Abelian varieties with group action

By Herbert Lange, S. Recillas. 2001

71 citations recorded by the source corpus.

Let G be a finite group acting on a smooth projective curve X. This induces an action of G on the Jacobian JX of X and thus a decomposition of JX up to isogeny. The most prominent example of such a situation is the group G of two elements. Let X --> Y denote the corresponding quotient map. Then JX is isogenous to the product of JY with the Prym variety of X/Y. In this paper some general results on group actions on abelian varieties are given and applied to deduce a decomposition of the jacobian JX for arbitrary group actions. Several examples are given.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2f26e8ed9268052837fa53c09e7c6127b587ebf3

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Topological methods in moduli theory (2014, background): “The above discussion does however leave open the following question (see [265] for some partial results), which should have an affirmative answer (in the sense that each family of complex structures should contain some Abelian variety).”
Group actions on Jacobian varieties (2003, methodology): “As a corollary, we obtain that all of these dimensions are positive if the genus of the orbit surface S/G is greater or equal than 2, which was obtained by different methods in [9], and conditions characterizing the situation when the orbit surface has genus 1.”