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Modular Forms and Special Cycles on Shimura Curves (Annals of Mathematics Studies)by Stephen S. Kudla
Synopses & ReviewsPublisher Comments: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. Synopsis: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.
Synopsis: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.
About the AuthorStephen S. Kudla is at the University of Maryland. Michael Rapoport is at the Mathematisches Institut der Universitat, Bonn, Germany. Tonghai Yang is at the University of Wisconsin, Madison.
Table of ContentsAcknowledgments ix
Chapter 1. Introduction 1 Bibliography 21 Chapter 2. Arithmetic intersection theory on stacks 27 Chapter 3. Cycles on Shimura curves 45 Chapter 4. An arithmetic theta function 71 Chapter 5. The central derivative of a genus two Eisenstein series 105 Chapter 6. The generating function for 0cycles 167 Chapter 6 Appendix. The case p = 2, p  D (B) 181 Chapter 7. An inner product formula 205 Chapter 8. On the doubling integral 265 Chapter 9. Central derivatives of Lfunctions 351 Index 371 What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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