Bad reduction of genus three curves with complex multiplication

Bouw, I., Cooley, J., Lauter, K., Lorenzo Garcia, E., Manes, M., Newton, R. ORCID: https://orcid.org/0000-0003-4925-635X and Ozman, E. (2015) Bad reduction of genus three curves with complex multiplication. In: Bertin, M. J., Bucur, A., Feigon, B. and Schneps, L. (eds.) Women in Numbers Europe: Research Directions in Number Theory. Association for Women in Mathematics Series, 2 (2364-5733). Springer, pp. 109-151. ISBN 9783319179865 doi: 10.1007/978-3-319-17987-2

Abstract/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 p of M such that the stable reduction of C at p contains three irreducible components of genus 1.