2025/04/04 by Scott Ahlgren, Nickolas Andersen, Ahlgren, Scott +3 · 1 citation
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.2504.03954
openalex publication_date 2025/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the 1960s Atkin discovered congruences modulo primes ℓ≤ 31 for the partition function p(n) in arithmetic progressions modulo ℓ Q3, where Q≠ ℓ is prime. Recent work of the first author with Allen and Tang shows that such congruences exist for all primes ℓ≥ 5. Here we consider (for primes m≥ 5) the m-colored generalized Frobenius partition functions cϕm(n); these are natural level m analogues of p(n). For each such m we prove that there are similar congruences for cϕm(n)\pmod ℓ for all primes ℓ outside of an explicit finite set depending on m. To prove the result we first construct, using both theoretical and computational methods, cusp forms of half-integral weight on Γ0(m) which capture the relevant values of cϕm(n) modulo~ℓ. We then apply previous work of the authors on the Shimura lift for modular forms with the eta multiplier together with tools from the theory of modular Galois representations.