On elliptic curves over function fields of characteristic two
Elliptic curves in characteristic \(2\) (and \(3\)) present special features different from and more complicated than those in other characteristics. For example, if \(K\) is a local field of characteristic \(2\), there exist elliptic curves \(E/K\) with arbitrarily large conductors. NEWLINENEWLINENEWLINEEven more complicated is the possible behavior of curves \(E\) over global fields of characteristic \(2\). For fixing ideas, let \(K\) be the rational function field \(k_0(T)\), and \(E/K\) elliptic and non-constant. Quite generally, the conductor of \(E/K\) (a divisor on the associated projective line) has degree at least four, and there is an abundance of papers and results about the classification of such \(E\) (or, what is essentially the same, of the corresponding elliptic surfaces) with conductor of degree exactly four, where \(k_0 = \mathbb C\), say. NEWLINENEWLINENEWLINEThe paper under consideration deals with the same problem, but where \(k_0 = \mathbb F_{2^r}\) is a finite field of characteristic \(2\). Marvellously, the author is able to give a complete list of isomorphism classes of elliptic curves \(E/\mathbb F_{2^r}(T)\) with conductors of degree four, through explicit equations, and including the isogenies between them. Techniques used are rather involved and range from the ideal theory of quaternion algebras over \(K\) and the arithmetic and geometry of Drinfeld modular curves to the analysis of Tate's algorithm and explicit calculation of some cases. The strategy is discussed in the introduction: NEWLINENEWLINENEWLINE(1) Find the places of supersingular reduction of the wanted elliptic curves (the number of such places is finite!), which restricts the possible \(j\)-invariants. NEWLINENEWLINENEWLINE(2) Use Tate's algorithm to determine the conductors of ``untwisted'' elliptic curves with these \(j\)-invariants. NEWLINENEWLINENEWLINE(3) Describe the affect of twisting on the conductor. NEWLINENEWLINENEWLINE(4) Find the isogenies between the curves identified by (1), (2), (3). NEWLINENEWLINENEWLINEApart from its intrinsic interest, the value of these results is in providing very concrete examples for a multitude of open questions. For example, they give an affirmative answer to a question raised by W. Lang and H. Ito: NEWLINENEWLINENEWLINEEvery extremal elliptic surface over the algebraic closure of \(\mathbb F_2\) is unirational.
scientific article; zbMATH DE number 1594760