Alexandre Gélin, Everett W. Howe, and Christophe Ritzenthaler: Principally polarized squares of elliptic curves with field of moduli equal to Q, pp. 257–274 in: ANTS XIII: Proceedings of the Thirteenth Algorithmic Number Theory Symposium (R. Scheidler and J. Sorenson, eds.), the Open Book Series 2, Mathematical Sciences Publishers, Berkeley, 2019.

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 E2, where E is an elliptic curve with complex multiplication by the imaginary quadratic order of discriminant Δ. Each curve C is a double cover of its corresponding elliptic curve E, and the involution of C associated to this double cover is given by (x, y) → (d/x, d3/2 y/x3), where d is as in the table.

ΔdEquation for curve
−8 1 y2 = x5 + x
−11 (−11)1/3 y2 = 2 x6 + 11 x3 − 2·11
−19 −19 y2 = x6 + 1026 x5 + 627 x4 + 38988 x3 − 627·19 x2 + 1026·192 x − 193
−43 −43 y2 = x6 + 48762 x5 + 1419 x4 + 4193532 x3 − 1419·43 x2 + 48762·432 x − 433
−67 −67 y2 = x6 + 785106 x5 + 2211 x4 + 105204204 x3 − 2211·67 x2 + 785106·672 x − 673
−163 −163 y2 = x6 + 1635420402 x5 + 5379 x4 + 533147051052 x3 − 5379·163 x2 + 1635420402·1632 x − 1633
−20 51/2 y2 = x5 + 5 x3 + 5 x
−24 21/2 y2 = 3 x5 + 8 x3 + 3·2 x
−40 51/2 y2 = 9 x5 + 40 x3 + 9·5 x
−52 131/2 y2 = 9 x5 + 65 x3 + 9·13 x
−88 21/2 y2 = 99 x5 + 280 x3 + 99·2 x
−148 371/2 y2 = 441 x5 + 5365 x3 + 441·37 x
−232 291/2 y2 = 9801 x5 + 105560 x3 + 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.