Root number of the twists of an elliptic curve

Desjardins, Julie

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Journal Article

Root number of the twists of an elliptic curve

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Desjardins, Julie
Max Planck Institute for Mathematics, Max Planck Society;

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http://jtnb.centre-mersenne.org/item?id=JTNB_2020__32_1_73_0
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arXiv:1801.05262.pdf
(Preprint), 303KB

Desjardins_Root number of the twists of an elliptic curve_2020.pdf
(Publisher version), 775KB

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Citation

Desjardins, J. (2020). Root number of the twists of an elliptic curve. Journal de Théorie des Nombres de Bordeaux, 32(1), 73-101.

Cite as: https://hdl.handle.net/21.11116/0000-0007-A150-8

Abstract

We give an explicit description of the behaviour of the root number in the
family given by twists of an elliptic curve $E$ by the rational values of a
polynomial $f(T)$. In particular, we give a criterion (on $f$ depending on $E$)
for the family to have a constant root number over $\mathbb{Q}$. This completes
a work of Rohrlich: we detail the behaviour of the root number when $E$ has bad
reduction over $\mathbb{Q}^{ab}$ and we treat the cases $j(E)=0,1728$ which
were not considered previously.