Modular Forms and Special Cycles on Shimura Curves

Erscheinungsjahr 2006
ISBN: 978-0-691-12551-0
Verlag: Princeton University Press

Buch, Englisch, 384 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 588 g

Reihe: 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.

Autoren/Hrsg.

Kudla, Stephen S.

Rapoport, Michael

Yang, Tonghai

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Mathematische Analysis Harmonische Analysis, Fourier-Mathematik
• Mathematik | Informatik Mathematik Mathematische Analysis Moderne Anwendungen der Analysis

Inhaltsverzeichnis

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

Über Autor(innen)

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