl form a fundamental system of modular seminvariants of A@ + Aly (since these characterize the classes represented by A@ and Aly). The corresponding functions for Fl’ = PO&X + PoA2y are PoAl and {1 -
= (1 - A1+)PoA2
(P0A3p+}PoA2
= P1A2
(mod P).
Hence the theorem shows that, if p > 2, (24)
Ao,
=
202
form a fundamental f2, these are (249
2a0,
4Ad2
-
A12,
system of modular
PO&,
of F2.
For
let n = pg.
By
aeminvariants
S2 = adz2 - a12, POal,
8. Order a Multiple
PA2
of the Modulus.-Next,
Plan.
Fermat’s theorem, xP - xyp’ and hence (25)
C#B = AO(xp - zyr’)*
is unaltered modulo p by any transformation (1). Hence if, for each value of the seminvariant Ao, we separate the forms (26)
INVARIANTS
AND
NUMBER
23
THEORY.
into classes under (l), multiply each form by y and add 9, we obtain the classes of forms F, for this value of Ao. Hence, iif n & o?iv&ible by p a fundamental system of modular seminvariants of F, is given by A0 and a fundamental system for F-1.
For example, if n = p = 2, FI
=
(AI
+
Adz
+
A2y
can be transformed into z or A2y by (l), according as Ao + AI = 1 or 0 (mod 2). Adding C$= A&? - xy) to xy and Azy2, we obtain representatives of the classes of forms F2. Hence the 6 classes are completely characterized by the seminvariants A0 and those (0 7) of Fl, and hence by Ao,
(27)
AI,
J = (1 + Ao + A&&.
9. Seminvarianf.s of the Binary algebraic forms f3 are a&
(2%
3ady+
(29)
+
Cub&~ Form.-The
3a0C1S2a$
+
ia14S13ti,
classes of
a0-2S3y3,
3a2xy2,
a&,
where the S’s are given by (8) and (101). The discriminant D of fa is given by (13). As in 0 3, ao, S2, &, D form a fundamental system of seminvariants of fa; they are connected by the syzygy (13). Henceforth, let the coefficients of fa be integers taken modulo p, the excluded case p = 3 being treated in 0 15. If p > 3, the classes are again (28) and (29), and a fundamental system of seminvariants is given by ao, 82, 83, D, Pm, Pm. (30) It is instructive to compare this result with that obtained by the method of 0 7. Forming the functions (24) for f2’
=
Pofs/2/ = 3po%z + 3P!lwY
+ Poa3Y2
(mod PI,
and deleting the factor 3 from the first and second, we get* P0a1,
6 = Po(4ala8
* They characterize
-
3a22) =
h&3,
the classes (29) of ja with
Pla2,
P2a3.
a0 = 0 and may be so derived.
24
THE
MADISON
COLLOQUIUM.
Hence, if p > 3, these four functions and a~, St, Sa form a fundamental system of modular seminvariants of fa. We may drop POUI since P--s PQ&~ Sa = f 2Pgz$ = f 2Poal (31) (mod PI. Hence a fundamental system of seminvariants of $3 for p > 3 is
Pla2, P2a3. It is easy to deduce 6 from the old set (30), and D from this new (32)
SZ, SS, 6 =
a0,
PO&~,
set.* Finally, let p = 2. By 4 7, a fundamental system of seminvariants fort8 is given by ao, Sz, S3 and a fundamental system for f2’. The latter system is derived from (27) by replacing Ao, Al, A2 by Poa~, Poaz, Pea,, and hence is (I+
(1 +
a0h
We may drop (1 +
UO)UI
(1 + a0>(1 = (1 + a&!!&.
10. The Binary Qua& Form. Fs = Ad
+ al + aph.
a0la2,
+ (Ao +
For p = 2, we have
A2)+/
+
Asq.i2
+
A&,
whose seminvariants are obtained from those offs at the end of Q 9. They with A0 give a fundamental system of seminvariants of Fd:
Ao, AI,
AlAa + Ao +
AlA, + AlAs(Ao + An equivalent
(33)
Az),
K
fundamental
Ao, AI, Au44
* D = a~“-‘(&
=
Cl+
A2, A)(1
+
AM, Ao
+
A2
+
AaM-
system is? A2 + At,
+
Cl+
AoA2
+ 4W)
+
(1 + AI) At, AA,
- 6Sa
K.
(mod PI.
For, if a0 ( 0, then 6 E 0 and this relation follows from (13); while, if uo = 0, D = alWl~ = a13 = - S& Conversely, 6 can be expressed in terms of the functions (30). The above relation givea s&. The product of this by SP iscongruentto6if&+O. Also~=Oifa~+O. Thereremainsthecaee in which SI = 0, a0 = 0, whence al = 0, 1 = - 3a2 = - 3(Plar)*. t Annals of Mat-h, ser. 2, vol. 15, March, 1914. I there give also a completeset of linearly independentinvaxiants and of linear ~~variants
INVARIANTS
AND
NUMBER
25
THEORY.
For p > 3, fs’ is obtained from f3 by replacing ao, al, a2, as by 4alP0,
&PO,
&P0,
a4P0,
respectively. Making this replacement in the second set of seminvariants of f3 in Q 9, we obtain P@I, which may be dropped in view of (31), and the last five functions (34). Hence,for p > 3, a fundamental
system of modular
(34)
Sat
a0,
82,
84,
P&13,
seminvariants
PO&~,
P824,
of f4 ti given by P2aa,
Paa4.
Here the three i!? n.-Instead of 11. Explicit Fundamental employing the above step by step process, we can obtain directly a fundamental system of modular seminvariants of fn when the modulus p exceeds the order n of the binary form (22). Consider a particular f,, in which ok is the first non-vanishing coefficient: II n ap?&-'Yi (ak $: 0). 2=x =(.> 2 To this we apply transformation (1) and obtain
where we have replaced j by I-
i and set
Take k < n and give to t the value (2). n ckl 0 Akz = ((k + ~)a,)zwky ak
(35)
Thus
26
THE
MADISON
COLLOQUIUM.
In particular, gkk
=
1,
uk,,+l
=
0,
(roz
g (- 1)“” (r)
=
aO+zl%i,
the last being the algebraic seminvariant designated earlier by SZ. It is obtained from the expansion of (a0 - al)’ by replacing a single a0 in each term by ai. Except for a numerical factor not divisible by p, (TkZ(for 0 < k < I - 1) equals the SkZ in (10) and in (38) below. The classes ck of forms f,, in which ak is the first non-vanishing a are distinguished from each other by the value of a, if k = n, and if k < n by the values of the parameters ak, UkZ (I! = k -i- 2, * * *9 n). Employing the notation (19), we shall verify that Pk-lak and PklCTkZ are modular seminvariants of f,,. They vanish for a form Cj (j 2 k - 1) since then 1 - a,“’ = 0. For ck, they reduce to the parameters ak and rkZ of that class. For a0 = 0, se., ok = 0, the first is zero and the second is the expression for (Tkl when ak = 0, whose non-vanishing terms (given by i = k and i = k + 1) are constant multiples of a&:; but uk+l is constant for any class Cj (j > k). It foilows also that the parameter awl in a class ck+l is determined by the seminvariants PklakZ (I = k -I- 2, k -I- 3), provided k + 3 s n. But u,~ and a,, not so determined, are found from Pklak (k = n - 1, n). Hence a fundamental system of modular
seminvariunts
of fn, for p > n, is given by
a0,
n),
(k = 1, .*., n - 2; I = k-t- 2, a.., n),
PJ+lukZ
(36)
(I = 2, - - -,
Qoz
Pn-2%-l,
Pn-l&b.
For n = 2, 3, 4, these are (24’), (32), (34), respectively, except for the difference of notation indicated above. For n = 5, we system of modular seminvwiants of fat see that a fundumental for p > 5, ti a0,
82,
83,
S4,
SK,
PO&,
PO&~,
PO&S,
(37) pls24,
.
..---~--._--------.-
PlSPE,
p2&51
P$4,
-. _..”-..-”.-........-----.----__^_I
p4a5,
----_
--.--
---
- -~- J
INVARIANTS
AND
NUMBER
27
THEORY.
in which the symbols are defined by (@-(lo),
(19) and
SSI= ao4a3- 5agsa1a4+ 10a02a12as- lOadQa2 + 4a16, &S = 16a13as- 40a12a2a4+ 40ala22a3 - 15h4, s 45a2a3ad+ 20a33, 25 = 27a32a3-
(33)
S35 = 8a3a5 - 5aa2. 12. Another Methodfor the Case p > n.-We may formulate the method of 0 7 so that it shall be free from the induction process. The classes of forms (23) with P,& + 0, and hence the classes of forms F, with A,, = 0, AI + 0, are characterized by the seminvariants given by the products of PO by the functions Q’, * . . obtained from ~2, aa, . . ., u-1 by increasing the subscript of each Ai by unity and replacing n by n - 1; indeed, P,2 = PI (mod p). When the process of deriving (23) from (20) is applied to (23), we get r:-,
= 11 - (PoA1)~‘lF:-,/y
= (1 - Ar1)P8,/y2
= P1F,/y2 = P~A~x”-~ + P1A3x”~y
(39)
+ - - . + P~A,Y”~
(mod PI.
The class of forms (39) with PlA3 + 0, and hence the classes of forms F, with A0 = AI = 0, A3 + 0, are characterized by the seminvariants given by the products of PI by the functions ua”, . . . obtained from IJ~‘, . . . , u-2’ by increasing the subscript of each Ai by unity and replacing n by n - 1. Finally, we obtain P,3A,,-lx + P,3A,y, characterized by the seminvariants ,+r and P-IA,. The earlier P+IAs may be dropped Pa-2A (0 11). For example, if n = 3, p > 3, the fundamental system of F3 is
Ao,
~2,
~3,
Pouz’
= Po(4AA2
-
A22),
Pd2,
P2Aa.
Changing the notation from F3 to f3, we see that ~2’ becomes 3(4ala3 - 3az2), so that the resulting seminvariants are (32). We may of course apply the method directly to fa; in S2 we replace a~, al, a2 by 3aI, #a2, a3 and obtain $(4ala3 - 3az2).
28
THE
MADISON
COLLOQUIUM.
Again, to find a fundamental system of f4 for p > 3, we take ao, SZ, &, Sq and the products of PO by the functions $S’lz and 16S14 obtained from Sz and S’s by replacing ao, al, a2, aa by 4al, 6 . 6a2, Q * 4a3, ad; then the product of PI by the function 2524 obtained from Sz by replacing a~, al, a2 by 6~2, 4 * 4as, ah; then P2a.3 and Paah, to characterize P2(4asx + say). We again have (34). 13. Number of Linearly Independent Seminvariants.-Let p > n and employ the notations of 0 11. In the classes C’k(k < n). Akk = ak (;> has p - 1 values, f&+1 = 0, while &k+?, * * a, Akn take independently the values 0, 1, . . . , p - 1. In the classes C,, a, has p values. Hence there are r-1
p+k~O(p-l)Pn-~l=P+P”-l
distinct classes of forms f,,. Thus by 0 11 of Lecture I, there are exactly p” -I- p -
1 linearly
independent
modular
seminvariants
of
f,whenp>n. I)ERIVATION*
OF MODULAR
IN-VARIANTS
FROM
SEMINVARIANTS,
$5 14-15 14. Invarianti Any polynomial
of the Binary
Form.-First, let p=2. (27) is a linear function of
Q?.idratic
in the seminvariants
Aoh, J, AoJ = AoAIAz, 1, Ao, A, since (A0 + A1)J = 0. Since there were six classes, these six seminvariants form a co.mplete set of linearly independent seminvariants. Now a seminvariant is an invariant if and only if it is symmetrical in A0 and AZ. But I=(l-Ao)(l-Ar)(l-A+(l-Ao)(J+l+Al)
Thus 1, AI, AoJ and I are invariants. --___
(mod 2). By subtracting constant
*While this method is usually longer than the method of Lecture I, it requires no artificea and makes no use of the technical theory of numbers. Moreover, it leads to the actual expressions of the invariants in terms of the semiuvariants of a fundamental system, thus yielding material of value in the construction of covariants.
-
-_ ”
)----~.
.-.-.--.----“-l_-~
-.
~~.-.
INVARIANTS
AND
NUMBER
29
THEORY.
multiples of these four, any seminvariant can be reduced to cAO+ &A,& which is an invariant only when identically zero. Hence 1, AI, A&AZ and I form a complete set of linearly huhpendent invariants of Fz m&do 2. Next, let p > 2. The discriminant of f2 is D = 5’2. Any polynomial in the four fundamental seminvariants (24’) is a linear function of a2Dj,
Poa14,
Plazi
(i, j = 0, 1, . . ., p - l),
since the product of Peal or Plas by a0 is zero, that of PlaP by Pea, or D is zero, while DPoal = - Pals. Further, PO = 1 - aoF’,
Po[Dj -
(-
aoPIDi
Pl = PO - P&-I,
a12)j] = 0, s Di _ (-
l)ipoa12i,
modulo p. Hence any seminvariant is a linear function of P-l , aoiDj (i = 0, 1, . . ., p - 2; j = 0, 1, . . ., p - l), (40) ao (k= 1, ..*,p-1). Pod, Plazk The number of these is p2 + p - 1. Hence (0 13) they form a complete set of linearly independent modular seminvariunt8 of f2 for p > 2. The invariant A = A1 in Q 6 of Lecture I becomes for two variables (41)
where p = (p - 1)/2. (4)
ao@)) (l-
A= {ao“+a~“(l-
DP-I) = ao’(l-
By the expansion of Dpl,
Dp1)+PIa2”, we get*
A = (sop + as“) ( 1 - 2 ao(,2;a12@-2i> . t=o
* Transactions of the American Mathematical Society, vol. 10 (1909), p. 132. To give a direct proof of the identity of the final expression (41) and (42), note that the product of the ikal factor in (42) by D equala a&t - (a&r+1 algebraically, 80 that the product AD is divisible by p. But the product of (41) by D is evidently divisible by p. It therefore remains only to treat the case D = 0. Replacing aI* by acat, we see that the Cnal factor in (42) becomes Hence (41) and (42) are now identical if 1 - G( + l)ao”aac. aOfiaP (ao* - azw) = 0 But, if aDal + 0, aOpaylr = a+
(mod PI. = 1, aOp = a2c = * 1.
30
THE
MADISON
COLLOQUIUM.
Since (42) is therefore a seminvariant and is symmetrical in uo and a2 and since the weight of every term is divisible by By (41), P - 1, A is an absolute invariant. A2 = up (1 - 0”1) + P&2P, (43)
LP+ LP-l-
(1 - a#--1)~“1 = PO@-‘,
l= - IO, IO= (l-ua,p-‘)(l-ualq(l-uP).
Hence also IO is an absolute invariant. lo=l-u~~-‘-~~oa~~‘-P~u2~‘,
A,
Subtracting Dj
o’=O,
multiples of 1, . . . . p-l),
we may reduce any seminvariant to a linear function of the expressions (40) other than Pru$‘+, P&‘, Dj (j = 0, * * *, p - 1). The resulting linear function L is not an invariant. For example, ifp= 3, it is (a, . . ., f constants).
L=u4z02+bu0+cu0D+du~2+ePoul+fP~~2
Interchange
a0 and 1x2,and change the sign of al.
We get
UU~~-I- but + cu2D + du2D2 + (1 - a~~)(ful - euJ.
This is to be identically congruent to the invariant L. Taking = 0, we see that e = f = a = b = 0, c = d. Then L a2 = cuoaz(uo+ u2) + CCZ~~U~~U~ is not symmetric in a0 and ~22. Hence L = 0. For any p, a like result may be proved by considering separately the terms of L of constant weights module P - 1. Hence in accord with 0 11 of Lecture I, a complete set of linearly independent invariants of f2, for p > 2, i-9 given by IO, A and the powers of D. In place of DO = 1, we may use A2, in
view of (43). 15. Invariants of th Binary Cubic Modulo 3.-A fundamental system of seminvariants of F3 modulo 3 is given by A0 and a fundamental system of F2 = Ad
+ (Ao + A2)zy + A3y2.
Hence, by (24), a fundamental Ao,
AI,
system for F3 is given by
t = AlA3 -
(Ao +
D = (1 -
A12)[1 -
A212,
(Ao +
(1 - A12>(Ao + A2j2]A3.
Az),
INV.QRIANTS
AND
NUMBER
THEORY.
31
In place of the fourth and third we may evidently use x=
(1 -
u=
h2M2,
&43+&A*-
A?A*2 = t + A02 + x2.
Here u is the discriminant of Fa for p = 3. By 0 13 there are 11 classes of forms i;~. Hence, by Q8, there are 3 * 11 classes of forms R. Thus there are exactly 33 linearly independent seminvariants of Fa. Since AJ = A@ = 0, ax = AOF, p(u + AI)*) = 0, p(x + Ao) = 0,
(1 - &*)a = A&
modulo 3, any polynomial in the seminvariants Ao, Al, u, X, p of the fundamental system is congruent to a linear function of
(44) Ao’Alj, Aoiak, Ao’Ad,
Ao’Xk, A&”
(i, j= 0, 1, 2; k= 1, 2).
Hence these 33 functions form a complete set of linearly pendent seminvariants of Fa. The seminvariants
inde-
P = 1 - A? - X2 = (1 - A12)(1 - As*), (45)
IO = (I - Ao*)(P - 4) = & (1 - Ai%
E = A&(u
- a*) + Aop = A,,As(A,,Az.- AlAa+ AI*- AZ*)
are seen to be invariants as follows.* The weights of the terms of each are all even or all odd. Moreover, under the substitution (A&(AIA2), induced upon the coefficients of Fs by the interchange of z and y, the functions u, P and 10 are unaltered, while E is changed in sign. Hence u, P, 10 are absolute invariants, while E is an invariant of index unity. We now have 7 linearly independent invariants (46) Noting that (47)
IO, E,
E,
us 9,
P,
1.
E = Ao2p2+ A,,*(u - f + X2) - A&
*Or by general theorems, Transactions of the Ammixn Mathematicd Society, vol. 8 (X107), pp. 206-207. Note that E ia the eliminant of Fa = 0, 2’ = 2,2/’ 5 g (mod 3).
32
THE
MADISON
COLLOQUIUM.
we may employ the functions (46) to delete from (44) 4,
AOP, A OPJ 2 2 QJ o-2, x2, 1
in turn (no one of these terms being reintroduced at a later stage). There remain 11 seminvariants of odd weight (48)
A&Al,
AOiAlU,
AoiAd,
p,
A&
(i = 0: 1, 2),
and 15 of even weight (49) A,,, Ao2, AoiA12, Aou, Ad,
A,,%, A,,W,AoIX, AoX2, Ao2xz,Ati2.
Now the weight and index of a seminvariant of F3 modulo 3 are both even or both odd.* A linear combination of the functions (48) which is changed in sign by the substitution (AoAs) (AIA2) is seen to be identically zero (it suffices to set Aa = 0, A2 = 0 in turn). A linear combination of the functions (49) which is unaltered by that substitution is seen similarly to be identically zero. Hencet a complete set of hearty independent invariants of Fa modulo 3 is gigen by (46). * When the sign of y is changed, a seminvariant is unaltered or changed in sign according 88 ita weight is even or odd. t Another proof, using the clof Fa under the group of all binary linear transformations of determinant unity module 3, and involving a use of more technical theory of numbers, is given in Tranaadims of the Ammicun Mathe ctz&kd Society, vol. 10 (lQOQ), pp. 144-154. The case of any modulus p ia there treated.
LECTURE
III
INVARIANTS OF A MODULAR GROUP. FORMAL INVARIANTS AND COVARIANTS OF MODULAR FORMS. APPLICATIONS
INVARIANTS
OF CERTAIN
MODULAR
GROUPS,
$0
14
1. Introduction.-Let G be any given group of g linear homogeneous transformations on the indeterminates x1, . . ., x,,, with integral coefficients taken modulo p, a prime. Hurwitz* raised the question of the existence of a finite fundamental system of invariants of G. For the relatively unimportant case in which g is not divisible by p, he readily obtained an affirmative answer by use of Hilbert’s well known theorem on a set of homogeneous functions, but emphasized the difficulty of the problem in the general case. In 0 5 I shall consider the relation of this question to that of modular covariants and formal invariants of a system of forms and incidentally answer the above question for special groups of orders divisible by p. I shall, however, first present a simplification of my own work on the total group. Its invariants are universal covariants, i. e., covariants of any system of modular forms (§ 13). It was from the latter standpoint that I was led to the subject of invariants of a modular group independently of Hurwitz’s paper, in the title of which the word invariant does not occur. 2. Invariants of the Total Binary Group.-Consider the group G of all modular linear homogeneous transformations with integral coefhcients of determinant unity: (1)
=
+ dy,
9’ =
+ ey, be - cd = 1
(mod P>. The term point will be used in the sense of homogeneous coordinates, so that (2, y) = (ax, ay), while (0, 0) is excluded. X’
by
* Archiv elm Mathematik
cx
und Phytik,
(3), vol. 5 (1903), p. 25.
33
34
THE
MADISON
COLLOQUIUM.
We do not restrict the coordinates to be integers, but permit their ratio to be a root of any congruence with integral coefficients modulo p. A point is called real if the ratio of its coordinates is rational. A point (x, y) is invariant under a transformation (1) if 5’ = px, y’ = py, or (2)
(b - p)x + cly = 0, cx + (e - p)y = 0
If these congruences hold identically dscco, and the transformation (3)
(mod
PI.
as to x, y, then
bzes+l
(mod
P)
is one of the transformations
2’ G f 2,
y'
E f
y
(mod
P),
which leave every point invariant. A special point is one invariant under at least one transformation (1) not of the form (3). There are p(p2 - 1) transformations (1). We shall assume in the text that p > 2 (relegating to foot-notes the modifications to be made when p = 2). Then there are two transformations (3). Hence any non-special point is one of exactly* w = &p(p” - 1)
(4)
conjugate points under the group G, while a special point is one of fewer than o conjugates. Let (x, y) be a special point and let (1) be a transformation, not of the form (3), which leaves it invariant. Thus the conThe determinant of their gruences (2) are not both identities. coefficients must therefore be divisible by p. Hence p is a root of the characteristic congruence (in which (Y = b + e) (5)
P2 -
0p+1=0
(mod
PI.
