Date of Award

5-2013

Level of Access Assigned by Author

Campus-Only Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

Advisor

Andrew Knightly

Second Committee Member

Robert Franzosa

Third Committee Member

William Snyder

Abstract

Over the past 60 years, an important problem in algebraic number theory has been to understand the group GL2(F), where F is a finite-degree field extension of Q P.

One way to study this group is through its complex representations. It is a theorem of Casselman that every irreducible, admissible representation of GL2(F) has a conductor. This thesis focuses on the supercuspidal representations of GL2(F) of minimal conductor. These representations are explicitly constructed, and then the construction is used to compute the matrix coefficient of a supercuspidal representation, using the new vector. This computation is of interest in the study of modular forms in number theory.

To construct such a supercuspidal representation, the starting point is a cuspidal representation of the group GL2(k), where k is the finite residue field of F. This representation is extended to a representation of K = GL2(0) in a natural way. This representation is extended to a representation of ZK by letting the center Z act by a complex character. Finally, the resulting representation is compactly-induced to GL2(F). This representation of GL2(F) is irreducible, admissible, and supercuspidal.

Thus, in order to understand these supercuspidal representations, it suffices to understand the cuspidal representations of GL2(k), where k is the finite field with q elements. These representations are defined so that no vector is fixed by the unipotent subgroup. Such a representation will automatically be of dimension q — 1. Furthermore, the restriction of a cuspidal representation to the mirabolic subgroup of GL2(k) will be isomorphic to the representation constructed by inducing any nontrivial character from the unipotent subgroup to the mirabolic subgroup.

Using this isomorphism, the cuspidal representations of GL2(k) are constructed explicitly. This explicit construction allows for a formula for the matrix coefficients of these representations in terms of the entries of a given matrix g G GL2{k). Using this computation, a formula for the matrix coefficient of supercuspidal representations of minimal conductor is obtained.

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