Hypergeometric functions over finite fields

Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields consequently study in a manner that is parallel to classical functions. Using comparison between gamma function its field analogue Gauss sum, give systematic way obtain certain types transformation evaluation formulas interpret them geometrically using Galois representation perspective. As an application, few analogues identities, quadratic higher formulas, formulas. We further apply these compute number rational points varieties.