论文标题
Ramanujan Rankin-Cohen类型和双曲三角形的系统
Ramanujan systems of Rankin-Cohen type and hyperbolic triangles
论文作者
论文摘要
在本文的第一部分中，我们表征了一阶非线性微分方程的某些系统，其解决方案是$ \ mathfrak {sl} _2（\ Mathbb {C}）$ - 模块。我们证明，这样的系统称为Rankin-Cohen类型的Ramanujan系统，具有特殊的形状，并且正是其解决方案空间允许Rankin-Cohen结构的系统。在本文的第二部分中，我们考虑三角形$δ（n，m，\ infty）$。通过模块化的嵌入，我们将各种非线性ODES的系统与该溶液具有代数独立的扭曲模块化形式相关联。特别是，所有这样的系统的解决方案（rankin-cohen型）生成了$δ（n，m，\ infty）$上的所有理性权重模块化形式。作为推论，我们发现在上半平面上的功能上评估的高斯超几何函数的新关系。为了在非经典环境中证明我们的方法的力量，我们从非线性ODES溶液中构建了$δ（2,5，\ Infty）$的整体权重扭曲模块形式的空间。
In the first part of the paper we characterize certain systems of first order nonlinear differential equations whose space of solutions is an $\mathfrak{sl}_2(\mathbb{C})$-module. We prove that such systems, called Ramanujan systems of Rankin-Cohen type, have a special shape and are precisely the ones whose solution space admits a Rankin-Cohen structure. In the second part of the paper we consider triangle groups $Δ(n,m,\infty)$. By means of modular embeddings, we associate to every such group a number of systems of non linear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on $Δ(n,m,\infty)$ are generated by the solutions of one such system (which is of Rankin-Cohen type). As a corollary we find new relations for the Gauss hypergeometric function evaluated at functions on the upper half-plane. To demonstrate the power of our approach in the non classical setting, we construct the space of integral weight twisted modular form on $Δ(2,5,\infty)$ from solutions of systems of nonlinear ODEs.
