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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Explicit root numbers of abelian varieties


Author: Matthew Bisatt
Journal: Trans. Amer. Math. Soc. 372 (2019), 7889-7920
MSC (2010): Primary 11G10, 11G40
DOI: https://doi.org/10.1090/tran/7926
Published electronically: September 6, 2019
MathSciNet review: 4029685
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Abstract: The Birch and Swinnerton-Dyer conjecture predicts that the parity of the algebraic rank of an abelian variety over a global field should be controlled by the expected sign of the functional equation of its $L$-function, known as the global root number. In this paper, we give explicit formulae for the local root numbers as a product of Jacobi symbols. This enables one to compute the global root number, generalising work of Rohrlich, who studies the case of elliptic curves. We provide similar formulae for the root numbers after twisting the abelian variety by a self-dual Artin representation. As an application, we find a rational genus two hyperelliptic curve with a simple Jacobian whose root number is invariant under quadratic twist.


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Additional Information

Matthew Bisatt
Affiliation: Howard House, University of Bristol, Bristol, BS8 1SD, United Kingdom
MR Author ID: 1263481
Email: matthew.bisatt@bristol.ac.uk

Received by editor(s): May 15, 2018
Received by editor(s) in revised form: February 18, 2019
Published electronically: September 6, 2019
Article copyright: © Copyright 2019 American Mathematical Society