平均场方程、超椭圆曲线和模形式：II
Mean field equations, hyperelliptic curves and modular forms: II
摘要：
对于每个$n \in \Bbb N$引入权重$\tfrac{1}{2} n(n + 1)$的预模形式$Z_n(σ; τ)$，其中$(σ, τ) \in \Bbb C \times \Bbb H$，使得对于$E_τ= \Bbb C/(\Bbb Z + \Bbb Z τ)$，$Z_n(σ; τ)$的每个非平凡零，即$σ\not\in E_τ[2]$，对应于具有奇异强度$ρ= 8πn$的平坦环面$E_τ$上的平均场方程\begin{equation} \tag{MFE} \triangle u + e^u = ρ\, δ_0 \end{equation}的解（的标度族）。在第I部分（Cambridge J. Math. 3，2015）中，构建了与MFE相关的超椭圆曲线$\bar X_n(τ) \subset {\rm Sym}^n E_τ$，即Lamé曲线。我们构造的$Z_n(σ; τ)$依赖于对超椭圆投影和加法映射的对应关系$\Bbb P^1 \leftarrow \bar X_n(τ) \to E_τ$的详细研究。作为权10预模形式$Z_4(σ; τ)$的显式的应用，在附录中给出了有限单值的Lamé方程$n = 4$的计数公式. C. Chou）。
英文摘要：
A pre-modular form $Z_n(σ; τ)$ of weight $\tfrac{1}{2} n(n + 1)$ is introduced for each $n \in \Bbb N$, where $(σ, τ) \in \Bbb C \times \Bbb H$, such that for $E_τ= \Bbb C/(\Bbb Z + \Bbb Z τ)$, every non-trivial zero of $Z_n(σ; τ)$, namely $σ\not\in E_τ[2]$, corresponds to a (scaling family of) solution to the mean field equation \begin{equation} \tag{MFE} \triangle u + e^u = ρ\, δ_0 \end{equation} on the flat torus $E_τ$ with singular strength $ρ= 8πn$. In Part I (Cambridge J. Math. 3, 2015), a hyperelliptic curve $\bar X_n(τ) \subset {\rm Sym}^n E_τ$, the Lamé curve, associated to the MFE was constructed. Our construction of $Z_n(σ; τ)$ relies on a detailed study on the correspondence $\Bbb P^1 \leftarrow \bar X_n(τ) \to E_τ$ induced from the hyperelliptic projection and the addition map. As an application of the explicit form of the weight 10 pre-modular form $Z_4(σ; τ)$, a counting formula for Lamé equations of degree $n = 4$ with finite monodromy is given in the appendix (by Y.-C. Chou).
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