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arXiv · 1801.05262

Root number of the twists of an elliptic curve

Julie Desjardins
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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.

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Julie Desjardins. 2018-01-16. Root number of the twists of an elliptic curve. https://arxiv.org/abs/1801.05262

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