4,995 research outputs found
Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians
In this article, we show that in each of four standard families of
hyperelliptic curves, there is a density- subset of members with the
property that their Jacobians have adelic Galois representation with image as
large as possible. This result constitutes an explicit application of a general
theorem on arbitrary rational families of abelian varieties to the case of
families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit
examples of hyperelliptic curves of genus and over whose
Jacobians have such maximal adelic Galois representations.Comment: 24 page
Pointless Hyperelliptic Curves
In this paper we consider the question of whether there exists a hyperelliptic curve of genus g which is defined over but has no rational points over for various pairs . As an example of such a result, we show that if p is a prime such that is also prime then there will be pointless hyperelliptic curves over of every genus
Group law computations on Jacobians of hyperelliptic curves
We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form
Hyperelliptic Szpiro inequality
We generalize the classical Szpiro inequality to the case of a semistable
family of hyperelliptic curves. We show that for a semistable symplectic
Lefschetz fibration of hyperelliptic curves of genus , the number of
non-separating vanishing cycles and the number of singular fibers satisfy
the inequality .Comment: LaTeX2e, 27 page
- …