Supercongruences arising from Ramanujan-Sato Series

TL;DR

This paper develops a general p-adic supercongruence theorem linked to CM hypergeometric elliptic curves, providing p-adic analogues of Ramanujan-Sato series for 1/π and constructing explicit examples.

Contribution

It introduces a new p-adic supercongruence theorem connecting Ramanujan-Sato series with CM hypergeometric elliptic curves, expanding the understanding of these series in p-adic contexts.

Findings

Proved a general p-adic supercongruence theorem.

Constructed explicit p-adic analogues of Ramanujan-Sato series.

Generated new examples related to existing Ramanujan-Sato series.

Abstract

Recently, the authors with Lea Beneish established a recipe for constructing Ramanujan-Sato series for $1/\pi$, and used this to construct 11 explicit examples of Ramanujan-Sato series arising from modular forms for arithmetic triangle groups of non-compact type. Here, we use work of Chisholm, Deines, Long, Nebe and the third author to prove a general $p$-adic supercongruence theorem through an explicit connection to CM hypergeometric elliptic curves that provides $p$-adic analogues of these Ramanujan-Sato series. We further use this theorem to construct explicit examples related to each of our explicit Ramanujan-Sato series examples.