**CBMS Regional Conference Series in Mathematics**

Volume: 34;
1977;
96 pp;
Softcover

MSC: Primary 14;
Secondary 22; 33; 42

Print ISBN: 978-0-8218-1684-4

Product Code: CBMS/34

List Price: $25.00

Individual Price: $20.00

**Electronic ISBN: 978-1-4704-2394-0
Product Code: CBMS/34.E**

List Price: $25.00

Individual Price: $20.00

# Lecture Notes on Nil-Theta Functions

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*L. Auslander*

A co-publication of the AMS and CBMS

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.

#### Readership

#### Table of Contents

# Table of Contents

## Lecture Notes on Nil-Theta Functions

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Table of Contents v6 free
- I. Foundations 110 free
- 1. Bilinear Forms and Presentations of Certain 2-Step Nilpotent Lie Groups 110
- 2. Discrete Subgroups of the Heisenberg Group 615
- 3. The Automorphism Group of the Heisenberg Group 1423
- 4. Fundamental Unitary Representations of the Heisenberg Group 1726
- 5. The Fourier Transform and the Weil-Brezin Map 2231
- 6. Distinguished Subspaces and Left Action 3443

- II. Jacobi Theta Functions and the Finite Fourier Transform 4857
- III. Abelian Varieties, Nil-Theta and Theta Functions 6978
- Bibliography 95104
- Back Cover Back Cover1106