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 = E2 with E an elliptic curve and that A/G ≃ ℙ2 in all cases. Moreover, for fixed E, there are only finitely many pairs (E2, 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.