Abstract This paper concerns the study of the Schwartz differential equation { h , τ } = s ⁢ E 4 ⁡ ( τ ) {\{h,\tau\}=s\operatorname{E}_{4}(\tau)} , where E 4 {\operatorname{E}_{4}} is the weight 4 Eisenstein series and s is a complex parameter. In particular, we determine all the values of s for which the solutions h are modular functions for a finite index subgroup of SL 2 ⁡ ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} . We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of SL 2 ⁡ ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} . This also leads to the solutions to the Fuchsian differential equation y ′′ + s ⁢ E 4 ⁡ y = 0 {y^{\prime\prime}+s\operatorname{E}_{4}y=0} .

中文翻译：

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自守施瓦兹方程

摘要 本文研究 Schwartz 微分方程 { h , τ } = s ⁢ E 4 ⁡ ( τ ) {\{h,\tau\}=s\operatorname{E}_{4}(\tau)} ，其中 E 4 {\operatorname{E}_{4}} 是权重 4 艾森斯坦级数，s 是一个复参数。特别地，我们确定 s 的所有值，其解 h 是 SL 2 ⁡ ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} 的有限索引子群的模函数. 我们使用复上半平面上的等变函数理论以及对 SL 2 ⁡ ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}} )}。这也导致了 Fuchsian 微分方程 y ′′ + s ⁢ E 4 ⁡ y = 0 {y^{\prime\prime}+s\operatorname{E}_{4}y=0} 的解。