Lectures on Nilpotent Groups and Abelian Varieties

Abstract

A. A. Albert, in an immense burst of creative energy succeeded in solving the “Riemann matrix problem.” Although this is one of the great mathematical achievements of our century, there are few systematic accounts of Albert's work. Perhaps, C. L. Siegel's account [6] comes the closest to providing us with a view of this marvelous achievement. Albert's and Siegel's treatment are difficult because their arguments are based on matrix calculations. Because a coordinate system has been chosen, there is a hidden identification of a vector space with its dual and matrices play the role of both linear transformations and bilinear forms.

In these notes, we will present a way of using nilpotent groups to formulate the ideas of Abelian varieties and present part of the existence theorems contained in Albert's work. A full treatment of the existence part of Albert's work will appear in [4]. Our approach rests on nilpotent algebraic groups. This enables us to present a matrix-free treatment of the Riemann matrix problem. We hope this approach will reawaken admiration for, and interest in, Albert's achievement.