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Preprint (Vorabdruck) zugänglich unter
On the tate modules of elliptic curves over a local field of characteristic two
URN: urn:nbn:de:bsz:291-scidok-44928
URL: http://scidok.sulb.uni-saarland.de/volltexte/2012/4492/
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Dokument 1.pdf (174 KB)
Institut:
Fachrichtung 6.1 - Mathematik
DDC-Sachgruppe:
Mathematik
Dokumentart:
Preprint (Vorabdruck)
Schriftenreihe:
Preprint / Fachrichtung Mathematik, Universität des Saarlandes
Bandnummer:
127
Sprache:
Englisch
Erstellungsjahr:
2004
Publikationsdatum:
15.02.2012
Kurzfassung auf Englisch:
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.
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