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

Kilicer, P., Lauter, K., Lorenzo Garcia, E., Newton, R., Ozman, E. and Streng, M. (2020) A bound on the primes of bad reduction for CM curves of genus 3. Proceedings of the American Mathematical Society, 148. p. 2843. ISSN 0002-9939

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To link to this item DOI: 10.1090/proc/14975

Abstract/Summary

We give bounds on the primes of geometric bad reduction for curves of genus three of primitive 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 two, 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 two. 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.

Item Type:Article
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:67539
Publisher:American Mathematical Society

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