Balanced modular parameterizations

TL;DR

This paper constructs theta quotient sets for prime levels 5 to 19 that generate modular form rings for (p), revealing symmetries and applications including a new representation for t-core partitions and connections to Kleinian groups.

Contribution

It introduces explicit (p) permutation representations of theta quotients and explores their applications and symmetries, including a novel approach to t-core partitions.

Findings

Constructed (p)-permuted theta quotient sets for prime levels 5 to 19.

Established explicit permutation representations and symmetry properties.

Linked symmetries at levels 5, 7, 11 to Kleinian automorphism groups.

Abstract

For prime levels $5 \le p \le 19$, sets of $\Gamma_{0}(p)$-permuted theta quotients are constructed that generate the graded rings of modular forms of positive integer weight for $\Gamma_{1}(p)$. An explicit formulation of the permutation representation and several applications are given, including a new representation for the number of $t$-core partitions. The $\Gamma_{0}(p)$-action induces coefficient symmetries within representations for modular forms and invariance subgroups for coupled systems of differential equations. The symmetry for levels $p = 5,7,11$ is linked to the Kleinian automorphism groups.