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

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- by Pınar Kılıçer, Kristin Lauter, Elisa Lorenzo García, Rachel Newton, Ekin Ozman and Marco Streng PDF
- Proc. Amer. Math. Soc.
**148**(2020), 2843-2861 Request permission

## 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
- MR Author ID: 619019
- ORCID: 0000-0002-1320-696X
- Email: klauter@microsoft.com
**Elisa Lorenzo García**- Affiliation: IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
- ORCID: 0000-0001-7360-1411
- 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
- MR Author ID: 975652
- Email: r.d.newton@reading.ac.uk
**Ekin Ozman**- Affiliation: Mathematics Department, Boğazı̇çı̇ University, Faculty of Arts and Sciences, Bebek, Istanbul, 34342, Turkey
- MR Author ID: 955558
- Email: ekin.ozman@boun.edu.tr
**Marco Streng**- Affiliation: Mathematisch Instituut, Universiteit Leiden, PO Box 9512, 2300 RA Leiden, The Netherlands
- MR Author ID: 831652
- Email: streng@math.leidenuniv.nl
- 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 Boğazı̇çı̇ 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
- © Copyright 2020 American Mathematical Society
- 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
- MathSciNet review: 4099774