Modular Forms and Special Cycles on Shimura Curves. (AM-161) [Paperback]

Modular Forms and Special Cycles on Shimura Curvesis 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.

Stephen S. Kudlais at the University of Maryland.Michael Rapoportis at the Mathematisches Institut der Universit?t, Bonn, Germany.Tonghai Yangis at the University of Wisconsin, Madison. This book represents a major milestone for research at the intersection of arithmetic geometry and automorphic forms. The results will shape the research in this area for some time to come. ---Jens Funke,Mathematical Reviewl£.