Root number of twists of an elliptic curve

Julie Desjardins

doi:10.5802/jtnb.1112

Julie Desjardins ¹

¹DH-3062, University of Toronto Mississauga 3359 Mississauga Road Mississauga, ON L5L 1C6, Canada

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

Abstract (Original language)
Résumé

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

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

Article information
Cite this article

Received: 2019-01-10
Revised: 2020-03-20
Accepted: 2020-04-25
Published online: 2020-08-21

DOI: 10.5802/jtnb.1112

Classification: 11G05, 11G07
Keywords: elliptic curve, root number, twist

Author's affiliations:

Julie Desjardins ¹

¹ DH-3062, University of Toronto Mississauga 3359 Mississauga Road Mississauga, ON L5L 1C6, Canada

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

Julie Desjardins. Root number of twists of an elliptic curve. Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 73-101. doi: 10.5802/jtnb.1112

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