Date of Completion

5-9-2016

Embargo Period

5-6-2016

Keywords

Gauss sums, p-adic analysis, Gross-Koblitz formula, Stickelberger's congruence

Major Advisor

Keith Conrad

Associate Advisor

Alvaro Lozano-Robledo

Associate Advisor

Kyu-Hwan Lee

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

In 2005 Blache studied certain generalized Gauss sums and established an analogue for them of Stickelberger's congruence for classical Gauss sums over finite fields. We improve on Blache's work in two ways: (i) simplify Blache's proof and give a second proof that works for a larger family of generalized Gauss sums, and (ii) give a p-adic lifting of Stickelberger's congruence for the larger family of generalized Gauss sums that is partial progress towards a version of the Gross-Koblitz formula for these sums. In addition, we study this larger family of generalized Gauss sums, prove a formula for them which simplifies computations, prove Stickelberger-type congruences for power series representations of these sums, make a conjecture for their degree over the p-adic numbers and prove cases of it. We conclude by making a family of conjectures for generalized Gross-Koblitz formulas regarding generalized Gauss sums and power series representations of them.

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