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Transactions of the American Mathematical Society

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Characterizations of Jacobians of curves with automorphisms


Authors: Esteban Gómez González, José M. Muñoz Porras, Francisco J. Plaza Martín and Rubí E. Rodríguez
Journal: Trans. Amer. Math. Soc. 362 (2010), 5373-5394
MSC (2010): Primary 14H42, 14H37; Secondary 37K10, 14K25
DOI: https://doi.org/10.1090/S0002-9947-2010-05029-9
Published electronically: May 19, 2010
MathSciNet review: 2657684
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Abstract: We obtain a characterization of theta functions of Jacobian varieties of curves with automorphisms among theta functions of principally polarized abelian varieties (p.p.a.v.). We first give a characterization in terms of finite dimensional orbits for a suitable action in the Sato Grassmannian. Secondly, the introduction of formal Baker-Akhiezer functions and formal $ \tau$-functions attached to a p.p.a.v. (for the multipuncture case) allows us to characterize, in terms of bilinear identities, those Baker-Akhiezer functions that are Baker-Akhiezer functions of Jacobians of curves with automorphisms. Further, in the case of automorphisms with fixed points, we rewrite the previous result as a hierarchy of partial differential equations for the $ \tau$-function of a p.p.a.v. Finally, since Baker-Akhiezer and $ \tau$ functions are written in terms of theta functions, these results give rise to characterizations of p.p.a.v. in terms of their theta functions.


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Additional Information

Esteban Gómez González
Affiliation: Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain
Email: esteban@usal.es

José M. Muñoz Porras
Affiliation: Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain
Email: jmp@usal.es

Francisco J. Plaza Martín
Affiliation: Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain
Email: fplaza@usal.es

Rubí E. Rodríguez
Affiliation: Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306-22, Santiago, Chile
Email: rubi@mat.puc.cl

DOI: https://doi.org/10.1090/S0002-9947-2010-05029-9
Keywords: Principally polarized abelian varieties, Jacobian varieties, Schottky problem, curves with automorphisms, theta functions, KP and KdV hierarchies, completely integrable systems
Received by editor(s): September 22, 2008
Published electronically: May 19, 2010
Additional Notes: This work was partially supported by the research contracts MTM2006-07618 of DGI and SA112A07 of JCyL, and Fondecyt grant 1060742
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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