First, suppose that (5) has an integral root p. For this value of p, one of the congruences (2) is a consequence of the other, and the ratio x : y is uniquely determined as an integer modulo p. * For p = 2, o is to be replaced
by 2(2* -
1) = 6.
INVARIANTS
AND
NUMBER
35
THEORY.
Hence only real special points are invariant under a transformation [other than (3)] whose characteristic congruence has an integral root. Moreover, all real points are conjugate under the group G. Indeed, x’ = bx,
y’ = x + b-ly,
and
x’ = - y,
y’ = x
replace (1, 0) by (b, 1) and (0, 1) respectively. Hence if an invariant of G vanishes for one of the real points, it vanishes for all and has the factor p--l (6) L = Ygcx - uy) = xpy - xyp (mod PI, the congruence following from Fermat’s theorem. Obviously, any transformation of G replaces a real point by a real point, and therefore L by kL. The constant k is in fact unity and L is an invariant of G. Indeed, for (7)
x=aX+bY,
y=cX+dY
where a, se., d are integers of determinant
(mod P>,
A = ad - bc,
Next, suppose that (5) has no integral root and therefore two Galois imaginary roots. By (2), each root p uniquely determines a point (x, y) with y =!=0. We may therefore take y = 1, whence cx = p - e. The resulting two special points are therefore imaginary points of the form (rp + 8, l), where r and a are integers modulo p, and r is not divisible by p. The imaginaries introduced* by new transformations are expressible linearly in terms of this p. Indeed, (2~ - a)” = A, where A = (y2 - 4 is a quadratic non-residue of p (i. e., is not the remainder when the square of any integer is divided by p). Thus A = a2v, where v is a fixed non-residue of p. Hence the roots of all congruences (5) having no integral roots are expressible in the form k + 16, where k and 1 are integers. * There
tie no new onea if p = 2, since a = 0 (mod 2).
36
THE
MADISON
COLLOQUIUM.
Hence the special points invariant under transformations whose characteristic congruences have no integral roots are all of the form (rp + S, l), where T and a are integers, r not divisible by p, while p is a fixed root of a particular one of these congruences (5). We next show that these p2 - p imaginary special points are all conjugate under the group G. It suffices to prove that they are all conjugate with (p, I), which is invariant under 2’ s cyx - y, Now transformation
y’ = 5.
(1) replaces (p, 1) by (R, l), where
We are to prove that there exist integers b, c, d, e satisfying be - cd=
(9)
1
(mod P>,
such that R = rp + s, where r and s are any assigned integers for which r is not divisible by p. Denote the second root of (5) by p’ and multiply the numerator and denominator of R by cp’ + e. Using (9), we get
j&X!? Q
’
N = bc + de + &a,
q = 8 -I- ace -I- e2.
We first show* that we can choose integers c and e such that q E i (mod p), where i is any assigned integer not divisible by p.
If i is a quadratic residue of p, we may take c = 0. Next, let i be a quadratic non-residue of p. Taking c + 0, e = kc, we have q = St(k), f(k) = 1 + ak + k2. Now f(k) 3 f(K) if and only if K = k or K = the p - 1 values of k other than - 42 give valueoff( Thusfork = 0, . . ..p - l,f(k) incongruent values, no one a multiple of p * If p = 2, then a = 0; taking
- a! - k. Hence by pairs the same takes 1 + *(p-l) [since (5) has no
c = 1, e = 0, we have p = 1 = i (mod 2).
INVARIANTS
AND
NUMBER
THEORY.
37
integral root], and consequently a value which is a quadratic non-residue of p. Then, by choice of c, q can be made congruent to any assigned non-residue. Having made q = i (mod p) by choice of c and e, we proceed to choose integral solutions b and d of (9) such that N will be congruent to any assigned integer j. If c = 0, so that e + 0, we take d = j/e. If c + 0, we eliminate d from N by use of (9) and obtain N+bq-
e-m),
q=c2+ace+e2.
Since q $ 0, we may make N = j by choice of b. We have therefore proved that there are exactly p2 - p imaginary special points, viz., (rp + s, l), r + 0, and that they are all conjugate under the group G. Hence any invariant of G which vanishes for an imaginary special point has the factor
(10) Indeed, the numerator of the first fraction vanishes for x= rp+s, y = 1, since (rp + 8)PZ = rppD + 8,
p+ = p
(mod ~1,
the last congruence* being a case of Galois’s generalization of Fermat’s theorem. We have divided out L, which vanishes for the real points (8, 1) and (1, 0). Since any transformation of G replaces one of our imaginary points by another, it replaces Q by kQ. The constant k is in fact unity and Q is an invariant of G. Indeed, (8) holds if .we replace the exponents p by p2. Hence the quotient Q is invariantt under all transformations (7). * It may be proved by noting
that (5) implies
(d - ap + 1)p = p*’ - apP + 1 = 0 (mod PI, 80 that pp ia the second root of (5). By the same argument, (pp)p is a root, distinct from p*, and hence identical with p. t I gave the notation & to the invariant (10) since it is the product of all of the binary quadratic forma z* + +. . which are irreducible modulo p. Indeed, the latter vanishes for two point8 of the form (rp + s, 1) and (rp’ + s, l), where p and p’ are the roots of (5) and T, s are integers, r + 0, and conversely.
38
THE
MADISON
COLLOQUIUM.
We are now ready to prove that any rational integral invariant I, with integ+ral coejbients, of the group G ti a rational integral function of L and Q with integral coe+nb. After removing possible factors L and Q, we may assume that I vanishes for no special point. If I is not a constant, it vanishes at a point (c, d) and hence at the w distinct points conjugate with (c, d) under the group G. The invariants* P+l
tip--l)
q=QT,
(11) are of degree o.
l=Lz
The constant 7, determined by
q(c, d) + 7 * Z(c, d) = 0
(mod PI, is a root of a congruence of a certain degree t with integral coefficients and irreducible modulo p. Now q + 71 is a factor of I. Since q, I and I have integral coefficients, I has also the factors (12)
q+rp1,
q+&,
‘-a)
q+Fl.
For, by Galois’s theorem mentioned above, 7,
7P*
TP’
f
. . .,
TP”
are the roots of our irreducible congruence of degree t. Since the conditions which imply that q + zl shall be a factor of I are congruences satisfied when z = r, they are satisfied when z = r@. Hence if we multiply q + 71 by the product of the invariants (12), we obtain an invariant T with integral coefficients module p. Since L and Q have no common factor, no two of the functions q + rl and (12) have a common factor. Hence T is a factor of I. Proceeding in like manner with I/T, we arrive finally at the truth of the theorem.? 3. Invarianti of Smaller Binary Groups.-We shall later need the theorem that a fun&amentalsystem of rational integral invarianta * If p = 2, we omit the divisor 2 in the exponents. t Proved less simply in !&ansactions of the Avnericun Mathmatkal Society, vol. 12 (1911), p. 1. Still simpler ia the proof that various coe&icients of an invariant are zero, QuuT~~~ Journd of Mathematic+ 1911, p. 158.
INVARIANTS
AND
NUMBER
39
THEORY.
of the group composed of the p powers of tk tran$oormation 2’ = 2 + y, . G given by y and X, where
y’ = y
(13)
(14)
x = x(x+y)(x+ay)*
* ’ (x+p-ly)
(mod
P>
=xp-qpl
(mod
PI-
Now (1, 0) is the only special point, being the only point unaltered by (13) or its kth power, k < p. Hence an invariant not having a factor y or X vanishes at imaginary points falling into sets each of p points conjugate under our group. As at the end of 9 2, the invariant is a product of factors yp + A so related that the product equals a polynomial in yP and X with integral coefficients. Other results will be merely stated, since they are not presupposed in what follows. Within the group G of all transformations (l), any subgroup of order a multiple of p is conjugate with one containing (13) and transformations exclusively of the form 2’ = tx + ly,
(15)
y’ = t-ly
(mod
P),
and having y and X as a fundamental system of invariants.* The invariants of any subgroup whose order is prime to p have been f0und.t 4. Invariants of the Total Group on m Variables.-The QP=+-l
. . .
QP”
g&P=-’
. . .
. xlP--”
W-3
L =
.
. . . .
.
&&PC’ .
.
. . .
&P
Xl
a-*
&
.
.
&P” .
.
p*+l %??+’
. 9
xlP
functions
Qm= :;p-, : : : hp.-, . . . . . 2
. ..
%I
are seen, by a generalization of (8), to be invariants of index 1 and 0 respectively of the group I?, of all linear homogeneous transformations on x1, 9. . , G with integral coefficients modulo p. + BuL?e.t~n of the America n Mathematical Society, vol. 20 (1913), pp. 132-4. t American Journal of Mathentatics, vol. 33 (1911), p. 175.
40
THE MADISON COLLOQUIUM.
Since L,,, is an invariant of I?,,,and has the factor 51, it follows from an examination of its diagonal term that* (17)
L
=
ii k=l
2 y=o
(ac
+
%+1ac+1+
- * * +
G&J
(mod
P>,
in which occurs one of each set of proportional linear forms modulo p. A like proof shows that the numerator of Q,,,* is divisible by each of the linear functions (17) and hence by L,, modulo p. Making use of the theorem in 0 2, I have proved by inductiont that the m invariants L,,,, &,,,I, +. ., &,,,,,,-I are independent and form a fundamental system of rational integral invariants of I?,. A fundamental system of invariants of the group of all modular linear transformations on two sets of two cogredient variables has been obtained very recently by Dr. W. C. Krathwohl in his Chicago dissertati0n.S FORMAL INVARIANTS
5. Formal Modular
AND SEMINVARIANTS
00 5-13 Invariants.-Consider
OF MODULAR
FORMS,
a binary form
f(x, Y) = a0f + as?% + - - - + c~ry: in which x, y, ao, * a0, a, are arbitrary variables. The transformation (7) with integral coefficients, whose determinant A is not divisible by the prime p, replaces f by a form c$(X, Y) = AoX’ + AlX’-‘Y
+ * * - + k&Y’,
in which (18)
Aa = f(a, c), A1 = Pa)-%a0 + . . a, . . .,
A, = f(b, d).
A polynomial P(ao, * . *, a;) with integral coefficients is called a formal invariant modulo p of index X off under the transformaSociety, vol. 2 (1896), * E. H. Moore, BdZetin of the Americu n Mathematical p. 189. His proofs do not use the iuvariantive property. A like remark ia true of the proof that the product (17), iu the case zn = 1, is congruent to a determimmt of order m - 1, then obviously equal to L,,,, by R. Levavasseur, M.z%wirea de l’Acm%mie des Seienees ok Twbuse, ser. 10, vol. 3 (1903), pp. 39-48; compka R-endw, 135 (1902), p. 949. t Transadms of the America n Mdhenatical Society, vol. 12 (1911), p. 75. $ American Journal of Mat~ics, October, 1914.
TNVe4RIANTS
tion
AND
NUMBER
41
THEORY.
(7) if
(19)
P(Ao,
Al,
* * -9
A,) = A’P(ao,
al, . . . , a,)
(mod P>,
identically as to ao, . +. , a,, after the A’s have been replaced by their values (18) in terms of the ui. If P is invariant modulo p under all transformations (7), it is called a formal invariant modulo p off. The term formal is here used in connection with a formf whose coefficients are arbitrary variables in contrast to the case, treated in the earlier Lectures, in which the coefficients are undetermined integers taken modulo p. In the latter case, (19) necessarily becomes an identical congruence in the u’s only after the exponent of each a is reduced to a value less than p by means of Fermat’s theorem UP = a (mod p). The functions (18) are linear in a~, . * a, a,. It is customary to say that relations (18) define a linear transformation on uo, “.,a, which is induced by the binary transformation (7). Let r be the group of all of the transformations (18) induced by the group of all of the binary transformations (7). Making no further use of the form f, we may state the above problem of the determination of the formal invariants off in the following terms. We desire a fundamental system of invariants of group r. This problem is of the type proposed in $1; the group r is a special group of order a multiple of p. Here and below the term invariant is restricted to rational integral functions of uo, . . ., a,. A theory of formal invariants has not been found. For no form f has a fundamental system of formal invariants been published. Some light is thrown upon this interesting but di5cult problem by the following complete treatment of a binary quadratic form, first for the exceptional case p = 2 and next for the case p > 2, and preliminary treatment of a binary cubic form. 6. Formal Write (20)
Invarianti
Moddo
2 of a Binary
f = ad + bxy + cg2,
&uudrutic
Fornz.-
42
THE
MADISON
where a, b, c are arbitrary
COLLOQUIUM.
variables.
x = 2’ + y’,
(21)
Under the transformation y = y’,
f becomes f’, in which the coefficients are a’ e a, b’ = b, c’ = a + b + c
cm
(mod 2).
By $3, the only invariants under d’ = d, c’ = c + d, modulo 2, are the polynomials in d and c(c + d). Take d = a + b. Hence the only seminvariunh off are the polynomials in a, b and a = c(c + a + b).
(23)
Such a polynomial is an invariant of f if and only if it is unaltered by the substitution (ac) induced by (q). Thus (24)
b, k = as, q = b(a + c) -I- a2 + UC+ c2 = s + ab +
a2
are invariants off. Introducing q in place of s, we see that any seminvariant is a polynomial in a, b, q. Consider an invariant of this type. Since its terms free of a are invariants, the sum of its terms involving a is an invariant with the factor a and hence also the factors c and a + b + c, the last by (22). Hence this sum has the factor k, and its quotient by k is an invariant. By induction we have the theorem: Any rational integral formal invariant of f equal a rational iniegral function* of b, q, k. 7. Formal Seminvarianh of a Binary Quadratic Form for p > 2. Write (25)
f = d
+ 2bxy + cy”,
where a, b, o are arbitrary variables. Under the transformation (21), f becomes?, whose coefficients are (2’3 * Replace
a’ = a, b’ = a + b, c’ = a + 2b + c. 01, zg ZS, of 0 4 by a, b, c ; then
La - bkk(k + W,
&:a = b’ + bk + P,
&a~=
Pq’ + bq7c + b% + V.
INVARIANTS
Evident (27)
AND
formal seminvariants P-1 f9=p+b)“bp-baPl
NUMBER
43
THEORY.
are a, A = be - ac, and (mod P>,
(28)
Indeed, the linear function under the product sign in (28) is transformed by (26) into the function derived from it by replacing t by t + 1. As in (27), (29)
[-ykJa=cl= cp - a-1
(mod PI.
Let S(a, b, c) be a homogeneous rational integral seminvariant with integral coefficients. Then, by (26), S(0, b, c) = S(0, b, 2b + c)
(mod PI.
Thus, by 0 3, S(0, b, c) equals a polynomial Hence, by (29), S(a, b, 4 = da,
h 4 + $4, ok)
where u and tj are polynomials b2’ = A’ + a(
),
in b, CC’- cP. (mod P),
in their arguments. bpf2’ = @A; + a(
Now
).
Hence (30)
S = ah(a, b, c) + #(&A,
Yk) + ‘p~~pdgb2i+1ykeJ,
where X and $ are polynomials in their arguments, and di is an integer. When y is multiplied by a primitive root p of p, a, b, c are Hence p is multiplied by p, multiplied by 1, p, p2, respectively. while, by (29), Tk and A are multiplied by p2. If therefore we attribute the weights 0, 1,2 to a, b, c, respectively, and the weight 8 + 2t to a’b*c’, we see that the weight of every term of rk is congruent to 2 modulo p - 1. We can now prove that every di is divisible by p. For, if not, the seminvariant S - # has a term of odd weight, so that every
44
THE
MADISON
COLLOQUIUM.
term of X is of odd weight and hence has the factor b. Thus S - $ has the factor b and therefore the factor p, so that its terms free of a have the factor bp. But this is impossible, since 2i + 1 < p and (29) does not have the factor b. Hence S - $I has the factor a and the quotient is a seminvariant of the form ax’ + #‘. Proceeding in this way, we obtain the theorem : Any seminvariant & a polynomial in a, A, fi and any single Yk. Of these, @alone is of odd weight. Hence any seminvariant is a polynomial in a, A, Yk, @*or the product of such a polynomial by ,f3. But P-1 p2= aPyo+A(A T - az’12> (31) (mod PI. To prove this, it suffices to show that the second member is divisible by b and hence by & and being of even weight therefore by p2, and to remark that each member of (31) reduces to b2p for a = 0. Now hola=o = ‘Q (t’a + c) = c ( “g
(Pa + c) )” p-1 s ~(0” -
aP[yO]b,O = ac( (-
ac)’
-
(-
aP1j2
p--l a)“12
(mod P>,
(mod P>.
But A reduces to - ac for b = 0. Hence the second member of (31) has the factor b. We therefore have the theorem: For p > 2, any formal seminvariant of a binary quadratic form is a polynomial in a, A, y. or the product of such a polynomial by /3. 8. Formal
Invariants
of a Binary
Quadratic Form for p > 2.
The product (32)
r
=
UYk k
(k
ranging over the quadratic non-residues of p)
is an absolute invariant of f under the group G of all binary transformations with integral coefficients taken module p of
INVARL4NTS
AND
NUMBER
45
THEORY.
determinant unity. It suffices to prove that this seminvariant is unaltered by the sub&it ution a’ = c, c’ = a,
(33)
b’ = - b,
induced by the transformation x = y’, y = - x’. the general factor in (28) is replaced by
Under (33),
(t”-k){(T2-K)a+2Tb+c}, where k K = (p - k)2’
T=&
Hence K is quadratic non-residue of p when k is. Also, p@~-k)=-k(i~j$k-t~)}2--k(;‘-l)2=-4k
(mod P>
if k is a non-residue. To show that the product of the resulting numbers - 4k is congruent to unity, we set x = 0 in p-1
(34)
v(x-
k) = x2+
1
(modp),
and note that 2P-l = 1. Hence (32) is unaltered by (33) and is an absolute invariant off under G. It is very easy to verify that J = aTo
(35)
is unaltered by (33), so that J is an invariant of f under G. If an invariant has the factor /3, it has the factor (36)
B= B&r
(r ranging over the quadratic residues of p).
For, under the substitution (33), b+Ta (T/O) becomes T(c-b/T). By choice of r, we reach c + 2tb, where t is any assigned integer not divisible by p. This is a factor of Yk where k = P. The fact that B is an invariant may be verified as in the case of (32) or deduced from the fact that P-1 @nYk = am - 131’ k=O
46
THE
MADISON
COLLOQUIUM.
is an invariant, being the product of all non-proportional linear functions of a, b, c with integral coefficients modulo p. Hence any invariant is the product of a power of B by an invariant which is a polynomial P in a, A, +yo. Since rk is a seminvariant not divisible by /3, it equals a polynomial in a, A, r. (9 7). But if a = 0, 7k = y. (mod p), by (29), and A = b2 is free of c, so that ok is not a polynomial in a and A only. Hence (37)
rrn = YO+ gda, A)
(mod PI.
For p = 3, the polynomial P therefore equals a polynomial in a, A, 7~ = T. Now an invariant 4(a, A, I’) differs from the invariant +(O, A, I’) by an invariant with the factor a and hence the factor (35). Treating the quotient similarly, we ultimately obtain the following theorem for the case p = 3: A fundumental system of formal invarianti of the binary quadratic form f modulo p, p > 2, is given by the discriminani A and I’, J, B, defined by (32), (35), (36). The product of the last three is congruent module p to the product of all the non-proportional linear function8 of th4 coe~ienh off. To prove the theorem for p > 3, note first, by (37), that I’, given by (32), differs from ~0% by a polynomial in 70, a, A of = (p - 1) 12. Hence a polynomial in degree n - linyo,wheren a, A, 70 equals a polynomial in a, A, 70, I’ of degree at most n - 1 in yo. Subtract from each the terms of the latter involving only the invariants A, I’. We have therefore to investigate invariants of the type n-1 s-1 C CCYO’P~@,r) + C y0Vi(a, A, r), in which the ci are integers, while Pi and t#~iare polynomials in their arguments, and +i has the factor a. If every cl = 0, the invariant has the factor a and hence the factor aye = J, and the quotient by J is an invariant which may be treated similarly. The theorem will therefore follow if we show that a contradiction
INVARIANTS
ANB
NUMBER
47
THEORY.
is involved in the assumption that a certain cj is not divisible by p. First, the remaining ci are divisible by p. For if also ci + 0, let &A’PJ be the term of Pi of highest degree in A. Since +yoand I? are of degrees p and np, and of weights = 2 and 0 l), y&Pi is of degree P; + 2ri + s;np and of weight bc+= 2i+ 21-i (mod p - 1). But p = 1 (mod n). Hence i + 2Ti s j + 2Tjp 2i + 2ri s 2j + 2rj
(mod 4,
so that i = j (mod n). But i and j are positive integers < n. Hence i = j. Multiplying our invariant by a suitably chosen integer, we have the invariant (39)
rojPj(A,
I’> +
A, I’),
EJ~o~&(u,
Pj = A’I”
+ *. l .
Now - (c - ka)b- is the term of highest degree in b in Tk. Hence . . .. **a, r = apb-O+ (40) Yo = -cbpl+ (41) u = ${-
(c - ka)) = (- c)” + (- a)”
(mod p),
where k ranges over the non-residues of p, the last following from (34) for 2 = c/u. Since 70 and I’ are of even weights, only even powers of b enter (39). Hence an invariant (39) is symmetrical in a and c. We shall prove that this is not the case for the terms of highest degree in b. For TaPj this term is (42)
(- c)ju*b8, /I = j(p - 1) + 2r + sn(p - 1).
Let C&Af~P be one of the terms of & in which the exponent of b is a maximum. Then in y&i the highest power of b occurs in the terms (43)
C,xf~( - c) Wbb i2
Pi=
vi
+
g;n(p
-
1) +
i(P - 1).
Since the weight and degree is the same as for (42), 2i + fli s 2j + /3
(mod P - 11, ei + i + gsn + pi = j + 8?2 + 8.
45
THE
MADISON
COLLOQUIUM.
