MAGMA Computational Algebra System

Introduction

This chapter deals with curves of genus one that are given by equations in a particular form. Most of the functionality involves invariant theory of these models, and applications of this to arithmetic problems concerning genus one curves over number fields.

Geometrically (viewed over an algebraically closed field) a genus one model of degree n is an elliptic curve embedded in P^{n - 1} via the linear system |n.O|. In general, a genus one model of degree n is a principal homogeneous space for an elliptic curve (of order n in the Weil-Chatelet group) which is embedded in P^{n - 1} in an analogous way. Such models are sometimes called genus one normal curves. Not every element of order n in the Weil-Chatelet group admits such an embedding, although it does if it is everywhere locally soluble.

Genus one models may be defined in Magma over any ring. The degree n can be 2, 3, 4 or 5. Genus one models have their own type, ModelG1, in Magma, which is not a subtype of any other type. In particular, these objects are not curves or even schemes. The data that defines a genus one model is either a multivariate polynomial (for degree 2 or 3), a pair of multivariate polynomials (degree 4), or a matrix of linear forms (degree 5). Each of these are now described in detail.

A genus one model of degree 2 in Magma is defined by either a binary quartic g(x, z) (referred to as a model without CrossTerms), or more generally an equation y² + f(x, z) y - g(x, z) where f and g are homogeneous of degrees 2 and 4, and the variables x, z, y are assigned weights 1, 1, 2 respectively. A binary quartic g(x, z) defines the same model as the equation y² - g(x, z). (The implicit map is the projection to P¹_{x, z}, which in this case is not an embedding, but rather has degree 2.)

A genus one model of degree 3 in Magma is defined by a cubic form in 3 variables. (in other words, the equation of a projective plane cubic curve).

A genus one model of degree 4 in Magma is defined by a sequence of two homogeneous polynomials of degree 2 in 4 variables. This represents an intersection of two quadric forms in P³ (which is the standard form in which Magma returns curves that are obtained by doing FourDescent on an elliptic curve).

A genus one model of degree 5 in Magma is defined by a 5 by 5 alternating matrix whose entries are linear forms in 5 variables. The associated subscheme of P⁴ is cut out by the 4 by 4 Pfaffians of the matrix. It is known that every genus one normal curve of degree 5 arises in this way.

It's important to note that degenerate cases are allowed, which means that the scheme associated to a genus one model is not always a smooth curve of genus 1 (or even a curve).