Root number of twists of an elliptic curve

Julie Desjardins

doi:10.5802/jtnb.1112

Julie Desjardins ¹

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

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

Résumé
Abstract

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.

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.

Reçu le : 2019-01-10
Révisé le : 2020-03-20
Accepté le : 2020-04-25
Publié le : 2020-08-21

DOI : 10.5802/jtnb.1112

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

Affiliations des auteurs :

Julie Desjardins ¹

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Root number of twists of an elliptic curve},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {73--101},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {1},
     year = {2020},
     doi = {10.5802/jtnb.1112},
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Julie Desjardins. Root number of twists of an elliptic curve. Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 73-101. doi : 10.5802/jtnb.1112. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1112/

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