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arXiv · 1407.3589

Bad reduction of genus $3$ curves with complex multiplication

Irene Bouw·Jenny Cooley·Kristin Lauter·Elisa Lorenzo Garcia·Michelle Manes·Rachel Newton·Ekin Ozman
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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 $1$.

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Irene Bouw, Jenny Cooley, Kristin Lauter, Elisa Lorenzo Garcia, Michelle Manes, Rachel Newton, Ekin Ozman. 2014-07-14. Bad reduction of genus $3$ curves with complex multiplication. https://arxiv.org/abs/1407.3589

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