Theta functions, fourth moments of eigenforms and the sup-norm problem

Paul Nelson (ETH Zurich)

02-Dec-2020, 13:00-14:00 (3 years ago)

Abstract: I will discuss joint work with Raphael Steiner and Ilya Khayutin in which we study the sup norm problem for GL(2) eigenforms in the squarefree level aspect. Unlike the standard approach to the problem via arithmetic amplification following Iwaniec--Sarnak, we apply a method, introduced earlier in other aspects by my collaborators, which consists of identifying a fourth moment over a family of eigenforms evaluated at the point of interest with the L^2-norm of a theta function defined using the correspondence of Eichler, Shimizu and Jacquet--Langlands. After solving some counting problems (involving both "linear" sums as in traditional approaches and new "bilinear" sums), we obtain a bound comparable to the fourth root of the volume, improving upon the trivial square root bound and the nontrivial cube root bound established by Harcos--Templier and Blomer--Michel. I will describe the proof in the simplest case.

algebraic geometryalgebraic topologygroup theorynumber theoryrepresentation theory

Audience: researchers in the topic


Queen Mary University of London Algebra and Number Theory Seminar

Organizer: Shu Sasaki*
*contact for this listing

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