Plane quartics over $\mathbb {Q}$ with complex multiplication

Abstract

We give examples of smooth plane quartics over Q with complex multiplication over Q by a maximal order with primitive CM type. We describe the required algorithms as we go; these involve the reduction of period matrices, the fast computation of Dixmier-Ohno invariants, and reconstruction from these invariants. Finally, we discuss some of the reduction properties of the curves that we obtain.

A bound on the primes of bad reduction for CM curves of genus 3

“…The hyperelliptic locus is, however, of codimension 1, and there is indeed an invariant among the Dixmier-Ohno invariants whose vanishing locus is the locus of hyperelliptic curves and decomposable Jacobians. Numerical experiments of Kılıçer, Labrande, Lercier, Ritzenthaler, Sijsling and Streng [10] suggest that the analogues of Theorems 1.5 and 1.7 for arbitrary CM curves of genus three are false. More research is needed in that direction.…”

Isogenous hyperelliptic and non-hyperelliptic Jacobians with maximal complex multiplication

“…Given the even theta-null values, we can determine a model of X over C to the given precision, either by using the Rosenhain invariants as in [1] or by using the Weber model from [27]. We can then calculate a normalized weighted representative I of the corresponding invariants (using the Shioda invariants in the hyperelliptic case and the Dixmier-Ohno invariants in the non-hyperelliptic case).…”

“…The consideration of general non-hyperelliptic CM curves was taken up in the context of work of Kılıçer, who in her thesis [26] determined the full list of genus-3 CM curves with field of moduli Q. Explicit defining equations of these curves were found in [21] in the hyperelliptic case and [27] in the non-hyperelliptic case, which included 19 non-hyperelliptic, non-Picard CM curves. Reduction properties in the general case were studied in the works [7,28].…”

“…The explicit defining equations in the works [21,27,46] mentioned above are heuristic, in the sense that while the results obtained are exceedingly likely to be correct, their rigorous verification in practice (available in theory by using the methods in [11]) is still open. The current work, which will obtain defining equations and invariants of CM curves over Q, will also take this heuristic approach throughout.…”

Constructing Picard curves with complex multiplication using the Chinese remainder theorem

“…However, invariants do exist for the classes of hyperelliptic curves and non-hyperelliptic plane quartics separately. By making restrictions on the type of genus 3 curves considered, algorithms for constructing genus 3 curves with complex multiplication have been presented in [36], [23], [25], [4], and [20]. All these papers take a complex analytic approach to The first author was partially supported by National Science Foundation grants DMS-1056703 and CNS-1617802.…”