Mean field equations, hyperelliptic curves and modular forms: II
[Équations de champ moyen, courbes hyperelliptiques et formes modulaires : II]
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 557-593.

Nous introduisons une forme pré-modulaire Z n (σ;τ) de poids 1 2n(n+1) pour chaque n, avec (σ,τ)×, de sorte que pour E τ =/(+τ), tout zéro non trivial de Z n (σ;τ), c’est-à-dire que σ n’est pas de 2-torsion dans E τ , correspond à une (famille de) solution de l’équation

u+eu=ρδ0,(MFE)

sur le tore plat E τ avec ρ=8πn.

Dans la partie I [1], nous avons construit une courbe hyperelliptique X ¯ n (τ)Sym n E τ , la courbe de Lamé, associée à l’équation (MFE). Notre construction de Z n (σ;τ) s’appuie sur une étude détaillée de la correspondance 1 ()X ¯ n (τ)E τ 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 (σ;τ) 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 (σ;τ) of weight 1 2n(n+1) is introduced for each n, where (σ,τ)×, such that for E τ =/(+τ), every non-trivial zero of Z n (σ;τ), i.e. σ is not a 2-torsion of E τ , corresponds to a (scaling family of) solution to the equation

u+eu=ρδ0,(MFE)

on the flat torus E τ with singular strength ρ=8πn.

In Part I [1], a hyperelliptic curve X ¯ n (τ)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 of the correspondence 1 ()X ¯ n (τ)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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DOI : https://doi.org/10.5802/jep.51
Classification : 33E10,  35J08,  35J75,  14H70
Mots clés : Courbe de Lamé, fonction de Hecke, forme pré-modulaire
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Lin, Chang-Shou; Wang, Chin-Lung. Mean field equations, hyperelliptic curves and modular forms: II. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 557-593. doi : 10.5802/jep.51. http://www.numdam.org/articles/10.5802/jep.51/

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