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

Fabrizio Andreatta[1]

  • [1] Università La Sapienza, Dipartimento di Matematica - Instituto G. Castelnuovo, Piazzale Aldo Moro 2, 00185 Roma (Italie)

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 1, page 69-106
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Andreatta, Fabrizio. "The small Schottky-Jung locus in positive characteristics different from two." Annales de l’institut Fourier 53.1 (2003): 69-106. <http://eudml.org/doc/116038>.

@article{Andreatta2003,
abstract = {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.},
affiliation = {Università La Sapienza, Dipartimento di Matematica - Instituto G. Castelnuovo, Piazzale Aldo Moro 2, 00185 Roma (Italie)},
author = {Andreatta, Fabrizio},
journal = {Annales de l’institut Fourier},
keywords = {Schottky-Jung relations; theta functions; Mumford's uniformization},
language = {eng},
number = {1},
pages = {69-106},
publisher = {Association des Annales de l'Institut Fourier},
title = {The small Schottky-Jung locus in positive characteristics different from two},
url = {http://eudml.org/doc/116038},
volume = {53},
year = {2003},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 1
SP - 69
EP - 106
AB - 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.
LA - eng
KW - Schottky-Jung relations; theta functions; Mumford's uniformization
UR - http://eudml.org/doc/116038
ER -

References

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  13. D. Mumford, An analytic construction of degenerating abelian varieties over complete rings, Comp. Math 24 (1972), 239-272 Zbl0241.14020MR352106
  14. D. Mumford, Prym varieties 1, Contributions to analysis (1974), 325-350, Acad. Press, New York Zbl0299.14018
  15. G. Van Steen, The Schottky-Jung theorem for Mumford curves, Ann. Inst. Fourier (Grenoble) 39 (1989), 1-15 Zbl0658.14015MR1011975
  16. G.E. Welters, The surface C - C in Jacobi varieties and second order theta functions, Acta Math 157 (1986), 1-22 Zbl0771.14012MR857677
  17. G.E. Welters, Polarized abelian varieties and the heat equations, Comp. Math 49 (1983), 173-194 Zbl0576.14042MR704390

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