Based on the developments of many people including Evans, Greene, Katz, McCarthy, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions over finite fields in a manner that is parallel to the classical hypergeometric functions. Using a comparison between the classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of transformation and evaluation formulas over finite fields and interpret them geometrically using a Galois representation perspective. We will also discuss the applications of these formulas.

This is a joint work with Jenny Fuselier, Ravi Ramakrishna, Holly Swisher and Fang-Ting Tu.