JHU Number Theory Seminar, Spring 2004

Abstract of Talks

February 11, 3:00pm--4:00pm

Ling Long, On cusp forms of noncongruence subgroups
Theory of modular form of noncongruence subgroups is far less developed compared to theory of congruence modular form. The Atkin-Swinnerton-Dyer conjecture is one of the major conjectures for cusp forms of noncongruence subgroups. While there has been significant progress, for example, the weak Atkin-Swinnerton-Dyer conjecture proved by Scholl, the full version of the conjecture still remain to be open.
We are going to discuss a particular situation where some spaces of cusp forms of nongongruence subgroups admit bases satisfying the Atkin-Swinnerton-Dyer congruence relations with certain newforms of congruence subgroups. Consequently, the Atkin-Swinnerton-Dyer conjecture for these spaces are established.

March 3, 4:00pm--5:00pm

Steven Miller, Random Matrix Theory and Families of Elliptic Curves: Evidence for the Underlying Group Symmetries
Random Matrix Theory began in the 50s, when physicists used matrix ensembles to model energy levels of nuclei. Since then, the theory has been generalized and applied to zeros of L-functions. Based on the function field case, we expect every family of L-functions has an associated symmetry group (one of the classical compact groups) controlling the distribution of its zeros: the behavior of zeros near the central point is the same as that of eigenvalues near 1 for that group. We describe two families, Dirichlet Characters and Elliptic Curves.
The first family is straightforward to analyze, and illustrates the techniques of the subject; the variation of the conductors in the second shows some of the complications that can arise. For elliptic curves, we expect to see SO(even) (SO(odd)) if every curve is even (odd). By studying the n-level density (defined by summing a test function over the zeros), we obtain a statistic which is different for each of the candidate groups. Previous investigations calculated the 1-level density for functions supported in (-1,1), where the orthogonal groups are indistinguishable. In this talk we calculate the 2-level density in a restricted range, often for families with forced rank over Q(t). We show the three orthogonal groups have different 2-level densities for test functions supported in an arbitrarily small neighborhood of the origin. Assuming standard conjectures, this allows us to observe the expected orthogonal group, and obtain evidence supporting the Birch and Swinnerton-Dyer conjecture.

March 10, 4:00pm--5:00pm

William Stein, Visibility of Shafarevich-Tate Groups at Higher Level
I will begin by discussing the Birch and Swinnerton-Dyer conjecture in the context of abelian varieties attached to modular forms. I will then introduce Mazur's notion of visibility of Shafarevich-Tate groups and explain some of the basic facts and theorems about it. Cremona, Mazur, Agashe, and myself carried out large computations about visibility for modular abelian varieties of level N in J_0(N). These computations addressed the following question: If A is a modular abelian variety of level N, how much of the Shafarevich-Tate group of A is modular of level N, i.e., visible in J_0(N). The results of these computations suggest that often much of the Shafarevich-Tate group is not modular of level N. This suggests asking if every element is modular of level N*m, for some auxiliary integer m, and if so, what can one say about the set of such m? I will finish the talk with some new data and thoughts about this last question, which is still very much open.

April 7, 4:00pm--5:00pm

Jordan Ellenberg, Counting number fields of bounded discriminant
An old conjecture holds that the number N_n(X) of degree n number fields with discriminant less than X is asymptotic to c_n X when X grows and n is fixed. This conjecture was proved by Davenport and Heilbronn for n = 3, and recently for n = 4,5 by Bhargava. For general n, however, the best known upper bound, due to Schmidt, was N_n(X) << X^{(n+2)/4}. We prove the much stronger bound N_n(X) << X^{exp(C sqrt(log n))}. While the theorem appears number-theoretic, the arithmetic input is minimal, and the method is purely algebro-geometric; the main idea is to relate the problem of counting number fields to the problem of counting integral points on certain carefully chosen varieties related to Hilbert schemes of points on affine space. We will also describe function-field analogues of the main theorem, and speculate about relations between the theorems proved here and the Batyrev-Manin heuristics for rational points on Fano varieties. This is joint work with Akshay Venkatesh.

April 21, 4:00pm--5:00pm

Gergely Harcos, The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points
We shall review the latest developments in a program initiated by Kowalski, Michel and Vanderkam and further pursued by Michel. The goal is to prove a nontrivial bound in the level aspect for the L-function of the product of two classical Maass forms of arbitrary weights, one of the forms being fixed. In joint work with Michel, we succeed in proving such a bound as long as the fixed form obeys Selberg's conjecture or at least does not violate it too much. Another case where we succeed is when the nebentypus character of the varying form has small conductor. We pay special attention to polynomial uniformity in all other parameters. The bound implies the equidistribution of small Galois orbits of Heegner points on certain Shimura curves.