Modular forms and special cycles on Shimura curves /

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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Bibliographic Details
Main Author: Kudla, Stephen S., 1950-
Other Authors: Rapoport, M., 1948-, Yang, Tonghai, 1963-
Format: Electronic eBook
Language:English
Published: Princeton : Princeton University Press, 2006.
Series:Annals of mathematics studies ; no. 161.
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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-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
Item Description:EBSCO eBook Academic Comprehensive Collection North America
Books at JSTOR Evidence Based Acquisitions
Physical Description:1 online resource (vii, 373 pages)
Bibliography:Includes bibliographical references and index.
ISBN:9781400837168
1400837162
1299401023
9781299401020