Modular Forms and Special Cycles on Shimura Curves
Seiten
2006
Princeton University Press (Verlag)
978-0-691-12550-3 (ISBN)
Princeton University Press (Verlag)
978-0-691-12550-3 (ISBN)
- Titel ist leider vergriffen;
keine Neuauflage - Artikel merken
Presents a 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.
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.
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.
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.
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
| Erscheint lt. Verlag | 24.4.2006 |
|---|---|
| Reihe/Serie | Annals of Mathematics Studies |
| Zusatzinfo | 1 line illus. 3 tables. |
| Verlagsort | New Jersey |
| Sprache | englisch |
| Maße | 152 x 235 mm |
| Gewicht | 652 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie | |
| ISBN-10 | 0-691-12550-3 / 0691125503 |
| ISBN-13 | 978-0-691-12550-3 / 9780691125503 |
| Zustand | Neuware |
| Haben Sie eine Frage zum Produkt? |
Mehr entdecken
aus dem Bereich
aus dem Bereich
Buch | Softcover (2024)
De Gruyter Oldenbourg (Verlag)
CHF 83,90
