Mean field equations, hyperelliptic curves and modular forms: II

Chang-Shou Lin; Chin-Lung Wang

doi:10.5802/jep.51

Mean field equations, hyperelliptic curves and modular forms: II
[Équations de champ moyen, courbes hyperelliptiques et formes modulaires : II]

Chang-Shou Lin ¹ ; Chin-Lung Wang ²

¹ Department of Mathematics and Center for Advanced Studies in Theoretic Sciences (CASTS), National Taiwan University No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan 106
² Department of Mathematics and Taida Institute of Mathematical Sciences (TIMS), National Taiwan University No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan 106

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

Résumé
Abstract

Nous introduisons une forme pré-modulaire $Z_{n} (σ; τ)$ de poids $\frac{1}{2} n (n + 1)$ pour chaque $n \in ℕ$ , avec $(σ, τ) \in ℂ \times ℍ$ , de sorte que pour $E_{τ} = ℂ / (ℤ + ℤ τ)$ , 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 + e^{u} = ρ δ_{0}, (MFE)

sur le tore plat $E_{τ}$ avec $ρ = 8 π n$ .

Dans la partie I [1], nous avons construit une courbe hyperelliptique ${\bar{X}}_{n} (τ) \subset {Sym}^{n} E_{τ}$ , 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 $ℙ^{1} (ℂ) \leftarrow {\bar{X}}_{n} (τ) \to E_{τ}$ 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 $\frac{1}{2} n (n + 1)$ is introduced for each $n \in ℕ$ , where $(σ, τ) \in ℂ \times ℍ$ , such that for $E_{τ} = ℂ / (ℤ + ℤ τ)$ , 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 + e^{u} = ρ δ_{0}, (MFE)

on the flat torus $E_{τ}$ with singular strength $ρ = 8 π n$ .

In Part I [1], a hyperelliptic curve ${\bar{X}}_{n} (τ) \subset {Sym}^{n} E_{τ}$ , the Lamé curve, associated to the MFE was constructed. Our construction of $Z_{n} (σ; τ)$ relies on a detailed study of the correspondence $ℙ^{1} (ℂ) \leftarrow {\bar{X}}_{n} (τ) \to E_{τ}$ 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).

Reçu le : 2016-09-21
Accepté le : 2017-05-14
Publié le : 2017-05-30

MR Zbl

DOI : 10.5802/jep.51

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

Affiliations des auteurs :

Chang-Shou Lin ¹ ; Chin-Lung Wang ²

¹ Department of Mathematics and Center for Advanced Studies in Theoretic Sciences (CASTS), National Taiwan University No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan 106
² Department of Mathematics and Taida Institute of Mathematical Sciences (TIMS), National Taiwan University No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan 106

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Chang-Shou Lin and Chin-Lung Wang},
     title = {Mean field equations, hyperelliptic curves and modular forms: {II}},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {557--593},
     publisher = {\'Ecole polytechnique},
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Chang-Shou Lin; Chin-Lung Wang. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.51/

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