Mean field equations, hyperelliptic curves and modular forms: II

Lin, Chang-Shou; Wang, Chin-Lung

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 557-593.

Nous introduisons une forme pré-modulaire $Z_n(\sigma ;\tau )$ de poids $\frac{1}{2}n(n+1)$ pour chaque $n\in \mathbb{N}$, avec $(\sigma ,\tau )\in ℂ\times ℍ$, de sorte que pour $E_{\tau }=ℂ/(\mathbb{Z}+\mathbb{Z}\tau )$, tout zéro non trivial de $Z_n(\sigma ;\tau )$, c’est-à-dire que $\sigma$ n’est pas de $2$-torsion dans $E_{\tau }$, correspond à une (famille de) solution de l’équation

$$▵u+e^u=\rho {\delta }_0,\left(\mathrm{MFE}\right)$$

sur le tore plat $E_{\tau }$ avec $\rho =8\pi n$.

Dans la partie I [1], nous avons construit une courbe hyperelliptique ${\overline{X}}_n\left(\tau \right)\subset {\mathrm{Sym}}^n{E}_{\tau }$, la courbe de Lamé, associée à l’équation (MFE). Notre construction de $Z_n(\sigma ;\tau )$ s’appuie sur une étude détaillée de la correspondance ${\mathbb{P}}^1\left(ℂ\right)\leftarrow {\overline{X}}_n\left(\tau \right)\to E_{\tau }$ induite par la projection hyperelliptique et l’application d’addition.

Dans l’appendice, Y.-C. Chou donne, comme application de l’expression explicite de la forme $Z_4(\sigma ;\tau )$ pré-modulaire de poids $10$, une formule de comptage pour les équations de Lamé de degré $n=4$ avec monodromie finie.

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 $(\sigma ,\tau )\in ℂ\times ℍ$, such that for $E_{\tau }=ℂ/(\mathbb{Z}+\mathbb{Z}\tau )$, every non-trivial zero of $Z_n(\sigma ;\tau )$, i.e. $\sigma$ is not a $2$-torsion of $E_{\tau }$, corresponds to a (scaling family of) solution to the equation

$$▵u+e^u=\rho {\delta }_0,\left(\mathrm{MFE}\right)$$

on the flat torus $E_{\tau }$ with singular strength $\rho =8\pi n$.

In Part I [1], a hyperelliptic curve ${\overline{X}}_n\left(\tau \right)\subset {\mathrm{Sym}}^n{E}_{\tau }$, the Lamé curve, associated to the MFE was constructed. Our construction of $Z_n(\sigma ;\tau )$ relies on a detailed study of the correspondence ${\mathbb{P}}^1\left(ℂ\right)\leftarrow {\overline{X}}_n\left(\tau \right)\to E_{\tau }$ 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).

Classification : 33E10, 35J08, 35J75, 14H70
Keywords: Lamé curve, Hecke function, pre-modular form
Mot clés : Courbe de Lamé, fonction de Hecke, forme pré-modulaire

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