We give an explicit description of the behaviour of the root number in the family given by the twists of an elliptic curve by the rational values of a polynomial . 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 has bad reduction over and we treat the cases 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 par les valeurs rationnelles d’un polynôme . 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 a mauvaise réduction sur et nous traitons les cas qui n’étaient pas considérés précédemment.
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Keywords: elliptic curve, root number, twist
@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 AU - Julie Desjardins TI - Root number of twists of an elliptic curve JO - Journal de théorie des nombres de Bordeaux PY - 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/ DO - 10.5802/jtnb.1112 LA - en ID - JTNB_2020__32_1_73_0 ER -
%0 Journal Article %A Julie Desjardins %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://jtnb.centre-mersenne.org/articles/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/
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