Finding elliptic curves defined over the rationals and with complex multiplication (CM) is equivalent to the Gauss’ class number one problem for imaginary quadratic fields and was solved by K. Heegner [Math. Z. 56, 227–253 (1952; Zbl 0049.16202)], A. Baker [Mathematika 13, 204–216 (1966; Zbl 0161.05201)] and H. M. Stark [Mich. Math. J. 14, 1–27 (1967; Zbl 0148.27802)]. The similar problem for genus 2 is considered here and completes a series of papers given each a part of the full answer.
The problem actually comes in various flavors. First the authors restricts to non-biquadratic quartic CM fields which corresponds to absolutely simple abelian varieties (for the left case, partial results are available in [A. Gélin et al., Open Book Ser. 2, 257–274 (2019; Zbl 1517.11063)] and a preprint by F. Narbonne [“Polarized products of elliptic curves with complex multiplication and field of moduli \(\mathbb Q\)”, Preprint, arXiv:2203.11982] for maximal orders).
For these ones, the case where there exists a curve over \(\mathbb{Q}\) with Jacobian having CM by an order in a quartic number field only happens for cyclic extensions with class number 1. [N. Murabayashi and A. Umegaki, RIMS Kokyuroku 1160, 169–176 (2000; Zbl 0969.14502)] already found the 19 ones corresponding to maximal orders and [F. Bouyer and M. Streng, LMS J. Comput. Math. 18, 507–538 (2015; Zbl 1397.11103)] showed that this list is complete under this restriction. Two new curves with non-maximal orders were given in [G. Bisson and M. Streng, Math. Res. Lett. 24, No. 2, 247–270 (2017; Zbl 1417.11114)]. It remained to prove that the list is now complete which is done here by enumerating the 20 cyclic quartic CM fields with class number 1 (the abelian variety also depends on the type of the CM field, which explains why one can have 21 curves for 20 fields only).
Then, the authors extend the problem by finding 301 (resp. 231) \(\overline{\mathbb{Q}}\)-isomorphism classes of genus \(2\) curves such that their Jacobian has CM by an order (resp. the maximal order) in a quartic CM field and with field of moduli contained in the reflex field (a field naturally associated to the CM field and type). On top of this theoretical results, numerical computations have been done to find models for some of the curves (at least to find all Igusa invariants), although the authors warned about the fact that these computations are not certified.
In order to achieve their classification, a lot of efforts is necessary to obtain the list of CM fields satisfying a variation of the class number one problem (called CM class number 1 in the article). Available bounds on the discriminants of such fields are too large and sieving techniques are deployed to make the list manageable. Nice results are also showed on the way about the possible splitting type of ramified primes.