Supercongruences using modular forms

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 ).

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BibTeX RIS

BibTeX

@article{beukers2026supercongruences,
  title = {{Supercongruences using modular forms}},
  author = {Frits Beukers},
  journal = {Algebra \& Number Theory},
  year = {2026},
  month = oct,
  doi = {10.2140/ant.2026.20.1727},
  url = {https://doi.org/10.2140/ant.2026.20.1727},
  note = {CC BY},
}

RIS

TY  - JOUR
TI  - Supercongruences using modular forms
AU  - Frits Beukers
PY  - 2026
DA  - 2026/10/19
JO  - Algebra & Number Theory
DO  - 10.2140/ant.2026.20.1727
UR  - https://doi.org/10.2140/ant.2026.20.1727
AB  - 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 ).
SN  - 1937-0652
SN  - 1944-7833
ER  -

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