Ramanujan type congruences for quotients of Klein forms

Abstract

In this work, Ramanujan type congruences modulo powers of primes $p \ge 5$ 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 $Γ_{1}(p)$ are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for $Γ_{1}(p)$ of weights $1\le k \le 5$ for primes $5\le p \le 19$ 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 $5\le p \le 19$ are constructed by summing weight one Hecke Eisensten series of levels $5\le p \le 19$ and expressing the result as a quotient of Klein forms. Generating sets for the graded algebras of modular forms for $Γ_{1}(p)$ 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 $Γ_{1}(p)$ as a direct sum of subspaces of modular forms for $Γ(p)$ of the form $q^{r/p}\Bbb Z[[q]]$. 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.

Theorems & Definitions (84)

• Remark 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1: KL
• Lemma 2.2
• proof
• Corollary 2.3
• proof