Journal fu¨r die reine und angewandte Mathematik

J. reine angew. Math. 577 (2004), 153—169

( Walter de Gruyter Berlin
New York 2004

The rank of elliptic surfaces in unramified abelian towers By Joseph H. Silverman at Providence

Abstract. Let E ! C be an elliptic surface defined over a number field K. For a finite covering C 0 ! C defined over K, let E 0 ¼ E C C 0 be the corresponding elliptic surface over C 0 . In this paper we give a strong upper bound for the rank of E 0 ðC 0 =KÞ in the case of geometrically abelian unramified coverings C 0 ! C and under the assumption that the Tate conjecture is true for E 0 =K. In the case that C is an elliptic curve
 and the map C 0 ¼ C ! C is the multiplication-by-n map, the bound for rank E 0 ðC 0 =KÞ takes the form Oðn k=log log n Þ, which may be compared with the elementary bound of Oðn 2 Þ.

1. Introduction It is a longstanding problem to describe the variation of the rank of the Mordell-Weil group in families of elliptic curves. There are many variations on this theme. One can study elliptic curves over number fields or over function fields and one can study the rank for a fixed base field and varying elliptic curve or for a fixed elliptic curve and varying base field. In this note we begin with a number field K, a curve C=K defined over K, and a (nonconstant) elliptic curve E defined over the function field KðCÞ of C. Equivalently, we consider an elliptic surface E ! C defined over K with generic fiber
 E. Our main result is the following (conditional) upper bound for the rank of E KðC 0 Þ for certain coverings C 0 ! C. Theorem 1. With notation and assumptions as above, let C 0 ! C be a finite unramified covering defined over K, and assume that the geometric automorphism group A ¼ AutK ðC 0 =CÞ of the covering is abelian. Notice that GK=K acts on A. Assume that the Tate conjecture is true for the elliptic surface E 0 ¼ E C C 0 , and let NðE 0 Þ denote the conductor of E 0 . Then
 ðNumber of GK=K orbits of AÞ
 rank E KðC 0 Þ e
jNðE 0 Þj þ 4g 0  4 ; jAj where g 0 is the genus of the curve C 0 . This research supported by NSF DMS-9970382.

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0 Remark 1. The geometric (relative) automorphism
 group AutK ðC =CÞ may be 0 identified with the geometric Galois group Gal KðC Þ=KðCÞ , and the action of GalðK=KÞ on AutK ðC 0 =CÞ comes from the exact sequence



  
0 ! Gal KðC 0 Þ=KðCÞ ! Gal KðC 0 Þ=KðCÞ ! Gal KðCÞ=KðCÞ ! 0         AutK ðC 0 =CÞ

GalðK=KÞ:

Remark 2. There is an elementary geometric upper bound for the rank, coming from cohomological considerations, which says that ð1Þ


 rank E KðC 0 Þ e jNðE 0 Þj þ 4g 0  4:

Thus the gain in Theorem 1 comes from nontrivial action of GK=K on the automorphism group of the field extension KðC 0 Þ=KðCÞ, or equivalently from nontrivial action of GK=K on the group of deck transformations of the finite unramified covering C 0 ! C. As an interesting special case of the theorem, we take C ¼ C 0 to be an elliptic curve and consider the coverings ½n : C ! C given by the multiplication-by-n maps. Then A ¼ C½n is the group of n torsion points on C and Serre’s theorem tells us that the action of GK=K on A is highly nontrivial as n increases. Combining this with the theorem gives our second main result. Theorem 2. Let C=K be an elliptic curve defined over a number field, let E=KðCÞ be an elliptic curve, and for each n f 1, let Kn be the extension field of KðCÞ corresponding to the multiplication-by-n map ½n : C ! C. Assume that Tate’s conjecture is true for the elliptic surface En associated to E=Kn . Then there is a k > 0 and an n0 ¼ n0 ðK; C; dÞ such that for all n > n0 , rank EðKn Þ e jNðEn Þj k=log log n ¼ jn 2
NðEÞj k=log log n : We also show in this situation that the average rank of EðKn Þ is smaller than a multiple of the logarithm of its conductor. (See Theorem 16.) Thus in an unramified abelian tower over an elliptic base, the rank grows much more slowly than indicated by the elementary geometric bound. There are many interesting questions one might ask, for example:

. Can the rank go to infinity in an unramfied abelian tower? . What is the best upper bound for the rank in terms of the conductor? In the case that the number field K is replaced by a finite field Fq , results of Shioda [19], Brumer [2] and Ulmer [25] provide a definitive answer. Theorem 3. Let E be a nonconstant elliptic curve defined over a function field Fq ðCÞ over a finite field.

