2,473 research outputs found
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
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
Explicit isogeny descent on elliptic curves
In this note, we consider an l-isogeny descent on a pair of elliptic curves
over Q. We assume that l > 3 is a prime. The main result expresses the relevant
Selmer groups as kernels of simple explicit maps between finite- dimensional
F_l-vector spaces defined in terms of the splitting fields of the kernels of
the two isogenies. We give examples of proving the l-part of the Birch and
Swinnerton-Dyer conjectural formula for certain curves of small conductor.Comment: 17 pages, accepted for publication in Mathematics of Computatio
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
Splitting full matrix algebras over algebraic number fields
Let K be an algebraic number field of degree d and discriminant D over Q. Let
A be an associative algebra over K given by structure constants such that A is
isomorphic to the algebra M_n(K) of n by n matrices over K for some positive
integer n. Suppose that d, n and D are bounded. Then an isomorphism of A with
M_n(K) can be constructed by a polynomial time ff-algorithm. (An ff-algorithm
is a deterministic procedure which is allowed to call oracles for factoring
integers and factoring univariate polynomials over finite fields.)
As a consequence, we obtain a polynomial time ff-algorithm to compute
isomorphisms of central simple algebras of bounded degree over K.Comment: 15 pages; Theorem 2 and Lemma 8 correcte
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