This monograph consists of three chapters covering the following topics: Foundations, (1) Bilinear forms and presentations of certain 2-step nilpotent Lie groups, (2) Discrete subgroups of the Heisenberg group, (3) The automorphism group of the Heisenberg group, (4) Fundamental unitary representations of the Heisenberg group, (5) The Fourier transform and the Weil-Brezin map, (6) Distinguished subspaces and left action: Jacobi theta functions and the finite Fourier transform, (1) Nil-theta functions and Jacobi-theta functions, (2) The algebra of the finite Fourier transform; Abelian varieties, nil-theta and theta functions, (1) A general construction and algebraic foundations, (2) Nil-theta functions associated with a positive definite H-morphism of an Abelian variety, (3) The relation between nil-theta and classical theta functions.

In presenting the material, the author has attempted to lay a careful foundation and has stressed low-dimensional examples and special computations when proving general results by general techniques.

This an expository piece of work, although some of the results are new.

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