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This book aims first to prove the local Langlands conjecture for GLn over a p-adic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the "simple" Shimura varieties. These two problems go hand in hand. The results represent a major advance in algebraic number theory, finally proving the conjecture first proposed in Langlands's 1969 Washington lecture as a non-abelian generalization of local class field theory. The local Langlands conjecture for GLn(K), where K is a p-adic field, asserts the existence of a correspondence, with certain formal properties, relating n-dimensional representations of the Galois group of K with the representation theory of the locally compact group GLn(K). This book constructs a candidate for such a local Langlands correspondence on the vanishing cycles attached to the bad reduction over the integer ring of K of a certain family of Shimura varieties. And it proves that this is roughly compatible with the global Galois correspondence realized on the cohomology of the same Shimura varieties. The local Langlands conjecture is obtained as a corollary. Certain techniques developed in this book should extend to more general Shimura varieties, providing new instances of the local Langlands conjecture. Moreover, the geometry of the special fibers is strictly analogous to that of Shimura curves and can be expected to have applications to a variety of questions in number theory.
Mathematics --- Shimura varieties. --- MATHEMATICS / Number Theory. --- Varieties, Shimura --- Arithmetical algebraic geometry --- Math --- Science --- Abelian variety. --- Absolute value. --- Algebraic group. --- Algebraically closed field. --- Artinian. --- Automorphic form. --- Base change. --- Bijection. --- Canonical map. --- Codimension. --- Coefficient. --- Cohomology. --- Compactification (mathematics). --- Conjecture. --- Corollary. --- Dimension (vector space). --- Dimension. --- Direct limit. --- Division algebra. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Embedding. --- Equivalence class. --- Equivalence of categories. --- Existence theorem. --- Field of fractions. --- Finite field. --- Function field. --- Functor. --- Galois cohomology. --- Galois group. --- Generic point. --- Geometry. --- Hasse invariant. --- Infinitesimal character. --- Integer. --- Inverse system. --- Isomorphism class. --- Lie algebra. --- Local class field theory. --- Maximal torus. --- Modular curve. --- Moduli space. --- Monic polynomial. --- P-adic number. --- Prime number. --- Profinite group. --- Residue field. --- Ring of integers. --- Separable extension. --- Sheaf (mathematics). --- Shimura variety. --- Simple group. --- Special case. --- Spectral sequence. --- Square root. --- Subset. --- Tate module. --- Theorem. --- Transcendence degree. --- Unitary group. --- Valuative criterion. --- Variable (mathematics). --- Vector space. --- Weil group. --- Weil pairing. --- Zariski topology.
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This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.
Curves, Algebraic. --- Finite fields (Algebra) --- Modular fields (Algebra) --- Algebra, Abstract --- Algebraic fields --- Galois theory --- Modules (Algebra) --- Algebraic curves --- Algebraic varieties --- Abelian group. --- Abelian variety. --- Affine plane. --- Affine space. --- Affine variety. --- Algebraic closure. --- Algebraic curve. --- Algebraic equation. --- Algebraic extension. --- Algebraic function. --- Algebraic geometry. --- Algebraic integer. --- Algebraic number field. --- Algebraic number theory. --- Algebraic number. --- Algebraic variety. --- Algebraically closed field. --- Applied mathematics. --- Automorphism. --- Birational invariant. --- Characteristic exponent. --- Classification theorem. --- Clifford's theorem. --- Combinatorics. --- Complex number. --- Computation. --- Cyclic group. --- Cyclotomic polynomial. --- Degeneracy (mathematics). --- Degenerate conic. --- Divisor (algebraic geometry). --- Divisor. --- Dual curve. --- Dual space. --- Elliptic curve. --- Equation. --- Fermat curve. --- Finite field. --- Finite geometry. --- Finite group. --- Formal power series. --- Function (mathematics). --- Function field. --- Fundamental theorem. --- Galois extension. --- Galois theory. --- Gauss map. --- General position. --- Generic point. --- Geometry. --- Homogeneous polynomial. --- Hurwitz's theorem. --- Hyperelliptic curve. --- Hyperplane. --- Identity matrix. --- Inequality (mathematics). --- Intersection number (graph theory). --- Intersection number. --- J-invariant. --- Line at infinity. --- Linear algebra. --- Linear map. --- Mathematical induction. --- Mathematics. --- Menelaus' theorem. --- Modular curve. --- Natural number. --- Number theory. --- Parity (mathematics). --- Permutation group. --- Plane curve. --- Point at infinity. --- Polar curve. --- Polygon. --- Polynomial. --- Power series. --- Prime number. --- Projective plane. --- Projective space. --- Quadratic transformation. --- Quadric. --- Resolution of singularities. --- Riemann hypothesis. --- Scalar multiplication. --- Scientific notation. --- Separable extension. --- Separable polynomial. --- Sign (mathematics). --- Singular point of a curve. --- Special case. --- Subgroup. --- Sylow theorems. --- System of linear equations. --- Tangent. --- Theorem. --- Transcendence degree. --- Upper and lower bounds. --- Valuation ring. --- Variable (mathematics). --- Vector space.
