论文标题
体重2形式的自动施瓦兹方程和积分
Automorphic Schwarzian equations and integrals of weight 2 forms
论文作者
论文摘要
在本文中，我们研究了Schwarz微分方程的非模块化解决方案$ \ {f，τ\} = SE_4（τ）$，其中$ e_4（τ）$是重量4 Eisenstein系列和$ S $是一个复杂的参数。特别是，我们为每个$ s =2π^2（n/6）^2 $提供明确的解决方案，并提供$ n \ equiv 1 \ mod 12 $。这些溶液作为Meromormormormormortic重量2模块化形式的积分获得。结果，我们找到了微分方程的明确解决方案$ \ displayStyle y''+\ frac {π^2n^2} {36} \，e_4 \，y = 0 $，每个$ n \ equiv 1 \ equiv 1 \ mod 12 $概括了Hurwitz和Klein对CASE $ N = 1 $ n = 1 $ n = 1 $。我们的调查依赖于对等函数在复杂的上半平面上的作用理论。本文补充了先前的工作，我们确定上述Schwarzian方程具有模块化解决方案的所有参数$ s $。
In this paper, we investigate the non-modular solutions to the Schwarz differential equation $\{f,τ\}=sE_4(τ)$ where $E_4(τ)$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we provide explicit solutions for each $s=2π^2(n/6)^2$ with $n\equiv 1\mod 12$. These solutions are obtained as integrals of meromorphic weight 2 modular forms. As a consequence, we find explicit solutions to the differential equation $\displaystyle y''+\frac{π^2n^2}{36}\,E_4\,y=0$ for each $n\equiv 1\mod 12$ generalizing the work of Hurwitz and Klein on the case $n=1$. Our investigation relies on the theory of equivariant functions on the complex upper half-plane. This paper supplements a previous work where we determine all the parameters $s$ for which the above Schwarzian equation has a modular solution.
