Hyper-Kloosterman Sums in Supercharacter Theories
Madeleine Bulkow · Scholarship - Claremont (Claremont Colleges) · 2014
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.