Weighted averages of $p$-adic hypergeometric functions and traces of Frobenius of elliptic curves

Abstract

In this paper, we aim to study traces of Frobenius of certain one parameter families of elliptic curves and their relationships with $p$-adic hypergeometric functions. For example, we consider a DIK family of curves and establish the trace of Frobenius as weighted averages of special values of certain families of $p$-adic hypegeometric functions, where the average is taken over the arrays of parameters. Moreover, we consider Jacobi curves and express the trace of Frobenius as a special values of $p$-adic hypergeomtric functions. As a consequence of these results we obtain four summation identities for the $p$-adic hypegeometric functions that arise from the DIK family. Furthermore, we obtain $p$-adic analogous of Euler and Pfaff transformations for certain $p$-adic hypergemetric functions.