**Memoirs of the American Mathematical Society**

2014;
145 pp;
Softcover

MSC: Primary 14; 22; 32;

Print ISBN: 978-0-8218-9857-4

Product Code: MEMO/231/1088

List Price: $79.00

AMS Member Price: $47.40

MAA Member Price: $71.10

**Electronic ISBN: 978-1-4704-1724-6
Product Code: MEMO/231/1088.E**

List Price: $79.00

AMS Member Price: $47.40

MAA Member Price: $71.10

# Special Values of Automorphic Cohomology Classes

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*Mark Green; Phillip Griffiths; Matt Kerr*

The authors study the complex geometry and coherent cohomology of nonclassical Mumford-Tate domains and their quotients by discrete groups. Their focus throughout is on the domains \(D\) which occur as open \(G(\mathbb{R})\)-orbits in the flag varieties for \(G=SU(2,1)\) and \(Sp(4)\), regarded as classifying spaces for Hodge structures of weight three. In the context provided by these basic examples, the authors formulate and illustrate the general method by which correspondence spaces \(\mathcal{W}\) give rise to Penrose transforms between the cohomologies \(H^{q}(D,L)\) of distinct such orbits with coefficients in homogeneous line bundles.

#### Table of Contents

# Table of Contents

## Special Values of Automorphic Cohomology Classes

- Cover Cover11 free
- Title page i2 free
- Introduction 18 free
- Chapter I. Geometry of the Mumford-Tate domains 1724
- Chapter II. Homogeneous line bundles over the Mumford-Tate domains 3744
- Chapter III. Correspondence and cycle spaces; Penrose transforms 4956
- III.A. Introduction 4956
- III.B. Basic definitions and examples 4956
- III.C. The basic example 5966
- III.D. The Penrose transform in the compact case 6875
- Appendix to section III.D: Arithmetic aspects of the Penrose transform in the compact case 7380
- III.E. The Penrose transform in the first example 7784
- III.F. The Penrose transform in the second example 8693

- Chapter IV. The Penrose transform in the automorphic case and the main result 93100
- IV.A. Cuspidal automorphic cohomology 93100
- IV.B. Picard and Siegel cuspidal automorphic forms 98105
- IV.C. Arithmetic structures on vector spaces 102109
- Appendix to section IV.C: Explicit canonical models for the two examples 108115
- IV.D. Special values of cuspidal automorphic cohomology classes 110117
- Appendix to section IV.D: An alternate method for evaluating cohomology classes and a question 113120
- IV.E. CM points on correspondence spaces 117124
- IV.F. On a result of Carayol 123130
- Appendix to section IV.F: Geometric construction of 𝐾-types and discussion of totally degenerate limits of discrete series 130137

- Bibliography 143150
- Back Cover Back Cover1158