2021/12/17 by Scott Ahlgren, Ahlgren, Scott, Patrick B. Allen +3 · 2 citations
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2112.09481
openalex publication_date 2021/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p(n) be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form p( Q3 ℓ n+β)≡0\pmodℓ where ℓ and Q are prime and 5≤ ℓ≤ 31; these lie in two natural families distinguished by the square class of 1-24β\pmodℓ. In recent decades much work has been done to understand congruences of the form p(Qmℓ n+β)≡ 0\pmodℓ. It is now known that there are many such congruences when m≥ 4, that such congruences are scarce (if they exist at all) when m=1, 2, and that for m=0 such congruences exist only when ℓ=5, 7, 11. For congruences like Atkin's (when m=3), more examples have been found for 5≤ ℓ≤ 31 but little else seems to be known. Here we use the theory of modular Galois representations to prove that for every prime ℓ≥ 5, there are infinitely many congruences like Atkin's in the first natural family which he discovered and that for at least 17/24 of the primes ℓ there are infinitely many congruences in the second family.