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.
R. Auffarth, “A note on Galois embeddings of abelian varieties”, Manuscripta Math., vol. 154, no. 3–4, pp. 279–284, 2017.
R. Auffarth and G. Lucchini Arteche, “Smooth quotients of abelian varieties by finite groups”, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5), vol. 21, pp. 673–694, 2020.
V. Popov. Discrete complex reflection groups, Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, 15, Netherland: Rijksuniversiteit Utrecht, 1982.
G. C. Shephard and J. A. Todd, “Finite unitary reflection groups”. Canad. J. Math., vol. 6, pp. 274–304, 1954.
O. V. Å varcman, “A Chevalley theorem for complex crystallographic groups that are generated by mappings in the affine space C^2” (Russian), Uspekhi Mat. Nauk, vol. 34, no.1(205), pp. 249–250, 1979.
S. Tokunaga and M. Yoshida.“Complex crystallographic groups. I.”, J. Math. Soc. Japan, vol. 34, no. 4, pp. 581–593, 1982.
H. Yoshihara, “Galois embedding of algebraic variety and its application to abelian surface”, Rend. Semin. Mat. Univ. Padova, vol. 117, pp. 69–85, 2007.