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Quantum Ergodicity and Subconvexity of L-functions

Published in Courses
Wednesday, 09-25-2019

Topic and importance

The interplay between mathematical physics and automorphic forms led to the development of the area of Arithmetic Quantum Chaos.  With pioneering work by Sarnak, Rudnick, and Luo, and culminating with work of Lindenstrauss and Soundararajan,  it immediately became known that questions such as non-concentration of masses of eigenforms are intimately related to the problem of subconvexity of L-functions.

This also blossomed in the last twenty years under Michel, Venkatesh,  et al.  to become one of the centre points of the analytic theory of automorphic forms.

Currently an area of great activity, subconvexity results and their consequences have become central in the analytic study of automorphic forms; however, this is a difficult area for students to penetrate.  The purpose of this mini-course is to provide participants an entrance to this area by using a concrete, accessible set-up, where the solutions are known.


After introducing the notion of Quantum Ergodicity for chaotic systems, we look at the spectral decomposition of (arithmetic) hyperbolic surfaces,  the (non-holomorphic) Eisenstein series, and Maass forms. We discuss the L-functions of Maass cusp forms and Rankin-Selberg convolutions. We explain the significance of subconvexity of L-functions to Quantum Unique Ergodicity (QUE) and prove the subconvex results needed for QUE of Eisenstein series: Weyl's bound for zeta on the critical line, estimates near the edge of the critical strip,  subconvexity bounds for the L-functions of Maass cusp forms.


Lecture 1

Lecture 2

Problems 1

Lecture 3

Lecture 4

Problems 2

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