Modular Forms and Special Cycles on Shimura Curves (Annals of Mathematics Studies)

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.

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