Vol. 2, 2019

 Recent Volumes 4: ANTS XIV 3: Hillman: Poincaré Duality 2: ANTS XIII 1: ANTS X
 The Open Book Series All Volumes About the Series Ethics Statement Purchase Printed Copies Author Index MSP Books and Monographs Other MSP Publications
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 $\overline{ℚ}$, with models over $ℚ$, 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 $\overline{ℚ}$ with field of moduli $ℚ$ 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