A preprint version is also available: arXiv:1806.03826 [math.NT].

We give equations for 13 genus-2 curves over the algebraic closure of **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 the algebraic closure of **Q**
with field of moduli **Q** whose Jacobians are isomorphic to the square
of an elliptic curve with complex multiplication by a maximal order.

Equations for the 13 curves are given below. Each curve *C* has Jacobian isomorphic (as an unpolarized abelian surface) to *E ^{2}*,
where

Δ | d | Equation for curve |
---|---|---|

−8 | 1 | y^{2} = x^{5} + x |

−11 | (−11)^{1/3} | y^{2} = 2 x^{6} + 11 x^{3} − 2·11 |

−19 | −19 | y^{2} = x^{6} + 1026 x^{5} + 627 x^{4} + 38988 x^{3} − 627·19 x^{2} + 1026·19^{2} x − 19^{3} |

−43 | −43 | y^{2} = x^{6} + 48762 x^{5} + 1419 x^{4} + 4193532 x^{3} − 1419·43 x^{2} + 48762·43^{2} x − 43^{3} |

−67 | −67 | y^{2} = x^{6} + 785106 x^{5} + 2211 x^{4} + 105204204 x^{3} − 2211·67 x^{2} + 785106·67^{2} x − 67^{3} |

−163 | −163 | y^{2} = x^{6} + 1635420402 x^{5} + 5379 x^{4} + 533147051052 x^{3} − 5379·163 x^{2} + 1635420402·163^{2} x − 163^{3} |

−20 | 5^{1/2} | y^{2} = x^{5} + 5 x^{3} + 5 x |

−24 | 2^{1/2} | y^{2} = 3 x^{5} + 8 x^{3} + 3·2 x |

−40 | 5^{1/2} | y^{2} = 9 x^{5} + 40 x^{3} + 9·5 x |

−52 | 13^{1/2} | y^{2} = 9 x^{5} + 65 x^{3} + 9·13 x |

−88 | 2^{1/2} | y^{2} = 99 x^{5} + 280 x^{3} + 99·2 x |

−148 | 37^{1/2} | y^{2} = 441 x^{5} + 5365 x^{3} + 441·37 x |

−232 | 29^{1/2} | y^{2} = 9801 x^{5} + 105560 x^{3} + 9801·29 x |

We used several Magma and Maple programs to find these curves:

- Data.m: A file with the results of our computations.
- Polarizations.m: The programs in this file will compute reduced forms for all of the positive definite unimodular binary Hermitian forms over an imaginary quadratic maximal order that does not contain any roots of unity other than 1 and -1. Each reduced form is labeled as being either split (decomposable) or nonsplit (indecomposable), and the automorphism group of the form is provided (as a list of matrices).
- Pol.m: A list of all the polarizations computed by the programs in Polarizations.m.
- FindCurves.m: The main part of our code, once the 1226 polarizations have been computed.
For each polarization, we check if there exist matrices
*P*that demonstrate that the field of moduli of the associated curve is**Q**. If there are such matrices, in the second step we recover the rational invariants of the curve. We also try to recover a rational model when it is possible. - Reduction.maple: Maple code for reducing period matrices. The output is stored in RedPerMat.m to make it easy to import the results into Magma.
- RedPerMat.m: Magma code to find the period matrix corresponding to a given polarization, using the output from Reduction.maple.