675 research outputs found
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
On the height of Gross-Schoen cycles in genus three
We show that there exists a sequence of genus three curves defined over the
rationals in which the height of a canonical Gross-Schoen cycle tends to
infinity.Comment: 26 pages; v2: referee's remarks taken into accoun
On Using Expansions to the Base of -2
This short note investigates the effects of using expansions to the base of
-2. The main applications we have in mind are cryptographic protocols, where
the crucial operation is computation of scalar multiples. For the recently
proposed groups arising from Picard curves this leads to a saving of at least
7% for the computation of an m-fold. For more general non-hyperelliptic genus 3
curves we expect a larger speed-up.Comment: 5 page
The geometry of some parameterizations and encodings
We explore parameterizations by radicals of low genera algebraic curves. We
prove that for a prime power that is large enough and prime to , a fixed
positive proportion of all genus 2 curves over the field with elements can
be parameterized by -radicals. This results in the existence of a
deterministic encoding into these curves when is congruent to modulo
. We extend this construction to parameterizations by -radicals for
small odd integers , and make it explicit for
The arithmetic of Prym varieties in genus 3
Given a curve of genus 3 with an unramified double cover, we give an explicit
description of the associated Prym-variety. We also describe how an unramified
double cover of a non-hyperelliptic genus 3 curve can be mapped into the
Jacobian of a curve of genus 2 over its field of definition and how this can be
used to do Chabauty- and Brauer-Manin type calculations for curves of genus 5
with an unramified involution. As an application, we determine the rational
points on a smooth plane quartic with no particular geometric properties and
give examples of curves of genus 3 and 5 violating the Hasse-principle. We also
show how these constructions can be used to design smooth plane quartics with
specific arithmetic properties. As an example, we give a smooth plane quartic
with all 28 bitangents defined over Q(t).Comment: 21 page
Generalised Elliptic Functions
We consider multiply periodic functions, sometimes called Abelian functions,
defined with respect to the period matrices associated with classes of
algebraic curves. We realise them as generalisations of the Weierstras
P-function using two different approaches. These functions arise naturally as
solutions to some of the important equations of mathematical physics and their
differential equations, addition formulae, and applications have all been
recent topics of study.
The first approach discussed sees the functions defined as logarithmic
derivatives of the sigma-function, a modified Riemann theta-function. We can
make use of known properties of the sigma function to derive power series
expansions and in turn the properties mentioned above. This approach has been
extended to a wide range of non hyperelliptic and higher genus curves and an
overview of recent results is given.
The second approach defines the functions algebraically, after first
modifying the curve into its equivariant form. This approach allows the use of
representation theory to derive a range of results at lower computational cost.
We discuss the development of this theory for hyperelliptic curves and how it
may be extended in the future.Comment: 16 page
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