First, let /3i = /3. Then i = j, ei = 0 (mod n), whence i = j. Thus the exponent of a in any term (42) or (43) is divisible by n, while the exponent of c is not, being congruent to j modulo n. Hence the coefficient of bS in the sum of (42) and the various terms (43), with i = j, is not symmetrical in a and c, unless identically zero. But (43) has the factor a while (42) does not. Hence the greatest & exceeds 8. Next, consider a set of terms (43) and a set of terms of like form with i replaced by k, all being of equal degree in b. Then fli = /3k. By (441), 2i + Pi = 2k + &, i = k. Consider finally terms (43) with /3i constant. In them the residue modulo n of ei is a constant + i. For, if ei = i, then 2i + & = j + @ (mod n) by (44,), so that j = 0 (mod n) by (441). Hence these terms (43) are not symmetric in a and c and yet do not cancel.* Our fundamental invariants are connected by a syzygy; for P = 3, B2 = A312 + J(J - A2)2. (45) 9. Formal
Invariants
of a Binary
Cubic
Form
for
p + 3.-
We have seen that the theory of formal invariants of a binary quadratic form is dominated by the invariantive products of linear functions of the coefficients. While these products depended upon the classification of integers into the quadratic residues and the non-residues of p, we shall find that for a cubic form it is a question not merely of cubic residues and non-residues of p, but of the larger classes of reducible and irreducible congruences. Write
f
= ax3 + Sba+y + 3cxy2 + dy3,
thus taking p + 3. Under transformation whose coefficients are given by (26) and (46)
(21), f becomes f’,
d’ = a + 3b + 3c + d.
* If two are of lie degree in c, their g’s are equal and hence their fs are equal; then, if of like degree in a, their e’s are equal. But then we have the same term of +i.
INVARIANTS
Hence riants;
AND
NUMBER
49
THEORY.
/3 and ok, given by (27) and (28), are again seminva-
a,
also, p-1
(47)
si,=~{(tS-3kt--~a+3(P-k)b+3tc+a~ (j, k = 0, - * -, p - 1).
Indeed, if Ft(a, b, c, d) is the function in brackets, Ft(a’, b’, G’, 8) = Fwl(a, b, c, 4. Any invariant
with the factor a has the factor
(48) a&o = a’5 (Pa + 3t2b + 3tc + d) = f(1, 0)&t, t=o
l),
whose vanishing is the condition that one of the points (z, 9) represented by f = 0 shall be one of the existing p + 1 real points (1, 0), (t, 1) of the modular line. To verify algebraically that the seminvariant (48) is an invariant,* note that it is unaltered modulo p by the substitution a’ =
(49)
-
a,
dt = a,
b’ =
C,
d = - b,
which is induced on the coefficients of f by x = y’, y = - x’. The product P of the ajh in which j and k are such that x = t3 -3kt-j is irreducible modulo p is a formal invariant. The substitution (49) replaces the general factor of (47) by - a + 3tb - 3(P - k)c + Xd = X((T3 - 3KT - J)a + 3(T2 - K)b + 3Tc + d}, where T-k-t2--
x
,
h=
K=;,
g = l2 + k2a+ tj,
J=;,
- 2k3 + 6k2t2+ 3ktj + t3j + Jo.
*For any form, e.ee Transactions vol. 8 (X307), pp. 207-208.
of tk
American
Mathemuticd Society,
50
THE
MADISON
COLLOQUIUM.
We are to show that there is no integral solution z of x3-3KxMultiply
J=O
(mod P>.
this by X3 and set kr = y. y3 - 3gy - h = 0
Then (mod P).
But the negative of the left member is the result of substituting ?-+a=
- t,
rs = - y - 2k
in the expansion of the product (T3 - 3kr - 31(s3 - 3ks - j). The latter is congruent to zero modulo p for no values of r and s which are integers or the roots of an irreducible quadratic congruence with the integral coefficients t, - y - 2k. For p = 2, P = 611. For p = 5, P is the product of two invariants* (50)
b1822832~41,
&3624&4643,
neither of which is a product of invariants. is true also of the following invariants: 71603,
74602,
The last property
y2~04&2~30~20~42,
(51) ~3~01'%0~2S~33~40,
@~0&4~21831~44.
The product of these seven invariants and a&o equals the product of all the linear functions of a, b, c, d, not proportional module 5. For p = 2, each of the 15 linear functions is a factor of just one of the following invariants (no one with an invariant factor) : (52)
USOO, 611, ProSo~, K = b + c, (a + b + c)&o.
For any p + 3, the cubic form has the formal invariant (53)
G = 3(bcp - bpc) - (a&’ - cd),
* In those linear factors of the limt which lack c, the product of the coefficients of a and b is a quadratic non-residue of 5; in those of the second invariant, a quadratic residue.
INVARIANTB
AND
NUMBER
and an absolute formal invariant*
51
THEORY.
K of degree p - 1. For
P = 5,
(54)
K = b4 + c4 - b2d2 - a2c2 -
bc2d -
ab2c + ad2 + a2bd.
Thus, for p = 5, K and the discriminant D are invariants of degree 4, and weights = 0, 2 (mod 4), while a&o and G are of degree 6 and weight = 3 (mod 4). It follows from 0 10 that there are no further invariants of degree less than 8. Now the first and second invariants (51) are of degree 10 and weight = 1 (mod 4). Hence if either is expressible as a polynomial in invariants of lower degrees, it must be the product of D by a linear function of a&o and G. This is seen to be impossible either by a consideration of the terms of degree 2 5 in d or by noting that D has no linear factor. Thus y1& or cy,& occurs in a fundamental system of invariants. Invariantive products of linear functions of the coefficients of the cubic form therefore play an important r81e in the theory of its formal invariants. Whether or not they play as dominant a r81e as in the case of the quadratic form is not discussed here. We shall however treat more completely the seminvariants. 10. Formal Seminnariants of a Binary Cubic for p > 3.-We shall first determine the character of the function to which any seminvariant S(a, b, c, d) reduces when a = 0. Set A = 3b, 2B = 3c, C = d. Then (26) and (46) give A’ = A,
B’ = A + B,
C’ = A + 2B + C
(when a = 0).
Any function unaltered by this transformation is (0 7) a polynomial in A, B2 - AC, TO’, or the product of such a polynomial by /3’, where 70’ and /3’ are the functions yo and @ written in capitals. But p-1 ~0’ = 9 (3cb + 3~ + a) = [~jo]o=o,
* Tranaactiona of the America n Mahvnutieal vol. 10 (lQOQ), p. 154, foot-note. Bulletin Society, vol. 14 (1908), p. 316. Cf. Humitz,
Society, vol. 8 (1907), p. 221; Ma#mnutical 1. c.
of the American
52
THE
modulo p.
MADISON
COLLOQUITJM.
Hence S = m4a, b, c, 4 + ykV(b
(55)
q,
(e = 0 or l),
Sj0)
where k, j may be given any assigned integral values and q = c2 - +ba,
(56)
D being the discriminant off. S2 = -
(57)
b2 + ac,
- wq = [D]a=o, We use the seminvariants (II, 0 2)
6’3 = 2b3 + a(ad -
First, let p = 5. Then q = 8 + 2bd. seminvariants* u3
3bc).
We have the formal
= bq - abb + 24,
u4= K-S522=q2+a(abd-2ac2+b2c+c#), us= bg+a(-aaaus = qa+
bc@+ 3&i + ab$ - 2b3c + a9b), - 2bcd3- CW + absa - ab%d + a3bd+
a(ad4
2~4
- b2cS- 2aV- + ab’),
(58) a7
=
no
+
a(2(b2 - ac)d4 + a2bd3 - b&P - 2c4d2 + 2a2cW
- 2ac(b2 -
ac)# -
(b2 - ac)2d2 - 2a4d2 + 2abc3d
+ 2a3bcd + 2ab4c + 3(b2 - ac)c4 - a4b2 + 2aV),
while 2G differs from bya by a multiple of a. By (55)-(58), differs from a polynomial in the seminvariants (59)
a,
D,
S2,
Ss,
~a,
K,
us,
us,
~7,
G,
YO,
S
600
by a function aX + pb&, + uq6&, in which p and u are constants at least one of which is zero (in view of the degree of the terms). But the increment to b&, under transformation (26), (46), is * Aa the terma with the factor a were taken all of the proper degree and weight; then a term common to a combination of the seminvarianta (59) was deleted. Finally the coefficients were found by a proceaa equivalent to the use of a (non-linear) annihilator, Tranmctima of the Am&can Mathematical Society, vol. 8 (1907), p. 205. Expansions were made in powers of d and the terma involving d rechecked. As each remaining term in~olvw a new coefficient, there is no doubt ae to the existence of covariante of type US, ~6, 01, though the terms free of d were not rechecked.
1. INVARIANTS
AND
NUMBER
THEORY.
53
CL& with the term acZ50,while d does not occur to this power in the increment to a function X of degree 5g. Again, the increment while the increment to a function to qS$ has the term ~cu-P+~~, X of degree 5h + 1 is of smaller degree in d. Hence p = u = 0. Then in ah, X is a seminvariant which may be treated as was the initial S. A fundamental system of formal eeminvariante of the binary cubic form module 5 is given by the functions (59). 11. For p = 2, the method of 0 10 fails. In place of c we now introduce the seminvariant K = b + c. Then the transformation (26), (46), becomes (60)
a’ = a, K’=K,
b’=a+b,
d’=a+K+d.
By 0 3, any seminvariant S(a, K, b, d) becomes for a = 0 a polynomial in K, b, d(K + d), In place of the last we may use 600. Hence S = a~ + M, K, 6001, 600= d(a + K + 4. We make use of the seminvariants A = ad + bc = 600+ 601, p = b2 + ub, (61)
/Y+A=bK+a(b+d).
Hence S differs from a polynomial in K, 600,A, /3 by a function up + bT(P, 600). Let (60) replace p by p’. Then p + p’ = r (mod 2). Take a = K = 0; then (60) is the identity and 0 = T(b2, d2) identically in b, d. Hence the function ~(8, S) is identically zero. Thus up and hence p is a seminvariant. Hence a, K, 600,A, /3form a fundamental system of formal eeminvariants of th cubic module 2. Note that A2 is the discriminant, so that A is an invariant. The invariants (52) may be expressed in terms of our seminvariants: (62)
811 = I + A,
PYOSOI= B@ + K2 + UK) (A + So,),
(a+KNo= (a+K>(a2+I)=a600+KI, where I = a2 + aK + 600is an invariant.
54
THE
MADISON
COLLOQUIUM.
12. Miss Sanderson’s Theorem.*-Given a modular invariant i of a system of forms under any modular group G, we can construct a formal modular invariant I of the system of forms under G such that I = i (mod p) for all integral values of the coefficients of the forms. As the proof does not give a simple method of actually constructing I from i, it is in place here to give a very interesting illustration of the theorem with independent verification. Take as i the fundamental seminvariant (- l)=P,+ra, of a binary form f (Lecture II). Then 1 is the quotient L&L,,,, where Lkr is given by (16) or (17) with 21, . . . , Z~ replaced by the first m coefficients a0, al, . . . , a-l of the binary form f. Now z = 2’ + y’, y = y’, replaces f(z, y) by a form in which the coefficient aj’ is a linear function of ao, . * a, aj. Hence Lj is a formal seminvariant of f modulo p. First, La
sop
alp
a0
al
i a0 = aOP1al - alp
-=
Ll
I
I
is a formal seminvariant which reduces to - Peal for integral values of ao, al, where PO = 1 - aopl. Compare (27). Next, aOps a/ La = a# a0
a#
alp
a.2P ,
al
a2
C = L3/L2 = a# - a$& -I- a&P
(mod PI,
where, as in (lo), - aoa1p4 = $ Q = aoPsal L2
ao’(p+Ul’j
(8 = p -
1).
For integral values of the a’s, we have L2 = 0,
Q = a$ + al* + (p - l)aoaa~* = 1 - PI, PI = (1 - a#)(1 - al’),
module p, since each term of Q, with j + 0, j =l=p, is congruent * Transadms
of the Americun
Mathemdicd
Socktg,
vol. 14 (1913), p. 499.
INVARIANTS
AND
to ao8ala. Hence C = PIaz. L4fL1=
NUMBER
55
THEORY.
Similarly,
- mp’ + a/Q32
- apQ31
+ diP1
(mod PI,
where the Q’s are defined by (16) and are congruent to* &al = Q(L&P-1
with Q as above.
+
Lpgp,
&a
=
(L/L2P-’
Q”,
+
Hence for integral values of the a’s,
Qal G (1 - P1)Plap+=
0,
&a2 = 1 - PI (1 - a$‘-‘)
= 1 - P2,
L4/L3 = - P2a,,
13. Modular Covarianta.-Extending the usual definition of a covariant of an algebraic form f to the case in which the group is the set of all linear transformations with integral coefficients taken modulo p, we obtain the concepts modular covariants or formal modular covariants according as the coefficients of f are integers taken modulo p or are indeterminates. The contrast is the same as in 0 5. The universal covariants obtained in Q2 and 0 4 do not involve the coefficients off and hence are forma1 covariants. I have recently proved? that all rational integral modular covariante of any system of modular forms are rational integral functions of a finite number of these covariants. In the same paper I proved that a fundamental system of modular covarianta of the binary quadratic form (25) modulo 3 is given by the form f it&f, its diwriminunt A, the m&versa1 covariants L and Q, together u&!hS q=
(a+c)(b2+ac-l),
f4=&+bz3y+bzy3+cy4,
(63) Cl = (a2b - b3)z2 + 2(b2 + ac) (c -
a)zy + (b3 -
b$)y2,
CZ = (A + a”)9 - 2b(a -I- c)zy + (A + 8)y”. Here f4 is a formal covariant, which is congruent to f for integral * T~ama&nw of the Americun Mathemakd 8ocietg, vol. 12 (IgIl), p. 77. t !i?a?hsa&ons of the American Mathematical Society, vol. 299-310. The extension to cogredient seta of variablea has by Professor F. B. Wiley, and will be published in hi Chicago $ No one of the eight is a rational integral function of the even in the case of integral coefficients a, b, c taken modulo
14 (1913), pp. since been made dissertation. remaining seven 3.
56
THE
MADISON
COLLOQUIUM.
values of z, y. Also C; and (as here written) Cr are formal covariants. Note that - q is the invariant (42) of Lecture II. When q is made homogeneous by replacing - a - c by - a* - cs, we obtain the formal invariant I? = 79, given by (32). The resulting eight formal covariants off do not form a fundamental system of formal covariants; not all the formal invariants are polynomials in A and I’ (0 8). No instance of a fundamental system of formal covariants has yet been published. The method of proof will be here illustrated by the new and simpler case of a binary quadratic form (20) with integral coefficients modulo 2. By 0 6 any invariant off is a polynomial in (247
b, ah
q=
(b+l)(a+c)+w
to which the formal invariants (24) reduce modulo 2. Such a polynomial is congruent to a linear function of these three and unity, since bq=abc (mod .2). Further, any seminvariant is a polynomial in a, b and q (0 S), and hence is a linear function of 1, a, b, ab, q, abc. For, aq=a+ab+abc
(mod 2).
These results are in accord with those obtained otherwise in 0 14 of Lecture II. We shall now prove the following theorem: Every rational integral covariant K of th.ebinary quudratic form f modulo 2 6 a rational integral function off, its invariants b and q, the universal covarianta
Q = 9 + q/ + y”, L =
2~
+ XI/“,
and the linear covariant
l=
(a+b)x+
(b+c)y,
P=f+bQ
(mod 2).
The leading coefficient 8 of K is a seminvariant and hence is of the form I + ra + sab, where r and s are constants, and I is an invariant, a linear combination of the invariants (24’) and unity.
INVARIANTS
AND
NUMBER
First, let K be of even order 2n.
Then
K1 = K - IQ” - rf” is a covariant has the factor a covariant of Next, let K
57
THEORY.
8bs(
in which the coefficient of x2” is zero and hence y. Thus K1 has the factor L and the quotient is order 2n - 3 to which the next argument applies. be of odd order: K = i3~9”+~+ S#‘y+
sm..
After subtracting from K constant multiples of IQ” and bl&“, in which the coefficients of $‘+I are a + b and ab + b, respectively, we may assume that S is an invariant. After also subtracting from K a constant multiple of IL&“-l, where I is a linear combination of the invariants (24’) and unity, we may assume that S1 = /3ra + &c, where the S’s are functions of b only. Then the covariance of K with respect to the transformation (21) gives sx’2m+‘+$~x’my’+ . . . GKE Sx@‘+l+ (S+&)x’2ny’+
...
(mod 2),
where Sr’ denotes the function Sr formed for the new coefficients (22). Hence Sl’ - SI = Btb + b) must equal the invariant S. Since &b is a function of the invariant b, &a must be an invariant, so that & = 0. Thus S = 0 and K has the factor L as before. Hence the theorem is true for covariants of order w if true for those of order w - 3. But it was proved true for those of order zero. By a similar method I obtain the following theorem: A fundamental system of covariants of the binary quadratic form f, given by (20), and the linear form X = a2x + sly module 2 is given by f, X, I, h =
(aa2
+
j>z
+
(cm
+ j>y,
Q, L and f/w invuriunte b, q, (al - l)(a2 - 1) and j =
(a +
b)al
+
(b +
4~2.
58
THE MADISON COLLOQUIUM. I
Since aI and a2 are cogredient with x and y, the function j obtained from the covariant t off is an invariant off and X. The reverse of the last process is important. If we adjoin to a system of binary forms in the variables z’ and y’ the linear form yx’ - xy’, any modular invariant of the enlarged system, formal as to x, y, is a modular covariant of the given system with CC’,y’ replaced by x, y. The theorem of 0 12 therefore proves the existence of certain formal covariants. * APPLICATIONS
OF INVARIANTS
OF A MODULAR
GROUP, 50 14,15
14. Form Problem for the Total Binary Modutur Group I?.This group is composed of all binary linear transformations (7) with integral coefficients taken modulo p whose determinant A is not divisible by p. By (S), 034) Lb, Y) =AL(Z Y>, &(x9 Y> = &(X9 y> (mod PI, so that L@ and Q are absolute invariants of I’. Hence, of the functions (ll), Qis invariant under I?, while t is unaltered by certain transformations and changed in sign by others. Thus a homogeneous function of q and t having a term which is a power of q is a relative invariant of I’ only when an absolute invariant. Hence if p > 2, it involves only even powers of t, and by the homogeneity, only even powers of q. Hence any absolute invariant of r is a product of powers of LPI and Q by a polynomial in q7, t7, where 7 = 1 if p = 2, 7 = 2 if p > 2. In particular, LPI and Q form a fundamental system of absolute invariants of r. The so-called form problem for the group I’ requires the determination of all pairs of values of the variables x and y for which Lpl and Q are congruent modulo p to assigned values X and p, either integers or imaginary roots of congruences modulo p. We have therefore to solve the system of congruences (65)
IL&, Y>P-’ = A,
Qb, Y) = P
(mod p>.
*After these lecturea were delivered, I saw a manuscript by Professor 0. E. Glenn, containing tables of formal concomitants for forms of low ordera and moduli 2 and 3. He employs transvection between the form and the covariant L of 0 2.
___-. .__ ~~-,.-__“---__--____.-“._ .,... --._-_-.-__^____ -.. I
INVARIANTS
AND
NUbfBER
59
THEORY.
First, let X + 0. For z = x or z = y, we have 0s
2P’
gP2
ZP’
XP
yp
ZP = LZP” -
x
Y
z
QLzp + LPZ
(mod ~1.
Hence x and y are roots of (66)
F(z) = zp* - pzp + AZ = 0
(mod
P>.
Having no double root, this congruence has p2 distinct integral or imaginary roots. These roots are (67)
eX + fY
(e,f = 0, 1, . . ., p - l),
where X and Y are particular roots linearly independent modulo p. For, F(eX + fY) = eF(X) + fW7. (68) Hence any pair of solutions x, y of (65) is of the form (7), where a, a--, d are integers, whose determinant A is not divisible by p, in view of (641) and X + 0. Conversely, if X and Y are fixed linearly independent solutions of (66), any pair of linear functions of X and Y with integral coefficients, whose determinant is not divisible by p, gives a solution of (65). Indeed, by (68), x and y are solutions of (66). From the two resulting identities, we eliminate X and p in turn and get P = Q(x, Y), IUx, ~11” = Wx, Y). Since X and Y are linearly independent modulo p, L(X, Y) is not divisible by p [cf. (6)]. Thus L(x, y) + 0 by (64). Hence (65) hold. Hence, for X + 0, the form problem has been reduced to the solution of congruence (66). The latter will be discussed here in the simple but typical* case in which X and p are integers. Now the problem to find the real and imaginary roots of a con* For the general case, see Tran.mtims Society, vol. 12 (1911), p. 87.
of the Am&can
Mathmatieal
60
THE
MADISON
COLLOQUIUM.
gruence with integral coefficients is at bottom the problem to factor it into irreducible congruences with integral coefficients. When v is an integer, zp - vz is a factor of (66) if and only if v is a root of the characteristic* congruence u2 -p+x=o
(6%
(mod P).
Such a binomial is a prod&t of binomials zd - 6, irreducible modulo p, whose degree d is the exponent to which the integer v belongs modulo p. Since 2p - 1 < p”, the function (66) has an irreducible factor d(z) of degree D > 1, not of the preceding type ti - 6, and hence with a root T such that P/T is not congruent to an integer. Thus every root of (66) is of the form C~T+C~P, where the c’s are integers. The irreducible factors of (66) are of degree D except those, occurring only when (69) ha.s an integral 6, where d is a ditior of D. To find D, note that by raising (66) to the powers p, p2, . . . , we can express ZP”as a linear function It of ZP and z. Now D is the least value of t for which 4 = z. But the coefficients of Ir
root, of thefor?72 Zd-
are the elements of the first row of the matrix of S”,
* Note the analogy
of (66) with the linear differential
having the solution z = ert if u is a root Make dz/dt correspond to zp and hence equation corresponds to (66), and the 2.p = uz. t Let f(z) be an irreducible factor of
The latter (,d)z+
Thus
6 = rd is an integer.
equation
of u2 - PU + X = 0. Also, (68) holds. d%/dP to (zp)p. Thus the differential integral z = ear (viz., dz/dt = vz) to degree d.
