Modular Forms and Special Cycles on Shimura Curves. (AM-161)
Stephen S. Kudla, Michael Rapoport, & Tonghai Yang

Paper | 2006 | $49.95 / £29.95
384 pp. | 6 x 9 | 1 line illus. 3 tables.

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Chapter 1 [PDF]

Modular Forms and Special Cycles on Shimura Curves is a thorough study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "M" attached to a Shimura curve "M" over the field of rational numbers. These generating functions are shown to be the q-expansions of modular forms and Siegel modular forms of genus two respectively, valued in the Gillet-Soulé arithmetic Chow groups of "M". The two types of generating functions are related via an arithmetic inner product formula. In addition, an analogue of the classical Siegel-Weil formula identifies the generating function for zero-cycles as the central derivative of a Siegel Eisenstein series. As an application, an arithmetic analogue of the Shimura-Waldspurger correspondence is constructed, carrying holomorphic cusp forms of weight 3/2 to classes in the Mordell-Weil group of "M". In certain cases, the nonvanishing of this correspondence is related to the central derivative of the standard L-function for a modular form of weight 2. These results depend on a novel mixture of modular forms and arithmetic geometry and should provide a paradigm for further investigations. The proofs involve a wide range of techniques, including arithmetic intersection theory, the arithmetic adjunction formula, representation densities of quadratic forms, deformation theory of p-divisible groups, p-adic uniformization, the Weil representation, the local and global theta correspondence, and the doubling integral representation of L-functions.

Stephen S. Kudla is at the University of Maryland. Michael Rapoport is at the Mathematisches Institut der Universität, Bonn, Germany. Tonghai Yang is at the University of Wisconsin, Madison.

Review:

"This book represents a major milestone for research at the intersection of arithmetic geometry and automorphic forms. The results will shape the research in this area for some time to come."--Jens Funke, Mathematical Reviews

Table of Contents:

Acknowledgments ix
Chapter 1. Introduction 1
Bibliography 21
Chapter 2. Arithmetic intersection theory on stacks 27
Chapter 3. Cycles on Shimura curves 45
Chapter 4. An arithmetic theta function 71
Chapter 5. The central derivative of a genus two Eisenstein series 105
Chapter 6. The generating function for 0-cycles 167
Chapter 6 Appendix. The case p = 2, p | D (B) 181
Chapter 7. An inner product formula 205
Chapter 8. On the doubling integral 265
Chapter 9. Central derivatives of L-functions 351
Index 371

Another Princeton book by Michael Rapoport:

• Period Spaces for p-divisible Groups (AM-141).

Series:

• Annals of Mathematics Studies
  Phillip A. Griffiths, John N. Mather, and Elias M. Stein, Editors

Subject Area:

• Mathematics

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