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Some new aspects of mass equidistribution

Citation

Nelson, Paul D. (2011) Some new aspects of mass equidistribution. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:06042011-162333825

Abstract

This thesis presents two new results concerning the limiting behavior of families of automorphic forms. First, let F be a totally real number field of degree n, and let f traverse a sequence of nondihedral holomorphic eigencuspforms on GL(2,F) of trivial central character and full level. We show that the mass of $f$ equidistributes on the Hilbert modular variety as any one of the weight components of f tends to infinity. Our result answers affirmatively a natural analogue of a conjecture of Rudnick and Sarnak (1994). Our proof generalizes the argument of Holowinsky-Soundararajan (2008) who established the case F = Q. The essential difficulty in doing so is to adapt Holowinsky's bounds for the Weyl periods of the equidistribution problem in terms of manageable shifted convolution sums of Fourier coefficients to the case of a number field with nontrivial unit group. Second, let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing squarefree level q tending to infinity. We prove that the pushforward of the mass of f to the modular curve of level 1 equidistributes with respect to the Poincare measure. Our result answers affirmatively the squarefree level case of a conjecture spelled out by Kowalski, Michel, and VanderKam (2002) in the spirit of a conjecture of Rudnick and Sarnak (1994). Our proof follows the strategy of Holowinsky and Soundararajan (2008) who showed that newforms of level 1 and large weight have equidistributed mass. The new ingredients required to treat forms of fixed weight and large level are an adaptation of Holowinsky's reduction of the problem to one of bounding shifted sums of Fourier coefficients, a refinement of his bounds for shifted sums, an evaluation of the p-adic integral needed to extend Watson's formula to the case of three newforms where the level of one divides but need not equal the common squarefree level of the other two, and some additional technical work in the problematic case that the level has many small prime factors.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:number theory, automorphic forms, L-functions, Riemann hypothesis, quantum chaos, quantum unique ergodicity
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Ramakrishnan, Dinakar
Thesis Committee:
  • Mantovan, Elena
  • Flach, Matthias
  • Jorza, Andrei
  • Ramakrishnan, Dinakar (chair)
Defense Date:26 May 2011
Record Number:CaltechTHESIS:06042011-162333825
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:06042011-162333825
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6499
Collection:CaltechTHESIS
Deposited By: Paul Nelson
Deposited On:06 Jun 2011 23:36
Last Modified:23 Dec 2013 22:09

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