378 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
Jacobian Nullwerte, Periods and Symmetric Equations for Hyperelliptic Curves
We propose a solution to the hyperelliptic Schottky problem, based on the use
of Jacobian Nullwerte and symmetric models for hyperelliptic curves. Both
ingredients are interesting on its own, since the first provide period matrices
which can be geometrically described, and the second have remarkable arithmetic
properties.Comment: To appear in "Annales de l'Institut Fourier
Quadratic Points on Modular Curves
In this paper we determine the quadratic points on the modular curves X_0(N),
where the curve is non-hyperelliptic, the genus is 3, 4 or 5, and the
Mordell--Weil group of J_0(N) is finite. The values of N are 34, 38, 42, 44,
45, 51, 52, 54, 55, 56, 63, 64, 72, 75, 81.
As well as determining the non-cuspidal quadratic points, we give the
j-invariants of the elliptic curves parametrized by those points, and determine
if they have complex multiplication or are quadratic \Q-curves.Comment: Some improvements and corrections suggested by the referee are
incorporated. Magma programs used to generate the data are now available with
this arXiv versio
Computing canonical heights using arithmetic intersection theory
For several applications in the arithmetic of abelian varieties it is
important to compute canonical heights. Following Faltings and Hriljac, we show
how the canonical height on the Jacobian of a smooth projective curve can be
computed using arithmetic intersection theory on a regular model of the curve
in practice. In the case of hyperelliptic curves we present a complete
algorithm that has been implemented in Magma. Several examples are computed and
the behavior of the running time is discussed.Comment: 29 pages. Fixed typos and minor errors, restructured some sections.
Added new Example
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