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(a) The geometric rank of E is bounded by 
rank E Fq ðCÞ e jNðEÞj þ 4g  4: Further, there exist examples with conductor of arbitrarily high degree for which this bound is sharp. (See [19], [25].) (b) The arithmetic rank of E is bounded by !  jNðEÞj þ 4g  4
jNðEÞj þO
rank E Fq ðCÞ e 2 : 2 logq jNðEÞj logq jNðEÞj (See [2].) Further, there exist examples with conductor of arbitrarily high degree for which the main term in this bound is sharp. (See [25].) See [26], Section 4.3, for a detailed discussion of these results. We also observe that Brumer’s proof of the upper bound (Theorem 3) is modeled after a result of Mestre [11] that deals with elliptic curves over Q. Returning now to the case of an elliptic curve E over KðCÞ when K is a number field, we note that the geometric bound given in (1) holds more generally when the number field K is replaced by its algebraic closure, that is,
 rank E KðC 0 Þ e jNðE 0 Þj þ 4g 0  4: This is analogous to the bound in Theorem
3(a).  Ulmer [26], Section 8, has asked if this 0 bound can be improved for the group E KðC Þ . For example, one might be tempted to make the following conjecture as the analog of the bound in Theorem 3(b) and of Mestre’s result [11]. Conjecture 4. Let KðCÞ be the function field of a curve over a number field, and let E=KðCÞ be a non-constant elliptic curve (i.e., jðEÞ B K). Then
 jNðEÞj rank E KðCÞ f ; logjNðEÞj where the implied constant depends on K and C. More precisely, there is an absolute constant a > 0 so that
 jNðEÞj þ 4g  4
logj2
DiscðK=QÞj: rank E KðCÞ e a logjNðEÞj It is unclear to what extent Theorems 1 and 2 should be considered as providing evidence for this conjecture and to what extent they simply suggest that the rank over unramified towers is unusually small. Remark 3. Conjecture concerning the structure
 4 raises interesting questions

of the fields L=K for which E LðCÞ is strictly larger than E KðCÞ . The structure of E KðCÞ as

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a GalðK=KÞ-module has been studied by Shioda in his theory of Mordell-Weil lattices [22], [20], and also by Kuwata [9], [10]. Remark 4. Doug Ulmer (private communication) has pointed out that our principal result (Theorem 1) is valid if the map C 0 ! C is allowed to be ramified outside of the points where the elliptic surface E ! C fibers, i.e., outside of the discriminant
has singular  locus. This gives a bound for rank E KðC 0 Þ for a larger set of extensions KðC 0 Þ=KðCÞ, albeit for a set of extensions that depends on the particular elliptic curve E. Acknowledgements. The author would like to thank Philippe Michel for his help in understanding the estimates provided by Deligne’s work, Doug Ulmer for encouraging the author to extend the work in [23] and for helpful comments on the first draft, Michael Rosen for numerous mathematical discussions, and the referee for many suggestions that greatly improved the readability of the paper.

2. Setup for a single elliptic surface 2.1. Notation. We set the following notation. K=Q qp

a number field. the norm of an ideal p of K.

C=K

a smooth projective curve of genus g.

E=K

a nonconstant elliptic surface E ! C defined over K. In particular, the assumption that jðEÞ B K implies that E does not split as a product even after base extension of C.

EðC=KÞ

the group of sections of E ! C defined over K.

NðE=CÞ

the conductor of the elliptic surface E ! C. The conductor is a divisor on C; we denote its degree by jNðE=CÞj.

E=KðCÞ

the generic fiber of E, an elliptic curve over the function field KðCÞ.

Remark 5. We fix a finite set of primes S such that for all p B S, the elliptic surface E ! C has good reduction at p. That is, E ! C has a smooth model over the ring of Sintegers of K. We enlarge S further so that for each prime p B S, the conductor of E=C=Fp P is the reduction modulo p of the conductor of E=C=K. We will write to mean the sum p

over all primes of K that are not in S. From time to time, we may enlarge the set S. 2.2. A rank estimate and a rank formula. An elementary upper bound for the rank of E can be obtained from the cohomology of EðCÞ. We call this the geometric bound and observe that it automatically provides an upper bound for the rank of EðC=KÞ over the number field K.

