156 research outputs found
Second p descents on elliptic curves
Let p be a prime and let C be a genus one curve over a number field k
representing an element of order dividing p in the Shafarevich-Tate group of
its Jacobian. We describe an algorithm which computes the set of D in the
Shafarevich-Tate group such that pD = C and obtains explicit models for these D
as curves in projective space. This leads to a practical algorithm for
performing 9-descents on elliptic curves over the rationals.Comment: 45 page
Visualising Sha[2] in Abelian Surfaces
Given an elliptic curve E1 over a number field and an element s in its
2-Selmer group, we give two different ways to construct infinitely many Abelian
surfaces A such that the homogeneous space representing s occurs as a fibre of
A over another elliptic curve E2. We show that by comparing the 2-Selmer groups
of E1, E2 and A, we can obtain information about Sha(E1/K)[2] and we give
examples where we use this to obtain a sharp bound on the Mordell-Weil rank of
an elliptic curve.
As a tool, we give a precise description of the m-Selmer group of an Abelian
surface A that is m-isogenous to a product of elliptic curves E1 x E2. One of
the constructions can be applied iteratively to obtain information about
Sha(E1/K)[2^n]. We give an example where we use this iterated application to
exhibit an element of order 4 in Sha(E1/Q).Comment: 17 page
Rank zero elliptic curves induced by rational Diophantine triples
Rational Diophantine triples, i.e. rationals a,b,c with the property that
ab+1, ac+1, bc+1 are perfect squares, are often used in construction of
elliptic curves with high rank. In this paper, we consider the opposite problem
and ask how small can be the rank of elliptic curves induced by rational
Diophantine triples. It is easy to find rational Diophantine triples with
elements with mixed signs which induce elliptic curves with rank 0. However,
the problem of finding such examples of rational Diophantine triples with
positive elements is much more challenging, and we will provide the first such
known example.Comment: 7 page
Exhibiting Sha[2] on hyperelliptic jacobians
We discuss approaches to computing in the Shafarevich-Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2-Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of genus 2, we also demonstrate a connection with degree 4 del Pezzo surfaces, and show how the Brauer-Manin obstruction on these surfaces can be used to compute members of the Shafarevich-Tate group of Jacobians. We derive an explicit parametrised infinite family of genus 2 curves whose Jacobians have nontrivial members of the Sharevich-Tate group. Finally we prove that under certain conditions, the visualisation dimension for order 2 cocycles of Jacobians of certain genus 2 curves is 4 rather than the general bound of 32
Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve
It has been conjectured that for sufficiently large, there are no
quadratic polynomials in with rational periodic points of period
. Morton proved there were none with , by showing that the genus~
algebraic curve that classifies periodic points of period~4 is birational to
, whose rational points had been previously computed. We prove there
are none with . Here the relevant curve has genus~, but it has a
genus~ quotient, whose rational points we compute by performing
a~-descent on its Jacobian and applying a refinement of the method of
Chabauty and Coleman. We hope that our computation will serve as a model for
others who need to compute rational points on hyperelliptic curves. We also
describe the three possible Gal-stable -cycles, and show that
there exist Gal-stable -cycles for infinitely many .
Furthermore, we answer a question of Morton by showing that the genus~
curve and its quotient are not modular. Finally, we mention some partial
results for
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