Bad reduction of genus $3$ curves with complex multiplication

Irene Bouw (1) , Jenny Cooley (2) , Kristin Lauter (3) , Elisa Lorenzo García (4) , Michelle Manes (5) , Rachel Newton (6) , Ekin Ozman (7)

1 HUB - Institut für Reine Mathematik
2 Department of Mathematics, University of Warwick
3 Cryptography group
4 Mathematical institute
5 UHM - University of Hawai‘i [Mānoa]
6 MPI - Max-Planck-Institut für Mathematik
7 University of Texas at Austin [Austin]

Let $C$ be a smooth, absolutely irreducible genus-$3$ curve over a number field $M$. Suppose that the Jacobian of $C$ has complex multiplication by a sextic CM-field $K$. Suppose further that $K$ contains no imaginary quadratic subfield. We give a bound on the primes $\mathfrak{p}$ of $M$ such that the stable reduction of $C$ at $\mathfrak{p}$ contains three irreducible components of genus $1$.

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

Irene Bouw, Jenny Cooley, Kristin Lauter, Elisa Lorenzo García, Michelle Manes, et al.. Bad reduction of genus $3$ curves with complex multiplication. Alina Bucur; Marie Jose Bertin; Brooke Feigon; Leila Schneps. Women in numbers Europe 2015: research directions in number theory, 2, Springer, pp.109-151, 2015, Association for Women in Mathematics Series, 9783319179865. ⟨hal-01619346⟩