Vol. 2, 2019

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Principally polarized squares of elliptic curves with field of moduli equal to $\mathbb Q$

Alexandre Gélin, Everett W. Howe and Christophe Ritzenthaler

Vol. 2 (2019), No. 1, 257–274
Abstract

We give equations for 13 genus-2 curves over ¯, with models over , whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the generalized Riemann hypothesis is true, there are no further examples of such curves. More generally, we prove under the generalized Riemann hypothesis that there exist exactly 46 genus-2 curves over ¯ with field of moduli whose Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order.

Keywords
genus-2 curves, abelian varieties, polarizations, fields of moduli, complex multiplication
Mathematical Subject Classification 2010
Primary: 11G15
Secondary: 14H25, 14H45
Milestones
Received: 28 February 2018
Revised: 15 September 2018
Accepted: 16 September 2018
Published: 13 February 2019
Authors
Alexandre Gélin
Laboratoire de Mathématiques de Versailles
Université de Versailles Saint-Quentin-en-Yvelines
Centre national de la recherche scientifique
Université Paris-Saclay
Versailles
France
Everett W. Howe
Center for Communications Research
Institute for Defense Analyses
San Diego, CA
United States
Christophe Ritzenthaler
Institut de recherche mathématique de Rennes
Université de Rennes 1
Campus de Beaulieu
Rennes
France