论文标题
MLDES和RAMANUJAN-EISENSTEIN的两个特定解决方案和弗里克组的身份
Two- & Three-character solutions to MLDEs and Ramanujan-Eisenstein Identities for Fricke Groups
论文作者
论文摘要
在这项工作中，我们通过调查了弗里克斯组的两种和三个字符的MLDE，扩展了ARXIV：2210.07186的研究。我们已经使用$ \ mathit {novel} $ serre-ramanujan类型衍生运算符构建了这些更高的MLDE，该运算符将$ k $ - forms映射到$（k+2）$ - 以$γ^{+γ{+} _ 0（p）$中的形式。我们发现，这个$ \ mathit {novel} $衍生构造使我们能够写下一般处方，以获取这些组的$ \ mathit {ramanujan-eisenstein} $身份。我们发现了几个$ \ MATHIT {NOVER} $单，二，二和三字符的弗里克集团的可允许的解决方案，价格为$ 2 $和$ 3 $，在解决了MLDE之后，我们已经在McKay-Thompson系列和其他方面就相应的Hecke组的模块化形式意识到了一些MLDES。在这些解决方案中，我们已经确定了有趣的非平凡双线性身份。此外，我们可以基于这些双线性配对来构建这些理论的$ \ mathit {Puthative} $分区功能，这些功能可能具有一系列的晶格解释。我们还提出并讨论了MLDE的模块化重新参数及其针对最优惠级的弗里克组的解决方案。
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 $Γ^{+}_0(p)$. 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.
