min_red_binary_form.m
This implements minimization and reduction for binary forms over Q,
together with a function that tries to produce a simple form of a
given morphism P1Q → X
(by modifying by a suitable automorphism of P1Q).
This code is now available in the Magma distribution.
SelChabDyn.magma:
this implements the algorithm described in Section 2
for curves of the form
y2 = x2g+1 + h(x)2
with h of degree at most g, integral coefficients
and such that h(0) is odd and h(1) is even.
diophtuples.magma:
this implements the algorithm described in Section 2 (assuming GRH)
for curves arising from diophantine quadruples.
diophtuples-verify.magma:
this verifies the results (if feasible, without assuming GRH),
given information on the elliptic curve that gives a “good”
rank bound.
ellQt.magma:
this implements the algorithm described in Section 5.
Computation of the Mordell-Weil group of Jacobians of genus 2 NOTE: A newer (and better) version of this is now included in Magma.
This includes the files
g2wrapper.tar.gz:
a compressed tarball containing all of the following files
Kum3-verification.magma:
a script that verifies a number of computational claims in the paper
G3Hyp.spec, G3Hyp.m,
G3HypHelp.m:
package files providing code for various tasks; do “AttachSpec("G3Hyp.spec");”
Berno Reitsma has code for computing the rational torsion subgroup
of genus 3 hyperelliptic Jacobians, which is based on the functionality
provided by the files above; see
here.
ratcycles.magma
This demonstrates how to determine the set of rational n-cycles
under z ↦ z2 + c for n = 1,2,3,4,5.
(Following Flynn, Poonen and Schaefer.)
kleinquartic.magma
This shows how to determine the set of rational points on the
Klein Quartic
x3y + y3z + z3x = 0.
(Idea stolen from Nils Bruin.)
ellchab.magma
This shows how to determine the set of rational points on
y2 = x6 + x2 + 1
using “elliptic curve Chabauty”.
(Example from Joe Wetherell's thesis, method by Nils Bruin.)
g2ff.magma
This demonstrates how you can work with genus 2 curves over
function fields.
Xdyn06.magma
This is the file providing the computational details for the paper
M. Stoll:
Rational 6-cycles under iteration of quadratic polynomials,
London Math. Soc. J. Comput. Math. 11, 367-380 (2008).
preimages.magma
This is the file providing the computational details for the paper
X. Faber, B. Hutz and M. Stoll:
On the number of rational iterated pre-images of the origin
under quadratic dynamical systems,
Int. J. Number Theory 7:7, 1781-1806 (2011).
chabauty-MWS.m
This is an implementation of the "Chabauty + Mordell-Weil Sieve"
approach to find all rational points on a genus 2 curve over Q whose
Jacobian has Mordell-Weil rank 1.
Just do “Attach("chabauty-MWS.m");”
have a look at the example at the beginning of the file to see how to use it.
(This is now contained in Magma.)
MWSieve-new.m
Here is an implementation of the Mordell-Weil Sieve that can be used
to verify that a given curve (of genus 2 over Q) does not have rational
points, see
N. Bruin, M. Stoll:
The Mordell-Weil Sieve: Proving non-existence of rational points on curves,
LMS J. Comput. Math. 13, 272-306 (2010).
See the beginning of the file for how to use it.