Modeling Elliptic Curve L-Functions with Random Matrix theory

Florian B OUGUET École Normale Supérieure de Cachan - Antenne de Bretagne Bruz, France Research work supervised by

Nina C. S NAITH University of Bristol Bristol, United Kingdom

Contents 1

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Theoretical basics 1.1 Elliptic curves . . . . . . . . . . . . . . . . . . . . 1.1.1 Definition . . . . . . . . . . . . . . . . . . 1.1.2 Rational solutions and rank . . . . . . . . . 1.1.3 The Birch and Swinnerton-Dyer conjecture 1.1.4 Conductor . . . . . . . . . . . . . . . . . . 1.2 L-functions . . . . . . . . . . . . . . . . . . . . . 1.2.1 Hasse-Weil L-functions . . . . . . . . . . 1.2.2 Quadratic twists . . . . . . . . . . . . . . 1.2.3 Functional equations . . . . . . . . . . . . 1.2.4 The Riemann hypothesis . . . . . . . . . . Modeling L-functions statistics 2.1 Choice of the model . . . . . . . . . . . . 2.1.1 Raw idea . . . . . . . . . . . . . 2.1.2 Refining the model . . . . . . . . 2.1.3 The excised orthogonal ensemble 2.2 Validity of the model . . . . . . . . . . . 2.2.1 Cut-off value . . . . . . . . . . . 2.2.2 Second zeros . . . . . . . . . . .

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Conclusion

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Acknowledgements

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Introduction It all began with the encounter of Hugh L. Montgomery and Freeman J. Dyson over a cup of tea, at the Princeton Institute for Advanced Studies in 1973. The former, then a young graduate student, showed his latest work on the Riemann zeta function to the latter, who recognized a formula he had already calculated for the density of eigenvalues of random unitary matrices. From this chance meeting, several important connections between random matrix theory and analytic number theory have been conjectured, sometimes proved. We will also deal with the Birch and Swinnerton-Dyer conjecture and the Riemann Hypothesis, two Millenium Prize Problems having a direct link with our subject. In this paper, we take an interest in a random matrix models recently suggested in [6], and we try to check its validity. In the first part, we are going to recall some theoretical background about elliptic curves and L-functions, in order to comprehend the subject. In the second part, we shall introduce the model, refine it and then verify whether it is suitable or not.

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1

Theoretical basics

1.1 1.1.1

Elliptic curves Definition

Let us begin by defining what an elliptic curve is: Definition 1.1 (Elliptic curve) An elliptic curve over Q is a non-singular curve E ⊆ R2 defined by an equation of the form: E : y 2 = x3 + ax + b, where a, b ∈ Q.

E : y 2 = x3 − x + 1

Such an equation is called a Weierstrass equation. Thus, we can define the discriminant of E by ∆E = −16(4a3 + 27b2 ). E is non-singular (which means it has no self intersections, cusps, nodes or isolated points) if, and only if, ∆E 6= 0. This is also equivalent to the polynomial x3 + ax + b having 3 distinct complex roots. We can also add that the curve has two components if ∆E > 0 and only one if ∆E < 0. We recall that a singular point P of E is: • a node if there are two different tangent lines to E at P • a cusp if there is a single tangent line to E at P

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1.1.2

Rational solutions and rank

A well-known issue in the subject of the elliptic curves is to describe the set of the rational solutions of E, that is to say the pairs (x, y) ∈ Q2 lying on E. To this end, we are going to supply a group law to E (see [11]). Firstly, we need to create a "point at infinity" called O, which lies on any vertical line. The definition of an elliptic curve is often vague in this area, and we will identify E and E ∪ {O} (the latter is sometimes called the projective version of E). O is destined to be the identity element of the group E. Let us now define a group law on E. Let P and Q be two points of E. Any line through P and Q meets E in another point, called R - counting the multiplicity and including the "point at infinity" O defined above. Counting the multiplicity means "if the line is tangent to the curve at a point, then it is counted twice". This statement is a consequence of the following theorem: Theorem 1.1 (Bezout’s theorem) If X and Y are two plane projective curves defined over a field K and assuming that X and Y do not have an infinite number of points in common, then the number of intersection points of X and Y with coordinates in K, counted with their multiplicity, is equal to the product of the degrees of X and Y . Corollary 1.1 A cubic curve meets a line in three points. The theorem holds for points with coordinates in C; indeed, we could easily imagine an horizontal line meeting E in only one point. But since the coefficients of E are real and P and Q have coordinates in R, so does R (they are the roots of a degree 3 polynomial with real coefficients). We can eventually define P + Q as the reflection of R about the x-axis.

