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Bad reduction of genus $2$ curves with CM jacobian varieties

Part of: Arithmetic algebraic geometry Curves Abelian varieties and schemes Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  11 September 2017

Philipp Habegger  and
Fabien Pazuki
Philipp Habegger
Affiliation:
Departement Mathematik und Informatik, Spiegelgasse 1, 4051 Basel, Switzerland email philipp.habegger@unibas.ch
Fabien Pazuki
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark email fabien.pazuki@math.u-bordeaux.fr IMB, Université de Bordeaux, 351, cours de la Libération, 33405 Talence, France
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Abstract

We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a known case of the Colmez conjecture, due to Colmez and Obus, that is valid when the CM field is an abelian extension of the rationals. It links the height and the logarithmic derivatives of an $L$ -function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use the reduction theory of genus $2$ curves as developed by Igusa, Liu, Saito, and Ueno to relate the contribution at the finite places with the stable bad reduction of the curve. The subconvexity bounds by Michel and Venkatesh together with an equidistribution result of Zhang are used to bound the infinite places.

Keywords

curveshyperelliptic jacobianscomplex multiplication

MSC classification

Primary: 14K22: Complex multiplication
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields 11G50: Heights 14H40: Jacobians, Prym varieties 14G40: Arithmetic varieties and schemes; Arakelov theory; heights
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 12 , December 2017 , pp. 2534 - 2576
Copyright
© The Authors 2017 

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