9 research outputs found
Computing the Cassels–Tate pairing on the 3-Selmer group of an elliptic curve
We extend the method of Cassels for computing the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent
Explicit n-descent on elliptic curves. III. Algorithms
This is the third in a series of papers in which we study the n-Selmer group
of an elliptic curve, with the aim of representing its elements as curves of
degree n in P^{n-1}. The methods we describe are practical in the case n=3 for
elliptic curves over the rationals, and have been implemented in Magma.
One important ingredient of our work is an algorithm for trivialising central
simple algebras. This is of independent interest: for example, it could be used
for parametrising Brauer-Severi surfaces.Comment: 43 pages, comes with a file containing Magma code for the
computations used for the examples. v2: some small edit
Computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian
We describe a method for computing the Cassels-Tate pairing on the 2-Selmer
group of the Jacobian of a genus 2 curve. This can be used to improve the upper
bound coming from 2-descent for the rank of the group of rational points on the
Jacobian. Our method remains practical regardless of the Galois action on the
Weierstrass points of the genus 2 curve. It does however depend on being able
to find a rational point on a certain twisted Kummer surface. The latter does
not appear to be a severe restriction in practice. In particular, we have used
our method to unconditionally determine the ranks of all genus 2 Jacobians in
the L-functions and modular forms database (LMFDB).Comment: 48 page
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Explicit Methods in Number Theory
The aim of the series of Oberwolfach meetings on ‘Explicit methods in number theory’ is to bring together people attacking key problems in number theory via techniques involving concrete or computable descriptions. Here, number theory is interpreted broadly, including algebraic and analytic number theory, Galois theory and inverse Galois problems, arithmetic of curves and higher-dimensional varieties, zeta and -functions and their special values, and modular forms and functions