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