Algebra and Number Theory Seminar Spring 2013

Tuesday, April 23

Christelle Vincent, Stanford
The Tate-Drinfeld module and the u-series expansions of Drinfeld modular forms

We consider the so-called Drinfeld setting, a function field analogue of some aspects of the theory of modular forms, modular curves and elliptic curves. After giving a short but hopefully complete introduction to the main objects of study of this setting, we will describe the construction of the Tate-Drinfeld module. We will then present some applications to the arithmetic of the coefficients of Drinfeld modular forms, and to the geometry of the Drinfeld modular curves.

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Tuesday, April 30

Andrew Fiori, McGill University
Characterization of Special Points on Orthogonal Shimura Varieties

We shall discuss a characterization of the special fields associated to the special points on the Shimura varieties attached to orthogonal groups O_q coming from quadratic forms q of signature (2,n). The low dimensional members of this family of Shimura variety includes the modular curve, certain Shimura curves and Hilbert modular surfaces. Shimura reciprocity tells us that the values of modular forms at special points will always be contained in the Hilbert class field of a field closely related to the special field of the point. The special points are characterized by the maximal rational algebraic tori T \subset O_q. In this talk, starting with the basics, we will discuss the structure of algebraic tori in orthogonal groups, the conditions under which certain algebraic tori may be embedded in an orthogonal group and finally what implications this has for the possible special fields associated to the Shimura varieties in which we are interested.

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Tuesday, May 14

Paul Garrett, University of Minnesota
Self-adjoint operators on spaces of automorphic forms

Revisiting work of Hejhal, Colin de Verdiere, and others from the viewpoint of Levi-Sobolev function spaces allows design of boundary-value problems in spaces of automorphic forms whose discrete spectrum, if any, is intimately related to zeros of zeta functions. Similar constructions apply to Epstein zeta functions, and, more generally, to compact periods of Eisenstein series. The connection to Epstein zetas, shown by Davenport-Heilbronn to have many off-the-line zeros, shows one limitation of this approach. Discretization of continuous spectra, e.g. following Lax-Phillips, proves a regularity of spacing incompatible with pair correlations, giving another limitation. Possible means to surmount these difficulties are sketched. (Part of this is joint work with E. Bombieri.)
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Thursday, May 16

Terry Gannon, University of Alberta
Much ado about Mathieu

A couple of years ago Eguchi et al observed coefficients 90, 462, 1540, ... in the elliptic genus of string theory compactified on K3 surfaces, and with that Mathieu Moonshine was born. The main conjecture is that a certain infinite sequence of class functions are all true representations of the largest Mathieu group. I'll explain how to prove this. The big things still missing are to identify a VOA interpretation underlying Mathieu moonshine, and a connection to geometry.

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Tuesday, June 4

Modular Symbols associated to Eisenstein Series

Vinayak Vatsal, University of British Columbia

Let $E$ and $f$ be an Eisenstein series and a cusp form, respectively, of the same weight $k\geq 2$ and of the same level $N$, both eigenfunctions of the Hecke operators, and both normalized so that $a_1 = 1$. The main result we prove is that when $E$ and $f$ are congruent mod a prime $\mathfrak{p}$ (which we take to be a prime of $\overline{Q}$ lying over a rational prime $p >2$), the algebraic parts of the special values $L(E,\chi ,j)$ and $L(f,\chi ,j)$ satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions,

\[ \frac{\tau (\bar{\chi })L(f,\chi ,j)}{(2 \pi i)^{j-1}\Omega _f^{\text{sgn}(E)}} \equiv  \frac{\tau (\bar{\chi })L(E,\chi ,j)}{(2 \pi i)^{j}\Omega _E} \pmod{\mathfrak{p}} \]

where the sign of $E$ is $\pm 1$ depending on $E$, and $\Omega _f^{\text{sgn}(E)}$ is the corresponding canonical period for $f$. Also, $\chi $ is a primitive Dirichlet character of conductor $m$, $\tau (\bar{\chi })$ is a Gauss sum, and $j$ is an integer with $0< j< k$ such that $(-1)^{j-1}\cdot \chi(-1) = \text{sgn}(E)$. Finally, $\Omega _E$ is a $\mathfrak{p}$-adic unit which is independent of $\chi $ and $j$. This is a generalization of earlier results of Stevens and Vatsal for weight $k=2$. The main point is the construction of a modular symbol associated to an Eisenstein series.

This is joint with Jay Heumann