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Mathematics Colloquium
2014 - 2015

Wednesday, November 12, 2014
4:00 p.m.  //  151 Sloan
Joseph Bernstein (University of Tel Aviv) -
Convexity and Subconvexity bounds for automorphic periods
and representation theory

Let Y be a compact Riemann surface with a Riemannian metric of constant curvature −1. Consider the corresponding Laplace-Beltrami 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 introduce 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 = Σ an exp(2πint).  The problem is to give good bounds for Fourier coefficients an when n tends to infinity.

Time permitting; I will explain the relation of this problem with the theory of L-functions. This is an exposition of my joint works with Andre Reznikov.   

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