Nonvanishing of L-Functions for GL(n)

Author: Nastasescu, Maria Monica

Year: 2016

Degree: Dissertation (Ph.D.)

Advisor: Ramakrishnan, Dinakar

Committee Members: Ramakrishnan, Dinakar; Mantovan, Elena; Zhu, Xinwen; Tsai, Pei-Yu

Option: Mathematics

DOI: 10.7907/Z9QN64R2

Abstract

In this thesis I study two different approaches towards proving average results on values of L-functions, with an interest toward establishing new results on automorphic L-functions, especially concerning the nonvanishing of L-functions of degree > 2 at the center of the critical strip (and at other points of the complex plane), and their applications, particularly to p-adic L-functions. In the first problem, I evaluate a twisted average of L-values using the approximate functional equation in order to prove a result on the determination of isobaric representations of GL(3, AQ) by certain L-values of p-power twists. I also provide an application to the adjoint p-adic L-function of an elliptic curve. More specifically, I show that if E is an elliptic curve over Q with semistable reduction at some fixed prime p, then the adjoint p-adic L-function of E evaluated at any infinite set of integers relatively prime to p completely determines up to a quadratic twist the isogeny class of E.

For the second problem, which is part of a long project, I present some results towards proving an average result for the degree 4 L-function on GSp(4)/Q at the center using the relative trace formula. More specifically, I consider a suitable relative trace formula such that the spectral side is an average of central L-values of genus 2 holomorphic Siegel eigenforms of weight k and level N twisted by some fixed character. I then work towards computing the corresponding geometric side.

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