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Intersections Of Arithmetic Hirzebruch Zagier Cycles
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Book Synopsis Intersections of Hirzebruch–Zagier Divisors and CM Cycles by : Benjamin Howard
Download or read book Intersections of Hirzebruch–Zagier Divisors and CM Cycles written by Benjamin Howard and published by Springer Science & Business Media. This book was released on 2012-01-06 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph treats one case of a series of conjectures by S. Kudla, whose goal is to show that Fourier of Eisenstein series encode information about the Arakelov intersection theory of special cycles on Shimura varieties of orthogonal and unitary type. Here, the Eisenstein series is a Hilbert modular form of weight one over a real quadratic field, the Shimura variety is a classical Hilbert modular surface, and the special cycles are complex multiplication points and the Hirzebruch-Zagier divisors. By developing new techniques in deformation theory, the authors successfully compute the Arakelov intersection multiplicities of these divisors, and show that they agree with the Fourier coefficients of derivatives of Eisenstein series.
Book Synopsis Modular Forms and Special Cycles on Shimura Curves. (AM-161) by : Stephen S. Kudla
Download or read book Modular Forms and Special Cycles on Shimura Curves. (AM-161) written by Stephen S. Kudla and published by Princeton University Press. This book was released on 2006-04-24 with total page 387 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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-Soulé 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.
Book Synopsis Heegner Points and Rankin L-Series by : Henri Darmon
Download or read book Heegner Points and Rankin L-Series written by Henri Darmon and published by Cambridge University Press. This book was released on 2004-06-21 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: Thirteen articles by leading contributors on the history of the Gross-Zagier formula and its developments.
Book Synopsis The Gross-Zagier Formula on Shimura Curves by : Xinyi Yuan
Download or read book The Gross-Zagier Formula on Shimura Curves written by Xinyi Yuan and published by Princeton University Press. This book was released on 2013 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations. The book begins with a conceptual formulation of the Gross-Zagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of L-series by means of Weil representations. The Gross-Zagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the Gross-Zagier formula is reduced to local formulas. The Gross-Zagier Formula on Shimura Curves will be of great use to students wishing to enter this area and to those already working in it.
Book Synopsis WIN -- Women in Numbers by : Alina Carmen Cojocaru
Download or read book WIN -- Women in Numbers written by Alina Carmen Cojocaru and published by American Mathematical Soc.. This book was released on 2011 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a collection of papers on number theory which evolved out of the workshop WIN-Women In Numbers, held November 2-7, 2008. It includes articles showcasing outcomes from collaborative research initiated during the workshop as well as survey papers aimed at introducing graduate students and recent PhDs to important research topics in number theory.
Book Synopsis Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change by : Jayce Getz
Download or read book Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change written by Jayce Getz and published by Springer Science & Business Media. This book was released on 2012-03-28 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the 1970s Hirzebruch and Zagier produced elliptic modular forms with coefficients in the homology of a Hilbert modular surface. They then computed the Fourier coefficients of these forms in terms of period integrals and L-functions. In this book the authors take an alternate approach to these theorems and generalize them to the setting of Hilbert modular varieties of arbitrary dimension. The approach is conceptual and uses tools that were not available to Hirzebruch and Zagier, including intersection homology theory, properties of modular cycles, and base change. Automorphic vector bundles, Hecke operators and Fourier coefficients of modular forms are presented both in the classical and adèlic settings. The book should provide a foundation for approaching similar questions for other locally symmetric spaces.
Book Synopsis Mathematisches Institut Georg-august-universität Göttingen, Seminars Summer 2003/2004 by : Yuri Tschinkel
Download or read book Mathematisches Institut Georg-august-universität Göttingen, Seminars Summer 2003/2004 written by Yuri Tschinkel and published by Universitätsverlag Göttingen. This book was released on 2004 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains lecture notes from the seminars [alpha]Number Theory", [alpha]Algebraic Geometry" and [alpha]Geometric methods in representation theory" which took place at the Mathematics Institute of the University of Göttingen during the Winter Term 2003-2004. Most contributions report on recent work by the authors.
Book Synopsis Argos Seminar on Intersections of Modular Correspondences by : Société mathématique de France
Download or read book Argos Seminar on Intersections of Modular Correspondences written by Société mathématique de France and published by . This book was released on 2007 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the written account of the Bonn Seminar on Arithmetic Geometry 2003/2004. It gives a coherent exposition of the theory of intersections of modular correspondences. The focus of the seminar is the formula for the intersection number of arithmetic modular correspondences due to Gross and Keating. Other topics treated are Hurwitz's theorem on the intersection of modular correspondences over the field of complex numbers and the relation of the arithmetic intersection numbers to Fourier coefficients of Siegel-Eisenstein series. Also included is background material on one-dimensional formal groups and their endomorphisms and on quadratic forms over the ring of $p$-adic integers.
