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arXiv · 2211.15369

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

Arpit Das·Naveen Balaji Umasankar
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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.

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BibTeXRIS

Arpit Das, Naveen Balaji Umasankar. 2022-11-22. Two- & Three-character solutions to MLDEs and Ramanujan-Eisenstein Identities for Fricke Groups. https://arxiv.org/abs/2211.15369

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