Graduation Year

Spring 2014

Document Type

Campus Only Senior Thesis

Department

Mathematics

Reader 1

Stephan Garcia

Reader 2

Winston Ou

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© 2014 Madeleine C. Bulkow

Abstract

In this thesis, we briefly review facts of representation and character theory in order to define supercharacter theories, which give a generalization of the characters on a given group. Supercharacter theories for the groups (Z/nZ)^2 and (Z/nZ)^m are constructed which contain respectively the Kloosterman sums and the Hyper-Kloosterman sums, their higher- dimensional analogue. The graphical properties of Hyper-Kloosterman sums are examined, and a conjecture made regarding a hypocycloid bound for the degree 2 Hyper-Kloosterman sums. Additionally, the orthogonality relations from a supercharacter table producing Hyper-Kloosterman sums are used to give a proof of Mordell’s bound, a previously known bound of Hyper-Kloosterman sums.

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