Appell series over finite fields and modular forms

Abstract

In this article, we express the special values of F4⁎, a finite field analogue of the classical Appell series F4, in terms of the Fourier coefficients of weight three modular forms. These modular forms are related to modular K3 surfaces. To obtain these results, we first prove the finite field analogues of classical identities connecting classical F23-hypergeometric series and F4-Appell series.