The construction of this law on E assures us that three points of E lying on a line sum to O. Now, if we want to consider the multiplicity, one of the following cases holds:

7

The tough property to show is the associativity of this law. Anyway, the following result holds: Proposition 1.1 (E, +) is an abelian group, with O as the identity element. Note that we can also present formulas to work the coordinates of the sum out: • if xP 6= xQ yQ − yP 2 ). xQ − xP yP (xP +Q − xQ ) − yQ (xP +Q − xP ) = . xQ − xP

xP +Q = −xP − xQ + ( yP +Q • if xP = xQ and yP = −yQ

P + Q = O. • if xP = xQ and yP = yQ 6= 0 3x2P + a 2 ) − 2xP . 2yP 3x2 + a )(xP − x2P ) − yP . = ( P 2yP

x2P = ( y2P

Let us come back to our main issue: finding the rational solutions of E. Let us define E(Q) = (E ∩ Q2 ) ∪ {O} = {(x, y) ∈ Q2 y 2 = x3 + ax + b} ∪ {O}. By the formulas above, we can easily check the following proposition: Proposition 1.2 (E(Q), +) is an abelian subgroup of (E, +). Because we want to describe the structure of E(Q), the following theorem is one of the most important results in the present section:

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Theorem 1.2 (Mordell’s theorem) Let E be an elliptic curve. Then E(Q) is a finitely-generated abelian group. This theorem is a special case of the Mordell-Weil theorem, holding for any abelian variety and any number field - which means any finite field extension of Q. By the fundamental theorem of finitely generated abelian groups, we can now write E(Q) = Zn ⊕ G, where G is a direct sum of primary cyclic groups. n is the rank of E(Q), that is to say the number of copies of Z in E(Q) or, equivalently, the maximal number of independent points of infinite order. 1.1.3

The Birch and Swinnerton-Dyer conjecture

Definition 1.2 (Rank of an elliptic curve) rank(E)=rank(E(Q)). Determining the rank of an elliptic curve is not easy. For instance, one conjectures it can be arbitrarily large. The biggest known rank for an elliptic curve is 18, and curves of rank at least 28 are known - but we do not exactly know their rank. In order to conclude this section about elliptic curves, let us introduce the Birch and SwinnertonDyer conjecture, which is one of the Millenium Prize Problems of the Clay Mathematics Institute. It is a special case of Hilbert’s tenth problem, which deals with Diophantine equations. Since 1976, this conjecture has been proved only in some special cases. . . Conjecture 1.1 (Birch and Swinnerton-Dyer conjecture) The rank of an elliptic curve E over Q is the order of the zero of its Hasse-Weil L-function LE (s) at s = 1/2. The conjecture asserts in particular that if LE (1/2) 6= 0, then rank(E)=0, which means the number of rational solutions of E is finite. 1.1.4

Conductor

Before introducing the Hasse-Weil L-function of E, we need to define a major component of E: its conductor QE . This is is an integer quantity that measures the arithmetic complexity of the curve. However, its definition is rather complicated, and we need some preliminary work before being able to define what the conductor is. Firstly, let us talk about the reduction of E at prime numbers. We define for any prime number p:

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Definition 1.3 (Reduction of E at p) We call reduction of E at p the curve in R2 defined by E˜ : y 2 = x3 + a ˜x + ˜b, where a ˜ and ˜b are the respective reductions of a and b modulo p (a and b are the coefficients of the original equation of E). There are different kinds of reduction: • E has a good reduction at p if E˜ is non-singular (otherwise it is said to have a bad reduction) • E has an additive reduction at p if E˜ has a cusp • E has a multiplicative reduction at p if E˜ has a node Since E˜ is non-singular if, and only if, ∆E˜ 6= 0, E has a good reduction at every prime number non divisor of ∆E . We can now define the quantity fp such as:  0 if E has a good reduction at p    1 if E has a multiplicative reduction at p fp = 2 if E has an additive reduction at p, p 6= 2 or 3    2 + δp otherwise. where δp is some rather complicated measure of how bad the reduction is - depending on wild ¯ ramification in the action of the inertia group at p of Gal(Q/Q). Definition 1.4 (Conductor of E) The conductor of an elliptic curve E is QE =

Y p∈P

where P denotes the set of prime numbers. We are now ready to define the L-functions.