Silverman, Rank of elliptic surfaces

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Theorem 5 (Geometric rank bound). rank EðC=KÞ e rank EðC=KÞ e jNðE=CÞj þ 4g  4: Proof. See [21], Corollary 2. r In order to improve the geometric bound, we use a local-global formula originally proposed by Nagao [14]. The following notation is needed in order to state the required result. For each prime ideal p of K and each point R A CðFp Þ, let ER denote the fiber of E over R and let  ap ðER Þ ¼

qp þ 1  jER ðFp Þj if ER =Fp is smooth; 0 if ER =Fp is singular.

If ER =Fp is smooth, then ap ðER Þ is the trace of Frobenius and satisfies the usual Weil bound pffiffiffiffiffi jap ðER Þj e 2 qp . The ‘‘average’’ of these values over the fibers will be denoted Ap ðE=CÞ ¼

1 P ap ðER Þ: qp R A CðFp Þ

 pffiffiffiffiffi
The Weil bound gives jAp ðE=CÞj e 2 qp 1 þ oð1Þ but a theorem of Deligne (with a further improvement by Michel) gives a much better estimate. Theorem 6.

For all but finitely many primes p, jAp ðE=CÞj e jNðE=CÞj þ 4g  4 þ

jNðE=CÞj pffiffiffiffiffi : qp

pffiffiffiffiffi (For our purposes, it would su‰ce to know that the error is Oð1= qp Þ.) Proof. The weaker upper bound jAp ðE=CÞj e 2ðK of singular fibersÞ þ 4g  4 follows in a relatively straightforward manner from an equidistribution result of Deligne [3], The´ore`me 3.5.3, see also [6], Theorem 3.6, and [8], (3.6.3). Each singular fiber contributes at least one to the conductor, so this yields jAp ðE=CÞj e 2jNðE=CÞj þ 4g  4: The upper bound comes from an explicit estimate for the monodromy action around the points at which the fiber is singular. For an elliptic surface, which is our situation, one can calculate the monodromy action quite explictly around points where the fiber has semistable (i.e., multiplicative) reduction. Michel [12], Section 4, explains how this calculation allows one to save one factor of the semistable part of the conductor, so the upper bound becomes

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Silverman, Rank of elliptic surfaces

pffiffiffiffiffi pffiffiffiffiffi qp jAp ðE=CÞj e ðKfsingular fibersg þ Kfunstable fibersg þ 4g  4Þ qp þ Kfsingular fibersg: This yields the stated result with a slightly stronger error term. See also Michel [13] for similar results and extensions to families of abelian varieties. r The following analytic version of a conjecture of Nagao [14] gives a local-global forpffiffiffiffiffi mula for the rank of EðC=KÞ. The Weil bound jAp ðE=CÞj ¼ Oð qp Þ shows that the series (2) converges in some halfplane, and Deligne’s estimate jAp ðE=CÞj ¼ Oð1Þ from Theorem 6 gives convergence for 1. It is proven in [15] that if the Tate conjecture is true for E=K, then the series has a meromorphic continuation to all of C. Theorem 7.

Assume that the Tate conjecture is true for the surface E=K. Then rank EðC=KÞ ¼ res

ð2Þ

P

s¼1 p

Ap ðE=CÞ

log qp ; qps

where the sum is over all prime ideals of K not in S. Proof. This is proven in [15], Theorem 1.3. r

3. Elliptic surfaces in unramified abelian towers We continue with the notation from the previous section. In particular, C is a curve and E ! C is an elliptic surface, both defined over a number field K. For any finite cover C 0 ! C, we obtain a new elliptic surface via pullback, E 0 ¼ E C C 0 ! C 0 : We consider covers C 0 ! C satisfying the following conditions:

. The curve C and the map C ! C are defined over K, so E =C is likewise defined over K. . The map C ! C is
a geometrically Galois covering whose geometric Galois group A ¼ Aut ðC =KÞ ¼ Gal KðC Þ=KðCÞ is abelian. . The map C ! C is unramified. Notice that this assumption ensures that E is 0

0

0

0

0

0

0

K

0

0

0

0

nonsingular. In general, one would let E be a smooth model for E C C . The conductor of an elliptic surface behaves nicely under an unramified pullback of the base curve. Proposition 8. Let E ! C be an elliptic surface and let f : C 0 ! C be an unramified map. (a) The conductor of the pullback E 0 ¼ E C C 0 is given by
 NðE 0 =C 0 Þ ¼ f  NðE=CÞ :

