Work

Congruences like Atkin's for generalized Frobenius partitions

Year: 2025

Type: article

Abstract: In the 1960s, Atkin discovered congruences modulo primes for the partition function in arithmetic progressions modulo , where is prime. Recent work of the first author with Allen and Tang shows that such congruences exist for all primes . Here we consider (for primes ) the ‐colored generalized Frobenius partition functions ; these are natural level analogues of . For each such we prove that there are similar congruences for for all primes outside of an explicit finite set depending on . To prove the result we first construct, using both theoretical and computational methods, cusp forms of half‐integral weight on which capture the relevant values of modulo . 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.

Source: Bulletin of the London Mathematical Society

DOI: 10.1112/blms.70224

Authors: Scott Ahlgren, Nickolas Andersen, Robert Dicks

Institutions: University of Illinois Urbana-Champaign, University of Illinois System, Brigham Young University, Clemson University

Language: English

Topic: Advanced Mathematical Identities

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Simons Foundation

Awards: G1600387322, G3696730568