TY - BOOK UR - http://lib.ugent.be/catalog/ebk01:2560000000080611 ID - ebk01:2560000000080611 ET - Course Book. LA - eng TI - Modular Forms and Special Cycles on Shimura Curves. (AM-161) PY - 2006 SN - 9781400837168 AU - Kudla, Stephen S., author. AU - Rapoport, Michael, author. AU - Yang, Tonghai, author. AB - 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. ER -Download RIS file
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020 | a 9781400837168 | ||
024 | 7 | a 10.1515/9781400837168 2 doi | |
035 | a (DE-B1597)446524 | ||
035 | a (OCoLC)803434031 | ||
040 | a IN-ChSCO b eng c IN-ChSCO e rda | ||
041 | a eng | ||
050 | 4 | a QA242.5 | |
050 | 4 | a QA242.5 b .K83 2006eb | |
072 | 7 | a MAT x 037000 2 bisacsh | |
072 | 7 | a MAT037000 2 bisacsh | |
082 | 4 | a 516.3/5 2 22 | |
082 | 4 | a 516.3/5 2 23 | |
100 | 1 | a Kudla, Stephen S., e author. | |
245 | 1 | a Modular Forms and Special Cycles on Shimura Curves. (AM-161) / c Stephen S. Kudla, Michael Rapoport, Tonghai Yang. | |
250 | a Course Book. | ||
264 | 1 | a Princeton, N.J. : b Princeton University Press, c [2006]. | |
264 | 4 | c ©2006. | |
300 | a 1 online resource (384 pages) : b illustrations. | ||
336 | a text 2 rdacontent | ||
337 | a computer 2 rdamedia | ||
338 | a online resource 2 rdacarrier | ||
347 | a text file b PDF 2 rda | ||
490 | a Annals of Mathematics Studies, x 0066-2313 ; v 161 | ||
505 | t Frontmatter -- t Contents -- t Acknowledgments -- t Chapter 1. Introduction -- t Chapter 2. Arithmetic intersection theory on stacks -- t Chapter 3. Cycles on Shimura curves -- t Chapter 4. An arithmetic theta function -- t Chapter 5. The central derivative of a genus two Eisenstein series -- t Chapter 6. The generating function for 0-cycles -- t Chapter 6 Appendix -- t Chapter 7. An inner product formula -- t Chapter 8. On the doubling integral -- t Chapter 9. Central derivatives of L-functions -- t Index. | ||
520 | a 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. | ||
533 | a Electronic reproduction. b Princeton, N.J. : c Princeton University Press, d 2006. n Mode of access: World Wide Web. n System requirements: Web browser. n Access may be restricted to users at subscribing institutions. | ||
538 | a Mode of access: Internet via World Wide Web. | ||
546 | a In English. | ||
588 | a Description based on online resource; title from PDF title page (publisher’s Web site, viewed October 27 2015) | ||
650 | a Arithmetical algebraic geometry. | ||
650 | a MATHEMATICS v Functional Analysis. | ||
650 | a MATHEMATICS v Geometry v Algebraic. | ||
650 | a Shimura varieties. | ||
650 | 4 | a Analysis. | |
650 | 4 | a Arithmetical algebraic geometry. | |
650 | 4 | a Arithmetische Geometrie. | |
650 | 4 | a Eisenstein-Reihe. | |
650 | 4 | a Géométrie algébrique arithmétique. | |
650 | 4 | a Mathematics. | |
650 | 4 | a Mathematik. | |
650 | 4 | a Shimura varieties. | |
650 | 4 | a Shimura, Variétés de. | |
650 | 4 | a Shimura-Kurve. | |
650 | 4 | a Thetafunktion. | |
700 | 1 | a Rapoport, Michael, e author. | |
700 | 1 | a Yang, Tonghai, e author. | |
773 | 8 | i Title is part of eBook package: d De Gruyter t Princeton Annals of Mathematics Backlist eBook Package z 978-3-11-049491-4 | |
773 | 8 | i Title is part of eBook package: d De Gruyter t Princeton eBook Package Backlist 2000-2013 z 978-3-11-044250-2 | |
773 | 8 | i Title is part of eBook package: d De Gruyter t Princeton eBook Package Backlist 2000-2014 z 978-3-11-045953-1 | |
773 | 8 | i Title is part of eBook package: d De Gruyter t Princeton Univ. Press eBook Package 2000-2013 z 978-3-11-041343-4 | |
830 | a Annals of Mathematics Studies ; v 161. | ||
856 | 4 | u https://doi.org/10.1515/9781400837168 | |
856 | 4 | 2 | 3 Cover u https://www.degruyter.com/doc/cover/9781400837168.jpg |
912 | a 978-3-11-041343-4 Princeton Univ. Press eBook Package 2000-2013 | ||
912 | a 978-3-11-044250-2 Princeton eBook Package Backlist 2000-2013 | ||
912 | a 978-3-11-045953-1 Princeton eBook Package Backlist 2000-2014 | ||
912 | a 978-3-11-049491-4 Princeton Annals of Mathematics Backlist eBook Package | ||
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