Introduction. Let g be a positive integer and let T be a complex symmetric matrix of degree g with a positive-definite imaginary part. The set of such matrices forms a convex open subset of the 1 * g (g + 1) -dimensional complex vector space and (as a complex manifold) it is called the Siegel upper-half plane of degree (or "genus") g. A standard notation for this is Sg. Observe that 25, is the upper-half of the ordinary complex plane. We know that the group Sp (g, R) operates transitively on (Big as