Explicit root numbers of abelian varieties

Matthew Bisatt*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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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.

Original languageEnglish
Pages (from-to)7889-7920
Number of pages32
JournalTransactions of the American Mathematical Society
Volume372
Issue number11
Early online date6 Sep 2019
DOIs
Publication statusPublished - 1 Dec 2019

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