Supercongruences using modular forms

Beukers, Frits

doi:10.2140/ant.2026.20.1727

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Supercongruences using modular forms

Frits Beukers

Vol. 20 (2026), No. 9, 1727–1768

DOI: 10.2140/ant.2026.20.1727

Abstract

Let $F (t)$ be the generating series of some combinatorially interesting numbers, like the Apéry numbers. For any prime $p$ , we let $F_{p} (t)$ be the sum of the terms of degree $< p$ (the $p$ -truncation of $F$ ). It is a classical fact that, for many $p$ and algebraic arguments $α$ , we have the congruence $F_{p} (α) \equiv 𝜃 (\mod p)$ , where $𝜃$ is a zero of the characteristic $p$ $ζ$ -function of an algebraic variety associated to $F$ and $α$ .

Surprisingly, very often these congruences turn out to hold modulo $p^{2}$ or even $p^{3}$ . We call such congruences supercongruences, and in the past 15 years an abundance of them have been discovered. Here we show that a large proportion of them can be explained by the use of modular functions and forms.

Keywords

supercongruences, Apéry-like sequences, modular forms

Mathematical Subject Classification

Primary: 11A07, 11B65, 11F03

Milestones

Received: 12 July 2024

Revised: 7 May 2025

Accepted: 11 August 2025

Published: 19 October 2026

Authors

Frits Beukers
Department of Mathematics Utrecht University Utrecht Netherlands

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Vol. 20, No. 9, 2026