Highly ramified pencils of elliptic curves in characteristic 2
This paper is devoted to the study of conductors and ramification of elliptic curves over a field of characteristic 2. Using quadratic twists, the author calculates explicitly conductors and recovers the unboundedness of them. The author studies also elliptic curves over the global field \(K= \mathbb{F}_q(T)\) of characteristic 2. He proves that, if there exists only one place with bad reduction, assumed to be \(\mathbb{F}_q\)-rational, then the \(j\)-invariant is in \(\mathbb{F}_q\); he proves also that, if there exists only one place of multiplicative reduction and only one place of additive reduction, twice supposed \(\mathbb{F}_q\)-rational, then the \(j\)-invariant is equal to \(cT^k\) with \(c\in \mathbb{F}^*_q\) and \(k\in\mathbb{N}\). The proof of this last statement uses the explicit Shimura-Taniyama-Weil correspondence. As a corollary, one has, up to \(K\)-isomorphisms or \(K\)-isogenies, the description of such elliptic curves by a particular family, denoted \((E_{\alpha,\beta})\). After that, the author studies the family \((E_{\alpha,\beta})\) over a local field (of characteristic 2). He proves that their conductors take all values \(n\in\mathbb{N}\) with \(n\geq 3\) and \(n\neq 2\pmod 4\). This makes precise the statement on the unboundedness of the conductors. Finally, elliptic curves over \(\mathbb{F}_2(T)\) are studied (i.e. \(q=2\)), and explicit statements concerning the family \((E_{\alpha,\beta})\) are given.
scientific article; zbMATH DE number 1054077
scientific article