# Explicit n-descent on elliptic curves, I. Algebra

@inproceedings{Cremona2006ExplicitNO, 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}, year={2006} }

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.

#### 63 Citations

Explicit n-descent on elliptic curves III. Algorithms

- Computer Science, Mathematics
- Math. Comput.
- 2015

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 we… Expand

Explicit second p-descent on elliptic curves

- Mathematics
- 2010

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 elliptic… Expand

Explicit n-descent on elliptic curves, II. Geometry

- Mathematics
- 2009

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 curves… Expand

Second p-descents on elliptic curves

- Mathematics, Computer Science
- Math. Comput.
- 2014

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

- Mathematics, Computer Science
- Math. Comput.
- 2013

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

AVERAGE RANK OF ELLIPTIC CURVES

- Mathematics
- 2015

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 for… Expand

Rational points on curves

- Mathematics
- 2010

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 of… Expand

On some algebras associated to genus one curves

- Mathematics
- Journal of Algebra
- 2019

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, and… Expand

Some bounds on the coefficients of covering curves

- Mathematics
- 2012

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 author… Expand

Elliptic Curves with Large Analytic Order of Ш(E)

- Mathematics
- 2009

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 curves… Expand

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