Book Synopsis Complex Analysis of Several Variables by : Yum-Tong Siu
Download or read book Complex Analysis of Several Variables written by Yum-Tong Siu and published by American Mathematical Soc.. This book was released on 1984 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents a collection of papers from the Symposium on Several Complex Variables held April 12-15, 1983 in Madison, Wisconsin. This book contains a selection of the presented papers as well as some contributed papers.
Book Synopsis The 1-2-3 of Modular Forms by : Jan Hendrik Bruinier
Download or read book The 1-2-3 of Modular Forms written by Jan Hendrik Bruinier and published by Springer Science & Business Media. This book was released on 2008-02-10 with total page 273 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book grew out of three series of lectures given at the summer school on "Modular Forms and their Applications" at the Sophus Lie Conference Center in Nordfjordeid in June 2004. The first series treats the classical one-variable theory of elliptic modular forms. The second series presents the theory of Hilbert modular forms in two variables and Hilbert modular surfaces. The third series gives an introduction to Siegel modular forms and discusses a conjecture by Harder. It also contains Harder's original manuscript with the conjecture. Each part treats a number of beautiful applications.
Book Synopsis Bulletin of the American Mathematical Society by :
Download or read book Bulletin of the American Mathematical Society written by and published by . This book was released on 1986 with total page 654 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Bulletin (new Series) of the American Mathematical Society by :
Download or read book Bulletin (new Series) of the American Mathematical Society written by and published by . This book was released on 1986 with total page 648 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Geometric and Arithmetic Volume of Shimura Varieties of Orthogonal Type by : Fritz Hörmann
Download or read book The Geometric and Arithmetic Volume of Shimura Varieties of Orthogonal Type written by Fritz Hörmann and published by American Mathematical Society. This book was released on 2014-11-05 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book outlines a functorial theory of integral models of (mixed) Shimura varieties and of their toroidal compactifications, for odd primes of good reduction. This is the integral version, developed in the author's thesis, of the theory invented by Deligne and Pink in the rational case. In addition, the author develops a theory of arithmetic Chern classes of integral automorphic vector bundles with singular metrics using the work of Burgos, Kramer and Kühn. The main application is calculating arithmetic volumes or "heights" of Shimura varieties of orthogonal type using Borcherds' famous modular forms with their striking product formula--an idea due to Bruinier-Burgos-Kühn and Kudla. This should be seen as an Arakelov analogue of the classical calculation of volumes of orthogonal locally symmetric spaces by Siegel and Weil. In the latter theory, the volumes are related to special values of (normalized) Siegel Eisenstein series. In this book, it is proved that the Arakelov analogues are related to special derivatives of such Eisenstein series. This result gives substantial evidence in the direction of Kudla's conjectures in arbitrary dimensions. The validity of the full set of conjectures of Kudla, in turn, would give a conceptual proof and far-reaching generalizations of the work of Gross and Zagier on the Birch and Swinnerton-Dyer conjecture. Titles in this series are co-published with the Centre de Recherches Mathématiques.
Book Synopsis Current Developments in Mathematics by :
Download or read book Current Developments in Mathematics written by and published by . This book was released on 2002 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Algebraic Cycles and Motives: Volume 1 by : Jan Nagel
Download or read book Algebraic Cycles and Motives: Volume 1 written by Jan Nagel and published by Cambridge University Press. This book was released on 2007-05-03 with total page 293 pages. Available in PDF, EPUB and Kindle. Book excerpt: This 2007 book is a self-contained account of the subject of algebraic cycles and motives.
Book Synopsis Analytic Number Theory by : William Duke
Download or read book Analytic Number Theory written by William Duke and published by American Mathematical Soc.. This book was released on 2007 with total page 270 pages. Available in PDF, EPUB and Kindle. Book excerpt: Articles in this volume are based on talks given at the Gauss-Dirichlet Conference held in Gottingen on June 20-24, 2005. The conference commemorated the 150th anniversary of the death of C.-F. Gauss and the 200th anniversary of the birth of J.-L. Dirichlet. The volume begins with a definitive summary of the life and work of Dirichlet and continues with thirteen papers by leading experts on research topics of current interest in number theory that were directly influenced by Gauss and Dirichlet. Among the topics are the distribution of primes (long arithmetic progressions of primes and small gaps between primes), class groups of binary quadratic forms, various aspects of the theory of $L$-functions, the theory of modular forms, and the study of rational and integral solutions to polynomial equations in several variables. Information for our distributors: Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).
Book Synopsis Higher Ramanujan Equations and Periods of Abelian Varieties by : Tiago J. Fonseca
Download or read book Higher Ramanujan Equations and Periods of Abelian Varieties written by Tiago J. Fonseca and published by American Mathematical Society. This book was released on 2023-01-18 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
