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Modular Forms and Special Cycles on Shimura Curves. (Am-161) by Stephen S. Kudla

Modular Forms and Special Cycles on Shimura Curves. (Am-161)

by Stephen S. Kudla
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Book cover type: Paperback
  • ISBN13: 9780691125510
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Princeton University Press
  • Publisher Imprint: Princeton University Press
  • Publication Date:
  • Pages: 373
  • Original Price: USD 115.0
  • Language: English
  • Edition: N/A
  • Item Weight: 535 grams
  • BISAC Subject(s): Geometry / Algebraic and Functional Analysis

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.

Stephen S. Kudla is at the University of Maryland. Michael Rapoport is at the Mathematisches Institut der Universität, Bonn, Germany. Tonghai Yang is at the University of Wisconsin, Madison.

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