F4-Appell series in p-adic settings and their connections to algebraic curves

Abstract

Motivated by an expression for the number of points on an algebraic curve in terms of $F_4$ Appell series over finite fields, we here define a p-adic analog for the $F_4$ Appell series. Consequently, we find a relation of the number of points on the algebraic curve with the p-adic $F_4$ Appell series, extending the earlier result to all primes. Finally, we deduce some transformation formulas for the p-adic $F_4$ Appell series analogous to their classical counterparts.