Throughout this work we deal with a natural number g ≥ 2 and with an algebraically closed field k whose characteristic differs from 2. A hyperelliptic curve of genus g over k is a smooth curve of genus g, that is a double cover of the projective line P. The Riemann-Hurwitz formula implies that this covering should be ramified at 2g + 2 points. Because of this explicit description, hyperelliptic...

We construct three-dimensional families of hyperelliptic curves of genus 6, 12, and 14, two-dimensional families of hyperelliptic curves of genus 3, 6, 7, 10, 20, and 30, and one-dimensional families of hyperelliptic curves of genus 5, 10 and 15, all of which are equipped with an an explicit isogeny from their Jacobian to another hyperelliptic Jacobian. We show that the Jacobians are genericall...

We construct unramified central simple algebras representing 2-torsion classes in the Brauer group of a hyperelliptic curve, and show that every 2-torsion class can be constructed this way when the curve has a rational Weierstrass point or when the base field is C1. In general, we show that a large (but in general proper) subgroup of the 2torsion classes are given by the construction. Examples ...

In this brief note, we prove the following two Artin-Schreier curves are supersin-gular.

The object of this paper is the theta divisor of the compactified Jacobian of a nodal curve of genus g. We determine its irreducible components and give it a geometric interpretation. A characterization of hyperelliptic irreducible curves in M g is appended as an application.

In this paper, we give a ‘direct’ construction of the endomorphism ring of supersingular elliptic curves over a prime field Fp from ‘ideal classes’ of Q( √−p). We use the result to prove that the result of Kaneko on ‘minimal’ CM liftings of such supersingular elliptic curves is a best possible result. We also prove that the result of Elkies on ‘minimal’ CM liftings of all supersingular elliptic...

in this note we review a simple criterion, due to ekedahl, for superspecial curves defined over finite fields.using this we generalize and give some simple proofs for some well-known superspecial curves.