#### 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 $\overline{ℚ}$, 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