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Primes Dividing Invariants of CM Picard Curves

Part of: Geometry of numbers Arithmetic problems. Diophantine geometry Curves Abelian varieties and schemes Computational aspects in algebraic geometry

Published online by Cambridge University Press:  07 May 2019

Pınar Kılıçer ,
Elisa Lorenzo García  and
Marco Streng
Pınar Kılıçer
Affiliation:
Johann Bernoulli Instituut voor Wiskunde en Informatica, Rijksuniversiteit Groningen, Nijenborgh 9, 9747 AGGroningen, Nederland Email: p.kilicer@rug.nl
Elisa Lorenzo García
Affiliation:
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France Email: elisa.lorenzogarcia@univ-rennes1.fr
Marco Streng
Affiliation:
Mathematisch Instituut, Universiteit Leiden, P.O. box 9512, 2300 RA Leiden, The Netherlands Email: streng@math.leidenuniv.nl
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Abstract

We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based, not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.

Keywords

Picard curvecurve invariantcomplex multiplciationHilbert class polynomialbad reduction

MSC classification

Primary: 14H45: Special curves and curves of low genus
Secondary: 14K22: Complex multiplication 11H06: Lattices and convex bodies 14G50: Applications to coding theory and cryptography 14H40: Jacobians, Prym varieties 14Q05: Curves
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 2 , April 2020 , pp. 480 - 504
Copyright
© Canadian Mathematical Society 2018

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Footnotes

Author E. L. G. was partially supported by a project PEPS-Jeunes Chercheur-e-s - 2017. Author P. K. was partially supported by DFG priority project SPP 1489.

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