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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Explicit root numbers of abelian varieties
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by Matthew Bisatt PDF
Trans. Amer. Math. Soc. 372 (2019), 7889-7920 Request permission

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
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 7889-7920
  • MSC (2010): Primary 11G10, 11G40
  • DOI: https://doi.org/10.1090/tran/7926
  • MathSciNet review: 4029685