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Bad reduction of genus three curves with complex multiplication. (English) Zbl 1397.11102
Bertin, Marie José (ed.) et al., Women in numbers Europe. Research directions in number theory. Based on the presentations at the WINE workshop, Luminy, France, October 13–18, 2013. Cham: Springer (ISBN 978-3-319-17986-5/hbk; 978-3-319-17987-2/ebook). Association for Women in Mathematics Series 2, 109-151 (2015).
Summary: 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.
For the entire collection see [Zbl 1329.11002].

##### MSC:
 11G15 Complex multiplication and moduli of abelian varieties 14K22 Complex multiplication and abelian varieties 14H40 Jacobians, Prym varieties
##### Keywords:
complex multiplication; Jacobian; CM-field
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