Root number of the twists of an elliptic curve
Abstract
We give an explicit description of the behaviour of the root number in the family given by twists of an elliptic curve $E$ by the rational values of a polynomial $f(T)$. In particular, we give a criterion (on $f$ depending on $E$) for the family to have a constant root number over $\mathbb{Q}$. This completes a work of Rohrlich: we detail the behaviour of the root number when $E$ has bad reduction over $\mathbb{Q}^{ab}$ and we treat the cases $j(E)=0,1728$ which were not considered previously.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2018
- DOI:
- 10.48550/arXiv.1801.05262
- arXiv:
- arXiv:1801.05262
- Bibcode:
- 2018arXiv180105262D
- Keywords:
-
- Mathematics - Number Theory;
- 11G05;
- 11G07
- E-Print:
- 19 pages, to appear in Journal de Th\'eorie des Nombres de Bordeaux