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Special Frobenius Traces in Galois Representations

Citation

Chiriac, Liubomir (2015) Special Frobenius Traces in Galois Representations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9JM27J6. http://resolver.caltech.edu/CaltechTHESIS:05292015-222022033

Abstract

This thesis studies Frobenius traces in Galois representations from two different directions. In the first problem we explore how often they vanish in Artin-type representations. We give an upper bound for the density of the set of vanishing Frobenius traces in terms of the multiplicities of the irreducible components of the adjoint representation. Towards that, we construct an infinite family of representations of finite groups with an irreducible adjoint action.

In the second problem we partially extend for Hilbert modular forms a result of Coleman and Edixhoven that the Hecke eigenvalues ap of classical elliptical modular newforms f of weight 2 are never extremal, i.e., ap is strictly less than 2[square root]p. The generalization currently applies only to prime ideals p of degree one, though we expect it to hold for p of any odd degree. However, an even degree prime can be extremal for f. We prove our result in each of the following instances: when one can move to a Shimura curve defined by a quaternion algebra, when f is a CM form, when the crystalline Frobenius is semi-simple, and when the strong Tate conjecture holds for a product of two Hilbert modular surfaces (or quaternionic Shimura surfaces) over a finite field.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Galois representations, Frobenius traces, automorphic forms.
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Prize For Excellence In Graduate Studies, 2013; Apostol Award For Excellence In Teaching In Mathematics, 2014; Apostol Award For Excellence In Teaching In Mathematics, 2015
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Ramakrishnan, Dinakar
Thesis Committee:
  • Ramakrishnan, Dinakar (chair)
  • Mantovan, Elena
  • Flach, Matthias
  • Hadian-Jazi, Majid
Defense Date:26 May 2015
Record Number:CaltechTHESIS:05292015-222022033
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:05292015-222022033
DOI:10.7907/Z9JM27J6
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8942
Collection:CaltechTHESIS
Deposited By: Liubomir Chiriac
Deposited On:01 Jun 2015 22:18
Last Modified:12 Apr 2016 22:46

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