Screen LibraryModular forms and special cycles on shimura curves. (am-161) /
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 fu...
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| Main Authors: | |
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| Corporate Authors: | |
| Group Author: | ; |
| Published: |
Princeton University Press,
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| Publisher Address: | Princeton, N.J. : |
| Publication Dates: |
[2006] ©2006 |
| Literature type: | eBook |
| Language: | English |
| Edition: | Course Book. |
| Series: |
Annals of mathematics studies;
161 |
| Subjects: | |
| Online Access: |
http://dx.doi.org/10.1515/9781400837168 http://www.degruyter.com/doc/cover/9781400837168.jpg |
| Summary: |
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-Soul 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. |
| Carrier Form: | 1 online resource (384 pages) : illustrations. |
| ISBN: | 9781400837168 |
| Index Number: | QA242 |
| CLC: | O187 |
| Contents: |
Frontmatter -- 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 -- Chapter 7. An inner product formula -- Chapter 8. On the doubling integral -- Chapter 9. Central derivatives of L-functions -- Index. |