Ramanujan type congruences for quotients of Klein forms

Authors

Timothy Huber, The University of Texas Rio Grande ValleyFollow
Nathaniel Mayes, The University of Texas Rio Grande Valley
Jeffery Opoku, The University of Texas Rio Grande Valley
Dongxi Ye

Document Type

Article

Publication Date

5-2024

Abstract

In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ1(p) of weights 1≤k≤5 for primes 5≤p≤19 and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. For each product satisfying a congruence, cyclic permutations of the exponents determine additional products satisfying congruences. Common forms among the exponent sets lead to products satisfying Ramanujan type congruences for a broad class of primes, including p>19. Canonical bases for modular forms of level 5≤p≤19 are constructed by summing weight one Hecke Eisenstein series of levels 5≤p≤19 and expressing the result as a quotient of Klein forms. Generating sets for the graded algebras of modular forms for Γ1(p) and Γ(p) are formulated in terms of permutations of the exponent sets. A sieving process is described by decomposing the space of modular forms of weight 1 for Γ1(p) as a direct sum of subspaces of modular forms for Γ(p) of the form qr/pZ[[q]]. Since the relevant bases generate the graded algebra of modular forms for these groups, the weight one decompositions determine series dissections for modular forms of higher weight that lead to additional classes of congruences.

Comments

Original published version available at https://doi.org/10.1016/j.jnt.2023.11.009

Recommended Citation

Huber, Timothy, Nathaniel Mayes, Jeffery Opoku, and Dongxi Ye. "Ramanujan type congruences for quotients of Klein forms." Journal of Number Theory 258 (2024): 281-333. https://doi.org/10.1016/j.jnt.2023.11.009

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Publication Title

Journal of Number Theory

DOI

10.1016/j.jnt.2023.11.009