志村曲線上のモジュラ形式と特殊サイクル Modular Forms and Special Cycles on Shimura Curves (Annals of Mathematics Studies)

Kudla, Stephen S./ Rapoport, Michael/ Yang, Tonghai

Princeton University Press（2006/04発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 392 p.
• 言語 ENG
• 商品コード 9780691125510
• DDC分類 512.7

基本説明

A thorough study of the generating functions constructed from special cycles, on the arithmetic surface μ attached to a Shimura curve Μ over the field of rational numbers.

Full Description

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-Soule 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.

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