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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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Arithmetic and Geometry presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures-which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria-provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings height and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach.The first course, taught by Peter Scholze, contains his recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces.The second course, taught by Umberto Zannier, addresses the famous Pell equation-not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians.The third course, taught by Shou-Wu Zhang, originates in the Chowla-Selberg formula, which was taken up by Gross and Zagier to relate values of the L-function for elliptic curves with the height of Heegner points on the curves. Zhang, X. Yuan, and Wei Zhang prove the Gross-Zagier formula on Shimura curves and verify the Colmez conjecture on average.

Arithmetical algebraic geometry. --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Number theory --- Abelian variety. --- Algebraic geometry. --- Algebraic independence. --- Algebraic space. --- Analytic number theory. --- Arbitrarily large. --- Automorphic form. --- Automorphism. --- Base change. --- Big O notation. --- Class number formula. --- Cohomology. --- Complex multiplication. --- Computation. --- Conjecture. --- Conjugacy class. --- Continued fraction. --- Cusp form. --- Diagram (category theory). --- Dimension. --- Diophantine equation. --- Diophantine geometry. --- Discriminant. --- Divisible group. --- Double coset. --- Eisenstein series. --- Endomorphism. --- Equation. --- Existential quantification. --- