Tata Lectures on Theta III

The present volume contains the final chapter of this work on theta func tions. Like the other chapters, it originated with the senior author's lectures at the Tata Institute of Fundamental Research during November 1978- March 1979. Excellent notes on these lectures were made by M. Nori, but, due to the shortage of time, not all the projected topics were covered. In the next few years, the ideas in this chapter were developed in various di rections while the other parts of these lectures were published. However, this final chapter was not completed until a collaboration with Peter Nor man beginning in 1988 infused new life into the project. At the same time, the interest of string oriented theoretical physicists in theta functions gave extra impetus to completing these notes. We are pleased that this joint eflFort has now made it possible to publish this volume. The idea behind this chapter was to bring together and clarify the interrelations between three ways of viewing theta functions: a) as classical holomorphic functions in the vector z and/or the period matrix T, b) as matrix coefficients of a representation of the Heisenberg and/or metaplectic groups, c) as sections of line bundles on abelian varieties and/or the moduli space of abelian varieties. Although equivalent on a deep level, superficially these three points of view look totally different and require quite different vocabularies.