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).

Reçu le :
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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
@article{JEP_2017__4__557_0,
     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},
     volume = {4},
     year = {2017},
     doi = {10.5802/jep.51},
     mrnumber = {3665608},
     zbl = {1376.33022},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.51/}
}
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/

[1] C.-L. Chai, C.-S. Lin & C.-L. Wang - “Mean field equations, hyperelliptic curves and modular forms: I”, Camb. J. Math. 3 (2015) no. 1-2, p. 127-274 | Article | MR 3356357

[2] Z. Chen, K.-J. Kuo, C.-S. Lin & C.-L. Wang - “Green function, Painlevé VI equation, and Eisentein series of weight one”, J. Differential Geometry (to appear)

[3] S. Dahmen - Counting integral Lamé equations with finite monodromy by means of modular forms, Utrecht University, 2003, Master Thesis

[4] S. Dahmen - “Counting integral Lamé equations by means of dessins d’enfants”, Trans. Amer. Math. Soc. 359 (2007) no. 2, p. 909-922 | Article | MR 2255201 | Zbl 1131.34060

[5] G.-H. Halphen - Traité des fonctions elliptique II, Gauthier-Villars, Paris, 1888

[6] R. Hartshorne - Algebraic geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, 1977 | Zbl 0367.14001

[7] B. Hassett - Introduction to algebraic geometry, Cambridge University Press, Cambridge, 2007 | Article | MR 2324354 | Zbl 1118.14002

[8] E. Hecke - “Zur Theorie der elliptischen Modulfunctionen”, Math. Ann. 97 (1926), p. 210-242 | Article | Zbl 52.0377.04

[9] C.-S. Lin & C.-L. Wang - “Elliptic functions, Green functions and the mean field equations on tori”, Ann. of Math. (2) 172 (2010) no. 2, p. 911-954 | Article | MR 2680484

[10] C.-S. Lin & C.-L. Wang - “A function theoretic view of the mean field equations on tori”, in Recent advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 11, Int. Press, Somerville, MA, 2010, p. 173-193 | MR 2648944

[11] C.-S. Lin & C.-L. Wang - “On the minimality of extra critical points of Green functions on flat tori”, Internat. Math. Res. Notices (2016), doi:10.1093/imrn/rnw176 | Article

[12] R. S. Maier - “Lamé polynomials, hyperelliptic reductions and Lamé band structure”, Philos. Trans. Roy. Soc. London Ser. A 366 (2008) no. 1867, p. 1115-1153 | Article | Zbl 1153.37425

[13] D. Mumford - Abelian varieties, Oxford University Press, Cambridge, 1974

[14] E. T. Whittaker & G. N. Watson - A course of modern analysis, Cambridge University Press, Cambridge, 1927 | Zbl 53.0180.04