Published by Princeton University Press 2006

Modular Forms and Special Cycles on Shimura Curves

Stephen S. Kudla , Michael Rapoport and Tonghai Yang

Volume 161 in the series Annals of Mathematics Studies

About this book

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.

Author / Editor information

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.

Reviews

"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

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	v
Acknowledgments Requires Authentication Unlicensed Licensed Download PDF	ix
Chapter 1. Introduction Requires Authentication Unlicensed Licensed Download PDF	1
Chapter 2. Arithmetic intersection theory on stacks Requires Authentication Unlicensed Licensed Download PDF	27
Chapter 3. Cycles on Shimura curves Requires Authentication Unlicensed Licensed Download PDF	45
Chapter 4. An arithmetic theta function Requires Authentication Unlicensed Licensed Download PDF	71
Chapter 5. The central derivative of a genus two Eisenstein series Requires Authentication Unlicensed Licensed Download PDF	105
Chapter 6. The generating function for 0-cycles Requires Authentication Unlicensed Licensed Download PDF	167
Chapter 6 Appendix Requires Authentication Unlicensed Licensed Download PDF	181
Chapter 7. An inner product formula Requires Authentication Unlicensed Licensed Download PDF	205
Chapter 8. On the doubling integral Requires Authentication Unlicensed Licensed Download PDF	265
Chapter 9. Central derivatives of L-functions Requires Authentication Unlicensed Licensed Download PDF	351
Index Requires Authentication Unlicensed Licensed Download PDF	371

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