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

TL;DR

This paper links traces of Frobenius of elliptic curve families to weighted averages of $p$-adic hypergeometric functions, deriving new identities and transformations in the $p$-adic setting.

Contribution

It establishes novel relationships between Frobenius traces and $p$-adic hypergeometric functions, including summation identities and transformation formulas.

Findings

Trace of Frobenius expressed as weighted averages of $p$-adic hypergeometric functions.

Derived four summation identities for $p$-adic hypergeometric functions.

Established $p$-adic analogues of Euler and Pfaff transformations.

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.