On the tate modules of elliptic curves over a local field of characteristic two

Abstract

Let K:=\mathbb{F}_{2^{f}}((T)) be the field of Laurent series over the finite field with 2^{f} elements. Every non-supersingular elliptic curve \mathcal{E} over K has a short Weierstraß form Y^{2}+XY=X^{3}+\alpha X^{2}+\beta with appropriate \alpha,\beta\in K. The Tate module of \mathcal{E} yields a two dimensional representation \pi'_{\alpha,\beta} of the Weil-Deligne group W'(K^{sep}/K). Contrary to characteristics different from two, arbitrarily high ramification may occur. If \beta is integral, the rational points of \mathcal{E} can be completely described in terms of periodic functions. As a consequence, \pi'_{\alpha,\beta} is completely known. We will deal with the case in which \beta is not integral. In this case we can consider \pi'_{\alpha,\beta} as a representation \pi_{\alpha,\beta} of the Weil group W(K^{sep}/K) of K. The aim of this article is to give an explicit description of \pi_{\alpha,\beta} and to determine the ramification properties. As a consequence, we will be able to calculate the conductor.