Adrien:

$\mathcal{H}$ & Poincaré half-plane

Alex :

$\overline{k}$ & The algebraic closure of the field $k$. \\
$E/K$ & Field extension $K \subset E$. \\
$\Gal(E/K)$ & The Galois group of $E/K$. \\

$G_n$ & The $n$-torsion subgroup of the group $G$.\\
$\Delta$ & The discriminant of an elliptic curve.\\

Vlad: on écrit E(K) pour les points K-rationnels d'une courbe elliptique E.\\

Kevin:

$\mathcal{H}^{\ast}$ the extended upper half-plane \\
$\sL$ the set of all latices  \\
$\Gamma(1)$ the modular group \\

Fabrizio:

$\widetilde{E}_p$ & The reduction of an elliptic curve $E$ modulo $p$\\
$\IP^2(k)$ & The projective space over $k$\\
$\widetilde{\Delta}_p$ & The discriminant of a reduced elliptic curve modulo $p$\\
$v_p$ & The $p$-addic valuation.\\
$Z_E(T)$ & The $Z$-function of $E$\\
$\xi_E(s)$ & The zeta -function of $E$\\
$L_E(s)$ & The $L$-function of $E$