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159

(b) The canonical divisors on C and C 0 are related by KC 0 ¼ f  ðKC Þ: (c) In particular,
 jNðE 0 =C 0 Þj þ 4g 0  4 ¼ degð f Þ jNðE=CÞj þ 4g  4 ; where g and g 0 are the genera of C and C 0 , respectively. Proof. (a) Immediate from the fact that a minimal equation for E over the local ring OR of a point R A C remains a minimal Weierstrass equation over the local ring OP for any point P A f 1 ðRÞ, since OP is an unramified extension of OR . (b) This is [7], Proposition IV.2.3, with trivial ramification divisor. (c) Immediate from (a), (b), and degðKC Þ ¼ 2g  2 [7], IV.1.3.3. r Each element of A is an automorphism a : C 0 ! C 0 . These automorphisms need not be defined over K, but the fact that C 0 is defined over K implies that GK=K acts on A. Since A is finite, we can choose a finite extension L=K so that GL=K acts on A. Note that GL=K need not be abelian. Example 1. If C is an elliptic curve, we can take C 0 ¼ C and use the multiplication by n map ½n : C ! C. In this case the group A is the group of n-torsion points C½n on C, and the action of GK=K on A is the usual Galois action on the n-torsion points of an elliptic curve. Example 2. More generally, we can embed C into its Jacobian variety C ! J and let C 0 be the pullback of C via the multiplication-by n map ½n : J ! J. Then C 0 ! C is an abelian unramified cover with group A ¼ J½n having the natural GK=K action. Example 3. There is a partial converse to the previous example. If C 0 ! C is any abelian unramified cover, then there is an isogeny F : J 0 ! J of their Jacobians so that C 0 ¼ F 1 ðCÞ and A ¼ kerðF Þ. See [16], Chapter VI, Section 12, Corollary to Proposition 11. Example 4. If we drop the requirement that C 0 ! C be unramified, then an interesting case is C 0 ¼ C ¼ P 1 with the map T ! T n . This is the situation studied in [23], where it was shown that the rank of E 0 ðC 0 =KÞ is bounded by a generalized divisors-of-n function. If one further puts various sorts of technical restrictions on the discriminant of E, then Shioda, Stiller [24], and Fastenberg [4], [5] have shown that the rank of E 0 ðC 0 =KÞ is bounded independently of n. The techniques in the present paper can be adapted to handle coverings C 0 ! C with limited ramification, but the resulting estimates are somewhat complicated, so we have opted to restrict attention to unramified coverings.

4. Elementary results about groups acting on sets In this section we recall and prove some elementary estimates that will be required later.

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Silverman, Rank of elliptic surfaces

Lemma 9. Let G be a finite group that acts on a finite set X . Then 1 P jfx A X : sðxÞ ¼ xgj ¼ ðNumber of G-orbits of X Þ: jGj s A G Proof. Let r be the permutation representation of G acting on X and let w be its character [17], Section 1.2. Then as in [17], Section 2.3, ðNumber of G-orbits of X Þ ¼ ðNumber of times r contains the unit representationÞ ¼

1 P wðsÞ: jGj s A G

This is the desired formula, since from the definition of the permutation representation, it is clear that wðsÞ is equal to the number of elements of X that are fixed by s. r Lemma 10. of G. Then

Let G be a finite group that acts on a finite set X , and let H be a subgroup

ðNumber of H-orbits of X Þ e ðG : HÞ
ðNumber of G-orbits of X Þ; with equality if and only if Hx ¼ Gx for every x A X . (Here Gx ¼ fs A G : sðxÞ ¼ xg is the stabilizer of x, and similarly for Hx .) Proof. ðNumber of H-orbits of X Þ ¼

P

1 1 P ¼ jHx j jHj x A X x A X jHxj

¼ ðG : HÞ

P 1 jHx j P jHx j ¼ ðG : HÞ
x A X jGj x A X jGxj jGx j P

1 ¼ ðG : HÞ
ðNumber of G-orbits of X Þ: x A X jGxj

e ðG : HÞ

This proves the desired inequality. Further, we have equality if and only if jHx j ¼ jGx j for every x A X . r

5. Subgroups of A F Aut(CO/C ) and intermediate curves Each subgroup B H A corresponds to a curve CB satisfying C 0 ! CB ! C

and AutK ðC 0 =CB Þ ¼ B:

Equivalently, CB is the quotient curve C 0 =B. Since by assumption A is abelian, every subgroup of A is normal, so the covering CB ! C is also geometrically Galois with geometric automorphism group naturally isomorphic to A=B.