Its roots are
. . ., Ted-’ r, 7-p = u-l-, ,.P= E &., where ud = 1, vz + 1, 0 < 2 < d. Thus d is a divisor p-1 - v = ZP-1 - $-P-l
has the factor zd - rd.
s
,,d-IT,
of p - 1.
has a root T in common c vd E 1.
Hence f(z) = zd - 6.
_-__-~----~-_-II-_
--_--
where
Hence
with f(z).
But
INVARIANTS
AND
NUMBER
61
THEORY.
But ID = z implies that &,+I = sp. The condition for the latter is therefore SD = 1. Hence D is the period of S. But (69) is the characteristic determinant of S. According as it has distinct roots rl and 2)2or equal roots v = 8~ = X*, a linear substitution of matrix S can be transformed linearly into one of matrix*
According m the characteristic congruence (69) has distinct (real or imaginary) roots or a double root, D is the least common multiple of the exponente to which the &tin& roots belong moddo p, or ia p timea the exponent to which thfs double root be,?onga. Finally, let X = 0. By (6), either y = 0 or x - uy = 0 (mod p), where a is an integer. In the fist case,
Q=
xPLP,
xPa
-
*P
=
0.
If j,L = 0, then x = y = 0. If p =I=0, the roots x are equal in sets of p and hence are cx1 (c = 0, 1, . . ., p - l), where x1’ is a particular root not divisible by p. In the second case x - uy = 0, we take x - uy as a new variable X and conclude from the absolute invariance of Q that
Q(x, Y) = &to, Y) =
ypLp.
We thus have the first case with y in place of x. Using similar methods, I have solved the form problem for the total group of modular linear transformations on m variableat 15. Invariantive (70)
Ck&$cation
of Forms.-Let
4(x, $4 = p+
***
(m > 1)
be a binary form irreducible modulo p and having unity as the coefficient of the highest power of x. Let G be the group of all modular binary linear transformations (1) with integral coef* In the second case we use the new variables x and x - uy. t Tranaudons of the American Mathem&& Society, vol. 12 (1911),
84-92.
pp.
62
THE
MADISON
COLLOQUIUM.
ficients of determinant unity. Let ~$1= 4, &, . . . , 4k: denote all the forms of type (70) which can be transformed into constant multiples of C#J by transformations of G. Evidently their product P = 4142 - - * C$kis transformed into c,P by any transformation t of G. The constant ct is easily seen* to be congruent to unity. Hence P is an absolute invariant of G. If m > 2, no C&vanishes for a special point. We now apply the theorem in the first part of 0 14. Hence, if m > 2, the absolute invariant P is an integral function with integral coemnte of the invariants q, 1, each exponent of q and 1 be&g even ij p > 2. In view of the definition of the &, this function of q and 1 is an irreducible function of those
arguments modulo p. Two binary forms shall be said to be equivalent if and only if one of them can be transformed into a constant multiple of the other by a transformation of G. A set of all forms equivalent to a given one shall be called a genus. Thus &, . . . , $k form a genus. All of the irreducible forms (70) separate into a finite number f of distinct genera; let PI, . . ., Pf denote the products of the forms in the respective genera. Thus ?r, = PI . . * Pf is the product of all of the binary forms z”’ + 0. . irreducible modulo p. Hence ?r, is a polynomial in q, 1 with integral coefficients. Hence the f genera of irreducible binary forms of degree m > 2 are characterized invariantively by the f irreducible factors P&7, 0 of dq, 0 moddo p. We shall see that ?r,,,(q, I) is easily computed. By finding its factors irreducible modulo p in the arguments q, Z, we shall have
invariantive criteria for the equivalence of two irreducible binary forms of degree m. For example, we shall prove that r3 = q - 1 if p = 2, so that all irreducible binary cubic forms module 2 are equivalent. Further, 1r3= q2 - F if p > 2, so that the irreducible cubic factors of q - I are all equivalent, also those of q + 1, while no factor of the former is equivalent to one of the latter. * T~an.w&ms of the Am&cm Mathematical Society,vol. 12 (1911), p. 3, Q4. The present section is an account of the simpler
topics there treated at length.
INVARIANTS
AND
NUMBER
63
THEORY.
In general, let m be a product of powers of the distinct prime numbers q1, . . *, q,,, and set Ft = (zq/ - qp”)/L. From the expression for ?r, due to Galois we readily obtain
in which the first product in the numerator extends over the +I-& - 1) combinations of ql, . . . , q,, two at a time, and similarly for the remaining products. By the first theorem of this section, and (1 1), n,,, is a polynomial in Jzqy=
QP+‘,
K=~Y=
Lpcpl)
(+y= 1 if p=2, y=2 if p>2).
We readily verify the recursion formula
Ft = QF:-, - KF;:,
(mod P>,
since F1 = 1, Fs = Q. In particular,
Fs = J - K, NOW
Fq = Q(F3p - KJp’).
IQ = Fa, ~4 = Fd/Q. Hence a3 = J - K,
T4 s Jp - Kp - KJp-1
(mod PI.
The first of these results was discussed above. Next, for = P 2, ~4 is the irreducible quadratic form q2 - P - lq, so that all quartic forms irreducible modulo 2 are equivalent. For p > 2, 7r4 vanishes for K = pJ, where pP=l-p Except for p = 9, pPs
(mod P>.
p is a quadratic Galois imaginary since E
1
-
pP
s
p
(mod P>.
Thus 7r4 is a product of J - 2K and +(p - 1) irreducible quadratic forms in J, K. Some of the latter yield a quartic in q and 1 which is irreducible; others yield a quartic which is a product of two irreducible quadratics modulo p. A simple discussion shows
64
THE
MADISON
COLLOQUIUM.
that the number of irreducible factors of T*(Q, I) is 6k + t + 1 if p = 8k + t (t = * 1 or - 3), but is 6k + 2 if p = 8k + 3. We have therefore the number f of genera of irreducible quart& modulo p. For quintics and septics, the analogous discussion is simple, for sextics laborious. We may utilize similarly the invariants (16) of the group on m variables, obtain expressions in terms of them of the product of all forms in m variables of specified types (as quadratic forms transformable into an irreducible binary form, non-vanishing ternary forms, nondegenerate ternary quadratic forms, etc.), and hence draw conclusions as to the equivalence of forms of the specified type. * * Tranmctions of the A merican i!hdmatti 8ociety, vol. 12 (lQll), pp. 92-98.
LECTURE MODULAR GEOMETRY AND QUADRATIC FORM IN
IV
COVARIANTIVE m VARIABLES
THEORY MODULO
OF
A
2
1. Introduction.-The modular form that has been most used in geometry and the theory of functions is the quadratic form
(1)
Qm(X) =
ZCijXiXj
+
(i, j = 1, * * *, m; i < 3)
ZbiXi2
with integral coefficients taken modulo 2. In accord with Lecture III, we shall use the term point to denote a set of m ordered elements, not all zero, of the infinite field Fa composed of the roots of all congruences module 2 with integral coefficients. We shall identify such a point (x1, * * ., z,,J with (~21, * * a, ~z,,J where p is any element not zero in Fz. The point is called real if the ratios of the x’s are congruent to integers modulo 2. Let the cij and bi in (1) be elements not all zero of the field Fs. Then the aggregate of the points (x) = (x1, * * 0, z,,J for which ~~(2) = 0 (mod 2) shall be called a quadric locus, in particular, a conic if m = 3. The locus is thus composed of an infinitude of points, a finite number of which are real. While our results are purely arithmetical, we shall find that the employment of the terminology and methods of analytic projective geometry is of great help in the investigation. Usually the proofs are given initially in an essentially arithmetical form. In case a preliminary argument is based upon geometrical intuition, a purely algebraic proof is given later. The geometry brings out naturally the existence of a linear covariant, which is important in the problem of the determination of a fundamental system of covariants. 2. Tlw Polar Locus.-The is on q(x) = 0 if (2)
point
(KYI
-j-
+7(Y) + Khp(Y, 2) + h*Q(z) = 0 65
&I,
- - a,
KY,,,
(mod 3,
+
AZ,,,)
66
THE
where (3)
P(y, 2) =
COLLOQUIUM.
MADISON
+
ZCij(y$j
(i, j = 1, ’ ’ ‘) m; i < j).
YjZi)
If (y) is a tlxed point, all points (8) for which P(y, z) = 0 (mod 2) are said to form the polar* locus of (y). For (2) = (y), each summand in (3) is congruent to zero modulo 2. Hence the polar of (y) passes through (y). If (z) is on the polar of (y), (2) has a double root K : X and the line joining (y) and (z) is tangent to q = 0. We may write (3) in the form (3’) where
P(y,z)
Ul =
=
c12zz
242= c1221
(4)
u3
=
cl321
urn =
cl&l+
+
u1y1+
***
urnynl,
+
k&3
+
Cl424
+
* * - +
Qdm,
+
c2323
+
c2d4
+
"
- +
cZm%,
+
c34z4
+
---
+
c37&n,
c23z2
C2&2
+
+
c3l&+
.**
+
G&-1+1.
There is a striking difference between the cases m odd and m even. 3. Odd Number
of Variables;
Apex; Linear
Tangential
Equation.
Let m be odd. Then the determinant of the coefficients in (4) is congruent modulo 2 to a skew-symmetric determinant of odd order and hence is identically congruent to zero. Hence we can find values of 21, * . . , z,,, not all congruent to zero such that Ul, . . ., u, are all zero modulo 2. Thus the polars of all points (y) have at least one point in common.
We shall limit attention (5)
Cl= [23 - * - m],
to the case in which the pfaffians
CZ= 1134 - - - m], - - -, C,= [12 - - - m-l]
are not all congruent to zero.
The point (C,, . . ., C,) shall be
*Take I = 1 and let (z) be a point not on p(z) = 0. Then (2) is a quadratic congruence in A with coefficients in Fa and hence haa two roots hl and X2 in that field. Now the points (u) and (z) are separated harmonically by (y + Alz) and (1/ + Ati) if and only if X1 = - AZ, that is, if A1 = An (mod 2). But the condition for a double root of (2) is P E 0 (mod 2).
INVARIANTS
AND
NUMBER
67
THEORY.
called the apex* of the locus a(r) = 0. Now each zbi = 0 if 21 = Cl, - - -, z,,, = C,,,. Hence, for m odd, the polars of all points pass through
the apex.
If (y) is any point not the apex, the line joining (y) to the apex is tangent to q(x) = 0 (0 2). Thus any Zim through the apex is tangent to q(x) = 0.
For m = 3, it is true conversely that, if the line ZUiXi
(6)
G
(mod 2)
0
is tangent to q(x) = 0, it passes through the apex, so that (7)
K =
I;C;Ui
is zero modulo 2. Taking, for example, u3 =l=0, we obtain by eliminating x3 from (6) and q(x) = 0 a quadratic equation in x1 and x2 whose left member is the square of a linear function module 2 if and only if the coefficient of ~1x2 is congruent to zero. But this coefficient is the product of K by a power of u3. Thus K
= 0 is the tangential
equation of q(x) = 0.
The last result is true for any odd m. The spread (6) is said to be tangent to q(x) = 0 if the locus of their intersections is degenerate. Taking u, =l=0, and eliminating G between (6) and q(x) = 0, we obtain a quadratic form whose discriminant, defined by (24), equals a product of K by a power of u,,,, and hence is degenerate if and only if K = 0. We thus have geometrical evidence that K ti a formal contravariant of q(x), i. e., an invariant of q(x) and Zuixi. To give an algebraic proof, note that K is unaltered when xi and xj are interchanged, while (8)
x1=21’+52’,
52=x2’,
-a-,
%?a= %I&'
replaces q(x) by q’(x’) in which the altered coefficients are (9)
bz’
=
bz +
h
+
~12,
C;j
=
C2j
+
Clj
(i = 3, - - *, m).
*After these lectures were delivered, I learned that Professor U. G. Mitchell had obtained, independently of me, the notion apex (“ outside point “) for the case m = 3, Princeton dissertation, 1910, printed privately, 1913.
68
THE
MADISON
COLLOQUIUM.
The pfaffians Cz, . . *, C,,, are unaltered modulo 2, while (10)
Cl’= c;+ cz, T.&Z’=%+?&I,
Hence K is unaltered modulo 2. 0
(11)
$G
Cl2 .
Cl2
Cl3
.
.
-
* ’ .
(mod 2).
Note that
”
C23
0 .
z&i’= UC (i* 2)
.
CM
Ul
C2?n ua . . .
Clm
C2m
C3m
.‘.
0
u,
Ul
u2
243
a**
u,
0
(mod 2).
We saw that Cl, . * ., C, are cogredient with ~1, * -0, z~. Tbis is evident from the fact that the apex is covariantively related to q(z). Hence if we substitute Cl for 21, . a, C,,, for z,,, in (l), we obtain the formal invariant l
(12)
If this invariant vanishes, the apex is on the locus, which is then a cone. Indeed, by (2), every point on the line joining (C) to a point on q(z) = 0 lies on the latter. Hence q(x) can be transformed into a form in m - 1 variables and hence has the discriminant zero. To argue algebraically, let new variables be chosen so that the apex becomes (0, . *, 0,l). The polar of any point (3) passes through the apex. Taking z1 = 0, * ., z,,,-1 = 0, &I&= 1 in (4), we see that the polar (3’) becomes clmyr + . + c,,+l,y,l, which must vanish for arbitrary y’s. Hence b,,,~~ is the only term of (1) involving z,,,. But the apex is on the locus. Hence b,,, = 0 and q(z) is free of G. The converse is obvious from (5). Whether m is odd or even, q(z) has the invariant l
l
l
(13)
A,,, =
lI(Cij+
1)
(i, j=
1, ***,m;i is on the cone, and, by (2), P(z, z) is congruent to zero identically in ~1, . . ., z~. Hence the linear functions (4) all vanish. Thus the line S meets the cone in its vertex, and h2 is the discriminant of Q+~(x), while xi2 is obtained from that discriminant by interchanging m and i. For example, if m=4, z42
=
cl2cl3c23
Z12 =
c23c24c34
+
b& +
+ b&
&A +
+ btda
b&, +
** -, ba&.
The product of the general form (1) by 6 = & + 1 is a quadratic form whose discriminant is zero modulo 2 and hence has the vertex (6~1, . . ., Sz,J, where z? has the value just given. Hence 6z12, * . . , Sk2 are cogredient with xl, . . . , x,,,. 6. Covariant Plane of a Degenerate Qua&G Surface.-The product of q4 by 6 = [1234] + 1 is a quaternary form f whose
72
THE
MADISON
COLLOQUIUM.
discriminant is zero and hence can be transformed into a form (14) free of x4. With this cone F = 0 is associated covariantively the plane I = 0, where I is the ternary covariant (19). Hence f has a linear covariant L which reduces to 1 when b4 = 0, ci4 2 0 (i = 1, 2, 3). Relying upon symmetry and the presence of the factor 6, we are led to conjecture that (27)
L = 6{h + 1 + (CIZ + 1)hs +
l>(c14+
l>h
+ -a -
+ 6Iba + 1 + (‘314 + l)(c24 + 1>(‘34 + 01x4. It is readily verified algebraically tbat L is a covariant of 44. There is a simple interpretation of L. If [1234] + 0 (mod 2), then 8 = 0 and L is identically zero. If [1234] = 0, q4 is degenerate and can be transformed into C#= x1x2 + x8* or a form involving only x1 and x2, In the former case, L = x1 + x2 + x8. Of the 15 real points in space, the seven (100x), (010x), (111x) and (0001) are on the cone t$ = 0, the two (001x) are on the invariant line S through the vertex (0001) of the cone and the apex (0010) of the conic cut out by 24 = 0, while the remaining six (Lola), (011x), (110x) lie on the plane L = 0. Hence with a degenerate quadric surface, not a pair of planes, is associated covariantively a plane, just as a line (19) is associated with a non-degenerate conic (14). Every linear covariant is of the form IL, where I is an invariant. Every quadratic covariant is a linear combination of the ILP and Iq4. 7. A Cmfiguratiun Defined by the Q&nary Surface.-A q6 whose discriminant is not zero modulo 2 can be transformed into F = x1x2 +
52x4
+ xs2.
The 15 real points on F = 0 (mod 2) are given in the last column of the table below. In addition to these and the apex (00001) of F, there are just 15 real points in space: l= (00011), 2= (OlOOl), 3= (OlOll), S=(OOllO),
7= (OlllO),
b= (lOllO),
c= (llOOO), d= (llOlO),
.--.._
._
~-
4= (OOlOl), 5= (OllOl),
8= (lOOOl), 9= (lOOll), e= (lllOO),
a= (lOlOl),
f=
~_..-..-
(11111).
INVARIANTS
AND
NUMBER
73
THEORY.
These lie by threes in exactly 20 straight lines, which occur in the columns of the table, with the heading “Sides.” With these lines we can form exactly 15 complete quadrilaterals, the three diagonals of each of which intersect* in a point on F = 0, given in the last column. The columns, with the heading “Plane,” give the equations defining the plane of the quadrilateral. In each case, the two equations of the plane have in common with F = 0 a single real point, the intersection of the diagonals. Thus the real points on F = 0 are its points of contact with these tangent planes. Sides
Diagonals
Plane .-
146 146 146 157 157 lab 28~ 28c 28c 29d 29d 347 347 356 356
157 lab lef lab lef lef 29d 2ae 2bf 2ae 2bf 38d 39c 38d 39c
356 49b
4df 5~4 58e 2ae 38d 5ac 78f 69a 49b 4df 49b 58e 5ac
347 69a 6de 7bc 78f 2bf 39c 58e 7bc 6de 4df 78f 7bc 6de 69a
13 19 Id lc 18
45 4a 4e 5b 5f 12 af 23 89 25 8a 27 8b 26 9e 24 9f 3f 48 3b 4c 3e 5d 3a 59
67 6b 6f 7a 7e be cd ce cf ad bd 7d 79 68 6c
z1=0, zr+zd+zs=O x,=0, za+zt+zs=O cl=zs=z:+z,+z~ zl+s~+~~==q+z~+z~=(l zt=zs, z~=q+z~+zs m=za, ZZ=Z~+Z,+Z~ x*=0, x~=x2+xs x,=0, xl=xt+xa x8=x+ z1=zp+zp+z~ xl=xa+xc=x2+xa x1=x+ xa=g+x,+x~ x,=x4, x~=x~+x,+x~ x,=x~+x~=x~+x~ xz=xa+xc, zl=xe+xa x~=x~+x~=x~+x~
I
Iutersection 01000 10000 11001 11011 10010 01010 00010 00100 00111 01111 01100 10100 11101 10111 11110
8. Certain Formal and Modular Covariants of a Conk-For conic (14), the polar form is
(28)
al
a2
a3
Yl
!I2
Y3
21
z2
83
l
Hence if two sets of variables yi and zi be transformed cogrediently with the set zi, this polar form (28) is a covariant of F and the two points (y), (z), in an extended sense of the term * The dual of the theorem of Veblen and Bueaey, ” Fiuite projective geometries,” Transa&ma of the Amenkan Mathematical Society, vol. 7 (1906), p. 245.
74
THE
MADISON
COLLOQTJIUM.
covariant. In particular, if we take (y) = (z), (II) = (.2”), we obtain a covariant of Fin the narrow sense used in these lectures. In particular, (29)
K
=
‘al
a2
a3
Xl
z2
23
Xl2
x22
x32
,
M
=
al
a9
a3
Xl
x2
Z3
Xl"
x24
sf
are formal covariants of F. While the discriminant A, given by (15), is a formal invariant, (16) is not. But A+A+l=a!
(30) (31)
ff
=
E&i+
(mod 3,
2d+
am
+
ala3
+
w3,
a! being a formal invariant of F. By (23), the B’s are contragredient to the x’s and hence to the a’s, so that (32)
Al = ZaiBi = E&b? + ZU& +
is a formal invariant. (33)
For integral values of aij bi,
Al s A s Zai(& + 1)
Any form with undetermined taken module 2, has, by (21) Thus (16) (cl+ U(c2 + 1) **-* of F. Likewise from (19) and F (34)
ama
J = &&,&,
(mod 2).
integral coefficients cl, CZ, . . ., of Lecture I, the invariant is an invariant of (7) and hence itself, we obtain the invariants
AJ = An(bi + 1).
In (6) we made use geometrically of x = u1x1+ %X2 + u3x3. (35) Now F + tx2 is congruent module 2 to the quadratic form derived from F by replacing each bi by bi + tuz. Making this replacement in A, we seethat the coefficient of t is congruent to Ke,where
K = am + a2u2 + a3ut (36) is therefore a formal invariant* of F and X. Making * Since (36) is a contravariant of F, za&X/axi) is a covariant Taking &s, &I, L as C, we get K, M, A, respectively.
the same of F if C is.
INVARIANTS
AND
replacement in J and taking as the coefficient of t = t2 w
=
p&ZuS
+
8$3u2
+
NUMBER t
I
and ug to be integers, we obtain
t&33%
(37)
+ +
a modular invariant
75
THEORY.
of F and X.
hu2uS &ulu3
+
83ulu2
+
UlU2U3,
By the theorem used above,
u = cm+ l>(u2 + wur+ 11 (38) is an invariant of A. In w + u + 1, we replace /% by the congruent value Bi + 1, and render the expression homogeneous in the u’s and B’s separately. We get
w = Z(BIB2 + B12 + B3’)~3~ + ZB12U3~3, (39) a formal invariant of F, X. For, it is unaltered by the substitution induced by (xizj), and by the substitution (23) and (10) induced by (8). Let the coefficients of F be integers not all even. Then (39) becomes wbB2 + 1>u32 + m&+ 1hu3. (39’) Its covariant I, is identically zero. Hence, by the table in 0 9, if w is not identically zero it can be transformed into ut + us2 + 241~2and hence vanishes for a single set of integral values of ~1, 212, 7.43. These are seen to be ui = pi + 1. Hence* the line L = 0 i.e the only line with integral line coordinates on the line locus (39). The invariant A for (39) is J (its discriminant is zero, as just seen). Thus a knowledge of any one of the concomitants L, J, w implies that of the other two. The covariance of K in (29) implies that
* Also thus: just EILIthe point conic F = 0 determines ita line equation (36) and hence ita apex (a), so the covariant line conic (39) determinea the point equation ZBh = 0, which is the line L = 0 for integral values of the coefficients.
