{
  "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) ≡ 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.-a p X + p.In this case, our congruence can be rephrased as H p (t 0 ) ≡ ϑ p (mod p).To avoid clutter, we have suppressed the dependence of a p and ϑ p on t 0 in our notation.It turns out that, for certain values of t 0 , we get H p (t 0 ) ≡ ϑ p (mod p 2 ) for infinitely many primes p.This is a congruence modulo p 2 instead of p.Here is an example: Theorem 1.1.We have:• For all odd primes p, H p (1) ≡ (-1) ( p-1)/2 (mod p 2 ).",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics, Utrecht University, Utrecht, Netherlands"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Frits Beukers",
      "orcid": "0000-0003-0366-8615"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2e4vzxh56gq8e.bib",
  "cited_by": 13,
  "doi": "10.2140/ant.2026.20.1727",
  "fulltext_url": null,
  "id": "2e4vzxh56gq8e",
  "issns": [
    "1937-0652",
    "1944-7833"
  ],
  "landing_url": "https://doi.org/10.2140/ant.2026.20.1727",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W4392575318",
  "pdf_url": "https://msp.org/ant/2026/20-9/ant-v20-n9-p02-s.pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": "Mathematical Sciences Publishers",
  "ris": "https://research.draconitic.com/paper/2e4vzxh56gq8e.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T13:20:59Z",
      "native_id": "W4392575318",
      "source": "openalex",
      "url": "https://doi.org/10.2140/ant.2026.20.1727"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Advanced Mathematical Identities",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Supercongruences using modular forms",
  "title_text": "Supercongruences using modular forms",
  "type": "peer-reviewed",
  "updated_at": "2026-11-09T13:21:00Z",
  "url": "https://research.draconitic.com/paper/2e4vzxh56gq8e",
  "venue": "Algebra & Number Theory",
  "versions": []
}