Two- & Three-character solutions to MLDEs and Ramanujan-Eisenstein Identities for Fricke Groups

TL;DR

This paper extends the study of modular linear differential equations (MLDEs) for Fricke groups at prime levels, introducing a novel derivative operator to find new solutions and identities related to Ramanujan-Eisenstein series, with potential lattice applications.

Contribution

It introduces a new Serre-Ramanujan type derivative operator for Fricke groups, enabling the derivation of novel MLDE solutions and Ramanujan-Eisenstein identities at prime levels 2 and 3.

Findings

Discovered new one-, two-, and three-character solutions for Fricke groups.

Constructed putative partition functions with lattice interpretations.

Identified non-trivial bilinear identities among solutions.

Abstract

In this work we extend the study of arXiv:2210.07186 by investigating two- and three-character MLDEs for Fricke groups at prime levels. We have constructed these higher-character MLDEs by using a $\mathit{novel}$ Serre-Ramanujan type derivative operator which maps $k$-forms to $(k+2)$-forms in $\Gamma^{+}_0(p)$. We found that this $\mathit{novel}$ derivative construction enabled us to write down a general prescription for obtaining $\mathit{Ramanujan-Eisenstein}$ identities for these groups. We discovered several $\mathit{novel}$ single-, two-, and three-character admissible solutions for Fricke groups at levels $2$ and $3$ after solving the MLDEs among which we have realized some in terms of Mckay-Thompson series and others in terms of modular forms of the corresponding Hecke groups. Among these solutions, we have identified interesting non-trivial bilinear identities. Furthermore, we could construct $\mathit{putative}$ partition functions for these theories based on these bilinear pairings, which could have a range of lattice interpretations. We also present and discuss modular re-parameterization of MLDE and their solutions for Fricke groups of prime levels.