Bad reduction of genus 2 curves with CM jacobian varieties

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Bad reduction of genus 2 curves with CM jacobian varieties

JournalArticle (Originalarbeit in einer wissenschaftlichen Zeitschrift)

ID	4237673
Author(s)	Habegger, Philipp; Pazuki, Fabien
Author(s) at UniBasel	Habegger, Philipp
Year	2017
Title	Bad reduction of genus 2 curves with CM jacobian varieties
Journal	Compositio Mathematica
Volume	153
Number	12
Pages / Article-Number	2534-2576
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 formula by Colmez and Obus specific to the CM case and valid when the CM field is an abelian extension of the rationals. This formula 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 results of Igusa, Liu, and Saito to show that the contribution at the finite places in our decomposition measures the stable bad reduction of the curve and subconvexity bounds by Michel and Venkatesh together with an equidistribution result of Zhang to handle the infinite places.
Publisher	Cambridge University Press
ISSN/ISBN	0010-437X ; 1570-5846
URL	https://arxiv.org/abs/1506.02485
edoc-URL	https://edoc.unibas.ch/59216/
Full Text on edoc	No
Digital Object Identifier DOI	10.1112/S0010437X17007424
ISI-Number	WOS:000413336600003
Document type (ISI)	Article

14/05/2024

Research Database / FORSCHUNGSDATENBANK