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161

Recall that we have fixed a finite extension L=K so that GL=K acts on A. All of the curves CB are defined over L. If GL=K normalizes B, that is, if and s A GL=K ) sðbÞ A B;

bAB

then the curve CB is defined over K. In subsequent sections, we plan to reduce the groups B and the curves CB modulo primes p of K. In order to do this, we observe that A has a natural structure as a group scheme over K, and that it becomes a constant group scheme over L, since the action of GalðK=KÞ on AðKÞ factors through GalðL=KÞ. Further, the subgroup schemes of A that are defined over K are precisely those that are normalized by GalðL=KÞ. (This is nicely illustrated by Example 3.) Each subgroup B of AðKÞ ¼ AðLÞ can be extended to a group scheme over the ring of S 0 -integers of L for some finite set of primes S 0 . We adjoin to the set S the primes of K lying below the primes of S 0 for every such subgroup B. It then makes sense to reduce not only the curves CB , but also the automorphism groups B ¼ AutK ðC 0 =CB Þ and A=B ¼ AutK ðCB =CÞ, modulo primes not lying above S. If GL=K normalizes B, then B extends to a group scheme over K, and we may reduce modulo primes of K not in S.

6. Abelian unramified covers over finite fields We next reduce the unramified abelian covering C 0 ! C and the elliptic surfaces E and E 0 modulo a prime ideal p B S of K. Adjoining finitely many primes to our set S of excluded primes, we may assume the following conditions:

. The reduced curves C =F and C=F are nonsingular. . The map C ! C over F is unramifed and geometrically Galois with group ðC =CÞ equal to A. . The elliptic surfaces E=F and E =F are nonsingular. . The conductor NðE=C=F Þ of the reduction of E modulo p is equal to the reduc0

p

0

AutFp

p

p

0

p

0

p

p

tion modulo p of the global conductor NðE=C=KÞ. More generally, we assume that this is true for each pullback E C CB for each subgroup B H A that is defined over Fp . In particular, it is true for E 0 , which corresponds to taking B ¼ A. Proposition 11. Let s ¼ sp be the Frobenius map, so s generates the Galois group of Fp =Fp . Let B be the subgroup of A defined by B ¼ fsðaÞ  a1 : a A Ag: (a) The group B is defined over Fp , and hence the curve CB ¼ C 0 =B is also defined over Fp . (b) The image of C 0 ðFp Þ in CðFp Þ is the same as the image of CB ðFp Þ in CðFp Þ.

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Silverman, Rank of elliptic surfaces

(c) Let R A CðFp Þ be a point in the image of C 0 ðFp Þ. Then there are exactly jAðFp Þj points of C 0 ðFp Þ that map to R. (d) Let R A CðFp Þ be a point in the image of CB ðFp Þ. Then there are exactly ðA : BÞ points of CB ðFp Þ that map to R. Proof. (a) It is clear that sðBÞ ¼ B, so the fact that s generates GalðFp =Fp Þ implies that B is defined over Fp . Since C 0 is defined over Fp by assumption, this in turn implies that the quotient C 0 =B is defined over Fp . More precisely, the curve CB is the maximal subcover of C 0 ! C that is Galois over C=Fp , in the sense that Fp ðCB Þ is Galois over Fp ðCÞ. (b) The composition of maps C 0 ðFp Þ ! CB ðFp Þ ! CðFp Þ shows that the image of C ðFp Þ is contained in the image of CB ðFp Þ. Next let Q A CB ðFp Þ and choose any point P A C 0 ðFp Þ that maps to Q. The fact that Q is defined over Fp and that CB ¼ C 0 =B means that there is an automorphism b A B such that sðPÞ ¼ bðPÞ. By definition of B, there is an automorphism a A A such that b ¼ sðaÞ  a1 , and hence 0



a1 ðPÞ ¼ sðaÞ1 sðPÞ ¼ s a1 ðPÞ : Thus a1 ðPÞ is fixed by s, so it is in C 0 ðFp Þ. Further, a1 ðPÞ and P both have the same image in CðFp Þ, which in turn is the same as the image of Q (since P A C 0 maps to Q A CB ). This proves that given any point Q A CB ðFp Þ, its image in CðFp Þ is also the image of a point in C 0 ðFp Þ, which gives the other inclusion and completes the proof of (b). (c) By assumption, there is at least one point P A C 0 ðFp Þ that maps to R. The full inverse image of R is the orbit AR ¼ faðRÞ : a A Ag. The assumption that C 0 ! C is unramified implies that AR consists of jAj distinct points, or equivalently, that only the identity element of A fixes R. Hence
 aðRÞ A C 0 ðFp Þ , s aðRÞ ¼ aðRÞ
 , a1  sðaÞ ðRÞ ¼ R , sðaÞ ¼ a , a A AðFp Þ: This completes the proof that there are exactly jAðFp Þj points of C 0 ðFp Þ that map to R. (d) Applying (c) to the map CB ! C, we see that jðA=BÞðFp Þj points of CB ðFp Þ are mapped to R. However, the definition of B implies that s fixes every point in ðA=BÞðFp Þ. Hence jðA=BÞðFp Þj ¼ jðA=BÞðFp Þj ¼ jAj=jBj ¼ ðA : BÞ; which completes the proof of (d). r Remark 6. The equalities in Proposition 11 deserve further comment. Notice that the maps C 0 ! CB ! C induce

