Smooth quotients of abelian surfaces by finite groups that fix the origin

Robert Auffarth; Giancarlo Lucchini Arteche; Pablo Quezada

doi:10.4067/S0719-06462022000100037

DOI:

https://doi.org/10.4067/S0719-06462022000100037

Abstract

Let \(A\) be an abelian surface and let \(G\) be a finite group of automorphisms of \(A\) fixing the origin. Assume that the analytic representation of \(G\) is irreducible. We give a classification of the pairs \((A,G)\) such that the quotient \(A/G\) is smooth. In particular, we prove that \(A=E^2\) with \(E\) an elliptic curve and that \(A/G\simeq\mathbb P^2\) in all cases. Moreover, for fixed \(E\), there are only finitely many pairs \((E^2,G)\) up to isomorphism. This fills a small gap in the literature and completes the classification of smooth quotients of abelian varieties by finite groups fixing the origin started by the first two authors.

Keywords

Abelian surfaces , automorphisms

Robert Auffarth Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Ñuñoa, Santiago, Chile.
Giancarlo Lucchini Arteche Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Ñuñoa, Santiago, Chile.
Pablo Quezada Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Macul, Santiago, Chile.

Pages: 37–51
Date Published: 2022-04-04
Vol. 24 No. 1 (2022)

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