Preprint version: 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

We used several Magma programs to find these curves. They will be available soon on this web page.

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 |