Silverman, Rank of elliptic surfaces

163

f0

 

 

C 0 ðFp Þ ! CB ðFp Þ fB

f

CðFp Þ: Since ðA : BÞ ¼ jAðFp Þj, the proposition says that 

K fB1 ðRÞ ¼ K f 1 ðRÞ for R A CðFp Þ: Thus these two sets have the same cardinality and f 0 maps one to the other, so it is natural to assume that f 0 induces a bijection between them, but this is not always the case. (The author thanks the referee for this observation.) Proposition 12. jAp ðE 0 Þj e

 jAðFp Þj
pffiffiffiffiffi
jNðE 0 =C 0 Þj þ 4g 0  4 þ Oð1= qp Þ: jAj

Remark 7. Notice that if GalðFp =Fp Þ acts trivially on A, then AðFp Þ ¼ AðFp Þ, so the upper bound in Proposition 12 reduces to the geometric upper bound provided by Theorem 6. However, when the action is nontrivial, then Proposition 12 may provide a significant strengthening of the geometric bound. Proof. Let B ¼ fsðaÞ  a1 : a A Ag be the subgroup of A described in Proposition 11. To ease notation, we let C 00 ¼ CB ¼ C 0 =B and E 00 ¼ EB ¼ E C CB : Thus we have a Cartesian diagram E 0 ðFp Þ ! E 00 ðFp Þ ! EðFp Þ ? ? ? ? ? ? ? ? ? y y y C 0 ðFp Þ ! C 00 ðFp Þ ! CðFp Þ: Proposition 11 tells us that the image of C 0 ðFp Þ in CðFp Þ is the same as the image of C 00 ðFp Þ in CðFp Þ and gives us the multiplicity of each map. We use the fact that if P A C 0 ðFp Þ maps to R A CðFp Þ, then the fiber EP0 is isomorphic to ER , and similarly if Q A C 00 ðFp Þ maps to R A CðFp Þ, then EQ00 G ER . We compute qp Ap ðE 0 Þ ¼

P

ap ðEP0 Þ ðby definition of Ap Þ

P A C 0 ðFp Þ

¼ jAðFp Þj

P

ap ðER Þ

ðfrom Proposition 11ðcÞÞ

R A Image½C 0 ðFp Þ!CðFp Þ

¼ jAðFp Þj

P

ap ðER Þ ðfrom Proposition 11ðbÞÞ

R A Image½C 00 ðFp Þ!CðFp Þ

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Silverman, Rank of elliptic surfaces

¼

jAðFp Þj P ap ðEQ00 Þ ðA : BÞ Q A C 00 ðFp Þ

¼ qp Ap ðE 00 Þ

ðfrom Proposition 11ðdÞÞ

since jAðFp Þj ¼ ðA : BÞ:

Applying the conductor estimate given in Theorem 6 to E 00 and using the elementary relation (Proposition 8c) between the conductors of E 00 and E 0 , we obtain the upper bound jAp ðE 0 Þj ¼ jAp ðE 00 Þj pffiffiffiffiffi e jNðE 00 =C 00 Þj þ 4g 00  4 þ Oð1= qp Þ ¼

 1
pffiffiffiffiffi
jNðE 0 =C 0 Þj þ 4g 0  4 þ Oð1= qp Þ jBj

¼

 ðA : BÞ
pffiffiffiffiffi
jNðE 0 =C 0 Þj þ 4g 0  4 þ Oð1= qp Þ jAj

¼

 jAðFp Þj
pffiffiffiffiffi
jNðE 0 =C 0 Þj þ 4g 0  4 þ Oð1= qp Þ: jAj

r

7. An upper bound for the rank Theorem 13. Let E ! C be an elliptic surface defined over a number field K, let C ! C be an unramified abelian covering defined over K and with automorphism group A, and let E 0 ¼ E C C 0 be the pullback of E via this covering. Assume that the Tate conjecture is true for E 0 =K. Then 0

rank E 0 ðC 0 =KÞ e

ðNumber of GK=K orbits of AÞ

jNðE 0 =C 0 Þj þ 4g 0  4 : jAj

Remark 8. Notice that if GK=K acts trivially on A, then we obtain nothing better than the geometric upper bound given in Theorem 5. However, if the Galois action is nontrivial, as tends to the case in a tower C C1 C2


