The small Schottky-Jung locus in positive characteristics different from two
Annales de l'Institut Fourier, Volume 53 (2003) no. 1, p. 69-106

We prove that the locus of Jacobians is an irreducible component of the small Schottky locus in any characteristic different from 2. The proof follows an idea of B. van Geemen in characteristic 0 and relies on a detailed analysis at the boundary of the q- expansion of the Schottky-Jung relations. We obtain algebraically such relations using Mumford’s theory of 2-adic theta functions. We show how the uniformization theory of semiabelian schemes, as developed by D. Mumford, C.-L. Chai and G. Faltings, allows the study of higher dimensional q-expansions simplifying the argument.

Nous prouvons que le lieu des jacobiens est une composante irréductible du petit lieu de Schottky en caractéristique différente de 2. La preuve repose sur une méthode introduite par B. van Geemen en caractéristique 0 et se base sur une analyse détaillée au bord du q-développement des relations de Schottky-Jung. Nous obtenons ces relations d’une façon algébrique en utilisant les fonctions thêta 2-adiques définies par Mumford. La théorie d’uniformisation des schémas semi-abéliens, due à D. Mumford, C.-L. Chai et G. Faltings, permet d’ étudier des q-développements en dimension supérieure en donnant une preuve plus simple.

DOI : https://doi.org/10.5802/aif.1940
Classification:  14H42
Keywords: Schottky-Jung relations, theta functions, Mumford's uniformization
@article{AIF_2003__53_1_69_0,
     author = {Andreatta, Fabrizio},
     title = {The small Schottky-Jung locus in positive characteristics different from two},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {1},
     year = {2003},
     pages = {69-106},
     doi = {10.5802/aif.1940},
     zbl = {1067.14025},
     mrnumber = {1973069},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_1_69_0}
}
Andreatta, Fabrizio. The small Schottky-Jung locus in positive characteristics different from two. Annales de l'Institut Fourier, Volume 53 (2003) no. 1, pp. 69-106. doi : 10.5802/aif.1940. http://www.numdam.org/item/AIF_2003__53_1_69_0/

[An] F. Andreatta; C. Faber, G. Van Der Geer And F. Oort, Eds. On Mumford's uniformization and Néron models of Jacobians of semistable curves over complete bases, Moduli of Abelian Varieties, Birkhauser (Progress in Math) Tome 195 (2001), pp. 11-127 | Zbl 01643994

[Be] A. Beauville Prym varieties and the Schottky problem, Invent. Math, Tome 41 (1977), pp. 149-196 | Article | MR 572974 | Zbl 0333.14013

[BLR] S. Bosch; W. Lütkebohmert; M. Raynaud Néron Models, Springer-Verlag, Ergebnisse der Mathematik und ihrer Grenzebiete, 3 Folge, Tome Band 21 (1990) | MR 1045822 | Zbl 0705.14001

[Br] L. Breen Fonctions thêta et théorème du cube, Springer-Verlag, Lecture Notes in Math, Tome 980 (1983) | MR 823233 | Zbl 0558.14029

[Ch] C.-L. Chai Compactification of Siegel moduli schemes, London Math. Soc. Lecture Notes Series, Tome 107 (1985) | MR 853543 | Zbl 0578.14009

[Do1] R. Donagi Big Schottky, Invent. Math, Tome 89 (1987), pp. 569-599 | Article | MR 903385 | Zbl 0658.14022

[Do2] R. Donagi; E. Sernesi, Ed. The Schottky problem, Theory of Moduli, Springer-Verlag (Lecture Notes in Math) Tome 1337 (1988), pp. 84-137 | Zbl 0676.14008

[FC] G. Faltings; And C.-L. Chai Degeneration of abelian varieties, Springer-Verlag, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge, Tome Band 22 (1990) | MR 1083353 | Zbl 0744.14031

[MB] L. Moret-Bailly Pinceaux de variétés abéliennes, Astérisque, Tome 129 (1985) | MR 797982 | Zbl 0595.14032

[Mu1] D. Mumford On the equations defining abelian varieties 1, 2, 3, Invent. Math, Tome 1 ; 3 (1966 ; 1967), p. 287-358 ; 71--135 ; 215--244 | Article | MR 219542 | Zbl 0219.14024

[Mu2] D. Mumford The structure of the moduli spaces of curves and abelian varieties, Actes Congrès Intern. Math., Tome Tome 1 (1970), pp. 457-465 | Zbl 0222.14023

[Mu3] D. Mumford An analytic construction of degenerating abelian varieties over complete rings, Comp. Math, Tome 24 (1972), pp. 239-272 | Numdam | MR 352106 | Zbl 0241.14020

[Mu4] D. Mumford Prym varieties 1, Contributions to analysis, Acad. Press, New York (1974), pp. 325-350 | Zbl 0299.14018

[vG] B. Van Geemen Siegel modular forms vanishing on the moduli space of curves, Invent. Math, Tome 78 (1984), pp. 329-349 | Article | MR 767196 | Zbl 0568.14015

[vS] G. Van Steen The Schottky-Jung theorem for Mumford curves, Ann. Inst. Fourier (Grenoble), Tome 39 (1989) no. 1, pp. 1-15 | Article | Numdam | MR 1011975 | Zbl 0658.14015

[We1] G.E. Welters The surface C-C in Jacobi varieties and second order theta functions, Acta Math, Tome 157 (1986), pp. 1-22 | Article | MR 857677 | Zbl 0771.14012

[We2] G.E. Welters Polarized abelian varieties and the heat equations, Comp. Math, Tome 49 (1983), pp. 173-194 | Numdam | MR 704390 | Zbl 0576.14042