On the degree of the $p$-torsion field of elliptic curves over $\mathbb Q_\ell $ for $\ell \not =p$
Volume 195 / 2020
Acta Arithmetica 195 (2020), 13-55
MSC: Primary 11G05.
DOI: 10.4064/aa181029-8-11
Published online: 29 April 2020
Abstract
Let $\ell $ and $p \geq 3$ be distinct prime numbers. Let $E/\mathbb Q _{\ell }$ be an elliptic curve with $p$-torsion module $E_p$. Let $\mathbb Q _{\ell }(E_p)$ be the $p$-torsion field of $E$. We provide a complete description of the degree of the extension $\mathbb Q _{\ell }(E_p)/\mathbb Q _{\ell }$ in terms of standard information on $E$.
