Bad reduction of genus 2 curves with CM jacobian varieties

Habegger, Philipp and Pazuki, Fabien. (2017) Bad reduction of genus 2 curves with CM jacobian varieties. Compositio Mathematica, 153. pp. 2534-2576.

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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.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Zahlentheorie (Habegger)
UniBasel Contributors:Habegger, Philipp
Item Type:Article, refereed
Article Subtype:Research Article
Publisher:Cambridge University Press
Note:Publication type according to Uni Basel Research Database: Journal article
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Last Modified:24 Jul 2020 13:37
Deposited On:24 Jul 2020 13:37

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