Root number of twists of an elliptic curve

Desjardins, Julie

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 73-101.

Nous donnons une description explicite du comportement du signe (root number) dans la famille des tordues d’une courbe elliptique $E/\mathbb{Q}$ par les valeurs rationnelles d’un polynôme $f\left(t\right)$. En particulier, nous présentons un critère pour que la famille ait un signe constant sur $\mathbb{Q}$. Ceci complète un travail de Rohrlich  : nous donnons les détails du comportement du signe lorsque $E$ a mauvaise réduction sur ${\mathbb{Q}}^{ab}$ et nous traitons les cas $j\left(E\right)=0,1728$ qui n’étaient pas considérés précédemment.

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

Classification : 11G05, 11G07
Mots clés : elliptic curve, root number, twist

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