Hecke grids and congruences for weakly holomorphic modular forms

TL;DR

This paper constructs infinite families of weakly holomorphic modular forms on Fricke groups and demonstrates how their properties lead to strengthened congruences related to Atkin operators, advancing understanding of modular form congruences.

Contribution

It explicitly constructs weakly holomorphic modular forms on Fricke groups and proves strengthened congruences for Atkin operator-related forms, extending previous conjectures.

Findings

Constructed infinite families of modular forms on Fricke groups.

Described the action of the Hecke algebra on these forms.

Established strengthened congruences for weakly holomorphic modular forms.

Abstract

Let $U(p)$ denote the Atkin operator of prime index $p$. Honda and Kaneko proved infinite families of congruences of the form $f|U(p) \equiv 0 \pmod{p}$ for weakly holomorphic modular forms of low weight and level and primes $p$ in certain residue classes, and conjectured the existence of similar congruences modulo higher powers of $p$. Partial results on some of these conjectures were proved recently by Guerzhoy. We construct infinite families of weakly holomorphic modular forms on the Fricke groups $\Gamma^*(N)$ for $N=1,2,3,4$ and describe explicitly the action of the Hecke algebra on these forms. As a corollary, we obtain strengthened versions of all of the congruences conjectured by Honda and Kaneko.