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Explicit determination of root numbers of abelian varieties


Authors: Armand Brumer, Kenneth Kramer and Maria Sabitova
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11G10; Secondary 11G25, 11S20
DOI: https://doi.org/10.1090/tran/7116
Published electronically: December 26, 2017
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Abstract: Let $ A$ be an abelian variety over a nonarchimedean local field of definition $ K$ and let $ W(A)$ be the root number of $ A$. Let $ F$ be a Galois extension of $ K$ over which $ A$ has semistable reduction, allowing $ F = K$. We analyze $ W(A)$ in terms of contributions from the toric and abelian variety components of the closed fibers of the Néron models of $ A$ over the ring of integers of $ K$ and of $ F$. In particular, our results can be used to calculate $ W(A)$ in two main instances: first, for abelian varieties with additive reduction over $ K$ and totally toroidal reduction over $ F$, provided that the residue characteristic of $ K$ is odd; second, for the Jacobian $ A = J(C)$ of a stable curve $ C$ over $ K$.


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Additional Information

Armand Brumer
Affiliation: Department of Mathematics, Fordham University, Bronx, New York 10458
Email: brumer@fordham.edu

Kenneth Kramer
Affiliation: Department of Mathematics, Queens College, Flushing, New York 11367—and—The Graduate Center CUNY, New York, New York 10016
Email: kkramer@qc.cuny.edu

Maria Sabitova
Affiliation: Department of Mathematics, Queens College, Flushing, New York 11367—and—The Graduate Center CUNY, New York, New York 10016
Email: maria.sabitova@qc.cuny.edu

DOI: https://doi.org/10.1090/tran/7116
Keywords: Abelian variety, root number, Weil--Deligne group
Received by editor(s): March 19, 2015
Received by editor(s) in revised form: February 3, 2016, and July 24, 2016
Published electronically: December 26, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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