Modular Forms and Special Cycles on Shimura Curves

Stephen S. Kudla, Michael Rapoport, Tonghai Yang

• Publisher: Princeton University Press
• Year: 2006
• Print ISBN: 9780691125510, 9780691125503
• eBook ISBN: 9781400837168
• Topic: Mathematics
• Subtopic: Functional Analysis
• Index: Mathematics

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.

Table of contents

• Cover
• Title
• Copyright
• Contents
• Acknowledgments
• Chapter 1. Introduction
• Chapter 2. Arithmetic intersection theory on stacks
• Chapter 3. Cycles on Shimura curves
• Chapter 4. An arithmetic theta function
• Chapter 5. The central derivative of a genus two Eisenstein series
• Chapter 6. The generating function for 0-cycles
• Chapter 6 Appendix. The case p = 2, p | D(B)
• Chapter 7. An inner product formula
• Chapter 8. On the doubling integral
• Chapter 9. Central derivatives of L-functions
• Index