3,154 research outputs found
Some remarks on Heegner point computations
We explain how to find a rational point on a rational elliptic curve of rank
1 using Heegner points. We give some examples, and list new algorithms that are
due to Cremona and Delaunay. These are notes from a short course given at the
Institut Henri Poincare in December 2004
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
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