The small Schottky-Jung locus in positive characteristics different from two

Andreatta, Fabrizio

Annales de l'Institut Fourier, Tome 53 (2003) no. 1, pp. 69-106.

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.

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.

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