{ "item_title" : "Modular Forms and Special Cycles on Shimura Curves", "item_author" : [" Stephen S. Kudla", "Michael Rapoport", "Tonghai Yang "], "item_description" : "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-Soularithmetic 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.", "item_img_path" : "https://covers3.booksamillion.com/covers/bam/0/69/112/551/0691125511_b.jpg", "price_data" : { "retail_price" : "115.00", "online_price" : "115.00", "our_price" : "115.00", "club_price" : "115.00", "savings_pct" : "0", "savings_amt" : "0.00", "club_savings_pct" : "0", "club_savings_amt" : "0.00", "discount_pct" : "10", "store_price" : "" } }

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