p-Adic hypergeometric functions and the trace of Frobenius of elliptic curves

2023/11/06 by Sulakashna, Barman, Rupam
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2311.03259

Abstract

Let p be an odd prime and q=p^r, r≥ 1. For positive integers n, let _nG_n[⋯]_q denote McCarthy's p-adic hypergeometric functions. In this article, we prove an identity expressing a ₄G₄[⋯]_q hypergeometric function as a sum of two ₂G₂[⋯]_q hypergeometric functions. This identity generalizes some known identities satisfied by the finite field hypergeometric functions. We also prove a transfomation that relates _n+2G_n+2[⋯]_q and _nG_n[⋯]_q hypergeometric functions. Next, we express the trace of Frobenius of elliptic curves in terms of special values of ₄G₄[⋯]_q and ₆G₆[⋯]_q hypergeometric functions. Our results extend the recent works of Tripathi and Meher on the finite field hypergeometric functions to wider classes of primes.

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