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Nonvanishing of L-Functions for GL(n)


Nastasescu, Maria Monica (2016) Nonvanishing of L-Functions for GL(n). Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9QN64R2.


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:L-function, automorphic forms, relative trace formula
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Prize for Excellence as a First-Year Graduate Student, 2012. Apostol Award for Excellence in Teaching in Mathematics, 2013. Scott Russell Johnson Prize for Excellence in Graduate Studies, 2014. Apostol Award for Excellence in Teaching in Mathematics, 2015. Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2016.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Ramakrishnan, Dinakar
Thesis Committee:
  • Ramakrishnan, Dinakar (chair)
  • Mantovan, Elena
  • Zhu, Xinwen
  • Tsai, Pei-Yu
Defense Date:1 June 2016
Record Number:CaltechTHESIS:06072016-183228371
Persistent URL:
Related URLs:
URLURL TypeDescription 10.1016/j.jnt.2014.12.012DOIArticle adapted for ch. 2
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9855
Deposited By: Maria Nastasescu
Deposited On:09 Mar 2017 17:50
Last Modified:04 Oct 2019 00:14

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