“--
----.--_l--.~
-...-“-
“.__.-.._--.______ -----
-.-l__;_l_--..-
- ----- ---
~.
76
THE
MADISON
COLLOQUIUM.
are contragredient with al, clz, a3 and hence with xl, x2, x3, and therefore cogredient with ~1, u2, u3. Thus (39) yields the formal covariant w’ = x(&B2 + IL2 + B22)+f32 + =lhE,. (41’) From this or (39’), we obtain the modular covariant w = %3482 + l>ta2 (41) In these notations (29) become
La
z@l
+
K = ZOJ&~,M = ZU&(Q~+
(42) Finally, variants (43)
+
by (16) of Lecture III,
=
4
x2
x3
x12
x22
x32
x4
224
x34
&I
=
l)bb.
x2x3
+
~3~).
we have the universal
ZX~~X~~
+
Zx14x2x3
+
co-
x12x22x3:32,
,
Q2 = 2x1~ + 2~1~x2~ +
XIX~X~ZX~.
The covariant line .L = 0 of a non-degenerate conic F = 0 is determined by the three (collinear) diagonal points of the complete quadrangle having as its vertices the apex (a) and the three intersections of F = 0 with its covariant cubic curve K = 0. FORM F,
FUNDAMENTALSYSTEMOFCOVARIANTSOFTHETERNARY
00 9-32 9. Invariants of F.-A fundamental system of invariants of F is given by A, A, J. It suffices to prove that they completely characterize the classes of forms F under the group of all ternary linear transformations with integral coefficients modulo 2. This is evident from the following table A
J
1
0
0
0
0
1
x1x2
0
0
0
Xl2
0
1
0
0
1
1
Class
21x2
x1x2
+
x32
+
Xl2
+
A
x22
0 ,
--
L
. a+
x2
+
0 Xl
+
x2
Xl
0
x3
-
INVARIANTS
AND
NUMBER
77
THEORY.
As to the classes, we saw in 0 4 that, if F is not the square of a linear function (i. e., not reducible to xl2 or 0), it can be transformed into 51x2+ bl~1~+ 132x2~ + Axs2 and hence into one of the f&t three classes of the table. By means of the relations (44)
AA=O,
AJ=O,
any polynomial
A2=A,
A2=A,
P=J
(mod 3,
in A, A, J equals a linear function of 1, A,
(455)
A,
These are linearly independent
J, sihce
AJ. there are five classes.
10. Leader of a &variant of F.-Let S be the coefficient of 58”’ in a covariant of order w of F. Writing (14) in the form (46)
F = f+ lx:,+bm2, f = bm*+
wm+
b2xz2, I= a2x1+am,
we see that the leader S is a function of bs and the invariants of the pair of forms f and I under the linear group on x1, x2. In the modular covariants forming a fundamental system for f (Q 13 of Lecture III), we replace x1 by al and 22 by a2 and obtain a fundamental system of modular invariants of the pair f and I: (47)
a3,
q=
ff1a2,
bd2+(bl+bdw,
j=
(bl+adal+(b2+ada2,
where CQ= ai + 1. By means of the relations a~cu2j = 0,
(48)
(mod 2),
qj = j + a3j
any polynomial in the four functions (47) can be reduced to a linear combination of (49)
1,
~3,
q,
a3q,
aa2,
w2a3,
wx2q,
w2a3q,
j,
These form a complete set of linearly independent* off, 1.
a3j.
invariants
*Instead of verifying as usual that these 10 functions are linearly independent, we may deduce that result from the fact that there are 10 clssstx: I = 21, f = uslw2 + asxa* or 2 = 0, Since (47) characterize
--.
f = 22 + 21x2 + x2,
QXl” + wwr x1x2,
x1* or
the classes, they form a fundamental
---._II_I_I.-..--____
--
+ wf,
_._.
0. system.
.____.,.-_- ,_._-.= . _.._..
78
THE
MADISON
COLLOQUIUM.
Hence S is a linear combination of the functions (49) and their products by bs. Moreover, S must remain unaltered modulo 2 when USand bl are replaced by aa’ s US+ al,
(50)
bl’ = bl + bs + az,
which are the only altered coefficients of the form obtained from F by the transformation (51)
x1 = Xl’,
=
x2
x2’,
x3
Both requirements are evidently 1,
(52) and any invariant
a1~12,
of F.
+ 21’
x3’
(mod 2).
met by the functions ba, b3am
We find that
A = ~@$!(@ + l), (53)
=
A =
+ j + aaba+ as,
crm’2U3
J = a~cuz(m + l)@s + 1) + btj + c&j + b3q+ mcuzq, AJ =
+ Wa + l)(q + lb
cw2(a3
From these and their products by b3, we see that
AJ, baJ, J, W,
(54)
kb
4
A
contain the respective terms bww~q,
hamq,
cumq,
hj,
bma3,
j,
cmm,
while no one involves an earlier one of these terms. Hence any linear combination of the functions (49) and their products by ba is a linear combination of the functions (52), (54) and (55)
as, ba,
q, bag, w,
hq,
a3.i
bmj,
mww.
A linear combination of the latter is of the form u=
ma3
+
m2q
+
m3a39
+ m4a3j +
mw2a39,
where ml, . - a, m4 are linear functions of bs, while m is a constant. The coefficient of a,bl is seen to be p
_I
-...----_I_
= m2 + m&t + ma + mbm(R + b3 + 11,
INVARIANTS
where R = ba + u2 is the increment to bl in (50). (56)
u=p&+ra~+&+t
Set
(p, *es, t independent
of as, bl).
(50) replace u by Q’. Then
Let the substitution (57)
79
AND NUMBER THEORY.
u’ - Q = pRas + palbl + palR + rul + sR.
This is zero for every ~3, bl if and only if (53)
pR = 0, pal = 0,
ral = sR
(mod 2).
For p= mn+ a.., pal = 0 gives m3 = 0, m4 z m. Then pR = 0 gives m2 = 0, rnba = 0, whence m = 0. Thus u = mlas, so that ml = 0. Hence the leader of a covariant of F has the form I + Ml +
(59)
where I and II are invariants,
cala2
+ okw&a,
c and d are constants.
COVARIANTS WHOSE LEADERS ARE NOT ZERO, @ll-19 11. Consider a covariant (60)
of odd order w :
c = Sxs” + s1x3w-1x1 + s2x3o-2x12 + * * *.
If S1’ is derived from S1 by the substitution (61)
fiy = s1+ us = i&-j- s
(50), then, by (51), (mod 2).
Give S1 the notation (56). Then S is given by (57) and has no term with the factor atbl. Now a3bl enters no term of (59) except J and AJ of I and* b,J of bsI1, and in these is multiplied by (62)
bm + WQ,
wa(b2 + l>(h + 11, bala2,
respectively. Since the latter are linearly independent, neither J nor AJ occurs in the I, II of the leader (59). Also, A and crlcv2occur only in the combinations A + 1, cvlcvz+ 1, since (57) has no constant term. The coefficients of xao in L”, AL*, (A + A)L” are respectively (63)
ba +
am2
* AJ is not retained
+ 1, Ah, in II, since b.dJ
A + Ah +bma2, = 0, AJ being
(34).
80
THE
MADISON
COLLOQTJIUM.
where L is the linear covariant (19). After subtracting from C a linear combination of these three covariants, we may set S = mdA + 1) + m2A + m&a + mb8ala2. Since /3&xrul(rs = 0, AJ = 0, the leader of the covariant
JC is
JS = mlAJ + md + m3b3J. Hence ml = ma = 0. The m2(a1a2+ b3) and must vanish pR by (57). Hence m2 = 0. mFLue2 has this same leader. C=
+
m(bma223
coefficient of a8 in S is now for b3 = a2 since it is of the form Thus S = rnbgxlas. For w > 1, For u = 1, b1a2wa
+
b2ww2),
which
satisfies (61) only when m = 0. Hence eeery linear is a linear function of L, AL, AL; evey covariant of odd oxler w > 1 difers from a linear combination of L”, AL”, AL”, FLWe2 by a covariant whose leader G zero. covariand
12. In the covariants of order 4n I&$‘,
(64)
IP”,
L4”,
F2”-‘L2
(I an invariant),
the coefficients of xz4” are respectively I,
&I,
bt +
cw~2
+
1,
kwxz.
Linear combinations of these give every leader (59). Hence every covariunt of OTO%T 4n dij’era from a linear combination of the covariants (64) by a covariant whose leader is zero. 13. In the covariants of order w = 4n + 2 IQt”F,
(65)
Q2”L2, AQ2”L2
(I an invariant),
the coefficients of xs* are respectively baI,
bs + WYZ + 1, A + bs(A + alaza3).
The sum of the third function and bt(A + A) is A -I- bgxlctrz. Hence any covariant C is of the form P + C’, where P is a linear
-.-
____----I
. .-.---
_____ l__l_--.--.
.
.-~ _J
INVARIANTO
AND
NUMBER
81
THEORY.
combination of the covariants (65), while C’ is a covariant whose leader is an invariant. For w = 2, C’ = Sx? + 51X3X1+ 5x12 + x29. This is transformed by (51) into a function having Sr as the coefficient of 21’~. Since S is an invariant, Sr = S. Thus every coefficient of C’ equals S. Then (51) transforms C’ into a function in which the coefficient of ~1~x2’is zero, so that S = 0. Hence every quadratic covaria& is a linear fwhon of F,
(66)
AF,
AF,
JF,
L2, AL2.
14. There remains the more difhcult case of covariants (60) of order w = 4n + 2 > 2. If Si’ is the function obtained from Si by the substitution (50), then 231’= Sl,
(67)
= s + 6%+
S2’
Now S1 is unaltered also by the substitutions
s2.
(22) and
as’ = ag + a2, b2’ = b2 + ba + al
um
(mod 21,
induced on the coefficients of F by the transformations x1 = Xl’,
(69)
x2 = x2’,
(8) and
53 = 23’ + x2’.
15. A fun&mental system of invariants of F, under t/w group I’ generated by the transformutiona (8), (51) and (69), is given by A, A, J, a2, h, am2 and (70)
P =
Wa
+
a2>.
It suffices to prove that these seven functions, which are evidently invariant under I’, completely characterize the classes of forms F under I’. There are six cases. (i) bt=a2= 1. Replacing x1 by x1 + ~1x2 and x8 by x3 +
w2,
we
get
F = ,dx12+ Axz2 + xa2+ ~1x3. (ii) b, = 1, a2 = 0, all = 1. Replacing x3 by z3 + asxr, we get F = AXI’ + b2xz2+ ~3~ + ~2x3.
82
THE
MADISON
If A = 0, then bz = J. and get
COLLOQUIUM.
If A = 1, we replace z1 by x1 + b2zz a2 +
a2 +
~2%
(iii) bs = 1, a2 = ala2 = 0. Replacing 23 by za + bm + b2x2, we get xa2 +&x2. (iv) bs = 0, a2 = 1. After replacing xS by x3 + u8x2, we obtain a form with also aa E 0. Taking this as F, and replacing xl by x1 + alxz, we get haa2 + ha2 + ~1~s. Replacing xa by x8 + blxr, we get Ax? + ~1x1. (v) bs 5 az = 0, alcu, = 1. Replacing x3 by ZQ+ ~3x1+ b2x2, we get px12+
X2X8.
(vi) bs = a+?E am = 0. Then F is the binary formf in (46). The effective part of I’ is now the subgroup rl generated by (8). Now @= bt
A + 1 = aa, J = B + (bl + l)cra,
B = br(bl + era).
These seminvariants bl, ea, B of f completely characterize the classes of formsf under I’l. For, if aa = bt f = blxr2 + Bxz2 + blxlq; while if a8 = bl + 1, we replace x1 by x1 + b2x2 and get bm2+
(bl+
1)x152.
16. The number of classes of forms F in the respective cases (i)-(vi) is 4, 3, 2, 2, 6. Hence there are exactly 19 linearly independent invariants of F under the group l?. As these we may take 1, a?, alcrz, A, A= (71)
bm+ B =
bar b,az, bsam,
*es, ad= bdbs +
a~>,
blalazf
bsA, ---,
azP = Msm,
----..--.---J
INVARIANTS
(71)
&=b~(ba+l)A, J-
blbtbs+
ANB
NUMBER
bA=b&m+ -se,
THEORY.
- - -, ahA=b&w2+
83 - - -,
asJ= blbzb3az+ s-e,
bsJ = b,b,bs(ala, + al + ar) +
l
l
a, AJ = blbzbaA + - - a.
These are linearly independent since the first eight do not involve bl, while all the terms with the factor bl in the next seven are given explicitly, likewise all with the factor blbzbs in the last four. Hence tha 19 functions (71) form a complete set of linearly independent invariants of F under the group I?. 17. Hence, in 0 14, Sr is a linear combination of the functions (71). By 03721, S + S 1 is of the form (57) if S2 be denoted by (56). Now aabl occurs in J, AJ, biJ, a2J, A/3, but in no further function (71). In the first three, asbl is multiplied by the linearly independent functions (62), respectively; in the last two by belat and alcu2(b3+ l), whose sum is congruent to the first function (62). Hence the part of S + Sr involving J, - - -, A/3 is a linear combination of (72)
(b3 + adJ = b&&w2
+ b&wwa,
(73)
J + baJ + 43 = @a+ lNdtcw~
+ bzA + A).
But bl occurs in just six of the functions (71) other than the five just considered. Thus the factor pal of bl in (57) is a linear combination of the coefficients of bl in (72), (73), S, a@, A, QA, bb, azb&. Now al is a factor of the coefficients of bl in all except the second, third and fourth, while in these the coefficients are (bs + Www2,
bs+e+
1, 43
and are linearly independent. Hence (73), 8, a@ do not occur in S + I!&. By (57), the latter has no constant term and hence involves 1, A only in the combination A + 1. This cannot occur since the total coefficient of a3 must be of the form pR and hence vanish for b3 = ~22. At the same time we see that the sum of the constant multipliers of A, aA, bb, Rbd is zero modulo 2. Hence S + SI is a linear combination of the functions
84
THE
MADISON
COLLOQUIUM.
~2, 58, bsscz,ala2, and the last six in (74) below. Lie (57), this combination must vanish for al = 0, ba = a. Since all but the first three of the ten functions then vanish, the sum of the multipliers of these three must be zero module 2. Hence S + S1 is a linear combination of (74)
b8 + fh2,
a2, NJ,
at(b, +
+ 11,
11,
a1a2,
Na2h
+
bw2, 11,
h-4 @a
+
az>J.
18. Without altering the invariant S, we may simplify S1 by subtracting from C constant multiples of L4*+ K and its product by A, where K is given by (29), and hence delete a2(ba + 1) and A(a&s + 1) from the terms (74) of &. Then s1 = S + mh
+ mA(bs + 1) + m&s + az)J + Mb3 + 4
+
m4am + m5baalcuz+ m&A.
The coefficient T of q”+ x2 in C is obtained from S1by applying the substitution (a&(b~b2) induced by (51~2). In view of the transformation (8), we see that T’ = T + 6’1, where T’ is derived from T by (22). Hence S = (m + ml)A + mdb + m&J + (m4 + m&)(a1a2 + a+
4
+
m&3
+
m&A.
Let L: be the sum of the second member and the function obtained from it by the substitution (ma3)(b2ba). Thus E = 0. Taking bs = b2, we get m4 = ma = 0. Then Z = (b2 + bt)I,
I = m9 + m2J + ma + meA.
Applying to Z the substitution (68), we get (b2 + al)1 = 0. Applying (alas)(b~bg) to the latter, we get (b2 + aa)1 = 0. Adding, we get (al + a3)I = 0. Applying (50), we see that aJ = 0. Then each ail= 0, so that I = gA, where g is a constant. By 2: = 0, g = 0. Thus ml, m2, ma, me are zero. Hence S = mA, ~‘31= mAa2. But (75)
E=F(L4+AF2)+(A+A)L8=Ax~6+~~~.
1 -
INVARIANTS
AND
NUMBER
THEORY.
Hence 12- &2”+E has the leader zero. Any covariant of &j = 4.n + 2 > 2 diJ$ers fTOV2 a linear of variante (65) and Qz“+E by a covmiant whose leader is zero.
85 order
co?nbination the co-
19. &?gukM and Irregular copa&&; Rank.-A covariant shall be called regular or irregular according as it has not or has the factor La, given by (43). The quotient of an irregular covariant by Ls is a covariant. Hence the determination of all irregular covariants reduces to that of the regular covariants. If a covariant has a linear factor it has as a factor each of the seven ternary linear functions incongruent modulo 2, whose product is LS. Hence a. regular covariant has a non-vanishing component involving only 21, 53. In a regular covariant C without terms z$’ (i. e., with leader zero), this component has the factors zl, 5 and (by the covariant property) also Al+ x3. The product of these three linear factors was denoted by & in (40). Let tSm be the highest power of .$ which is a factor of the component and let n be the degree of the quotient in the x’s, Then C may be given the notation &a,* =
(76)
&fit? +
xlx2x3:ad,
where, if n = 0, fz is a function of the a’s and b’s not identically zero, while, if n > 0, f2 is a function also of XI, x3 in which the coefficients of xrn and 23” are not zero; fi is a function of x2, x3; f3
of
Xl,
52.
The regular covariant (76) shall be said to be of rank m. In an irregular covariant the component free of x2 is zero and hence is divisible by an arbitrary power of &; it is proper and convenient to say that an irregular covariant is of infinite rank. Any covariant of rank zero differs from one of rank greater than zero by a polynomial in the known covariants (77)
A, A, J, F, 4
Q2.
This is a consequence of the theorems in Q§ 11-18, where the polynomial is given explicitly. Any product, of order w in the
86
THE MADISON COLLOQUIUM.
of powers of the the syzygies
covariants
(77) can be reduced by means of
JL = 0, AL2 = AF,
(A + A + J + l)(FL + K) = 0,
z’s,
(78)
AK = 0, FL2 + (A + A)L4 + AF2 + AQ, = LK, FS+ QzF = L3K + (A + JW + (A + 1)LG + (-4 + l)Q,,
to a sum of covariants of order w given in 00 11-18 and a linear function, with covariant coefficients, of K, &I and G
= Q2L -I-
(79)
L6 =
1)2s2+
x2[@3@1+
+
PlcB3
+
US121
x
(51x2
+
21x3
+
+
+ 1)x351
(h/33
w253[(Pl+
z2:253)
P2
+
z@i
+
+
P3
+
1)
lh?].
Here G and K, given by (42), are of rank 1, while &I= tt2+x2 ( ) is of rank 2. As this theorem is not presupposed in what follows, its proof is omitted. However, it led naturally to the important relations (75) and (79) and showed that no new combinations of the covariants (77) of rank zero yield covariants of rank > 0, a fact used as a guide in the investigation of the latter covariants. REGULAR
COVARIANTS RI,,,,, 00 20-22
20. A separate treatment is necessary for covariants (76) 0. Then each fi is a function of the coefficients ui, bi. withn= Since the factor E3’”of the partf3t3(11 of &o free of 23 is unaltered by every linear transformation on 21 and ~2, f3 is a linear combination of the functions (49) and their products by b3. Also, fa must be unaltered by (80)
zl
=
a$
+
~3’:
al’
=
al
+
~3,
ba’
=
b3
+
bl
+
~2.
Both conditions are evidently satisfied by the ternary invariants and by us and q, in (47). In view of (53), we may employ
AJ,
J,
4,
A, USJ, q4
A
to replace in turn b3wza3q,
b 3ala2a39
&j#
j,
a3b3%
ala2a3qt
wff2h
INVARIANTS
AND
NUMBER
87
THEORY.
since a term previously replaced is not introduced later. Thus fs is a linear combination of these seven functions, aa, q, asq, and wa2,
ww,
h,
hat,
be,
b3ala2,
kww,
b3j,
bmj.
Give to any linear function m~cv~acz + * - - of these the notation
Call e the increment bl + a2 to bain (80) and employ e to eliminate bl. Then u is unaltered by (80) if and only if cue= 0, aa3 = 0, /?u3= ye
(mod 2).
Since ba does not occur in q or j, nor al in q, we have (Y =
mscw2
+ m?ar2q+ m8(e + u2 + a3) + ms3(e + a2 + as).
Thus eye= 0 gives me = mr = 0, ms = me. Then gives m = 0. Now B=
mla2
+ m2a2q, y = ma +
m4a3
+
aas = 0
m5q,
and flag = ‘ye readily gives Q = 0. Any function of b3 and the invarianta (49) off and 1, which i.~ unaltered by (80), is a linear combination of the ternary invariant8 (45) and a3, q, a3q, aa,
asJ, qA. 21. For n = 0 and m even, there exists a covariant (76) in which fs is any function specified in the preceding theorem. For, if I is any ternary invariant, I&r”/” has f3 = I. By (42) and (41), I-P and PI2 are of the form (76) with fa = a3 and /392 + 1, respectively; they may be multiplied by any invariant. By (19) and (47), we have (81)
/W2
+ 1 = q + ad + A + 1, eq = aA + qA + a3J.
Hence we obtain q, then qA, qA, and therefore arq. Any covariunt with n = 0, m even, differ8 by an irregular covariant from a linear function of IQf’$I#“,
MP’~2
(I=l,A,
A, J, AJ; L=l,A,
J; &=l,A,
A).
88
THE
MADISON
COLLOQUIUM.
22. For n = 0 and m odd, we may delete the terms aJ1 from f* by use of IlK”. First, let m = 1 and apply transformation (51) ; we get 41 (82)
=
tl'+
R'=fl&'+fib'
4st, +
52 =
dfi
(2',
h
+fdEa'+
=
s,
(zlk2'd+
d2z2')+.
fa = fs’ gives
Thus + = 0. Since fs = I + Lq, condition fi + I
=
I2(&
+
a2b2
+
a&8
Add to this the relation obtained 1, 2. Thus 0 =
I2@1
+
+
ws
+
alad.
by permuting
b2 +
a2m
+
the subscripts
am).
