Convexity and subconvexity bounds for automorphic periods and special values of L-functions

Let Y be a compact Riemann surface with a Riemannian metric of constant curvature −1. Consider the corresponding Laplace-Beltramy operator in the space of functions on Y and fix its eigenfunction φ. Such function is called Maass form; study of such forms plays an important role in Geometry and Number Theory.

I will present a general method how to bound invariants arising from Maass forms. This method is based on representation theory of the group SL(2, R).

In my talk I will discuss the following concrete problem. Fix a closed geodesic C ⊂ Y , consider the restriction f of the Maass form φ to C and decompose it into its Fourier series
f =\sum  an exp(nt · 2πi). The problem is to give good bounds for Fourier coefficients an when n tends to infinity.

One of the goals of my talk is to describe the relation between automorphic periods and special values of L-functions; this relation provides highly non- trivial information about the asymptotic behavior of automorphic periods. In particular it suggests some initial bound of these periods (it is called convexity bound.)

The main goal of the talk is to show how to use representation theory to prove some stronger version of convexity bound for automorphic periods. I will also try to indicate how one can try to prove subcovexity bounds for this type of problems.

This is an exposition of my joint works with Andre Reznikov.