Universal deformations of even Galois representations and relations to Maass wave forms
Bockle, Gebhard
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https://hdl.handle.net/2142/21198
Description
Title
Universal deformations of even Galois representations and relations to Maass wave forms
Author(s)
Bockle, Gebhard
Issue Date
1995
Doctoral Committee Chair(s)
Ullom, Stephen V.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
The whole area of deformations of Galois representations started in 1986 with a ground-breaking paper by Barry Mazur where he established the first existence theorem and some examples in the case of odd two-dimensional Galois representations. Such representations arise from elliptic curves and they produce L-functions which allows one to link them to modular forms. This link was used by Andrew Wiles to establish Fermat's Last Theorem.
We investigate the analogous case of deformations of even Galois representations and their link to Maass wave forms. Currently the only known method of linking those is via L-functions. A method using Hecke algebras that has been applied successfully in the odd case is not available.
In chapter I, we present so-called converse theorems that guarantee the existence of a Maass wave form if one is given a pair of L-functions that satisfies certain functional equations--in certain cases a single L-function is sufficient. The main result was already proved by Jacquet and Langlands. We give a new, more elementary proof, and we adapt a result by Razar to our case.
In the next chapter we construct explicit universal deformations for even Galois representations that satisfy certain nice conditions using a method by Boston and Mazur. The main tool is the theory of pro-p Galois extensions.
In chapter III, we study the obstruction to a reasonable theory of pro-p extensions, the set $V\sb{S}$ as defined in chapter II. We establish a prime-to-p Galois descent for it, prove a statistical result on its vanishing in certain cases, and present some computer computations.
Chapter IV combines chapters I and II. First, we construct a family of nice examples for which the universal deformation can be computed explicitly, then we discuss L-functions that arise as specializations of the universal deformation, and finally we use the results from chapter I to link those to Maass wave forms. Our results indicate that Maass wave forms should be much more rigid than modular forms.
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