Arithmetic of hyperelliptic curves over local fields

Arithmetic of Hyperelliptic Curves

Elliptic and hyperelliptic curves over supersimple fields

We prove that if F is an infinite field with characteristic different from 2, whose theory is supersimple, and C is an elliptic or hyperelliptic curve over F with generic moduli then C has a generic F -rational point. The notion of generity here is in the sense of the supersimple field F .

On Hyperelliptic Modular Curves over Function Fields

We prove that there are only finitely many modular curves of Delliptic sheaves over Fq(T ) which are hyperelliptic. In odd characteristic we give a complete classification of such curves.

On Arithmetic Of Hyperelliptic Curves

In this exposé, Pell’s equation is put in a geometric perspective, and a version of Artin’s primitive roots conjecture is formulated for hyperelliptic jacobians. Also explained are some recent results which throw new lights, having to do with Ankeny-Artin-Chowla’s conjecture, class number relations, and Cohen-Lenstra heuristics.

Empirical optimization of divisor arithmetic on hyperelliptic curves over F2m

A significant amount of effort has been devoted to improving divisor arithmetic on low-genus hyperelliptic curves via explicit versions of generic algorithms. Moderate and high genus curves also arise in cryptographic applications, for example, via the Weil descent attack on the elliptic curve discrete logarithm problem, but for these curves, the generic algorithms are to date the most efficien...