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Arakelov theory --- 512.7 --- Arakelov geometry --- Arithmetical algebraic geometry --- Algebraic geometry. Commutative rings and algebras --- Arakelov theory. --- 512.7 Algebraic geometry. Commutative rings and algebras

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Ordered algebraic structures --- 517.544 --- Arakelov theory --- Capacity theory (Mathematics) --- Capacity of a set --- Pluripotential theory --- Arakelov geometry --- Arithmetical algebraic geometry --- Boundary properties of analytic functions. Boundary value problems --- 517.544 Boundary properties of analytic functions. Boundary value problems --- Arakelov theory. --- Arakelov, Théorie d' --- Arakelov, Théorie d'.

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The purpose of this book is to build the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for research on arithmetic geometry in several directions. By adelic curve is meant a field equipped with a family of absolute values parametrized by a measure space, such that the logarithmic absolute value of each non-zero element of the field is an integrable function on the measure space. In the literature, such construction has been discussed in various settings which are apparently transversal to each other. The authors first formalize the notion of adelic curves and discuss in a systematic way its algebraic covers, which are important in the study of height theory of algebraic points beyond Weil–Lang’s height theory. They then establish a theory of adelic vector bundles on adelic curves, which considerably generalizes the classic geometry of vector bundles or that of Hermitian vector bundles over an arithmetic curve. They focus on an analogue of the slope theory in the setting of adelic curves and in particular estimate the minimal slope of tensor product adelic vector bundles. Finally, by using the adelic vector bundles as a tool, a birational Arakelov geometry for projective variety over an adelic curve is developed. As an application, a vast generalization of Nakai–Moishezon’s criterion of positivity is proven in clarifying the arguments of geometric nature from several fundamental results in the classic geometry of numbers. Assuming basic knowledge of algebraic geometry and algebraic number theory, the book is almost self-contained. It is suitable for researchers in arithmetic geometry as well as graduate students focusing on these topics for their doctoral theses.

Algebraic geometry. --- Commutative algebra. --- Commutative rings. --- Functional analysis. --- Algebraic Geometry. --- Commutative Rings and Algebras. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Rings (Algebra) --- Algebra --- Algebraic geometry --- Geometry --- Arakelov theory. --- Arakelov geometry --- Arithmetical algebraic geometry

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The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof. It should enable mathematicians with a background in arithmetic algebraic geometry to understand some basic techniques in the rapidly evolving field of Arakelov-theory.

Algebraic geometry --- Algebraïsche meetkunde --- Geometry [Algebraic ] --- Géométrie algébrique --- Meetkunde [Algebraïsche ] --- Riemann-Roch theorema's --- Riemann-Roch thoerems --- Theoremes de Riemann-Roch --- Geometry, Algebraic. --- Riemann-Roch theorems. --- Theorems, Riemann-Roch --- Algebraic functions --- Geometry, Algebraic --- Geometry --- Addition. --- Adjoint. --- Alexander Grothendieck. --- Algebraic geometry. --- Analytic torsion. --- Arakelov theory. --- Asymptote. --- Asymptotic expansion. --- Asymptotic formula. --- Big O notation. --- Cartesian coordinate system. --- Characteristic class. --- Chern class. --- Chow group. --- Closed immersion. --- Codimension. --- Coherent sheaf. --- Cohomology. --- Combination. --- Commutator. --- Computation. --- Covariant derivative. --- Curvature. --- Derivative. --- Determinant. --- Diagonal. --- Differentiable manifold. --- Differential form. --- Dimension (vector space). --- Divisor. --- Domain of a function. --- Dual basis. --- E6 (mathematics). --- Eigenvalues and eigenvectors. --- Embedding. --- Endomorphism. --- Exact sequence. --- Exponential function. --- Generic point. --- Heat kernel. --- Injective function. --- Intersection theory. --- K-group. --- Levi-Civita connection. --- Line bundle. --- Linear algebra. --- Local coordinates. --- Mathematical induction. --- Morphism. --- Natural number. --- Neighbourhood (mathematics). --- Parameter. --- Projective space. --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Riemannian manifold. --- Riemann–Roch theorem. --- Self-adjoint operator. --- Smoothness. --- Sobolev space. --- Stochastic calculus. --- Summation. --- Supertrace. --- Theorem. --- Transition function. --- Upper half-plane. --- Vector bundle. --- Volume form.

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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.

Arithmetical algebraic geometry. --- Shimura varieties. --- Varieties, Shimura --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Arithmetical algebraic geometry --- Number theory --- Abelian group. --- Addition. --- Adjunction formula. --- Algebraic number theory. --- Arakelov theory. --- Arithmetic. --- Automorphism. --- Bijection. --- Borel subgroup. --- Calculation. --- Chow group. --- Coefficient. --- Cohomology. --- Combinatorics. --- Compact Riemann surface. --- Complex multiplication. --- Complex number. --- Cup product. --- Deformation theory. --- Derivative. --- Dimension. --- Disjoint union. --- Divisor. --- Dual pair. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Elliptic curve. --- Endomorphism. --- Equation. --- Explicit formulae (L-function). --- Fields Institute. --- Formal group. --- Fourier series. --- Fundamental matrix (linear differential equation). --- Galois group. --- Generating function. --- Green's function. --- Group action. --- Induced representation. --- Intersection (set theory). --- Intersection number. --- Irreducible component. --- Isomorphism class. --- L-function. --- Laurent series. --- Level structure. --- Line bundle. --- Local ring. --- Mathematical sciences. --- Mathematics. --- Metaplectic group. --- Modular curve. --- Modular form. --- Modularity (networks). --- Moduli space. --- Multiple integral. --- Number theory. --- Numerical integration. --- Orbifold. --- Orthogonal complement. --- P-adic number. --- Pairing. --- Prime factor. --- Prime number. --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Quadratic form. --- Quadratic residue. --- Quantity. --- Quaternion algebra. --- Quaternion. --- Quotient stack. --- Rational number. --- Real number. --- Residue field. --- Riemann zeta function. --- Ring of integers. --- SL2(R). --- Scientific notation. --- Shimura variety. --- Siegel Eisenstein series. --- Siegel modular form. --- Special case. --- Standard L-function. --- Subgroup. --- Subset. --- Summation. --- Tensor product. --- Test vector. --- Theorem. --- Three-dimensional space (mathematics). --- Topology. --- Trace (linear algebra). --- Triangular matrix. --- Two-dimensional space. --- Uniformization. --- Valuative criterion. --- Whittaker function.