JHU Number Theory Seminar, Spring 2008

Elliptic curves, quadratic twists and p-(in)divisibility of L-values

Let E be an elliptic curve over the rationals and L(E,s) the L-function associated to E. A famous theorem of Waldspurger states that there exists a quadratic discriminant d such that the central value L(E_d, 1) is nonzero, where E_d is the d'th quadratic twist of E. This theorem has many applications, most notably to the Birch and Swinnerton-Dyer conjecture in the case of rank 1 elliptic curves. I will explain a conjectural "mod p" version of Waldspurger's theorem and some related results that are obtained by studying the p-adic properties of the Shimura correspondence.

A Symplectic Test of the L-Functions Ratios Conjecture

Recently Conrey, Farmer and Zirnbauer conjectured formulas for the averages over a family of ratios of products of shifted L-functions. Their L-functions Ratios Conjecture predicts both the main and lower order terms for many problems, ranging from n-level correlations and densities to mollifiers and moments to vanishing at the central point. There are now many results showing agreement between the main terms of number theory and random matrix theory; however, there are very few families where the lower order terms are known. These terms often depend on subtle arithmetic properties of the family, and provide a way to break the universality of behavior. The L-functions Ratios Conjecture provides a powerful and tractable way to predict these terms. We test a specific case here, that of the 1-level density for the symplectic family of quadratic Dirichlet characters arising from even fundamental discriminants d < X. For test functions supported in (-1/3, 1/3) we calculate all the lower order terms up to size O(|X|^{-1/2 + epsilon}) and observe perfect agreement with the conjecture. Thus for this family and suitably restricted test functions, we completely verify the Ratios Conjecture's prediction for the 1-level density. If time permits we will discuss other families.

Note: the talk will begin with a review of L-functions and various statistics of their zeros. We will concentrate on explaining the ideas and techniques, and not the technical details.

Bounded gaps between products of primes with applications to class numbers and elliptic

In recent work, Goldston, Pintz, and Yildirim proved that $\liminf (p_{n + 1} - p_n) = 0$. (Here $p_n$ denotes the $n$th prime.) In follow-up work with S. Graham, they also proved the existence of bounded gaps between products of two primes. After an overview of their work and some of the history of this and related problems, we will present a similar bound for products of $r$ primes, for any $r \geq 2$, which covers the case when these primes are restricted to any Chebotarev set. As a consequence, we may apply results of Ono, Soundararajan, and Deligne-Serre to obtain "bounded gaps" results concerning class numbers, Fourier coefficients of modular forms, and ranks of elliptic curves.

Indivisibility of relative class numbers

For a totally real field $F$ and an odd prime number $p$, it is conjectured that there are infinitely many CM quadratic extension of $F$ whose relative class number is not divisible by $p$. I will discuss this problem for general totally real fields under a weak hypothesis on $p$.

On a geometric realization of Jacquet-Langlands correspondence

For a reductive group over a number field, it is expected, and known in some cases, that the representation theory is related to that of a quasi-split inner form (Jacquet-Langlands correspondence). In this talk, I will discuss a realization of this correspondence using mod p geometry of unitary Shimura varieties.