Alexandre Gélin, Everett W. Howe, and Christophe Ritzenthaler: Principally polarized squares of elliptic curves with field of moduli equal to Q. To appear in the Proceedings of the 13th Algorithmic Number Theory Symposium (Madison, WI 2018).

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 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.

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

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