El. knyga: 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 and consequently study hypergeometric functions over finite fields in a manner that is parallel to that of 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 hypergeometric transformation and evaluation formulas over finite fields and interpret them geometrically using a Galois representation perspective. As an application, we obtain a few finite field analogues of algebraic hypergeometric identities, quadratic and higher transformation formulas, and evaluation formulas. We further apply these finite field formulas to compute the number of rational points of certain hypergeometric varieties"--

Fuselier and colleagues 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 that of the classical hypergeometric functions. Using a comparison between the classical gamma function and its finite-field analog, the Gauss sum, they describe a systematic way to obtain certain types of hypergeometric transformation and evaluation formulae over finite fields, and interpret them geometrically using a Galois representation perspective. For applications, they obtain a few finite field analogues of algebraic hypergeometric identities, quadratic and higher transformation formulae, and evaluation formulae. They also apply these finite field formulae to compute the number of rational points of certain hypergeometric varieties. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)