The small Schottky-Jung locus in positive characteristics different from two
Annales de l'Institut Fourier, Volume 53 (2003) no. 1, pp. 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: 10.5802/aif.1940
Classification: 14H42
Keywords: Schottky-Jung relations, theta functions, Mumford's uniformization
Mot clés : relations de Schottky-Jung, fonctions theta, uniformisation à la Mumford
Andreatta, Fabrizio 1

1 Università La Sapienza, Dipartimento di Matematica - Instituto G. Castelnuovo, Piazzale Aldo Moro 2, 00185 Roma (Italie)
@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},
     pages = {69--106},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {1},
     year = {2003},
     doi = {10.5802/aif.1940},
     zbl = {1067.14025},
     mrnumber = {1973069},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1940/}
}
TY  - JOUR
AU  - Andreatta, Fabrizio
TI  - The small Schottky-Jung locus in positive characteristics different from two
JO  - Annales de l'Institut Fourier
PY  - 2003
SP  - 69
EP  - 106
VL  - 53
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1940/
DO  - 10.5802/aif.1940
LA  - en
ID  - AIF_2003__53_1_69_0
ER  - 
%0 Journal Article
%A Andreatta, Fabrizio
%T The small Schottky-Jung locus in positive characteristics different from two
%J Annales de l'Institut Fourier
%D 2003
%P 69-106
%V 53
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1940/
%R 10.5802/aif.1940
%G en
%F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.1940/

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

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

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

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

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

[Do1] R. Donagi Big Schottky, Invent. Math, Volume 89 (1987), pp. 569-599 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

Cited by Sources: