


Modular Forms and Special Cycles on Shimura Curves. (AM161) 
Modular Forms and Special Cycles on Shimura Curves is a thorough study of the generating functions constructed from special cycles, both divisors and zerocycles, 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 qexpansions of modular forms and Siegel modular forms of genus two respectively, valued in the GilletSoulé 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 SiegelWeil formula identifies the generating function for zerocycles as the central derivative of a Siegel Eisenstein series. As an application, an arithmetic analogue of the ShimuraWaldspurger correspondence is constructed, carrying holomorphic cusp forms of weight 3/2 to classes in the MordellWeil group of "M". In certain cases, the nonvanishing of this correspondence is related to the central derivative of the standard Lfunction 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 pdivisible groups, padic uniformization, the Weil representation, the local and global theta correspondence, and the doubling integral representation of Lfunctions. "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 Reviews Acknowledgments ix Another Princeton book authored or coauthored by Michael Rapoport: Series:
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