Mean field equations, hyperelliptic curves and modular forms: II

Abstract

A pre-modular form $Z_n(\sigma ;\tau )$ of weight $\frac{1}{2}n(n+1)$ is introduced for each $n\in \mathbb{N}$, where $(\sigma ,\tau )\in \mathbb{C}\times \mathbb{H}$, such that for $E_{\tau }=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$, every non-trivial zero of $Z_n(\sigma ;\tau )$, namely $\sigma \notin E_{\tau }[2]$, 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 $E_{\tau }$ with singular strength $\rho =8\pi n$. In Part I (Cambridge J. Math. 3, 2015), a hyperelliptic curve ${\bar{X}}_n(\tau )\subset {\mathrm{Sym}}^n{E}_{\tau }$, the Lamé curve, associated to the MFE was constructed. Our construction of $Z_n(\sigma ;\tau )$ relies on a detailed study on the correspondence ${\mathbb{P}}^1\leftarrow {\bar{X}}_n(\tau )\to E_{\tau }$ induced from the hyperelliptic projection and the addition map. As an application of the explicit form of the weight 10 pre-modular form $Z_4(\sigma ;\tau )$, a counting formula for Lamé equations of degree $n=4$ with finite monodromy is given in the appendix (by Y.-C. Chou).

32 pages. Part of content in previous versions is removed and published separately. One author is removed

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