bokomslag Modular Forms and Special Cycles on Shimura Curves
Vetenskap & teknik

Modular Forms and Special Cycles on Shimura Curves

Stephen S Kudla Michael Rapoport Tonghai Yang

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  • 392 sidor
  • 2006
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-Soule 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.
  • Författare: Stephen S Kudla, Michael Rapoport, Tonghai Yang
  • Format: Pocket/Paperback
  • ISBN: 9780691125510
  • Språk: Engelska
  • Antal sidor: 392
  • Utgivningsdatum: 2006-04-01
  • Förlag: Princeton University Press