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p fp ,

1.2 1.2.1

L-functions Hasse-Weil L-functions

The theory of L-functions is a large area of analytic number theory, with many open problems unsolved so far. Such functions generalize the Riemann zeta function, and can be really useful, in particular for studying elliptic curves. Definition 1.5 (Riemann zeta function) ζ(s) : C → C analytically continues the series ∞ X 1 . ns n=1

One may note that it converges if 1. The Riemann zeta function is the analytic continuation of this series to the whole complex plane. Definition 1.6 (L-function) L(s) : C → C is an L-function if there exists (an ) ∈ CN such as L(s) is the analytic continuation of the series ∞ X an . s n n=1 There are several ways to define L-functions (sometimes called zeta functions), but this is basic definition. The main types of L-functions are Dirichlet L-functions, Dedekind L-functions and Hasse-Weil L-functions, and we are going to study the latter. However, a lot of properties of L-functions are still conjectural. Definition 1.7 (Hasse-Weil L-function of an elliptic curve) The Hasse-Weil L-function of an elliptic curve E is the following function: Y
LE (s) = 1 − ap p−s + ε(p)p1−2s , p∈P

  ε(p) = 1 if E has a good reduction at p and 0 otherwise. ap = p + 1 − ](E(FP )). where  E(Fp ) = {(x, y) ∈ F2p y 2 ≡ x3 + ax + b (mod p)}.

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Proposition 1.3 • LE (s) is an L-function. • LE (s) converges by definition over { 3/2}. • LE (s) can be extended by analytic continuation to the whole complex plane.

1.2.2

Quadratic twists

Before introducing the family of L-functions that will interest us in the rest of the paper, we must firstly define the notions of fundamental discriminant and quadratic twist. Definition 1.8 (Fundamental discriminant) Let d be an integer. d is a fundamental discriminant, and we write d ∈ D, if one of the following statements holds: • d ≡ 1 (mod 4) and d is square-free • d = 4m and m ≡ 2 or 3 (mod 4) and m is square-free. Some fundamental discriminants around 0 are -20, -19, -15, -11, -8, -7, -4, -3, 1, 5, 8, 12, 13, 17, 21. . . Definition 1.9 (Quadratic twist) Given an elliptic curve over Q E : y 2 = x3 + ax + b and a fundamental discriminant d non-divisor of QE , we define the quadratic twist of E by d as Ed : dy 2 = x3 + ax + b. There is a relation between the conductors of Ed and E: QEd = QE d2 . Another important fact is that the Hasse-Weil L-function LEd (s) of Ed , also called the twisted L-function, is related to LE (s).

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Proposition 1.4 Let LE (s) be the Hasse-Weil L-function of E: LE (s) =

∞ X an n=1

ns

.

Then the following formula holds: LEd (s) = LE (s, χd ) =

∞ X an χd (n) n=1

where χd (n) =

d n




ns

,

is the Kronecker symbol.

This could be written for any d ∈ Z. However, assuming that d is a fundamental discriminant and d does not divide QE ensures us that χd (n) = 0, 1 or − 1. 1.2.3

Functional equations

The Hasse-Weil L-functions satisfy functional equations that may highlight some of their characteristics. Proposition 1.5 Let LE (s) be the Hasse-Weil L-function of an elliptic curve E.  −s If we write ϕE (s) = √2π Γ(s)LE (s), then ϕE (s) satisfies the functional equation QE ϕE (s) = wE ϕE (1 − s), where wE = ±1 is called the sign of the functional equation and depends only on E. Proposition 1.6 Let LE (s, χd ) be the Hasse-Weil L-function of the quadratic twist Ed , with d a fundamental discriminant.  −s 2π If we write ϕE (s, χd ) = |d|√ Γ(s)LE (s, χd ), then LE (s, χd ) satisfies the functional QE equation ϕE (s, χd ) = [wE χd (−QE )] ϕE (1 − s, χd ). Then there are two possibilities: • if wE χd (−QE ) = 1 then LE (s, χd ) is called even, and the order of its zero at s = 1/2 is an even number • otherwise, LE (s, χd ) is called odd, and the order of its zero at s = 1/2 is an odd number. In particular, according to the Birch and Swinnerton-Dyer conjecture, if wE χd (−QE ) 6= 1 then rank(E) ≥ 1.

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1.2.4

The Riemann hypothesis

Studying the zeros of L-functions is a very important issue in analytic number theory. It may lead to another Millenium Prize Problem, which is as well Hilbert’s eighth problem: the Riemann hypothesis. Conjecture 1.2 (Riemann hypothesis) The non-trivial zeros of ζ(s) have real part 1/2. So far, it has been proved that there are infinitely many zeros on the vertical line {