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:05292015222022033
Abstract
This thesis studies Frobenius traces in Galois representations from two different directions. In the first problem we explore how often they vanish in Artintype 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 a_{p} of classical elliptical modular newforms f of weight 2 are never extremal, i.e., a_{p} 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 semisimple, 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): 

Thesis Committee: 

Defense Date:  26 May 2015 
Record Number:  CaltechTHESIS:05292015222022033 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:05292015222022033 
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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