Root number of twists of an elliptic curve
Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 73-101.

We give an explicit description of the behaviour of the root number in the family given by the twists of an elliptic curve E/ 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 . This completes work by Rohrlich: we detail the behaviour of the root number when E has bad reduction over ab 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/ par les valeurs rationnelles d’un polynôme f(t). En particulier, nous présentons un critère pour que la famille ait un signe constant sur . Ceci complète un travail de Rohrlich  : nous donnons les détails du comportement du signe lorsque E a mauvaise réduction sur ab et nous traitons les cas j(E)=0,1728 qui n’étaient pas considérés précédemment.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1112
Classification: 11G05,  11G07
Keywords: elliptic curve, root number, twist
Julie Desjardins 1

1 DH-3062, University of Toronto Mississauga 3359 Mississauga Road Mississauga, ON L5L 1C6, Canada
@article{JTNB_2020__32_1_73_0,
     author = {Julie Desjardins},
     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},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1112/}
}
TY  - JOUR
TI  - Root number of twists of an elliptic curve
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2020
DA  - 2020///
SP  - 73
EP  - 101
VL  - 32
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1112/
UR  - https://doi.org/10.5802/jtnb.1112
DO  - 10.5802/jtnb.1112
LA  - en
ID  - JTNB_2020__32_1_73_0
ER  - 
%0 Journal Article
%T Root number of twists of an elliptic curve
%J Journal de Théorie des Nombres de Bordeaux
%D 2020
%P 73-101
%V 32
%N 1
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1112
%R 10.5802/jtnb.1112
%G en
%F JTNB_2020__32_1_73_0
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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1112/

[1] Bryan J. Birch; Nelson M. Stephens The parity of the rank of the Mordell–Weil group, Topology, Volume 5 (1966), pp. 295-299 | Article | MR: 201379 | Zbl: 0146.42401

[2] J. W. S. Cassels; Andrzej Schinzel Selmer’s conjecture and families of elliptic curves, Bull. Lond. Math. Soc., Volume 14 (1982) no. 4, pp. 345-348 | Article | MR: 663485 | Zbl: 0474.14010

[3] Ian Connell Calculating root numbers of elliptic curves over , Manuscr. Math., Volume 82 (1994) no. 1, pp. 93-104 | Article | MR: 1254143 | Zbl: 0805.14017

[4] Julie Desjardins Densité des points rationnels sur les surfaces elliptiques et les surfaces de del Pezzo de degré 1 (2016) (http://www.theses.fr/2016USPCC229) (Ph. D. Thesis)

[5] Julie Desjardins On the density of rational points on rational elliptic surfaces, Acta Arith., Volume 189 (2019) no. 2, pp. 109-146 | Article | MR: 3955695 | Zbl: 07087796

[6] Julie Desjardins On the variation of the root number of the fibers of families of elliptic curves, J. Lond. Math. Soc., Volume 99 (2019) no. 2, pp. 295-331 | Article | MR: 3939257 | Zbl: 07053663

[7] Julie Desjardins; Bartosz Naskrȩcki Geometry of del Pezzo surfaces of the form y 2 =x 3 +Am 6 +Bn 6 (2019) (https://arxiv.org/abs/1911.02684)

[8] Tim Dokchitser; Vladimir Dokchitser Elliptic curves with all quadratic twists of positive rank, Acta Arith., Volume 137 (2009) no. 2, pp. 193-197 | Article | MR: 2491537 | Zbl: 1275.11097

[9] Emmanuel Halberstadt Signes locaux des courbes elliptiques en 2 et 3, C. R. Math. Acad. Sci. Paris, Volume 326 (1998) no. 9, pp. 1047-1052 | Article | MR: 1647190 | Zbl: 0933.11030

[10] Zhizhong Huang Rational points on elliptic K3 surfaces of quadratic twist type (2018) (https://arxiv.org/abs/1806.07869)

[11] Eric Liverance A formula for the root number of a family of elliptic curves, J. Number Theory, Volume 51 (1995) no. 2, pp. 288-305 | Article | MR: 1326750 | Zbl: 0831.14012

[12] Ottavio G. Rizzo Average root numbers for a nonconstant family of elliptic curves, Compos. Math., Volume 136 (2003) no. 1, pp. 1-23 | Article | MR: 1965738 | Zbl: 1021.11020

[13] David E. Rohrlich Variation of the root number in families of elliptic curves, Compos. Math., Volume 87 (1993) no. 2, pp. 119-151 | Numdam | MR: 1219633 | Zbl: 0791.11026

[14] David E. Rohrlich Galois theory, elliptic curves, and root numbers, Compos. Math., Volume 100 (1996) no. 3, pp. 311-349 | Numdam | MR: 1387669 | Zbl: 0860.11033

[15] Anthony Várilly-Alvarado Density of rational points on isotrivial rational elliptic surfaces, Algebra Number Theory, Volume 5 (2011) no. 5, pp. 659-690 | Article | MR: 2889751 | Zbl: 1276.11114

Cited by Sources: