Archived Events

The main purpose of this talk is to explain the connection between vector-valued modular forms for SL(2,Z) and certain ordinary differential equations in the punctured unit disk, whose coefficient functions are holomorphic modular forms. An important tool arising in this context is a modular version of the familiar Wronskian from the classical ODE theory; I will discuss, for example, how this modular Wronskian provides a lower bound for the weight of a nonzero holomorphic form associated to a given representation of SL(2,Z).

In this talk we discuss two categories of integrable modules: modules with finite-dimensional weight spaces, and tensor modules. The latter modules form an interesting category on which the functor of algebraic dualization preserves the property of a module to have finite Loewy length. This is an unusual situation. Both categories are natural analogs of the category of finite-dimensional modules.

In this talk we will discuss the following problem: Given a triple of integers (r,s,t), which finite simple groups are quotients of the triangle group T(r,s,t)? This problem has many applications, especially concerning Riemann surfaces and Beauville surfaces. In the talk we will focus on the group theoretical aspects of this problem.

In this joint work with Stephan Baier, we prove a subconvexity bound for Godement-Jacquet $L$-functions associated with Maass forms for $SL(3,Z)$ The bound arrives from extending a method of M. Jutila (with new ingredients and innovations) on exponential sums with Fourier coefficients of holomorphic cusp forms for $SL(2,Z)$ to a $GL(3)$ setting.