The increment under (22) is I& + aa + a2cx8)= 0. Now I2 is of the form z + PA + z,A, where z, y, 2:are constants. From the terms in b&, we get y = 0. Then z = z = 0. The only oova~nte are therefore LK. Second, let m > 1. Then KW(-1)/2 is of the form (76) with fs = arq + aa, by (811). Hence we may set
fs=I+cq+&A
(c, d constants).
In R given by (76), let g denote the coefficient of (83)
*
3w2:22a
xpxp@+.
In the function derived from R by the transformation (51), the term corresponding to (83) has the coefficient g + fi, sinceby (82) the & parts contribute only one such term, that from
f G”-lEs’.
Now
fi = I + Cq‘+ 41’4
Q’ =
b2b3
+
@2
+
hi)W.
When g is given the notation (56), g’ - g = fi is the function (57). But a& occurs in fi only in J and AJ and in them with the linearly independent multipliers (62). Hence I = nl(A + 1) + mA. The coefficient of aa in fi is now nlala2
+
n&w2
+
bi3)
+
dQIcw2
=
p(h
+
az>.
.l_l_-^_____ll--l_____ - .-J
INVARIANTS
89
AND NUMBER THEORT.
Taking ba = as, we see that nl = nz = d = 0. Thus fr = cq’. By (57) for al = 0, bs = a2, we get c E 0. Any copariant wit6 ta = 0 and m oo?ddi$ers by an irregular covariant from a linear fun&m of Km, AK@‘, JK” and, if m > 1, KW(-l) D. COVARIANTS
OF RANK UNITY,
$5 23-26
23. Henceforth let m > 0, n > 0 in (76) and set
fi = Szg + 512pz1+ s$lpi$
(84)
+ **
l
(S
9
0).
Since S is unaltered by the group I? of 0 15, it is a linear combination of the functions (71). We may omit the functions az(ba + 1) and Aas(ba + l), since K”L” is of the form (76) with S = az(bs + 1). Thus (85)
S = I+ a& -5 bJ2 + hala2 + k2bsalcuz+kB+Sa28+lcsAB,
where I 1, A, A, First, and fi, function
is any invariant, II a linear function of 1, A, J; I2 one of J;ewhile /3 = bl(bs + ~2~). let m = 1. If T and B are the coefficients of x$’ in f3 transformation (51) replaces the covariant (76) by a in which, by (82), the coefficient of 21’x~~~+~is T + B = T’,
036)
where T’ is derived from T by the induced substitution (50). But T is obtained from S by the interchange [23] of subscripts, and B from T by [13]. We thus find by (86) that I = b& + (kl + k&t)(al + k&&l
+ am)
+ ah + a.& + am + am> + k&&h
+ aabs+ alas + aza3).
Let Z be the sum of the second member and the function obtained by applying (u2aa)(b2bs) to it. .InX=O,setb2= ba;weget {kl+k(+b3(k2+lea))(a2+as)crl=O,
h=kl,
Then Z = 0 may be written in the form (bz + b& = 0, X = 12 + kz(A + A + 1).
k4=k2.
90
THE
MADISON
COLLOQUIUM.
As in 0 18, X = 0. Thus Is and I are the products of A + A + 1 by ks, kl, so that S=
(h+Ws)@+A
+
1) +a2Il+k1(b1ba+blcvz+a1cun)
(87) +
knbdazh
+
~2)
+
kd(&
+
bl).
For n odd, S is the increment to SI under (50) and hence has no term containing a&l. If t is the coefficient of J in II, a& occurs in (87) only in tazJ and in the final part, being multiplied by fa?cu& and kgxla2(ba + l), respectively. Hence t = ka = 0. Since S is of the form (57), the coefficient of by must vanish if a1 = 0. Thus kl(ba + 4
+ kzbrae = 0, kl=
k2 = 0.
Now S = a2Il = as(u + VA) must vanish for al = 0, bs = az by (57) ; then A = az(b2+ us), so that u = v = 0, S = 0. Any covariant with m = 1 and n odd &few from on% of rank > 1 by a linear function of KL”, AKL”. 24. For m = 1, n = 4v, we may delete ~11 from (85) by use of 11KQ2”. Set fi = Bs” + - + &,~a”. Then (51) replaces (76) by l
R’ =
~2[S&’
l
+ S~X~‘-~XI + (SI +
+
(cl+
Esmd~a"+
+
Bn-lz2(ar1+
S2‘3)~3”-~ct5~
av-%'+ ar2cc1+
+ - . -1 + &fa
-*-> ***>I
+
cw2nl+
Z123J2w.
Since S1 is the increment of SZ, it is a linear combination of the functions (74). By use of Lw8Q1, LnssKa and their products by A and A, we may, without disturbing S, delete from SI ba+alcurr+l,
Ah,
A+bd+hwz,
a&+1),
wWa+l).
Hence we may set
SI = t&s + 4 + Mwm + t&a + t& + 4J. Applying (arap)(blb2) to S and &, we obtain B, and B-I. Let 1 be the coefficient of x2xan+ in c$, By the coefhcient of
IN-VARIANTS
x~x~q
- x2xP
AND
NTJMBER
THEORY.
91
in R’, we have B,+
B,r+l=
1’.
For 1 given by (56), B, + B ,+I is given by (57). By the coefficient of a&, we get ta 5 0. The coefficient of a3 must vanish for ba = a2. Hence km +
(k2
+
t2)wzz
+ kgzlcuzbz= 0, kl=
S = kzW + A + 1 +
a1cuz
ks = 0, ts = ks,
+ usbd.
The coefficient of kz equals that of 52x3” in
GFQ$‘-’ + AKL” + AKQ2y. Any covariant with m = 1, n = 4v, di$ers from one widh m 2 2 by a linear fun&m of KL”, AKL”, IKQZ”, GFQ2”l (I = 1, A, J). 25. For m = 1, n = 4v + 2, we may delete a211 from S, given by (87), by use of IlQ2’M. The coefficient of f2xan in Q2”G is d = B&-h + 1) = A + (bl +
Ww2
+ ba)+ haam.
The coefficient of kl in S equals d + Q,A + az(ba+ l), the final term of which was reached in 0 23, and a& above. The coefficients of ks and kz in S equal Ad and bM~l+az)+bd~zbz+
ws+azaa+az)
=Ad+dJ+l)+az(ba+l),
respectively. Any copariunt with m = 1, n = 4v + 2, d$ers from one with m 2 2 by a linear function of KL”, AKL”, IQS’G, I1Q2”M (I = 1, A, A; II = 1, A, J). For use in 0 26, we replace QsVM by Qz’FK, noting that @a
M = (F + L2)K
and that QvL2K ditfers from KL” by a covariant of rank 2.
26. By the last four theorems, any covariant of rank 1 differs from one of rank 2 2 by CK + DG, where C and D are known covariants of rank zero. Taking as CI and D1 arbitrary func-
92
THE MADISON COLLOQUIUM.
tions of the proper degree in the z’s, of the generators (77) of covariants of rank zero, I found the syzygies needed to reduce CrK + DIG to an expression differing from the above CK + DG by a covariant of rank 2 2, in which those of rank 2 are linear combinations of lP, KG, 62, W, Q1 and the new one (89) where
V = GP + AQzG + (A + J + 1)QzFK + ALsKz + ALa& = &2x&%+ * - -,
v=
(90)
a2 + b3(1 + UlcY2).
The only new syzygies needed for this reduction are
LG=QzL2+L6=
W, FLK=AW+AQl+(J+l)P,
(P + L4 + QX = (A +
(91)
W3,
(A + l>(FG + a4 + KQd + JKQz = AL&l +
wL3,
in which w is an invariant not computed. Proof need not be given of these facts since we presuppose below merely the existence of relation (89) which may be verified independently. Of course, the fact that V is the only new covariant of rank 2 was a guide in the later investigation. COVAHIANTS x3
OF EVEN RANK m = 2~ > 0, $527-29
27. First, let n be odd. In the covariant + XI. In view of (82), we get
R’ =
+
fi’fz”
f353’”
+ fi’(t? +
t32>”
+
(76) replace 58 by
(x1x223
Using the notation (84) for f2, we have Sr’ = Thus, as in 5 17, S is a linear combination of the Now QrPLn and its products by A and A + A are with S given by (63). Using also K”L”, in which and its product by A, we may set 8 = In
x1x2x3$,
+
kl(b3
a2)
+
k2b3ala2
+
k&z
+
kr(bs
let g be the coefficient of X1X2x3
* X22p-1X34w+n’2
=
(X22X34)
“X3”+xl.
+
x12x2)4’.
Sr + S in f2’. functions (74). covariants (76) S = ~3@3+l), +
an)J.
INVARIANTS
AND
NUMBER
93
THEORY.
Such a term occurs in neither of the first two parts of R’, since they are functions of only two variables. To obtain such a term from the third part of R’, we must omit terms with the factor Es2 (and hence s12) and take (~2s3~)~“in [12”, so as not to make the degree in x2 too high. Hence if T be the coefficient of zan in jr, g’ = g + T. Now (ala2)(bJ4 replaces S by T. The resulting T must be of the form (57). By the coefficient of U&I, kq = 0; cf. (72). By the coefficient ktcxlbs of as, ks = 0. Since T = 0 for al = 0, b3 = a2, we get kl= k2. Hence S = kg, where v is given by (90). For n = 1, f2 = 8x3 + &Xl. Thus S1 = klv’, where v’ is derived from v by interchanging the subscripts 1 and 3. Then SI’ = S1 + S gives kl 3 0. For n 2 3, Q1@-1Ln-3V is of the form (76) with S = v, since #f&v= 0. Any covariant with n odd, m = 2p > 0, differs from one of rank > m by a linear combination of IQlwLn (I = 1, A, A), K”L”, AKmLn and, if n > 1, Q1u-lLn-sV. 28. For m = 2~ > 0, n = 4v > 0, the coefficients of ~Z*XS” in
QI~&P~,Km&z", &?I-“,
(92)
Qf-1Q2-1G2,
Q,‘L”,
K”‘L”,
K-2Q$‘-W
are respectively 1,
a2,
bt,
Pa +
1,
a2@3
These may be multiplied 133 + AU% d +
+
1) A
+ +
(A 83
+
11,
d
=
P3@3
by any invariant.
1 + +
+
A
a2 + +
a2@
ad + a2b3 = nzblbo = a&
b3
l)b3 +
b3)
=
+
11,
a2d.
Now
w2,
+
ha2
+
A
=
br(b3
+
~2)
=
b3ala2, =
8,
Ad = Abl(b3 + 1) = A@.
Hence we have a covariant (76) in which the coefficient of [2"%3* is any linear combination of the functions (71). Hence ths
94
THE
MADISON
COILOQUIUM.
covariant differs from oozeof rank > m by a linear function of the covariante (92), the producte of the first three by any invarfuti except 1, the products of the fourth and jifth by A and the product of the eixth by A. 29. For m = 2~ > 0, n = 4v -I- 2, the coefficients of &“x$’ in MK”‘+Qs’,
(93)
K”‘L”,
GK-lQ$‘,
Fn’2Q2’,
L”QP
are respectively a, Linear
az(ba+ 11, aMb1 + 11, h,
b3 + am + 1.
combinations of products of these by invariants
a2, aA
azJ,
apba, Aazb,,
add,,
Iba,
am,
give*
A + bma.
Since S and SI are unaltered by the group I’ of 0 15, they are linear combinations of the functions (71). Deleting the above functions a2, a4, - - - from S, we have S = I + c/3 + e&3,
B = bdbs + m),
where c and e are constants, and I is an invariant. fl = BZ~” + BlX‘pX~
+ * - * + Bn-lx*x$--l
and call u the coefficient of
Set + Bnx3n,
.
w
X1X2& * x~4*+~~xp-1
in x~x~x&. Hence
The coefficientt
= (xgxpxpxl
of (94) in R’ of 0 27 is BI + u.
u’ - u = BI, if (50) replaces u by u’. Thus B1 must be of the form (57). For n = 2, Sz is derived from S by applying (alaJ(blba). Then (672) gives &. Applying (am)(bdd to Sl, we get B, = I + c(b&s + bzcvl+ bed + eA(bh
+ ba + U.
* For the last two, use the first two of the four equations in Q 28. t The first part of R’ is free of 21, the second of z8, while in the third psrt @ hag the factor ~9, and in fl’@r there is a single term (94) and it has the caefficient BI.
INVARIANTS
AND
NUMBER
95
THEORY.
Since this must be of the form (57), we get I = 0, c = e = 0. A covatiant with m = 2p, n = 2, di$ers from oozeof rank > m by a linear function of iMK”‘+,
K”L2,
AK”L2,
GK”-‘,
IF&l“,
L2Qic,
AL2Q1”
(i = 1, A, J; I = 1, A, A, J). For n > 2, we may delete A from the part I of S by use of E&I~&~-~, where E is given by (75). Without disturbing S we may delete as(ba + 1) and its product by A from &‘I by use of K!+tlLn+, since the term of t2”“f2 with the coefficient S1 is the term of highest degree in za in f22P+1(S15P + - * -). Since S + Sl is a linear combination of the functions (74), (95)
Sl = S + tdba + ad + tzam + t&ma2 + t&d
+ tA(ba + 1) + to@3+ 4J.
Apply (96)
+ t&-a
(alazas)(blb&~) to Br, of the form (57).
$1 = pm
+
pa&2
+
pazp
+
m
+
SP,
P =
Hence bl +
e.
Now a&s occurs in S only in the terms J, AJ of I and in the part of (95) after S only in the last term, given by (72). In these the factors of a&t are linearly independent. Hence to = 0, I = s(A + 1). The coefficient of al in Sr must vanish for br = as, and 6’1 itself if also ua = 0. Hence c= t2 = 2, t1 = ta = t* = t, Sl
=
%(A
+
1+
Ma
+
ha2
+
ala2
+
ta = x + t,
Aad + eAbl(ba + 1)
+ t(b3 + aft + hala + &A + bb + a&. Call E the coefficient in z1x2z8~ of XlZ2X.9
* xpx$‘+“-a
=
(xgxa4)
kT1x$ITp.
In R’ of 0 27, the coefficient of this product is e + B-1. Hence L&-r is of the form (57). Interchanging the subscripts 1 and 2 in B-1, we get Sr. Thus the coefficient of aa in Sl vanishes for bs = al. Hence S = Sl = 0. Any covariant with n > 2 difTer8
96
THE
MADISON
COLLOQUIUM.
from one of rank > m by a linear combination of iMKnrlQ~“,
jKmLn,
GKhlQzy,
IFn12QlM, jiY&l’,
EQ1@&2)‘1
(i = 1,. A, J; j = 1, A; I = 1, A, A, J). COVARIANTS
OF ODD
RANK m = 2~ + 1 > 1, $6 30-31
30. Replacing 23 by 23 + 51 in the covariant
R’ = fa’[zm + f3tsrn + fi’(& +
Es>”
+
(x1x2x3
(76), we get +
x12x2)4’.
In 2122x34, let g be the coefficient of (xlx22)(x22x3)“‘+x2n. The coefficient of the corresponding term of R’ is g’ = g + B, where B is that of ~2” in fi. Hence B is of the form (57). First, let n be odd. Then &‘I’ = S1 + S under (50), so that S is a linear combination of functions (74) with a2(b3 + 1) and its product by A deleted (0 23). Thus S is the sum of the terms (95) after the first. Applying (ala2as)(blb2b3) to B, of the form (57), we see that S is of the form (96). By these two results, S = t(b3 + a2 + baalcvz+ &A + b3L\+ ad). that in R’ is If I is the coe5cient of (z2z#%!3’L-1z1 in z1%2z3+, I?’= l+nB,. Hence, for n odd, B, is of the form (57). Interchanging the subscripts 1,2 in B,, we get S. Thus the coefficient of aa in S vanishes for bs = al, so that t = 0. Any cmariand with m and n odd di$eTs from one of rank > m by a linear function of K*P and AKmLn . 31. Finally, let m be odd and n even. According as n = 4u or 4~ + 2, KmQ2” or K”‘+MQ2’ is of the form (76) with a2 as the coe5cient of fzrnx3*. Hence we may delete the terms a211 in (85) and hence the terms aJr in B of $!23. But (0 30), B is of the form (57). Now a3bl occurs in J and AJ of I and in b2J of b212, having in these linearly independent multipliers. Hence I =x(A+l)+yA, I,=e+fA+gA. Since the coefficient of a3 in B shall vanish for b3 = a2, and B itself if also al = 0, we get kl = x = y = k3, k2 = f = g = 8.
INVARIANTS
AND
NUMBER
97
THEORY.
Thus S = x(A + 1 + A + alcvz + b&3 + l&g) + kraibrba
(97)
adba
+g(A+l+A+
+ kdh(be
+ 1).
First, let n = 4~ + 2 and write 2~ + 1 for m. Then GQl'Qc',
IPGQQ’-~Q~’
have d = @a(& + 1) and atd as the coefficients of &?x$‘. As in 0 25, the coefficients of x, kq, g, k5 in (97) equal respectively d + a@ + b3 + l),
a2d + azba, Ad + a& + a2J,
Ad.
The terms not containing d are combinations of the above a211 and a2(b3 + 1) of 5 23. Any covuriunt with m = 2p -I- 1 > 1, i ers f ram one of rank > m by a linear function of n=4v+2,d$ iK”L”,
11Km-IMQ2’,
IGQl’Q$‘,
K2GQPQ2”
(i = 1, A; 11 = 1, A, J; I = 1, A, A). Next, let n = 4~ > 0. In the last two covariants of the theorem below, the coefficients of f22P+1x+” are a&& + 1) and a = b&@r + 1). We had reached covariants in which the corresponding coefficients are ~221and c&b8 + 1)I. Thus we obtain the coefficient of kq in (97) and 6 + Aa2bs + ablb3, which equals the coefficient of g. We may therefore set kq = g = 0. Subtracting covariants of the fourth and fifth types in the theorem, we may take as S1 the function in 0 24, without disturbing S. Applying (a&(blb2) to S and S1, we get B, and B ,,-I. If b is the coefficient of x~x~~+~x~“+” in xrxsx&, its coefhcient in R’ of 0 30 is 1’ = I+ B, + B-1. Thus B, + B,,-1 is of the form (57). By the coefficient of a3bl, t4=0. Since the coefficient of a3 is zero for b3 = a2, we get z = k5 = ta = 0. Thus S= 0. Any covariant zoith m = 2p + 1 > 1, n = 4v > 0, di$ere from one of rank > m by a linear function of IPL”,
AK”‘L”,
IK*Qz”,
iLLR3Q1@+I, iL-3K2r+2, FGQ2-1Q~U
G2KQ2-1Ql”--l,
(i = 1, -4, A; I = 1, A, J).
98
THE
MADISON
COLLOQUIUM.
32. We have now completed the proof of the theorem: As a fundamental system of modular eovariants of the ternary quadratic form F with integral coejbienls mod& 2, we may take F, its invariants A, A, J, its linear covariant L, 6% “polar ” cubti covariant K, and the universal covarianti Qr, Q2, La.
Incidentally, we have obtained a complete independent covariants of each order and rank. find a complete set of independent syzygies. members are covariants of low rank are given in
set of linearly We might then Syzygies whose (78), (88), (91).
Geometry.-Gther aspects of the modular geometry of quadratic forms modulo 2 and, in particular, applications to theta functions have been considered by Cable.* For a treatment of non-homogeneous quadratic forms in x, y modulo p (p an odd prime), analogous to that of conies in elementary analytic geometry, but employing only real points on the modular locus, see G. AFnoux, Essai de GBom&rie analytique modulaire, Paris, 1911. The earlier paper by Veblen and Bussey was cited in 0 7. The paper by Mitchell was cited in 8 3. Applications of modular geometries have been made by Conwe1l.t The problem of coloring a map has been treated from the standpoint of modular geometry by Veb1en.S * Transactions of Ihe Ammimn Ma&m&cd Society, vol. 14 (1913), pp. 33. References
on Modular
241-276.
t Ann& $ Ann&
of Mathematics, ser. 2, vol. 11 (lQlO), pp. 66-76. of Mathematics, ser. 2, vol. 14 (1912), pp. 86-94.
LECTURE A THEORY OF PLANE CUBIC POINT VALID IN ORDINARY
V
CURVES WITH A REAL INFLEXION AND IN MODULAR GEOMETRY
1. Normal Form of Cubic.-Consider a ternary cubic form C(Z, y, z) with coefficients in a field F not having modulus 2 or 3. After applying a linear transformation with coefficients in F and of determinant unity, we may assume that (1, 0, 0) is an inflexion point. In particular, C lacks the term z3. If it lacks also $2 and $2, its first partial derivatives vanish for y = z = 0. But we shall assume that the discriminant of C is not zero. Hence the coefficient of 9 may be taken as the new variable y. At the inflexion point (1, 0, 0) the tangent y = 0 is to be an inflexion tangent, i. e., meet the cubic in a single point. Hence C lacks the term ~9. Thus c = x3/ + way2
+ By4 + 4(y, 2).
Replacing z by z - cuy- pz, we see that x!$ is now the only term involving Z. If y were a factor, the discriminant would be zero. Hence the term z3 occurs. Adding a suitable multiple of y to z, we get C = x2y + gy3 + hy2z + 6z3 (6 =I= 0). (1) 2. The Invariants s and L-The II=
Hessian of (1) is
- 36Sz - h2y3 + 9SgyZz + 36hyz2.
The sides of an inflexion triangle form a degenerate cubic belonging to the pencil of cubits kC + H. The latter has the factor z only when k = h = 0 and the factor y - lz only when kl = 36 (as shown by the terms in $), where k is a root of P + 186hk2 + 108cS2gk- 27a2h2 = 0. Before considering the factors involving 99
x, we note that the
100
THE
MADISON
COLLOQUIUM.
coefficients of this quartic equation are the values which relative invariants of a general cubic assume for the case of our cubic (1). Indeed, a linear transformation of determinant unity which replaces C by a cubic C’ must replace H by the Hessian H’ of C’, and hence replace the inflexion triangle of C given by a root k of the qua&c by that inflexion triangle of C’ which is given by the same number k. We denote the invariants by* S= - 36/z, t = - 108Pg.
