JHU Number Theory Seminar, Fall 2006

Gauss sums and Poincare sums

Let (g,(G,M)) be a simple setting, where (G,M) is a G-module and g is a group acting on (G,M) naturally. Gauss sums, Theta series and Eisenstein series are classical examples. I will discuss other cases including the wild ramification of number fields and arithmetic of SU(2).

Rank-one quadratic twists of elliptic curves

Using Heegner points on elliptic curves, we give a systematic way to find elliptic curves E whose algebraic rank and analytic rank are equal to one for a positive proportion of fundamental discriminants D.

Values of zeta functions at s=1/2

we discuss recent results about values of zeta functions at s=1/2. These are zeta functions of varieties over finite fields. After describing the main result, we will end with some comments about values at other half-integers.

On the rank of elliptic curves

We will review known results concerning the density of large rank elliptic curves, and we will discuss our current work from the point of view of diophantine geometry.

Analytic continuation of Langlands L-functions via boundary value distributions

The problem of analytically continuing Langlands L-functions has many consequences in number theory, including many deep cases of his functoriality conjectures. Historically there have existed two approaches: integral representations (the Rankin-Selberg method) and coefficients of Eisenstein series (the Langlands-Shahidi method). Many important successes have been obtained by these two methods, which have reached a degree of maturity. Though the two methods complement each other very well, they still often leave important gaps unresolved between them. We have developed a third method which, in preliminary cases, resolves these gaps. For example, so far we have given the full analytic continuation of the exterior square L-functions on GL(n,Z)\GL(n,R). I will discuss the basis of the method, which involves studying the boundary value distributions of automorphic forms, and then pairing them on flag varieties using open orbits. [Joint work with Wilfried Schmid, Harvard University]

On zeros of Eisenstein series for genus zero Fuchsian groups

Let $\Gamma\leq\SL_2(\mathbb{R})$ be a genus zero Fuchsian group of the first kind with $\infty$ as a cusp, and let $E_{2k}^{\Gamma}$ be the holomorphic Eisenstein series of weight $2k$ on $\Gamma$ that is nonvanishing at $\infty$ and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on $\Gamma,$ and on a choice of a fundamental domain $\mathcal{F}$, we prove that all but possibly $c(\Gamma,\mathcal{F})$ of the non-trivial zeros of $E_{2k}^{\Gamma}$ lie on a certain subset of $\{z\in\mathfrak{H}\,:\,j_{\gamma}(z)\in\mathbb{R}\}$. Here $c(\Gamma,\mathcal{F})$ is a constant that does not depend on the weight, $\mathfrak{H}$ is the upper half-plane, and $j_{\Gamma}$ is the canonical hauptmodul for $\GN.$

Motivating periods in renormalizable quantum field theories