Automorphic L-Functions, Geometric Invariants, and Dynamics

Author: Perozim de Faveri, Alexandre

Year: 2022

Degree: Dissertation (Ph.D.)

Advisor: Radziwiłł, Maksym

Committee Members: Ramakrishnan, Dinakar; Dunn, Alexander; Michel, Philippe R.; Radziwiłł, Maksym

Option: Mathematics

DOI: 10.7907/e1zd-7v46

Abstract

We address three different problems in analytic number theory.

In the first part, we show that the completed L-function of a modular form has Ω(Tδ) simple zeros with imaginary part in [-T, T], for any δ < 2⁄27. This is the first power bound for forms with non-trivial level in this problem, where previously the best result was Ω(log log log T). Along the way, we also improve the corresponding bound in the case of trivial level, and sharpen a certain zero-density result.

In the second part, we study the variance for the distribution of closed geodesics in random balls on the modular surface. A probabilistic model in which closed geodesics are modeled using random geodesic segments is proposed, and we rigorously analyze this model using mixing of the geodesic flow. This leads to a conjecture for the asymptotic behavior of the variance, and we prove this conjecture for sufficiently small balls.

In the third part, we prove Sarnak's Möbius disjointness conjecture for C1+ε skew products on the 2-torus over a rotation of the circle.

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