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

TL;DR

This paper classifies certain genus-2 curves over algebraic closures of rationals with Jacobians that are squares of CM elliptic curves, providing explicit equations for some and a classification under GRH.

Contribution

It explicitly constructs 13 genus-2 curves over Q with Jacobians as squares of CM elliptic curves and proves a classification result assuming GRH.

Findings

13 explicit genus-2 curves over Q with CM elliptic Jacobians

Under GRH, exactly 46 such genus-2 curves over Q exist

No further examples exist if GRH is true

Abstract

We give equations for 13 genus-2 curves over $\overline{\mathbb{Q}}$, with models over $\mathbb{Q}$, 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 $\overline{\mathbb{Q}}$ with field of moduli $\mathbb{Q}$ whose Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order.