P-Adic gamma function and the trace of Frobenius of elliptic curves

Abstract

In [12], McCarthy defined a function Gnn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function, and expressed the trace of Frobenius of elliptic curves in terms of special values of G22[⋯]. We establish two different expressions for the traces of Frobenius of elliptic curves in terms of the function G22[⋯]. As a result, we obtain two transformation formulas of the function G22[⋯] with different parameters.