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Algebraic geometry --- Number theory --- Arithmetical algebraic geometry. --- Hilbert modular surfaces. --- Forms, Modular. --- Moduli theory. --- Arithmetical algebraic geometry --- Forms, Modular --- Hilbert modular surfaces --- Moduli theory --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Modular surfaces, Hilbert --- Surfaces --- Modular forms --- Forms (Mathematics) --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine
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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.
Forms, Modular. --- Hilbert modular surfaces. --- Moduli theory. --- Hilbert modular surfaces --- Forms, Modular --- Moduli theory --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Algebra --- Theory of moduli --- Modular surfaces, Hilbert --- Modular forms --- Mathematics. --- Algebraic geometry. --- Number theory. --- Number Theory. --- Algebraic Geometry. --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Surfaces --- Forms (Mathematics) --- Geometry, algebraic. --- Algebraic geometry --- Number study --- Numbers, Theory of
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The notes in this volume correspond to advanced courses held at the Centre de Recerca Matemàtica as part of the research program in Arithmetic Geometry in the 2009-2010 academic year. The notes by Laurent Berger provide an introduction to p-adic Galois representations and Fontaine rings, which are especially useful for describing many local deformation rings at p that arise naturally in Galois deformation theory. The notes by Gebhard Böckle offer a comprehensive course on Galois deformation theory, starting from the foundational results of Mazur and discussing in detail the theory of pseudo-representations and their deformations, local deformations at places l ≠ p and local deformations at p which are flat. In the last section,the results of Böckle and Kisin on presentations of global deformation rings over local ones are discussed. The notes by Mladen Dimitrov present the basics of the arithmetic theory of Hilbert modular forms and varieties, with an emphasis on the study of the images of the attached Galois representations, on modularity lifting theorems over totally real number fields, and on the cohomology of Hilbert modular varieties with integral coefficients. The notes by Lassina Dembélé and John Voight describe methods for performing explicit computations in spaces of Hilbert modular forms. These methods depend on the Jacquet-Langlands correspondence and on computations in spaces of quaternionic modular forms, both for the case of definite and indefinite quaternion algebras. Several examples are given, and applications to modularity of Galois representations are discussed. The notes by Tim Dokchitser describe the proof, obtained by the author in a joint project with Vladimir Dokchitser, of the parity conjecture for elliptic curves over number fields under the assumption of finiteness of the Tate-Shafarevich group. The statement of the Birch and Swinnerton-Dyer conjecture is included, as well as a detailed study of local and global root numbers of elliptic curves and their classification.
Galois theory --- Curves, Elliptic --- Hilbert modular surfaces --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Modular surfaces, Hilbert --- Elliptic curves --- Mathematics. --- Algebra. --- Algebraic geometry. --- Number theory. --- Number Theory. --- Algebraic Geometry. --- Galois theory. --- Curves, Elliptic. --- Hilbert modular surfaces. --- Forms, Modular --- Surfaces --- Equations, Theory of --- Group theory --- Number theory --- Curves, Algebraic --- Geometry, algebraic. --- Mathematical analysis --- Algebraic geometry --- Geometry --- Number study --- Numbers, Theory of
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This volume contains original research articles, survey articles and lecture notes related to the Computations with Modular Forms 2011 Summer School and Conference, held at the University of Heidelberg. A key theme of the Conference and Summer School was the interplay between theory, algorithms and experiment. The 14 papers offer readers both, instructional courses on the latest algorithms for computing modular and automorphic forms, as well as original research articles reporting on the latest developments in the field. The three Summer School lectures provide an introduction to modern algorithms together with some theoretical background for computations of and with modular forms, including computing cohomology of arithmetic groups, algebraic automorphic forms, and overconvergent modular symbols. The 11 Conference papers cover a wide range of themes related to computations with modular forms, including lattice methods for algebraic modular forms on classical groups, a generalization of the Maeda conjecture, an efficient algorithm for special values of p-adic Rankin triple product L-functions, arithmetic aspects and experimental data of Bianchi groups, a theoretical study of the real Jacobian of modular curves, results on computing weight one modular forms, and more.
Forms, Modular -- Congresses. --- Hilbert modular surfaces -- Congresses. --- Mathematics. --- Forms, Modular --- Modular forms --- Algebra. --- Algebraic geometry. --- Algorithms. --- Number theory. --- Number Theory. --- Algebraic Geometry. --- Forms (Mathematics) --- Geometry, algebraic. --- Algebraic geometry --- Geometry --- Mathematics --- Mathematical analysis --- Algorism --- Algebra --- Arithmetic --- Number study --- Numbers, Theory of --- Foundations
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These notes introduce a new class of algebraic curves on Hilbert modular surfaces. These curves are called twisted Teichmüller curves, because their construction is very reminiscent of Hirzebruch-Zagier cycles. These new objects are analyzed in detail and their main properties are described. In particular, the volume of twisted Teichmüller curves is calculated and their components are partially classified. The study of algebraic curves on Hilbert modular surfaces has been widely covered in the literature due to their arithmetic importance. Among these, twisted diagonals (Hirzebruch-Zagier cycles) are some of the most important examples.
Mathematics. --- Geometry, algebraic. --- Differentiable dynamical systems. --- Number theory. --- Algebraic Geometry. --- Number Theory. --- Dynamical Systems and Ergodic Theory. --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Number study --- Numbers, Theory of --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Algebraic geometry --- Math --- Algebraic geometry. --- Dynamics. --- Ergodic theory. --- Teichmüller spaces. --- Curves, Algebraic. --- Hilbert modular surfaces. --- Modular surfaces, Hilbert --- Forms, Modular --- Surfaces --- Algebraic curves --- Algebraic varieties --- Spaces, Teichmüller --- Functions of several complex variables --- Riemann surfaces --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Algebra --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics
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