Curves in characteristic 2 with non-trivial 2-torsion

Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational 2-torsion point on their Jacobian. We extend their observation to curves given by Laurent polynomials with a fixed Newton polygon, provided that the polygon satisfies a certain combinatorial property. We also show that in each of these cases, the sufficiently general condition is implied by being ordinary. Our treatment includes many classical families, such as hyperelliptic curves of odd genus and Ca,b curves. In the hyperelliptic case, we provide alternative proofs using an explicit description of the 2-torsion subgroup

On Neron-Severi lattices of Jacobian elliptic K3 surfaces

We classify all Jacobian elliptic fibrations on K3 surfaces with finite automorphism group. We also classify all extremal Jacobian elliptic fibrations on K3 surfaces with infinite automorphism group and 2-elementary N\'eron-Severi lattice.Comment: 17 pages, 10 tables, 1 figur

Towers of 2-covers of hyperelliptic curves

In this article, we give a way of constructing an unramified Galois cover of a hyperelliptic curve. The geometric Galois-group is an elementary abelian 2-group. The construction does not make use of the embedding of the curve in its Jacobian and it readily displays all subcovers. We show that the cover we construct is isomorphic to the pullback along the multiplication-by-2 map of an embedding of the curve in its Jacobian. We show that the constructed cover has an abundance of elliptic and hyperelliptic subcovers. This makes this cover especially suited for covering techniques employed for determining the rational points on curves. Especially the hyperelliptic subcovers give a chance for applying the method iteratively, thus creating towers of elementary abelian 2-covers of hyperelliptic curves. As an application, we determine the rational points on the genus 2 curve arising from the question whether the sum of the first n fourth powers can ever be a square. For this curve, a simple covering step fails, but a second step succeeds

The 2-Ranks of Hyperelliptic Curves with Extra Automorphisms

This paper examines the relationship between the automorphism group of a hyperelliptic curve defined over an algebraically closed field of characteristic two and the 2-rank of the curve. In particular, we exploit the wild ramification to use the Deuring-Shafarevich formula in order to analyze the ramification of hyperelliptic curves that admit extra automorphisms and use this data to impose restrictions on the genera and 2-ranks of such curves. We also show how some of the techniques and results carry over to the case where our base field is of characteristic p \u3e 2