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

TL;DR

This paper classifies smooth quotients of abelian surfaces by finite groups fixing the origin, showing they are all isomorphic to the projective plane and that the surface is a product of elliptic curves.

Contribution

It completes the classification of smooth quotients of abelian surfaces by finite groups fixing the origin, especially when the analytic representation is irreducible.

Findings

A is isomorphic to E^2 with E an elliptic curve

A/G is isomorphic to the projective plane in all cases

Finitely many pairs (E^2,G) exist up to isomorphism for fixed E

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.