On the integral $2$-adic Tate module of elliptic curves

Abstract

We show that the integral $2$-adic Tate module of an elliptic curve over a complete discretely valued field of odd residue characteristic is determined by its $2$-torsion representation, together with the square class of $c$ and local information about the pairwise differences of the roots of $f$ in a model $E\colon y^2=cf(x)$, where $f$ is monic of degree $3$. The proof uses explicit halving formulae to determine the Galois action on the full tower of $2$-power torsion.