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Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections.
Tame algebras. --- Algebras, Tame --- Associative algebras --- Abhyankar property. --- Berkovich space. --- Galois orbit. --- Riemann-Roch. --- Zariski dense open set. --- Zariski open subset. --- Zariski topology. --- algebraic geometry. --- algebraic variety. --- algebraically closed valued field. --- analytic geometry. --- birational invariant. --- canonical extension. --- connectedness. --- continuity criteria. --- continuous definable map. --- continuous map. --- curve fibration. --- definable compactness. --- definable function. --- definable homotopy type. --- definable set. --- definable space. --- definable subset. --- definable topological space. --- definable topology. --- definable type. --- definably compact set. --- deformation retraction. --- finite simplicial complex. --- finite-dimensional vector space. --- forward-branching point. --- fundamental space. --- g-continuity. --- g-continuous. --- g-open set. --- germ. --- good metric. --- homotopy equivalence. --- homotopy. --- imaginary base set. --- ind-definable set. --- ind-definable subset. --- inflation homotopy. --- inflation. --- inverse limit. --- iso-definability. --- iso-definable set. --- iso-definable subset. --- iterated place. --- linear topology. --- main theorem. --- model theory. --- morphism. --- natural functor. --- non-archimedean geometry. --- non-archimedean tame topology. --- o-minimal formulation. --- o-minimality. --- orthogonality. --- path. --- pro-definable bijection. --- pro-definable map. --- pro-definable set. --- pro-definable subset. --- pseudo-Galois covering. --- real numbers. --- relatively compact set. --- residue field extension. --- retraction. --- schematic distance. --- semi-lattice. --- sequence. --- smooth case. --- smoothness. --- stability theory. --- stable completion. --- stable domination. --- stably dominated point. --- stably dominated type. --- stably dominated. --- strong stability. --- substructure. --- topological embedding. --- topological space. --- topological structure. --- topology. --- transcendence degree. --- v-continuity. --- valued field. --- Γ-internal set. --- Γ-internal space. --- Γ-internal subset.
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In this volume, the author covers profinite groups and their cohomology, Galois cohomology, and local class field theory, and concludes with a treatment of duality. His objective is to present effectively that body of material upon which all modern research in Diophantine geometry and higher arithmetic is based, and to do so in a manner that emphasizes the many interesting lines of inquiry leading from these foundations.
Group theory --- Finite groups --- Algebraic number theory --- 512.73 --- 512.66 --- Homology theory --- Number theory --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Groups, Finite --- Modules (Algebra) --- Cohomology theory of algebraic varieties and schemes --- Homological algebra --- 512.66 Homological algebra --- 512.73 Cohomology theory of algebraic varieties and schemes --- Groupes, Théorie des. --- Group theory. --- Homology theory. --- Finite groups. --- Algebraic number theory. --- Abelian group. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic extension. --- Algebraic geometry. --- Algebraic number field. --- Brauer group. --- Category of abelian groups. --- Category of sets. --- Characterization (mathematics). --- Class field theory. --- Cohomological dimension. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Composition series. --- Computation. --- Connected component (graph theory). --- Coset. --- Cup product. --- Dedekind domain. --- Degeneracy (mathematics). --- Diagram (category theory). --- Dimension (vector space). --- Diophantine geometry. --- Discrete group. --- Equivalence of categories. --- Exact sequence. --- Existential quantification. --- Explicit formula. --- Exponential function. --- Family of sets. --- Field extension. --- Finite group. --- Fundamental class. --- G-module. --- Galois cohomology. --- Galois extension. --- Galois group. --- Galois module. --- Galois theory. --- General topology. --- Geometry. --- Grothendieck topology. --- Group cohomology. --- Group extension. --- Group scheme. --- Hilbert symbol. --- Hopf algebra. --- Ideal (ring theory). --- Inequality (mathematics). --- Injective sheaf. --- Inner automorphism. --- Inverse limit. --- Kummer theory. --- Lie algebra. --- Linear independence. --- Local field. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Module (mathematics). --- Morphism. --- Natural topology. --- Neighbourhood (mathematics). --- Normal extension. --- Normal subgroup. --- Number theory. --- P-adic number. --- P-group. --- Polynomial. --- Pontryagin duality. --- Power series. --- Prime number. --- Principal ideal. --- Profinite group. --- Quadratic reciprocity. --- Quotient group. --- Ring of integers. --- Sheaf (mathematics). --- Special case. --- Subcategory. --- Subgroup. --- Supernatural number. --- Sylow theorems. --- Tangent space. --- Theorem. --- Topological group. --- Topological property. --- Topological ring. --- Topological space. --- Topology. --- Torsion group. --- Torsion subgroup. --- Transcendence degree. --- Triviality (mathematics). --- Unique factorization domain. --- Variable (mathematics). --- Vector space. --- Groupes, Théorie des --- Nombres, Théorie des
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