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A bound on the primes of bad reduction for CM curves of genus $ 3$


Authors: Pınar Kılıçer, Kristin Lauter, Elisa Lorenzo García, Rachel Newton, Ekin Ozman and Marco Streng
Journal: Proc. Amer. Math. Soc. 148 (2020), 2843-2861
MSC (2010): Primary 11G10, 11G15, 14H45, 14K22; Secondary 14J15, 14Q05
DOI: https://doi.org/10.1090/proc/14975
Published electronically: March 30, 2020
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Abstract: We give bounds on the primes of geometric bad reduction for curves of genus $ 3$ of primitive complex multiplication (CM) type in terms of the CM orders. In the case of elliptic curves, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least $ 2$, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus $ 2$.

In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are important for algorithmic construction of curves with given characteristic polynomials over finite fields.


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Additional Information

Pınar Kılıçer
Affiliation: Bernoulli Institute for Mathematics, Computer Science and AI, Nijenborgh 9, 9747 AG Groningen, The Netherlands
Email: p.kilicer@rug.nl

Kristin Lauter
Affiliation: Microsoft Research, Cryptography, One Microsoft Way, Redmond, Washington, 98052
Email: klauter@microsoft.com

Elisa Lorenzo García
Affiliation: IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
Email: elisa.lorenzogarcia@univ-rennes1.fr

Rachel Newton
Affiliation: Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, United Kingdom
Email: r.d.newton@reading.ac.uk

Ekin Ozman
Affiliation: Mathematics Department, Boğazıçı University, Faculty of Arts and Sciences, Bebek, Istanbul, 34342, Turkey
Email: ekin.ozman@boun.edu.tr

Marco Streng
Affiliation: Mathematisch Instituut, Universiteit Leiden, PO Box 9512, 2300 RA Leiden, The Netherlands
Email: streng@math.leidenuniv.nl

DOI: https://doi.org/10.1090/proc/14975
Received by editor(s): June 21, 2019
Received by editor(s) in revised form: November 19, 2019, and December 1, 2019
Published electronically: March 30, 2020
Additional Notes: The fourth author was supported by EPSRC grant EP/S004696/1. The fifth author was supported by Boğazıçı University Research Fund Grant Number 15B06SUP3 and by the BAGEP award of the Science Academy, 2016. The sixth author was supported by NWO Vernieuwingsimpuls
Communicated by: Rachel Pries
Article copyright: © Copyright 2020 American Mathematical Society