Introduction

This paper gives a brief and informal introduction to some applications of symmetric functions theory in random matrix theory. It is based on the lecture given as part of the program “Random matrix approaches in number theory” held at the Newton Institute from February to July of 2004; I would like to take this opportunity to thank the organizers for making this program such a wonderful and memorable one.

We begin, in section 2, by presenting a self-contained and fairly complete proof of the Weyl Integration Formula, the basic tool for averaging over U(n), following closely Weyl's original derivation. In the course of the proof we also provide a simple qualitative explanation for “quadratic repulsion” of the eigenvalues of unitary matrices. As a straightforward application of the Weyl Integration Formula we derive a simple but striking result of Rains on high powers of unitary matrices.

Following the organizers’ brief, in the exposition below I assume no prior knowledge of symmetric function theory. A crucial role in this theory is played by Schur functions and each section provides a glimpse of a different facet of their many-sided nature. In section 3 we introduce Schur functions and prove that they describe irreducible characters of the unitary group, following closely the derivation of Weyl in his classical book. In section 4 we prove asymptotic normality of traces of powers of random unitary matrices as a consequence of Schur-Weyl duality, following Diaconis and Shahshahani, who pioneered the symmetric functions theory approach.

Abstract

We discuss recent applications of analytic number theory to the study of ranks of elliptic curves.

Introduction

This article is meant to be a sampling of techniques and interesting results on the (analytic) ranks of elliptic curves. The main result discussed is an upper bound on the average rank of the family of all elliptic curves. The bound obtained is less than 2, which implies (by work of Kolyvagin) that a positive proportion of elliptic curves have finite Tate-Shafarevich group and algebraic rank equal to analytic rank. The synergy here between algebraic and analytic methods is extremely pleasant.

We also discuss the problem of showing that a large number of elliptic curve L-functions do not vanish at the central point. Many of the techniques used in bounding the average rank are useful in this direction.

Our exposition is meant to be somewhat colloquial. The interested reader should consult [Y1] and [Y2] for all technical details.

In this volume E. Kowalski has given a broad overview of what is known on ranks of elliptic curves in families. We shall refer to his article for general background knowledge on elliptic curves. We have attempted to minimize overlap with his article without loss of coherence of this paper.

We shall assume the Generalized Riemann Hypothesis throughout this article.

Acknowledgements

I would like to thank Henryk Iwaniec for supporting my last-minute decision to attend the workshops. I also thank the organizers of the Newton Institute program for inviting me to attend.

Abstract

We calculate the value distribution of the first derivative of characteristic polynomials of matrices from SO(2N + 1) at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. The connection between the values of random matrix characteristic polynomials and values of the L-functions of families of elliptic curves implies that this calculation in random matrix theory is relevant to the problem of predicting the frequency of rank three curves within these families, since the Birch and Swinnerton-Dyer conjecture relates the value of an L-function and its derivatives to the rank of the associated elliptic curve. This article is based on a talk given at the Isaac Newton Institute for Mathematical Sciences during the “Clay Mathematics Institute Special Week on Ranks of Elliptic Curves and Random Matrix Theory”.

Introduction

Random matrix theory and number theory

The connection between random matrix theory and number theory began with the work of Montgomery when he conjectured that the distribution of the complex zeros of the Riemann zeta function follows the same statistics as the eigenvalues of a random matrix chosen from U(N) generated uniformly with respect to Haar measure. This conjecture is supported by numerical evidence and also by further work suggesting that the same conjecture is true for more general L-functions. For all these L-functions there is a Generalized Riemann Hypothesis that the non-trivial zeros lie on a vertical line in the complex plane.

The group of rational points on an elliptic curve is a fascinating number theoretic object. The description of this group, as enunciated by Birch and Swinnerton-Dyer in terms of the special value of the associated L-function, or a derivative of some order, at the center of the critical strip, is surely one of the most beautiful relationships in all of mathematics; and it's understanding also carries a $1 million dollar reward!

Random Matrix Theory (RMT) has recently been revealed to be an exceptionally powerful tool for expressing the finer structure of the value-distribution of L-functions. Initially developed in great detail by physicists interested in the statistical properties of energy levels of atomic nuclei, RMT has proven to be capable of describing many complex phenomena, including average behavior of L-functions.

The purpose of this volume is to expose how RMT can be used to describe the statistics of some exotic phenomena such as the frequency of rank two elliptic curves. Many, but not all, of the papers here have origins in a workshop that took place at the Isaac Newton Institute in February of 2004 entitled “Clay Mathematics Institute Special week on Ranks of Elliptic Curves and Random Matrix Theory.” The workshop began with the Spittalsfield day of expository lectures, highlighted by reminiscences by Bryan Birch and Sir Peter Swinnerton-Dyer on the development of their conjecture. The week continued with a somewhat free-form workshop featuring discussion sessions, groups working on various problems, and spontaneous lectures.

When we have to study a number field or an elliptic curve defined over ℚ, some groups may appear which make the explicit computations more complicated and which are, in a way, not very “welcome”. These groups are the class groups of number fields and the Tate-Shafarevich groups of elliptic curves. A direct study of their general behavior is a very difficult problem. In, Cohen and Lenstra explained how to obtain precise conjectures for this purpose using a general fundamental heuristic principle. In, it is shown how to adapt the Cohen-Lenstra idea to Tate-Shafarevich groups using the analogy between number fields and elliptic curves. Understanding the behavior of Tate-Shafarevich groups is important in itself first but it may also be useful for studying the distribution of the special values of the L-functions L(E, s) attached to elliptic curves. Indeed, the Birch and Swinnerton-Dyer conjecture relates the value L(E; 1) to natural invariants of E including the order of the Tate-Shafarevich group. This paper sketches the Cohen-Lenstra philosophy in both cases of class groups and of Tate-Shafarevich groups. It is organized as follows:

In the first section, we describe the analogy between number fields and elliptic curves defined over ℚ. In the second section, we recall the Cohen-Lenstra heuristic for class groups. Using the analogy of the first section, we adapt, in the third section, the heuristic for Tate-Shafarevich groups. Finally, we restrict the heuristic to the case of families of quadratic twists of an elliptic curve. Acknowledgements.

Abstract

We give infinite families of elliptic curves over ℚ such that each curve has infinitely many non-isomorphic quadratic twists of rank at least 4. Assuming the Parity Conjecture, we also give elliptic curves over ℚ with infinitely many non-isomorphic quadratic twists of odd rank at least 5.

Introduction

Mestre showed that every elliptic curve over ℚ has infinitely many (non-isomorphic) quadratic twists of rank at least 2 over ℚ, and he gave several infinite families of elliptic curves over ℚ with infinitely many (non-isomorphic) quadratic twists of rank at least 3. Further, he stated that if E is an elliptic curve over ℚ with torsion subgroup isomorphic to ℤ/8ℤ × ℤ/2ℤ, then there are infinitely many (non-isomorphic) quadratic twists of E with rank at least 4 over ℚ.

In this paper (Theorems 3.2 and 3.6) we give additional infinite families of elliptic curves over ℚ with infinitely many (non-isomorphic) quadratic twists of rank at least 4. The family of elliptic curves in Theorem 3.2 is parametrized by the projective line. The family of elliptic curves in Theorem 3.6 is parametrized by an elliptic curve of rank one. In both cases, the twists are parametrized by an elliptic curve of rank at least one.

In addition, we find elliptic curves over ℚ that, assuming the Parity Conjecture, have infinitely many (non-isomorphic) quadratic twists of odd rank at least 5 (see Theorem 5.1 and Corollary 5.2). The proof relies on work of Rohrlich.