Equations defining abelian varieties

In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension d ≥ 2, however, it is no longer as straightforward to discuss such equations. There is a large classical literature on this question, which in a reformulation is, for complex algebraic geometry, a question of describing relations between theta functions. The modern geometric treatment now refers to some basic papers of David Mumford, from 1966 to 1967, which reformulated that theory in terms from abstract algebraic geometry valid over general fields. The only 'easy' cases are those for d = 1, for an elliptic curve with linear span the projective plane or projective 3-space. In the plane, every elliptic curve is given by a cubic curve. In P3, an elliptic curve can be obtained as the intersection of two quadrics. In general abelian varieties are not complete intersections. Computer algebra techniques are now able to have some impact on the direct handling of equations for small values of d > 1. The interest in nineteenth century geometry in the Kummer surface came in part from the way a quartic surface represented a quotient of an abelian variety with d = 2, by the group of order 2 of automorphisms generated by x → −x on the abelian variety. Mumford defined a theta group associated to an invertible sheaf L on an abelian variety A. This is a group of self-automorphisms of L, and is a finite analogue of the Heisenberg group. The primary results are on the action of the theta group on the global sections of L. When L is very ample, the linear representation can be described, by means of the structure of the theta group. In fact the theta group is abstractly a simple type of nilpotent group, a central extension of a group of torsion points on A, and the extension is known (it is in effect given by the Weil pairing).