Number theory seminar

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Speaker

Title

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* 19 March 2021
14:15-15:15

Dr. Raphael Steiner
ETH Zurich, Switzerland

Event Details

Title	Fourth moments and sup-norms with the aid of theta functions
Speaker, Affiliation	Dr. Raphael Steiner, ETH Zurich, Switzerland
Date, Time	19 March 2021, 14:15-15:15
Location	online, Zoom
Abstract	It is a classical problem in harmonic analysis to bound $L^p$-norms of eigenfunctions of the Laplacian on (compact) Riemannian manifolds in terms of the eigenvalue. A general sharp result in that direction was given by Hörmander and Sogge. However, in an arithmetic setting, one ought to do better. Indeed, it is a classical result of Iwaniec and Sarnak that exactly that is true for Hecke-Maass forms on arithmetic hyperbolic surfaces. They achieved their result by considering an amplified second moment of Hecke eigenforms. Their technique has since been adapted to numerous other settings. In my talk, I shall explain how to use Shimizu's theta function to express a fourth moment of Hecke eigenforms in geometric terms (second moment of matrix counts). In joint work with Ilya Khayutin and Paul Nelson, we give sharp bounds for said matrix counts and thus a sharp bound on the fourth moment in the weight and level aspect. As a consequence, we improve upon the best known bounds for the sup-norm in these aspects. In particular, we prove a stronger than Weyl-type sub-convexity result.

Fourth moments and sup-norms with the aid of theta functionsread_more

online, Zoom

* 26 March 2021
14:15-15:15

Dr. Subhajit Jana
Max Planck Institute

Event Details

Title	The Weyl bound for triple product $L$-functions
Speaker, Affiliation	Dr. Subhajit Jana, Max Planck Institute
Date, Time	26 March 2021, 14:15-15:15
Location	Zoom
Abstract	We prove the Weyl bound for the triple product $L$-functions of three unitary cuspidal representations for $\mathrm{SL}_2(\mathbb{Z})$, two of which are fixed and the remaining one has growing analytic conductor. We may also consider the varying representation to be an Eisenstein series to prove the $t$-aspect Weyl bound for the Rankin--Selberg $L$ functions. These results improve upon previous famous work of Bernstein--Reznikov. This is a joint work with Valentin Blomer and Paul Nelson.

The Weyl bound for triple product $L$-functionsread_more

Zoom

* 30 April 2021
14:15-15:15

Andrea Musso
ETH Zürich

Event Details

Title	Equidistribution of rational subspaces and their shapes
Speaker, Affiliation	Andrea Musso, ETH Zürich
Date, Time	30 April 2021, 14:15-15:15
Location	Zoom
Abstract	It is a classical problem in number theory to understand the distribution of integer points on the n-sphere. This talk will explore a recent result, obtained jointly with Menny Akka and Andreas Wieser, generalising a well known theorem concerning equidistribution of such integer points. We prove that triples consisting of a k-dimensional rational subspace of R^n with discriminant D and two associated lattice shapes (the shape of the lattice within the subspace and within its orthogonal complement) equidistribute simultaneously as the discriminant grows to infinity, under a congruence condition. The talk will begin by a short historical review of the problem, underlining the main results preceding our own. It will continue with a sketch of proof of our result with the goal of highlighting some core ideas and techniques in the interplay of dynamics and number theory.

Equidistribution of rational subspaces and their shapesread_more

Zoom

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