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Supersingular distribution, congruence class bias, and a refinement of strong multiplicity one

Citation

Walji, Nahid (2011) Supersingular distribution, congruence class bias, and a refinement of strong multiplicity one. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:02132011-055211033

Abstract

This thesis consists of four chapters, including an introduction.

In Chapter 2, we take an averaging approach to the question of the distribution of supersingular primes of degree one, for elliptic curves over a number field. We begin by modifying the Lang-Trotter heuristic to address the case of an abelian extension, then we show that it holds on average (up to a constant) for a family of elliptic curves by using ideas of David-Pappalardi.

In Chapter 3, we prove constructively that there exists an infinite number of (arbitrarily) thin families of rational elliptic curves for which the Lang-Trotter conjecture holds on average, in part by using techniques of Fouvry-Murty.

In Chapter 4, we obtain a result related to the strong multiplicity one theorem for non-dihedral cuspidal automorphic representations for GL(2), with trivial central character and non-twist-equivalent symmetric squares. Given a real algebraic number, we also find a lower bound for the lower density of the set of finite places for which the associated Hecke eigenvalue is not equal to that algebraic number.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:supersingular distribution, Lang-Trotter conjecture, strong multiplicity one.
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:
  • Ramakrishnan, Dinakar (chair)
  • Flach, Matthias
  • Makarov, Nikolai G.
  • Jorza, Andrei
Defense Date:6 January 2011
Record Number:CaltechTHESIS:02132011-055211033
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:02132011-055211033
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6242
Collection:CaltechTHESIS
Deposited By: Nahid Walji
Deposited On:29 Mar 2011 16:35
Last Modified:16 Apr 2013 23:33

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