Explicit n-descent on elliptic curves, I. Algebra

  title={Explicit n-descent on elliptic curves, I. Algebra},
  author={John Cremona and T. A. Fisher and C. O'Neil and Denis Simon and Michael Stoll},
Abstract This is the first 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 genus one normal curves of degree n. The methods we describe are practical in the case n = 3 for elliptic curves over the rationals, and have been implemented in MAGMA. 
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 genus one normal curves of degree n. The methods weExpand
Explicit second p-descent on elliptic curves
One of the fundamental motivating problems in arithmetic geometry is to understand the set V (k) of rational points on an algebraic variety V defined over a number field k. When V = E is an ellipticExpand
Explicit n-descent on elliptic curves, II. Geometry
Abstract This is the second in a series of papers in which we study the n-Selmer group of an elliptic curve. In this paper, we show how to realize elements of the n-Selmer group explicitly as curvesExpand
Second p-descents on elliptic curves
An algorithm is described 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, which leads to a practical algorithm for performing explicit 9-descents on elliptic curves over Q. Expand
Explicit isogeny descent on elliptic curves
The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finitedimensional F`-vector spaces defined in terms of the splitting fields of the kernels of the two isogenies. Expand
Bhargava and Shankar prove that as E varies over all elliptic curves over Q, the average rank of the finitely generated abelian group E(Q) is bounded. This result follows from an exact formula forExpand
Rational points on curves
This is an extended version of an invited lecture I gave at the Journees Arithmetiques in St. Etienne in July 2009. We discuss the state of the art regarding the problem of finding the set ofExpand
On some algebras associated to genus one curves
Haile, Han and Kuo have studied certain non-commutative algebras associated to a binary quartic or ternary cubic form. We extend their construction to pairs of quadratic forms in four variables, andExpand
Some bounds on the coefficients of covering curves
We compute bounds on the coefficients of the equations defining everywhere locally soluble n-coverings of elliptic curves over the rationals for n = 2, 3, 4. Our proofs use recent work of the authorExpand
Elliptic Curves with Large Analytic Order of Ш(E)
We present the results of our search for elliptic curves over \(\mathbb{Q}\) with exceptionally large analytic orders of the Tate-Shafarevich group. We exibit \(134\) examples of rank zero curvesExpand


Computing the Rank of Elliptic Curves over Number Fields
This paper describes an algorithm of 2-descent for computing the rank of an elliptic curve without 2-torsion, defined over a general number field. This allows one, in practice, to deal with fields ofExpand
Classical Invariants and 2-descent on Elliptic Curves
  • J. Cremona
  • Computer Science, Mathematics
  • J. Symb. Comput.
  • 2001
The results lead to some simplifications to the method first presented in Birch and Swinnerton-Dyer (1963), and can be applied to give a more efficient algorithm for determining Mordell?Weil groups over Q, as well as being more readily extended to other number fields. Expand
Jacobians of Genus One Curves
Abstract Consider a curve of genus one over a field K in one of three explicit forms: a double cover of P1, a plane cubic, or a space quartic. For each form, a certain syzygy from classical invariantExpand
The arithmetic of elliptic curves
  • J. Silverman
  • Mathematics, Computer Science
  • Graduate texts in mathematics
  • 1986
It is shown here how Elliptic Curves over Finite Fields, Local Fields, and Global Fields affect the geometry of the elliptic curves. Expand
Testing Equivalence of Ternary Cubics
  • T. Fisher
  • Mathematics, Computer Science
  • ANTS
  • 2006
A formula for the action of the 3-torsion of E on C is given, and it is explained how it is useful in studying the3-Selmer group of an elliptic curve defined over a number field. Expand
The Period-Index Obstruction for Elliptic Curves
Let K be a field and let E be an elliptic curve over K . Let GK be the absolute Galois group of K . The elements of the group HðGK ;EÞ are in one-to-one correspondence with isomorphism classes ofExpand
Computing the p-Selmer group of an elliptic curve
In this paper we explain how to bound the p-Selmer group of an elliptic curve over a number field K. Our method is an algorithm which is relatively simple to implement, although it requires data suchExpand
Jacobians of genus one curves
We introduce the notion of an “n-prepared curve,” which over a field containing nth roots of unity and where n is invertible is an embedding of a smooth genus one curve C in Pn−1 along with aExpand
Noncommutative cohomologies and Mumford groups
In this note we prove the “quadratic property” of the boundary mapping in the exact sequence of cohomologies. We present calculations for the action of the Galois group on the Mumford group.
How to do a p-descent on an elliptic curve
In this paper, we describe an algorithm that reduces the computation of the (full) p-Selmer group of an elliptic curve E over a number field to standard number field computations such as determiningExpand