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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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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: http://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.