(2)
The above quark
now becomes k4 - 6sk2 - tk - 3f = 0.
(3)
The discriminant A of C is such that 27A = P - 64~~.
(4)
There are four distinct roots of (3) since its discriminant is - 273A2. Our earlier results are that kC + H has the factor z only when k = s = 0 and the factor y - 36k% if k is a root =l=0 of (3). It has the factor x - ry - pz if and only if 3p2 = k,
9a2kr2 = a2+ tk/12,
kp2 - 66~7 = s,
66kpr - 962r2 - sk - t/4 = 0. These conditions are satisfied if and only if k is a root of (3) and p = k = 0, 36J2r2 = - t 3p2 = k,
6Skr = p(p -
38)
(k = 01, (k + 0).
3. The Four Injiexion Triangles.-First, let a = 0. Then t+Oby(4). Th e root lc = 0 gives the inflexion triangle with the sides z = 0, 2 = f 71y (366e,,2 = - t). (5) .* Wehaves = - 345, t = - 3eT, where S and T, given in Salmon’s Higher Plane Curves, p. 189, are the invarianta of the general cubic with multinomial coeflkients.
INVARIANTS
AND NUMBER THEORY.
101
Each root of k3 = t gives an inflexion triangle
Y’$, Next, let a =I=0.
x=
(3.366W=i),
Each root of (3) gives an inflexion triangle x=f
(7)
‘=n(Yf$)
k
4 3
‘+
k2 - 38 66k
4. The Parameter &-If we multiply x, y, z by p, p+, p, we obtain from (1) a cubic with 6 replaced by 6p3. If F is the field of all complex numbers, the field of all real numbers, or the finite field of the residues of integers modulo 3j + 2, a prime, every element is the cube of an element of the field [in the third case, e 5 (e-J)3], so that the parameter 6 may be taken to be unity. Although we do not use the fact below, it is in place to state here that for all further fields a new invariant is needed to distinguish the classes of cubits (1). Indeed, two cubits (l), with coefficients in F and with the same invariants a and t and discriminants not zero, are equivalent under a linear transformation with coefficients in F and having determinant unity if and only if the ratio of their 6’s is the cube of an element of F. CRITERIA
FOR 9, 3 OR 1 REAL INFLEXION
POINTS, $0 5-9
5. Infition Points when a = O.-Let K be a fixed root of k3 = t. Let ~1 and 72 be fixed roots of t.he equations at the end of (5) and (6). Then (71/72)2 = - 3 = (1 + 2@)2, 3 + w + 1 = 0. Choose w so that 71/72 = 1 + 2~. Denote the lines 2: = 0, x = 71y, x = - rly in (5) by L1, L2, La. For each value of i = 0, 1, 2, denote the three lines (6) with k = uoi by LG, Ls, Lsi, that with the lower sign being L3i. Then the 9 inflexion points and the subscripts of the 4 inflexion lines through each are given in the following table:
THE
MADISON
COLLOQUIUM.
(n, 1, 0) (--72, 1, 0)
(n,l,$)I( -d$)
1
1
2
3
20
30
li
li
11
21
31
2,i-
12
22
32
3,i-2
1
2,i-2 3,i--l
In the last two columns, i has the values 0, 1, 2; whilei - 1 or . z - 2 is to be replaced by the number 0, 1, 2 to which it is congruent module 3. When F is the field of all real numbers, K may be taken to be real, while just one of the numbers ~1 and 72 is real. Hence 3 and only 3 of the 9 inflexion points are real. The same result is true if F is the field of the p residues of integers modulo p, where p is a prime 3j + 2 > 2. For, K may be taken to be integral (0 4), while w is imaginary and hence - 3 is a quadratic non-residue of p. If - t is a quadratic residue, 71 is real and 72 imaginary. If - t is a non-residue, the reverse is true. Next, let p = 3j + 1, so that w is real and hence - 3 a quadratic residue. By (5) and (6), 71 and 72 are both real or both imaginary according as - t is a quadratic residue or non-residue of p. Hence all 9 inflexion points are real if and only if - t is both a square and a cube and hence a 6th power module p. If - t is a square but not a cube, only the first 3 inflexion points are real. If - t is a quadratic non-residue, (1, 0, 0) is the only real inflexion point. A with and and
cubic with integral coeficients taken modulo p, a prime > 3, at least one real inflexion point and with invariant e= 0 inva.riant t + 0, haa 9 real injlexion points if p = 3j + 1 - t ti a sixth power module p, a &gle real in$exion point if P = 3j + 1 and - t is a quadratic non-residue of p, and exactly 3 real inJEexion points in all of the remaining cases. For example, if p = 7 and s = 0, t + 0, there are 9 real inflexion points only when t = - 1. Taking 6 = 3, rl= - 2,
--
INVARIANTS
AND
NUMBER
103
THEORY.
l,K=1, we get o = 2. Thus a+$ - yJ + 328 = 0 has the 9 inflexion points (1, 0, 0), (1, 1, 0), (- 1, 1, 0), (-2,1, 3 * 23, (2, 1, 3 * 2i) (; = 0, 1, 2). 6. Inj?.eti and
Points when s =I=0, A $: O.-These
are (1, 0, 0)
(9) where k ranges over the roots of the qua&c (3). We seek the number of real roots k for which m is real. In order that the left member of (3) shall have the factors k2+wk+l,
(10)
k2-wk+m,
it is necessary and sufficient that (11) Let
1+ m - w2 = - 68, (I - m)w = t
+ 0 (for
t
= Osees9).
21= w2 - 6s + t/w,
Inserting
these values into (ll~), we get W6 -
12sw4
lm = - 3a2.
Thenw+Oand
(13)
(13)
t,
2m = w2 - 69 - t/w.
+ 4892w2- t2 = 0.
Set w2 = y + 4s. Then y= = t2 - 64sa = 27A.
(14)
For the rest of this section, let the field be that of the residues of integers module p, where p is an odd prime 3j + 2. Since any integer e has a unique cube root e-j modulo p, there is a single real root y of (14). First, let y + 49 be a quadratic residue of p. Then w is real and hence also 1 and m. The product of the discriminants of the quadratic functions (10) is seen by (111) and (11~) to equal (15)
(d - 4Z)(w2 - 4m) = - 3(~? - 48)2 = - 3y2
and hence is a quadratic non-residue of p. Thus a single one of the quadratics (lo), say the first, has a disc&&ant which is a
-.---- -----.-
_.._.--_-_.c^__-.“._“_ _“..-l-,-“-._~___. ----~..-~- -.
104
THE
MADISON
COLLOQUIUM.
quadratic residue and hence has real roots.
By (121),
41(ti - 4&t+ = - 2we - 6w3t + 36sw4 - 4t2 + 48dw - 144.#w2. Adding the vanishing quantity (16)
4l(w” - 4l)w2 = -
(13), we see that 3(w3
- &SW+ ty.
Since w2 - 41 is a quadratic residue and - 3 is a non-residue of p, it follows that 1 is a non-residue. Hence a single one of the roots of the first quadratic (lo), and hence a single one of the roots of the qua&c (3), is the negative of a quadratic residue. Thus just two of the inflexion points (9) are real. Next, let y + 49 be a quadratic non-residue of p. Then there is no factorizat.ion of the quartic (3) into real quadratic factors. Nor is there a real linear factor k - r and a real irreducible cubic factor. For, if so, the roots of the latter are of the form X, Xp, X** (cf. the first foot-note p. 37). Then (r-X)(r-X*)(r-Xp’),
(modp)
P=(x-XP)(xP--XP”)(XP*-X)~Ppp
are real, so that the discriminant of (3) is a quadratic residue. But this discriminant was seen to be - 3(81A)2, and - 3*is a non-residue. Hence (3) is irreducible modulo p. Thus (1, 0, 0) is the only real inflexion point. For p = 3j + 2 > 2, a cubic (1) with stA =I=0, )LLY exactly three real &flex&m
points
or a single one according residue or non-residue
number 3A4 -I- 49 W a quadratic
7. C&ic with &A =I=0, p = 3j + l.-Now residue of p and there are three real cube roots modulo p. In this section we shall assume that A is a Then there are three real roots yi of (14). At yi + 49 is a quadratic residue of p since
aa the real of p.
3 is a quadratic 1, o, w2 of unity cube modulo p. least one of the
a (yi + 48) = J/l3 + MS3 = t2. If yr + 49 is a quadratic residue, while y2 + 49 and y3 + 48
INVARIANTS
AND
NUMBER
105
THEORY.
are non-residues, there is a single factorization of quark (3) into real quadratics (10) and hence certainly not four real roots. The product (15) of the discriminants of the real quadratic factors is now a quadratic residue of p. If each were a residue, there would be four real roots. Hence each is a non-residue and there is no real root. There is a single real in$exion point if p = 3j -I- 1, &A + 0, A is a cube, and if the three values of 3A*$49 are not all quadratic
residues of p.
Next, let each yi + 4.9be a quadratic residue of p. Then there are three ways of factoring quartic (3) into real quadratics (10). But a root common to two distinct real quadratics is real. Hence all four roots are real. The discriminant of each quadratic (10) is therefore a quadratic residue of p. Hence, by (IS), I is a quadratic residue of p; similarly for the constant term of each quadratic factor. Thus the negatives of the four roots are all quadratic residues or all non-residues. To decide between these alternatives, we need the actual roots. In z0,2 = yi + 43, let the signs of the wi be chosen so that k2 -Wik+??Zi=
have a common root.
(i = 1, 2, 3)
0
As in (12), 2mi
= Wz -
68
For e =l=1, we find by subtraction that
-
t/Wi.
and cancellation of WI - we
2k = Wl + we + tl(wJc>.
Comparing the results for e = 2 and e = 3, we get WlW2WQ= t.
(17)
Hence* the roots of (3) are 3(wl+
w2
+
w3),
6.
w2 and wa, if necessary, we have WI2 +
cdw22+
w2w32
3
0
(mod
P>.
Conversely, if the w? are any quadratic residues satisfying (20) and if we define 9 and t by (19J and (17), we obtain a qua&c (3) with the four real roots (18). If we permute ~01, wa, 208 cyclically we obtain solutions of (20) leading to the same 8 and t and to the same four roots (18). Our first problem is therefore to find all sets of solutions of (20). To this end it is necessary to treat separately the cases - 1 a quadratic residue and - 1 a non-residue; viz., p = 12q+ 1 and p = 12q + 7 (since already p = 3j + 1). First, let p = 12q + 1. Then - 1s zY(mod p), where i is an integer. Set 2p = 201- iwwa, 2c = WI+- iuw3. Then (20) becomes 4pu = - cow22= (i&w#, so that pu must be a quadratic residue.
Hence we may take
INVARIANTS
AND
NUMBER
107
THEORY.
u = pF, where p and 1 are integers not divisible by p. Wl = p(l + P),
(21)
wq = 2iwpz,
wg = ?Yp(l
Then
- I”).
We must exclude the values of I which lead to equal values of two of the wi2, and hence to equal yi)s, since the roots of (14) are incongruent. Now if any two of the w? in (20) are congruent, all three are congruent. But wr2 = w22implies 1+12=
*2iwZ,
(Z=~=iw)~=w~,
I=
+b+eew2
(8s
1).
The values 12= 0, f 1 make one of the wi = 0. Hence we must exclude the 9 incongruent integral values of I: (22)
1 = 0,
f 1,
f ;., w2f iw,
- 02 f iw.
Using the values (21), we get (23) (24)
129 = p2{(1 - w)(l + Z4)- 6w2P}, t = 2p*Z(Z4 - l), =
Hwl+w2+w)
$Pu+ioa)(l+$&)p.
To make the negative of the last a. square, we must take (25)
p = -
2(1 + S)r2
(r + 0).
Now s, given by (23), is zero only when W-9
l=wzt.2,
-wzt&$.
The desired sets s, t are given by (23) integer not &tXble by p, while 1 i.9 any integers < p not congruent m0h.410 p to integers (22), (26). The minimum p is
and (25), where r is any one of the p - 13 positive one of tke 13 incongrwmt 37.
Second, let p = 12q + 7. Then X2 = - 1 (mod p) is irreducible. Its roots i and - i = ip are Galois imaginaries. Set (27)
7r=p+1,
o=p-1.
There exists a linear function R of i with integral coefficients such that R”” = 1, whiIe no lower power of R is unity. Any function of i is zero or a power of R and any integer is a power of
108
THE
R”, a primitive
MADISON
root of p.
&‘wt = R=“,
COLLOQUIUM.
Hence we may set
WI+ ww3i = R”,
where 0 = re = 2rq + +m The last condition is equivalent
(mod KU).
to
e = 27 + a/2 +ju
(2%
(0 2 j < T).
We have w2 = wR”“,
2wl= Re+Rpe,
2w2Zw1 = 2Rn”+
2w3 = -id’(R”-
(co2- iw)R”+
Rpa),
(u2+iu)R~*,
(co”- ;w)(w” + 720)= - 1, (cl.9- &,,)= = - 1, u2 - iw = R/c’/2 = ~R”‘I + &+f~ I2 _ R~doI2
(29) 2&zwI
=
~-3-pWkl)
R~bYf+l~l2l(R
12 +
(f odd),
Rm-d-W-1)
/2)2.
The last binomial is its own pth power and hence is real. We desire that the root @wI shall be the negative of a quadratic residue and hence a non-residue. Since R” is a primitive root of p, the condition is that j - cf + 1)/2 shall be odd: f = 21-
(30)
1, j - I=
odd.
We must exclude the values making w12= wz2: 0 = 2R~12(wl
=,= w2)
=
R2l)-hfjo
=,=
&,fp~k-~P
-
R%v-j~,
the second term having been simplified by use of R-12
=
-
1,
RP*
=
&-=.
Completing the square of the first two terms, we get (Rvl-~Wl)/2 F uRm--oj/2)2 = (o2 +
l)R2m-~j.
(c&o~)~,where c = 1 or - 1. Hence
NowW2+1=-U= Rv+‘(i+l)/2
=
(&
w +
gu2)Rm-ffi/2.
INVARIANTS
AND
NUMRER
109
THEORY.
But (u + ;w”)(u - iu2) = - 1, w + ,iw2= R”“‘2, (31)
a-h2=
-R-““/2
(v odd).
Hence we must exclude the four cases in which (32)
?J= j + 2p= v + l),
j + $(f
v + 7r + 1)
(mod 4,
these four values being incongruent. No one of the w’s in (29) is zero, since e is odd by (28), so that e + 0, a/2 (mod n). By (191) and (17), 489 = (1 - w)(R2” + R2p”) + 6W2R2=“, (33)
4t =
_
iR=l)(R26
_
PP”).
Finally, we must here exclude the cases in which a = 0. Combining Zwr2 = 0 with (20), we obtain the necessary and sufficient condition w12 = owa2 for s = 0. But w1 = f w2w3, in connection with (29), gives R”(1 f ;w) = R=‘“(-
1 f ;w),
R”(w f
3)”
= Rpb.
Thus, by (31), the condition is that e f vu = pe (mod nu) or e = f v (mod x). Then, by (28), q is congruent modulo ?r to one of the values (32) decreased by x/4. Hence the desired sets s, t are given by (33), subject to (28), in which the 8 incongruent $8 given by (32) and those values decreased by 7r/4 are excluded. In particular, p > 7.
For p = 19, the only admissible pairs are s = 2 - 22z, t = 6(-
2)3z (1 = 0, 1, . . ., 8).
For any 2,the negatives of the roots of quartic (3) are the products of - 3 = 42, 4, 7 = g2, - 8 = 72 by (- 2) z and hence are quadratic residues of 19 since - 2 = 62. For p = 31, the only pairs are CT= 321, t=5(-3)3l;
s=-3321,
t=13(-3)3’
(I = 0, * * a, 15),
the negatives of the roots of (3) being the products of 7, - 11, - 12, - 15 and - 3, 5, 9, - 11, respectively, by (- 3) z, and hence are quadratic residues of 31.
110
THE
MADISON
COLLOQUITJM.
8. Casep= 3j + 1, &A $1 0, A not a Cube.-The roots of (14) are now Galois imaginaries y, yp, 9~‘. As at the beginning of 0 7,
t2 = (y + 48)(yp
+
+f8)(yp*
+
48) =
(y +
48)““p”.
Raise each member to the power (p - 1)/2. We see that y + 48 is the square of an element, say w, of the Galois field of order ~3. The 6rst root (18) is $(w + wp -I- wP7 and equals its own pth power, and hence is real. This is not true of the remaining roots (18), since WP =I=w, or since a real quadratic factor would imply that w is real. Hence the quart& haa a single real root. For p = 7, the only cases in which the negative of the single real root is a quadratic residue are t = - 1 or 3, a = - 1, - 2,3; t = 2, a arbitrary =/=0. For p = 13, the only cases are +t= 4, 5, 6; a = - 1, - 3, 4 (83 = - 1); *t= and f
t
1, 5, 6; s = - 2, - 5, - 6
(8’ =
5);
= 3, - s equals one of the preceding six values of 8.
