Revista Matemática Iberoamericana

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Volume 23, Issue 2, 2007, pp. 397–420
DOI: 10.4171/RMI/500

Published online: 2007-08-31

Group actions on Jacobian varieties

Anita M. Rojas[1]

(1) Universidad de Chile, Santiago, Chile

Consider a finite group $G$ acting on a Riemann surface $S$, and the associated branched Galois cover $\pi_G:S \to Y=S/G$. We introduce the concept of \emph{geometric signature} for the action of $G$, and we show that it captures much information: the geometric structure of the lattice of intermediate covers, the isotypical decomposition of the rational representation of the group $G$ acting on the Jacobian variety $JS$ of $S$, and the dimension of the subvarieties of the isogeny decomposition of $JS$. We also give a version of Riemann's existence theorem, adjusted to the present setting.

Keywords: Jacobian varieties, Riemann surfaces, group actions, Riemann’s existence theorem, geometric signature

Rojas Anita: Group actions on Jacobian varieties. Rev. Mat. Iberoam. 23 (2007), 397-420. doi: 10.4171/RMI/500