TY - JOUR

T1 - A bound on the primes of bad reduction for CM curves of genus 3

AU - Kiliçer, Pinar

AU - Lauter, Kristin

AU - García, Elisa Lorenzo

AU - Newton, Rachel

AU - Ozman, Ekin

AU - Streng, Marco

N1 - Funding Information:
The fourth author was supported by EPSRC grant EP/S004696/1. The fifth author was supported by Bog˘azi¸ci University Research Fund Grant Number 15B06SUP3 and by the BAGEP award of the Science Academy, 2016. The sixth author was supported by NWO Vernieuwingsimpuls.
Publisher Copyright:
© 2020 American Mathematical Society.

PY - 2020/7

Y1 - 2020/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85085995685&partnerID=8YFLogxK

U2 - 10.1090/proc/14975

DO - 10.1090/proc/14975

M3 - Article

AN - SCOPUS:85085995685

SN - 0002-9939

VL - 148

SP - 2843

EP - 2861

JO - Proceedings of the american mathematical society

JF - Proceedings of the american mathematical society

IS - 7

ER -