9. Cubic z&h
t
= 0, s =i=O.-In
this case, (3) becomes
(P - 38)2 = 1282. If there be a real root k, 3 is a quadratic residue of p, and P=ls,
1=3*2&a
First, let p = 3j + 2, so that - 3 is a quadratic non-residue of p. Then - 1 must be a non-residue of p and hence p = 12~+ 11. The product of the two Z’s is - 3, so that a single value of P is a quadratic residue. Since the two real k’s are of opposite sign, there is a single real root k whose negative is a quadratic residue. For t = 0, s + 0, and p = 12r + 5, there is a tingle real in.xion points. point; for p = 12~ + 11, thme are just three real infiti Finally, let p = 3j + 1, so that - 3 is a quadratic residue of p. If p = 12r + 7, then 3 is a non-residue, and there is no real k and hence a single real inflexion point. If p = 127 + 1, the four roots k are all real or all imaginary. For p = 13, h? z - 28 or - 53, and - k is a quadratic residue if and only if k6 = 1, 83 G 8, s = 2, 5, 6. For p = 37, k2 = - 49 or loS, and - k is a residue if and only if s* = 1. The end
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extraneous material here!-only the theories, equations, and operations essential and immediately useful for radio work. Can be used as refresher, as handbook of applications and tables, or as full home-study course. Ranges from simplest arithmetic through calculus? series, and wave forms! hyperbolic trigonometry, simultaneous equations in mesh circuds, etc. Supplies applicahons right along with each math topic discussed. 22 useful tables of functions, formulas, logs, etc. Index. 166 exercises, 140 examples, all with answers. 95 diagrams. Bibliography. x + 336~~. 5% x 8. S141 Paperbound $1.75
Algebra, group theory, determinants, sets, matrix theory ALGEBRAS AND THEIR ARITHMETICS, 1. E. Dickson. Provides the foundation and background necessary to any advanced undergraduate or graduate student studying abstract algebra. Begins with elementary introduction to linear transformations, matrices, field of complex numbers; proceeds to order, basal units, modulus, quaternions, etc.; develops calculus of linears sets, describes various examples of algebras including invariant, difference, nilpotent, semi-simple. “Makes the reader marvel at his genius for clear and profound analysis,” Amer. Mathematical Monthly. Index. xii + 241~~. 5% x 8. S616 Paperbound $1.50
THEORY OF EGUATIONS WITH AH INTRODUCTION TO THE THEORY OF BINARY ALGEBRAIC FORMS, W. S. Burnside and A. W. Panton. Extremely thorough and concrete discussion of the theory of equations, with extensive detailed treatment of many topics curtailed in later texts. Covers theory of algebraic equations, properties of polynomials, symmetric functions, derived functions, Homer’s process, complex numbers and the complex variable, determinants and methods of elimination, invariant theory (nearly 100 pages), transformations, introduction to Galois theory, Abelian equations, and much more. Invaluable supplementary work for modern students and teachers. 759 examples and exercises. Index in each volume. TWO volume set. Total of xxiv + 604~~. 5% x 8. S714 Vol I Paperbound 1.85 S715 Vol II Paperbound 1.85 The set f 3.70 THE
COMPUTATIONAL METHODS OF LINEAR ALGEBRA, V. N. Faddaeva, translated by C. 0. Benster. First English translation of a unique and valuable work, the only work in English presenting a systematic exposition of the most important methods of linear algebra-classical and contemporary. Shows in detail how to derive numerical solutions of problems in mathematical physics which are frequently connected with those of linear algebra. Theory as well as individual practice. Part I surveys the mathematical background that Is indispensable to what follows. Parts II and Ill, the conclusion, set forth the most important methods of solution, for both exact and iterative groups. One of the most outstanding and valuable features of this work is the 23 tables, double and triple checked for accuracy. These tables will not be found elsewhere. Author’s preface. Translator’s note. New bibliography and index. x + 252~~. 5% x 8. S424 Paperbound $1.95 ALGEBRAIC EGUATIONS,, E. Oehn. Careful and complete presentation of Galois’ theory of algebraic equations; theorres of Lagrange and Galois developed in logical rather than historical form, with a more thorough exposition than in most modern books. Many concrete applications and fully-worked-out examples. Discusses basic theory (very clear exposition of the symmetric group); isomorphic, transitive, and Abelian groups; applications of Lagrange’s and Galois’ theories; and much more. Newly revised by the author. Index. List of Theorems. xi + 208~~. 5% x 8. S697 Paperbound $1.45
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ALGEBRAIC THEORIES. 1. E. Dickson. Best thorough introduction to classical topics in higher algebra develops theories centering around matrices, invariant!, groups. Higher algebra, Galois theory, finite linear groups, Klein’s icosahedron, algebraic invariants, linear transelementary divisors, Invariant factors; quadratic, bi-linear, Hermitian forms, formations, singly and in pairs. Proofs rigorous, detailed; topics developed lucidly, in close connection with their most frequent mathematical applications. Formerly “Modern Algebraic Theories.” 155 problems. Bibliography. 2 indexes. 285~~. 5% x 8. S547 Paperbound $1.50 LECTURES ON THE ICOSAHEDRON AND THE SOLUTION OF EQUATIONS OF THE FIFTH DEGREE. Felix Klein. The solution of quintics in terms of rotation of a regular icosahedron around its axes of symmetry. A classic & indispensable source for those interested in higher algebra, geometry, crystallography. Considerable explanatory material included. 230 footnotes, mostly bibliographic. 2nd edition, xvi + 289pp. 5% x 8. S314 Paperbound $2.25 LINEAR GROUPS, WITH AN EXPOSITION OF THE GALOIS FIELD THEORY, 1. E. Dickson. The classic exposition of the theory of groups, well within the range of the graduate student. Part I contains the most extensive and thorough presentation of the theory of Galois Fields available, with a wealth of examples and theorems. Part II is a full discussion of linear groups of finite order. Much material in this work is based on Dickson’s own contributions. Also Includes expositions of Jordan, Lie, Abel, Betti-Mathieu, Hermite, etc. “A milestone in the development of modern algebra,” W. Magnus, in his historical introduction to this edition. Index. xv + 312~~. 5% x 8. S482 Paperbound $l.SS INTRODUCTION TO THE THEORY OF GROUPS OF FINITE ORDER, R. Carmichael. Examines fundamental theorems and their application. Beginning with sets, systems, permutations, etc., it progresses in easy stages through important types of groups: Abelian, prime power, permutation, etc. Except 1 chapter where matrices are desirable, no higher math needed. 783 exercises, problems. Index. xvi + 447~~. 5% x 8. S3DO Paperbound $2.25 THEORY OF this is still independent itself, Abelian substitution,
GROUPS OF FINITE ORDER, W. Burnside. First published some 40 years ago, one of the clearest introductory texts. Partial contents: permutations, groups of representation, composition series of a group, isomorphism of a group with groups, prime power groups, permutation groups, invariants of groups of linear graphical representation, etc. 45pp. of notes. Indexes. xxiv + 512~~. 5H x 8. S38 Paperbound $2.75
CONTINUOUS GROUPS OF TRANSFORMATIONS, 1. P. Eisenhart. Intensive study of the theory and geometrical applications of continuous groups of transformations; a standard work on the subject, called forth by the revolution in physics in the 1920’s. Covers tensor analysis, Riemannian geometry, canonical parameters, transitivity, imprimitivity, differential invariants, the algebra of constants of structure, differential geometry, contact transformations,, etc. “Likely to remain one of the standard works on the subject for many years . . . prmcipel theorems are proved clearly and concisely, and the arrangement of the whole is coherent,” MATHEMATICAL GAZETTE. Index. 72-item bibliography. 185 exercises. ix + 301~11. 5% x 8. S781 Paperbound $2.00 THE THEORY OF GROUPS AND PUANTUM MECHANICS, H. Weyl. Discussions of Schroedinger’s wave equation, de Broglie’s waves of a particle, Jordan-Hoelder theorem, Lie’s continuous groups of transformations, Pauli exclusion principle, quantization of Maxwell-Dirac field equations, etc. Unitary geometry, quantum theory, groups, application of groups to quantum mechanics, symmetry permutation group, algebra of symmetric transformation, etc. ?nd revised edition. Bibliography. Index. xxii + 422~~. 5% x 8. S269 Paperbound $2.35 APPLIED GROUP-THEORETIC AND MATRIX METHODS, treatment of group and matrix theory for the physical easily-followed exposition of the basic ideas of group its applications in the various areas of physics and quantum theory, molecular structure and spectra, Includes rigorous proofs available only in works of figures, numerous tables. Bibliography. Index. xiii +
Bryan H&man. The first systematic scientist. Contains a comprehensive, theory (realized through matrices) and chem,stry: tensor analysis, relativity, and Eddington’s quantum relativity. a far more advanced character. 34 454~~. 53/e x 8% S1147 Paperbound $2.50
THE THEORY OF GROUP REPRESENTATIONS, Francis D. Murnaghan. A comprehensive introduction to the theory of group representations. Particular attention is devoted to those groups-mainly the symmetric and rotation groups-which have proved to be of fundamental significance for quantum mechanics (esp. nuclear physics). Also a valuable contribution to the literature on matrices, since the usual representations of groups are groups of matrices. Covers the theory of group integration (as developed by Schur and Weyl), the theory of 2-valued or spin representations, the representations of the symmetric group, the crystallographic groups, the Lorentz group, reducibility (Schur’s lemma, Burnside’s Theorem, etc.), the alternating group, linear groups, the orthogonal group, etc. Index. List of references. xi + 369pp. 5% x 8%. S1112 Paperbound $2.35 THEORY OF SETS, E. Kamke. Clearest, pendent study. Subdivision of main but emphasis is on general theory. and their cgrdinal numbers, ordered cardinal numbers. Bibliography. Key
amplest introduction in English, well suited for indetheory, such as theory of sets of points, are discussed, Partial contents: rudiments of set theory, arbitrary Sat.5 sets and their order types, well-ordered sets and thelr to symbols. Index. vii + 144~~. 5% x 8. S141 Paperbound $135
THE THEORY OF DETERMINANTS, MATRICES., AND INVARIANTS, H. W. Tumbull. Important study includes all salient features and maJor theories. 7 chapters on determinants and matrices cover fundamental properties, Laplace identities, multiplication, linear equations, rank and differentiation, etc. Sections on invariants gives general properties, symbolic and direct methods of reduction, binary and polar forms, general linear transformation, first fundamental theorem, multilinear forms. Following chapters study development and proof of Hilbert’s Basis Theorem, Gordan-Hilbert Finiteness Theorem, Clebsch’s Theorem, and include discussions of apolarity, canonical forms, geometrical Interpretations of algebraic forms, complete system of the general quadric, etc. New preface and appendix. Biblio raphy. xviii + 374pp. 5% x 8. S699 Paperboun f $2.25 AN INTRODUCTION TO THE THEORY OF CANONICAL MATRICES, H. W. Turnbull and A. C. Aitken. All principal aspects of the theory of canonical matrices, from definitions and fundamental roperties of matrices to the practical applications of their reduction to canonical form. 1 egrnnmg . with matrix multiplications, reciprocals, and partitioned matrices, the authors go on to elementary transformations and bilinear and quadratic forms. Also covers such topics as a rational canonical form for the collineatory group, congruent and conjunctive transformation for quadratic and hermitian forms, unitary and orthogonal transformations, canonical reduction of pencils of matrices, etc. Index. Appendix. Historical notes at chapter ends. Bibliographies. 275 problems. xiv + 200~~. 5% x 8. S177 Paperbound $1.55 A TREATISE ON THE THEORY OF DETERMINANTS,,T. Muir. Unequalled as an exhaustive compilation of nearly all the known facts about determmants up to the early 1930’s. Covers notation and general properties, row and column transformation, symmetry, compound determinants, adjugates, rectangular arrays and matrices, linear dependence, gradrents, Jacobians, Hessians, Wronskians, and much more. Invaluable for libraries of industrial and research organizations as well as for student, teacher, and mathematician: very useful in the field of computing machines. Revised and enlarged by W. H. Metzler. Index. 485 problems and scores of numerical examples. iv + 766~~. 5% x 8. S670 Paperbound $3.00 THEORY OF DETERMINANTS IN THE HISTORICAL ORDER OF DEVELOPMENT, Slr Thomas Muir. Unabridged reprinting of this complete study of 1,859 papers on determmant theory written between 1693 and 1900. Most important and orrginal sections reproduced, valuable commentary on each. No other work is necessary for determinant research: all types are coveredeach subdivision of the theory treated separately; all papers dealing with each type are covered; you are told exactly what each paper is about and how important its contribution is. Each result, theory, extension, or modification is assigned its own identifying numeral so that the full history may be more easily followed. Includes papers on determinants in general, determinants and linear equations, symmetric determinants, alternants, recurrents, determinants having invariant factors, and all other major types. “A model of what such histories ought to be,” NATURE. “Mathematicians must ever be grateful to Sir Thomas for his monumental work,” AMERICAN MATH MONTHLY. Four volumes bound as two. Indices. Bibliop raphies. Total of lxxxiv + 1977pp. 53,‘s x 8. S672-3 The set, Clothbound $12.50
Calculus and function theory, Fourier theory, infinite series, calculus of variations, real and complex functions FIVE VOLUME
“THEORY
OF FUNCTIONS’
SET BY KONRAO
KNOPP
This five-volume set, prepared by Konrad Knopp, provides a complete and readily followed account of theory of functions. Proofs are given concisely, yet without sacrifice of completeness or rigor. These volumes are used as texts by such umversities as M.I.T., University of Chicago, N. Y. City College, and many others. “Excellent introduction remarkably readable, concise, clear, rigorous,” JOURNAL 0~ THE AMERICAN STATISTICAL ASSOCIATION. ELEMENTS OF THE THEORY OF FUNCTIONS, Konrad Knopp. This book provides the student with background for further volumes in this set, or texts on a similar level. Partial contents: foundations, system of complex numbers and the Gaussian plane of numbers, Riemann sphere of numbers, mapping by linear functions, normal forms, the logarithm, the cyclometric functions and binomial series. “Not only for the young student, but also for the student who knows all about what is in it,” MATHEMATICAL JOURNAL. Bibliography. Index. 140~~. 5% x 8. S154 Paperbound $1.35
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Application and further development of general theory, special topics. Single valued functions, entire Weierstrass Meromorphtc functions. Riemann surfaces. Algebraic functions. Analytical con/iguration, Riimann surface. Bibliography. Index. X -I 150PP. 5% x 8. S157 Paperbound $1.35
THEORY
OF
FUNCTIONS,
PART
II,
Konrad
Knopp.
PROBLEM BOOK IN THE THEORY OF FUNCTIONS, VOLUME 1, Konrad Knopp. Problems In elementary theory, for ,use with Knopp’s THEORY OF FUNCTIONS, or any other text, arranged according to rncreasmg difficulty. Fundamental concepts, sequences of numbers and infinite series, complex variable, integral theorems, development in series, conformal mapping. 182 problems. Answers. viii + 126~~. 5% x 8. S158 Paperbound $1.35 PROBLEM BOOK IN THE THEORY OF FUNCTIONS, VOLUME 2, Konrad Knopp. Advanced theory of functions, to be used either with Knopp’s THEORY OF FUNCTIONS, or any other comparable text. Singularities, entire & meromorphic functions, periodic, analytic, continuation, multiple-valued functions, Riemann surfaces, conformal mapping. Includes a section of additional elementary problems. “ The difficult task of selecting from the immense material of the modern theory of functions the problems just within the reach of the beginner is here masterfully accomplished,” AM. MATH. SOC. Answers. 138~~. 5% x 8. S159 Paperbound $1.35
Vol. 2 part 2 S556 Paperbound $1.85 3 vol. set $8.26 MODERN THEORIES OF iNTEGRATION, H. Kestelman. Connected and concrete coverage, with fully-worked-out proofs for every step. Ranges from elementary definitions through theory of aggregates, sets of points, Rremann and Lebesgue integration, and much more. This new revised and enlarged edrtion contains a new chapter on Riemann-Stieltjes integration, as well as a supplementary section of 186 exercises. Ideal for the mathematician, student, teacher, or self-studier. Index of Definitions and Symbols. General Index. Bibliography. x + 310~~. 5% x 8%. S572 Paperbound $2.25 OF MAXIMA AND MINIMA, H. Hanoook. Fullest treatment ever written; only work in English with extended discussion of maxima and minima for functions of l! 2, or n variables, problems with subsidiary constraints, and relevant quadratic forms. Detarled proof of each important theorem. Covers the Scheeffer and von Dantscher theories, homogeneous quadratic forms, reversion of series, fallacious establishment of maxima and minima etc. Unsurpassed treatise for advanced students of calculus, mathematicians, economists, siatisticians. Index. 24 diagrams. 39 problems, many examples. 193pp. 5% x 8. S665 Paperbound $1.50 THEORY
AN ELEMENTARY TREATISE ON ELLIPTIC FUNCTIONS, A. Cayley. Still the fullest and clearest text on the theories of Jacobi and Legendre for the advanced student (and an excellent supplement for the beginner). A masterpiece of exposition by the great 19th century British mathematician (creator of the theory of matrices and abstract geometry), it covers the addition-theory, Landen’s theorem, the 3 kinds of elliptic integrals, transformations, the q-functions, reduction of a differential expression, and much more. Index. xii + 386~~. 5% x 8. S728 Paperbound $2.00 THE APPLICATIONS OF ELLIPTIC FUNCTIONS, A. G. Greenhill. Modern books forego detail for sake of brevity-this book offers complete exposition necessary for proper understanding, use of elliptic integrals. Formulas developed from definite physical, geometric problems; examples representative enough to offer basic information in widely useable form. Elliptic integrals, addition theorem, algebraical form of addition theorem, elliptic integrals of 2nd, 3rd kind, double periodicity, resolution into factors, series, transformation, etc. Introduction. Index. 25 illus. xi + 357~~. 5% x 8. S603 Paperbound $1.75 THE
THEORY
OF
FUNCTIONS
OF
REAL
VARIABLES,
James
Pierpont.
A
2-volume
authoritative
exposition, by one of the foremost mathematicians of his time. Each theorem stated with all conditions, then followed by proof. No need to go through complicated reasoning to discover conditions added without specific mention. Includes a particularly complete, rigorous presentation of theory of measure; and Pierpont’s own work on a theory of Lebesgue integrals, and treatment of area of a curved surface. Partial contents, Vol. 1: rational numbers, exponentials, logarithms, point aggregates, maxima, minima, proper integrals, improper integrals, multiple proper integrals, continuity, discontinuity, indeterminate forms. Vol. 2: point sets, proper integrals, series, power series, aggregates, ordinal numbers, discontinuous functions, sub-, infra-unrform convergence, much more. Index. 95 illustrations. 1229pp. 5% x 8. S558-9, 2 volume set, paperbound $5.20
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ELEMENTS OF THE THEORY OF REAL FUNCTIONS, 1. E. Littlewood. Based on lectures given at Trinity College, Cambridge, this book has proved to be extremely successful In introaucing graduate students to the modern theory of functions. It offers a full and concise covera e of classes and cardinal numbers, well-ordered series, other types of series, and elemen ‘i s of the theory of sets of points. 3rd revised edition. vi1 + 71pp. 5% x 8. S171 Clothbound 2.85 S172 Paperbound f 1.25 TRANSCENDENTAL AND ALGEBRAIC NUMBERS, A. 0. ttelfond. First English translation of work by leading Soviet mathematician. Thue-Siegel theorem, its p-adic analogue, on approximation of algebraic numbers by numbers in fixed algebraic field; Hermite-Lindemann theorem on transcendency of Bessel functions, solutions of other differential equations; Gelfond-Schneider theorem on transcendency of alpha to power beta: Schneider’s work on elliptic functions, ;&hxmgethod developed by Gelfond. Translated by L. F. Boron. Index. Bibliography. 200~~. S615 Paperbound $1.75 ELLIPTIC INTEGRALS, N. Hancock. Invaluable in work involving differential equations containing cubits or quartrcs under the root sign, where elementary calculus methods are inade. quate. Practical solutions to problems that occur in mathematics, engineering, physics: differential equations requiring integration of Lame’s, Briot’s, or Bouquet’s equations; determination of arc of ellipse, hyperbola, lemniscate; solutions of problems in elastica; motion of a projectile under resistance varying as the cube of the velocity; pendulums; many others. Exposition is in accordance with Legendre-Jacobi theory and includes rigorous dfscussion of Legendre transformations. 20 figures. 5 place table. Index. 104~~. 5% x 8. S484 Paperbound $1.25 LECTURES ON THE THEORY OF ELLIPTIC FUNCTIONS, H. Hancock. Reissue of the only book in English with so extensive a coverage, especially of Abel, Jacobi, Legendre, Weierstrasse, Hermite, Liouville, and Riemann. Unusual fullness of treatment, plus applications as well as theory, in discussing elliptic function (the universe of elliptic integrals originating in works of Abel and Jacobi), their existence, and ultimate meaning. Use is made of Riemann to provide the most general theory. 40 page table of formulas. 76 figures. xxiii + 498pp. S483 Paperbound $2.55
ALMOST PERIODIC FUNCTIONS, A. S. Besicovitch. This unique and important summary by a well-known mathematician covers in detail the two stages of development in Bohr’s theory of almost periodic functions: (1) as a generalization of pure periodicity, with results and proofs; (2) the work done by Stepanoff, Wiener, Weyl, and Bohr in generalizing the theory. Bibliography. xi + 180~~. 5% x 8. S18 Paperbound $1.75 THE ANALYTICAL THEORY OF HEAT, Joseph Fourier. This book, which revolutionized mathematical physics, is listed in the Great Books program, and many other listings of great books. It has been used with profit by generations of mathematicians and physicists who are interested in either heat or in the application of the Fourier integral. Covers cause and reflection of rays of heat, radiant heating, heating of closed spaces, use of trigonometric series in the theory of heat, Fourier integral, etc. Translated by Alexander Freeman. 20 S93 Paperbound $2.50 figures. xxii + 466~~. 5% x 8. AN INTRODUCTION TO FOURIER METHODS AND THE LAPLACE TRANSFORMATION, Philip Franklin. Concentrates upon essentials, enabling the reader with only a working knowledge of calculus to gain an understanding of Fourier methods in a broad sense, suitable for most applications. This work covers complex qualities with methods of computing elementary functions for complex values of the argument and finding approximations by the use of charts; Fourier series and integrals with half-range and complex Fourier series; harmonic analysis; Fourier and Laplace transformations, etc.; partial differential equations with applications to transmission of electricity; etc. The methods developed are related to physical problems of heat flow, vibrations, electrical transmission, electromagnetic radiation, etc. 828 problems with answers. Formerly entitled “Fourier Methods.” Bibliography. Index. x + 289pp. 5% x 8. S452 Paperbound $2.00 THE FOURIER INTEGRAL AND CERTAIN OF ITS APPLICATIONS, Norbert Wiener. The only booklength study of the Fourier integral as link between pure and applied math. An expansion of lectures given at Cambridge. Partial contents: Plancherel’s theorem, general Tauberian theorem, special Tauberian theorems, generalized harmonic analysis. Bibliography. viii -IS272 Paperbound $1.50 201~1. 5% x 8.
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INTRODUCTION TO THE THEORY OF FOURIER’S SERIES AND INTEGRALS, Ii. S. Carrlaw. 3rd revised edition. This excellent introduction is an outgrowth of the author’s courses at Cambrid e. Historical introduction, rational and irrational numbers, infinite sequences and series, f unctrons of a single variable, definite integral, Fourier series, Fourier integrals, and . similar topics. Appendixes discuss ractical harmonic analysis, periodogram analysis. Lebesfi rbllography. gue’s theory. Indexes. 84 examples, xii + 368~~. 5% x 8. S48 Paperbound $2.25 FOURIER’S SERIES AND SPHERICAL HARMONICS, W. E. Byerly. Continues to be recognized as one of most practical, useful expositions. Functions, series, and their differential equations are concretely explained in great detail; theory is applied constantly to practical problems, which are fully and lucidly worked out. Appendix includes 6 tables of surface zonal harmonics, hyperbolic functions, Bessel’s functions. Bibliography. 190 problems, a proximately half with answers. ix + 287~~. 5% x 8. S536 Paper 1 ound $1.75 INFINITE SEGUENCES AND SERIES, Konrad Knopp. First publication in any language! Excellent introduction to 2 topics of modern mathematics, designed to give the student background to penetrate farther by himself. Sequences & sets, real & complex numbers, etc. Functions of a real & complex variable. Sequences G series. Infinite series. Convergent power series. Expansion of elementary functions. Numerical evaluation of series. Bibliography. v + 186~~. 5% x 8. S153 Paperbound $1.75 TRI6ONOMETRICAL SERIES, Antoni Zygmund. Unique in any level. Contains carefully organized analyses of trigonometric, functions, with clear adequate descriptions of summability theory, conjugate series, convergence, divergence of Fourier Russian, Eastern European coverage. Bibliography. 329pp. 5% x
language on modern advanced orthogonal, Fourier systems of of Fourier series, proximation series. Especially valuable for 8. S290 Paperbound $2.00
DICTIONARY OF CONFORMAL REPRESENTATIONS, H. Kober. La lace’s equation in 2 dimensions solved in this unique book developed by the British Admira Pty. Scores of geometrical forms (P their transformations for electrical engineers, Joukowski aerofoil for aerodynamists. Schwarz-Christoffel transformations for hydrodynamics, transcendental functions. Contents classified according to analytical functions describing transformation. Twin diagrams show curves of most transformations with corresponding regions. Glossary. Topological index. 447 dlagrams. 244~~. 61/s x 9%. S160 Paperbound $2.00 CALCULUS OF VARIATIONS, A. R. Forsyth. Methods, solutions, rather than determination of weakest valid hypotheses. Over 150 examples completely worked-out show use of Euler, Legendre, Jacobi, Weierstrass tests for maxima, minima. Integrals with one ori inal dependent variable: with derivatives of 2nd order; two dependent variables, one in f ependent variable; double Integrals involving 1 dependent variable, 2 first derivatives; double integrals involving partial derivatives of 2nd order; triple integrals; much more. 50 diagrams. 678~~. S622 Paperbound $2.95 5% x 8%. LECTURES ON THE CALCULUS OF VARIATIONS, 0. Bolu. Analyzes in detail the fundamental concepts of the calculus of variations, as developed from Euler to Bilbert, with sharp formulations of the problems and rigorous demonstrations of their solutions. More than a score of solved examples; systematic references for each theorem. Covers the necessary and sufficient conditions; the contributions made by Euler, Du Bois-Reymond, Hilbert, Weierstrass, Legendre, Jacobi, Erdmann, Kneser, and Gauss; and much more. Index. Bibliography. xi + 271~~. 5% x 8. S218 Paperbound $1.65 A TREATISE ON THE CALCULUS OF FINITE DIFFERENCES, 6. Boole. A classic in the literature of the calculus. Thorough, clear discussion of basic principles, theorems, methods. Covers MacLaurin’s and Herschel’s theorems, mechanical quadrature, factorials, periodical constants, Bernoulli’s numbers, difference-equations (linear, mixed, and partial), etc. Stresses analogies with differential calculus. 236 problems, answers to the numerical ones. viii + 336~~. S695 Paperbound $1.55 5% x a.
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