164 research outputs found
Chabauty-Coleman experiments for genus 3 hyperelliptic curves
We describe a computation of rational points on genus 3 hyperelliptic curves
defined over whose Jacobians have Mordell-Weil rank 1. Using
the method of Chabauty and Coleman, we present and implement an algorithm in
Sage to compute the zero locus of two Coleman integrals and analyze the finite
set of points cut out by the vanishing of these integrals. We run the algorithm
on approximately 17,000 curves from a forthcoming database of genus 3
hyperelliptic curves and discuss some interesting examples where the zero set
includes global points not found in .Comment: 18 page
Linear quadratic Chabauty
We present a new quadratic Chabauty method to compute the integral points on
certain even degree hyperelliptic curves. Our approach relies on a nontrivial
degree zero divisor supported at the two points at infinity to restrict the
-adic height to a linear function; we can then express this restriction in
terms of holomorphic Coleman integrals under the standard quadratic Chabauty
assumption. Then we use this linear relation to extract the integral points on
the curve. We also generalize our method to integral points over number fields.
Our method is significantly simpler and faster than all other existing versions
of the quadratic Chabauty method. We give examples over \Q and
\Q(\sqrt{7}).Comment: 20 page
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
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