# Bad reduction of genus $3$ curves with complex multiplication

3 Cryptography group
Microsoft Research [Redmond]
Abstract : 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 $1$.
Document type :
Book sections
Domain :

https://hal.archives-ouvertes.fr/hal-01619346
Contributor : Marie-Annick Guillemer <>
Submitted on : Thursday, October 19, 2017 - 12:02:27 PM
Last modification on : Monday, April 2, 2018 - 9:12:01 AM

### Identifiers

• HAL Id : hal-01619346, version 1
• ARXIV : 1407.3589

### Citation

Irene Bouw, Jenny Cooley, Kristin Lauter, Elisa Lorenzo Garcia, 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〉

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