; then Theorem 13 often gives an upper bound that is considerably better than the geometric bound. We will see an example of this below (Theorem 16) in which for every e > 0, the upper bound for rank En ðCn =KÞ is smaller than jNðEn =Cn Þj e as n ! y. Proof. We apply the analytic rank formula in Theorem 7 to the elliptic surface E 0 ! C 0, rank E 0 ðC 0 =KÞ ¼ res

P

s¼1 p

Ap ðE 0 Þ

log qp : qps

Taking absolute values and using the estimate for jAp j provided by Theorem 12 yields !
 P jAðFp Þj log qp P log qp 0 0 0 0 0
: þ O res rank E ðC =KÞ e jNðE =C Þj þ 4g  4 res s¼1 p s¼1 p q sþ1=2 qps jAj p

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The series in the big-O term converges, so its residue is zero and it may be discarded. Next we observe that the size of AðFp Þ depends only on the action of p-Frobenius on A. In other words, if we choose an element s A GL=K , then jAðFp Þj is the same for every prime p such that s is in the p-Frobenius conjugacy class ðp; L=KÞ H GL=K . (As always, we have discarded the finitely many primes for which A has bad reduction.) More precisely, if s A ðp; L=KÞ, then jAðFp Þj ¼ jfa A A : sðaÞ ¼ agj is simply the number of elements of A fixed by s. We denote this last quantity by h 0 ðs; AÞ. We can thus rewrite the above sum as
 P h 0 ðs; AÞ res rank E 0 ðC 0 =KÞ e jNðE 0 =C 0 Þj þ 4g 0  4 jAj s¼1 s A GL=K

P p ðp; L=KÞ¼s

log qp : qps

The residue is the reciprocal of the degree of the extension L=K (cf. [23]), so
 rank E 0 ðC 0 =KÞ e jNðE 0 =C 0 Þj þ 4g 0  4

P h 0 ðs; AÞ : jGL=K j s A GL=K jAj 1

Finally, we apply Lemma 9 to obtain the desired bound, which completes the proof of Theorem 13. r 8. The rank in an elliptic tower We now consider the case that the base curve C=K is an elliptic curve and we take the unramified abelian covers ½n : C ! C given by the multiplication-by-n maps. In the notation of Theorem 13, we have C 0 ¼ C, but E 0 is most definitely not equal to E. We let En denote the pullback of E via the map ½n : C ! C. The automorphism group A in this case is the group A ¼ C½n of n torsion points of C, with the natural action of GK=K . As is clear from Theorem 13, the nontriviality of our bound for the rank of En ðC=KÞ depends on the degree of nontriviality of the action of GK=K on C½n. A famous theorem of Serre gives us control of that action. Theorem 14 (Serre [18]). Let C=K be an elliptic curve defined over a number field K. There is an integer I ðC=KÞ so that for every integer n f 1, the image of the representation rC; n : GK=K ! AutðC½nÞ G GL 2 ðZ=nZÞ has index at most IðC=KÞ. Remark 9. Conjecturally, the index bound in Theorem 14 can be chosen to depend only on the field K, and possibly only on the degree ½K : Q. We also need the following elementary fact concerning the natural action of the general linear group.

