{"product_id":"9780691125510-modular-forms-and-special-cycles-on-shimura-curves","title":"Modular Forms and Special Cycles on Shimura Curves","description":"\u003cmeta content=\"text\/html; charset=utf-8\" http-equiv=\"Content-Type\"\u003e\u003cp\u003e\u003cspan\u003eA 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.\u003cbr\u003eModular 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.\u003cbr\u003e\u003cbr\u003e\u003c\/span\u003e\u003c\/p\u003e","brand":"Rarewaves","offers":[{"title":"Default Title","offer_id":56581622301046,"sku":"9780691125510","price":97.69,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0092\/7504\/8033\/files\/orig_27114718.jpg?v=1762172217","url":"https:\/\/www.rarewaves.com\/products\/9780691125510-modular-forms-and-special-cycles-on-shimura-curves","provider":"Rarewaves.com","version":"1.0","type":"link"}