Petrow, Ian;
(2013)
Moments of Automorphic L-functions and Related Problems.
Doctoral thesis (Ph.D), Stanford University.
Preview |
Text
PetrowDissertation.pdf - Published Version Download (801kB) | Preview |
Abstract
We present in this dissertation several theorems on the subject of moments of automorphic L-functions. In chapter 1 we give an overview of this area of research and summarize our results. In chapter 2 we give asymptotic main term estimates for several different momentsof central values of L-functions of a fixed GL2 holomorphic cusp form f twisted by quadratic characters. When the sign of the functional equation of the twist L(s, f ⊗ χ_d) is −1, the central value vanishes and one instead studies the derivative L′(1/2, f ⊗ χ_d). We prove two theorems in the root number −1 case which are completely out of reach when the root number is +1. In chapter 3 we turn to an average of GL2 objects. We study the family of cuspforms of level q^2 which are given by f ⊗ χ, where f is a modular form of prime level q and χ is the quadratic character modulo q. We prove a precise asymptotic estimate uniform in shifts for the second moment with the purpose of understanding the off-diagonal main terms which arise in this family. In chapter 4 we prove an precise asymptotic estimate for averages of shifted con-volution sums of Fourier coefficients of full-level GL2 cusp forms over shifts. We find that there is a transition region which occurs when the square of the average overshifts is proportional to the length of the shifted sum. The asymptotic in this rangedepends very delicately on the constant of proportionality: its second derivative seems to be a continuous but nowhere differentiable function. We relate this phenomenon to periods of automorphic forms, multiple Dirichlet series, automorphic distributions, and moments of Rankin-Selberg L-functions.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Moments of Automorphic L-functions and Related Problems |
| Event: | Stanford University |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © Copyright 2013 by Ian Nicholas Petrow. This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 Unported License (http://creativecommons.org/licenses/by-nc/3.0/) |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10116981 |
Archive Staff Only
 |
View Item |