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Proposition 15. Let n f 1 and r f 1 be integers. Then the natural action of GLr ðZ=nZÞ on ðZ=nZÞ r has dðnÞ distinct orbits, where dðnÞ is the number of divisors of n. Proof. We begin with the case that n ¼ p e is a prime power. For any vector v ¼ ðv1 ; . . . ; vr Þ A ðZ=p e ZÞ r , let ord p ðvÞ ¼ minford p ðv1 Þ; . . . ; ord p ðvr Þ; eg: We claim that v and w have the same GLr -orbit if and only if ord p ðvÞ ¼ ord p ðwÞ. First, if v ¼ Aw for any integer matrix A, then it is clear from the definition that ord p ðvÞ f ord p ðwÞ. If v and w are in the same orbit, then A is invertible, so we get an equality ord p ðvÞ ¼ ord p ðwÞ. Next suppose that ord p ðvÞ ¼ ord p ðwÞ. If this common value is e, then v ¼ w ¼ 0 and there is nothing further to be said. So suppose that the common value is k with k < e. Then we can write v ¼ p k v 0 and w ¼ p k w 0 , where ord p ðv 0 Þ ¼ ord p ðw 0 Þ ¼ 0. It thus suffices to prove that any two vectors with ord p ¼ 0 are in the same GLr ðZ=p e ZÞ orbit, and for that it su‰ces to show that if ord p ðvÞ ¼ 0, then v is in the orbit of the unit vector e ¼ ð1; 0; 0; . . . ; 0Þ. We are thus reduced to showing that every vector with ord p ðvÞ ¼ 0 can be placed as the first column of a matrix in GLr ðZ=p e ZÞ. A matrix modulo p e is invertible if and only if its determinant is prime to p, so we are reduced to the case that e ¼ 1. Then the condition ord p ðvÞ ¼ 0 says simply the v E 0 ðmod pÞ, and the desired conclusion follows from the fact that a nonzero vector in a vector space can always be extended to a basis for the vector space. The vectors in this basis, lifted from ðZ=pZÞ r to ðZ=p e ZÞ r , form the desired matrix in GLr ðZ=p e ZÞ. This proves that ord p ðvÞ completely characterizes the orbit of v under the action of GLr ðZ=p e ZÞ. The quantity ord p ðvÞ is an integer between 0 and e, which proves that there are e þ 1 distinct orbits. Since dðp e Þ ¼ e þ 1, this proves the proposition when n ¼ p e is a prime power. Finally, the Chinese Remainder Theorem and the multiplicativity of dðnÞ gives the result for all integers n f 1. r Remark 10. Applying Lemma 9 with G ¼ GLr ðZ=nZÞ and X ¼ ðZ=nZÞ r and using Proposition 15 yields some amusing formulas. For example, take r ¼ 1, so G ¼ ðZ=nZÞ  , X ¼ Z=nZ, and the action is multiplication. Then for a A ðZ=nZÞ  , jfx A Z=nZ : ax 1 x ðmod nÞgj ¼ jfx A Z=nZ : njða  1Þxgj ¼ gcdða  1; nÞ: This yields P

gcdða  1; nÞ ¼ dðnÞfðnÞ:

0ea 0 we have rank En ðC=KÞ f jNðEn =CÞj e ; where the implied constant depends on K; C; E, and e, but is independent of n. Proof. Let rn : GK=K ! AutðC½nÞ G GL 2 ðZ=nZÞ be the representation of GK=K on the n-torsion of C. Serre’s theorem (Theorem 14) tells us that the index of rn ðGK=K Þ in GL 2 ðZ=nZÞ is at most IðC=KÞ, where I ðC=KÞ is independent of n. It follows from Proposition 10 and Proposition 15 that ðNumber of GK=K orbits in C½nÞ e I ðC=KÞ
ðNumber of AutðC½nÞ orbits in C½nÞ ¼ I ðC=KÞ
ðNumber of GL 2 ðZ=nZÞ orbits in ðZ=nZÞ 2 Þ ¼ I ðC=KÞdðnÞ: Applying our main result (Theorem 13) with A ¼ C½n and using the above bound for the number of GK=K orbits in C½n yields rank En ðC=KÞ e I ðC=KÞ
This completes the proof of (a).

dðnÞ
jNðEn =CÞj: n2

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Proposition 8 says that NðEn =CÞ ¼ n 2 NðE=CÞ, so we can rewrite the upper bound as rank En ðC=KÞ e I ðC=KÞ
dðnÞ
jNðE=CÞj: The divisor function dðnÞ has the properties (see [1], Theorem 3.3 and Theorem 13.12) P

dðnÞ @ x log x

and

lim sup n!y

nex

log dðnÞ ¼ log 2: log n=log log n

The first formula implies that 1 P dðnÞ x 2enex log n is bounded for all x f 2. Hence 1 P rank En ðC=KÞ 1 P I ðC=KÞ
dðnÞ
jNðE=CÞj 
e x nex logjNðEn =CÞj x nex log n 2 jNðE=CÞj is also bounded. This completes the proof of (b). Finally, let c1 ; c2 ; . . . denote absolute constants. Then using the second formula gives rank En ðC=KÞ e c1 I ðC=KÞjNðE=CÞjn c2 =log log n e c3 I ðC=KÞjNðE=CÞj jNðEn =CÞj c4 =log logjNðEn =CÞj ; which completes the proof of (c). r Added in proof. Jordan Ellenberg and Amilcar Pacheco have recently extended the results of this paper in various ways.

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Mathematics Department, Box 1917, Brown University, Providence, RI 02912 USA e-mail: [email protected] Eingegangen 6. Januar 2003, in revidierter Fassung 18. November 2003