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

Pinar Kiliçer (1) , Kristin Lauter (2) , Elisa Lorenzo García (3) , Rachel Newton (4) , Ekin Ozman (5) , Marco Streng (6)

1 OFFIS - Carl Von Ossietzky Universität Oldenburg = Carl von Ossietzky University of Oldenburg
2 Microsoft Corporation [Redmond]
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 UOR - University of Reading
5 Boǧaziçi üniversitesi = Boğaziçi University [Istanbul]
6 Universiteit Leiden = Leiden University

We give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of genus one, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least two, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus two. In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are essential for algorithmic construction of curves with given characteristic polynomials over finite fields.

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]

Pinar Kiliçer, Kristin Lauter, Elisa Lorenzo García, Rachel Newton, Ekin Ozman, et al.. A bound on the primes of bad reduction for CM curves of genus 3. Proceedings of the American Mathematical Society, 2020, 148 (7), pp.2843-2861. ⟨10.1090/proc/14975⟩. ⟨hal-01618616⟩