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Supercongruences arising from Ramanujan-Sato Series

Angelica Babei, Manami Roy, Holly Swisher, Bella Tobin, Fang-Ting Tu

TL;DR

The authors establish a general $p$-adic supercongruence framework for Ramanujan–Sato series tied to CM hypergeometric elliptic curves and triangle modular curves, extending prior works on $1/\pi$ series and $p$-adic analogues. They prove congruences for truncated hypergeometric products, linking them to the unit root of Frobenius on CM elliptic curves and, in many cases, to Fourier coefficients of weight $3$ CM modular forms, via modularity and periods. The paper provides 11 explicit Ramanujan–Sato series from WIN5 and 4 additional congruence instances, and develops a robust bridge between hypergeometric evaluations, CM theory, and modular forms, with several conjectures for higher powers of $p$. It also discusses the $p$-adic and complex-analytic structure of the associated families, including Beukers-type congruences and the role of Chowla–Selberg periods, thereby highlighting the arithmetic of hypergeometric motives and their modular interpretations. The work offers both concrete arithmetic results and a roadmap for further exploration of $p$-adic refinements (potentially modulo $p^3$) of Ramanujan–Sato phenomena and their connections to $L$-values of CM modular forms.

Abstract

Recently, the authors with Lea Beneish established a recipe for constructing Ramanujan-Sato series for $1/π$, 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.

Supercongruences arising from Ramanujan-Sato Series

TL;DR

The authors establish a general -adic supercongruence framework for Ramanujan–Sato series tied to CM hypergeometric elliptic curves and triangle modular curves, extending prior works on series and -adic analogues. They prove congruences for truncated hypergeometric products, linking them to the unit root of Frobenius on CM elliptic curves and, in many cases, to Fourier coefficients of weight CM modular forms, via modularity and periods. The paper provides 11 explicit Ramanujan–Sato series from WIN5 and 4 additional congruence instances, and develops a robust bridge between hypergeometric evaluations, CM theory, and modular forms, with several conjectures for higher powers of . It also discusses the -adic and complex-analytic structure of the associated families, including Beukers-type congruences and the role of Chowla–Selberg periods, thereby highlighting the arithmetic of hypergeometric motives and their modular interpretations. The work offers both concrete arithmetic results and a roadmap for further exploration of -adic refinements (potentially modulo ) of Ramanujan–Sato phenomena and their connections to -values of CM modular forms.

Abstract

Recently, the authors with Lea Beneish established a recipe for constructing Ramanujan-Sato series for , 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 -adic supercongruence theorem through an explicit connection to CM hypergeometric elliptic curves that provides -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.
Paper Structure (19 sections, 22 theorems, 180 equations, 5 tables)

This paper contains 19 sections, 22 theorems, 180 equations, 5 tables.

Table of Contents

  1. Introduction and Statement of Results
  2. Notations for main results
  3. Statement of main results
  4. Outline of the paper
  5. Acknowledgements
  6. Hypergeometric Elliptic Curves and Triangle Modular Curves
  7. Hypergeometric CM Elliptic Curves
  8. Implications of modularity
  9. Proof of Theorem \ref{['thm:main']}
  10. Examples
  11. Examples from
  12. Examples from
  13. Examples of Theorem \ref{['thm:WIN2a']}
  14. Connections with modular forms
  15. Special values and L-values.
  16. ...and 4 more sections

Key Result

Corollary 1.1

Let $\Gamma$ be a discrete subgroup of ${{\operatorname{SL}}}_2(\mathbb{R})$ commensurable with ${{\operatorname{SL}}}_2({\mathbb Z})$ such that $\left(\right) \in \Gamma$, and let $X(\tau)$ be a Hauptmodul of $\Gamma$. Let $Z(\tau)$ be a weight-$k$ modular form for $\Gamma$ such that $\frac{1}{2\pi Set $M_N(\tau):=Z(\tau)/Z(N\tau)$ for $N\in {\mathbb N}$ satisfying $\frac{a}{c}(1-N)\in {\mathbb Z

Theorems & Definitions (43)

  • Corollary 1.1: Corollary of WIN5
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Theorem 2.1: Chisholm, et al. WIN2
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 33 more