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Book synopsis

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.

Book details

Book category Books in English Mathematics & science Mathematics Calculus & mathematical analysis

• Full title: Modular Forms and Special Cycles on Shimura Curves. (AM-161)
• Author: Michael Rapoport, Stephen S. Kudla, Tonghai Yang
• Language: English
• Binding: Paperback
• Number of pages: 392
• EAN: 9780691125510
• ISBN: 0691125511
• ID: 04641107
• Publisher: Princeton University Press
• Weight: 608 g
• Dimensions: 235 × 152 × 23 mm
• Date of publishing: 24. April 2006

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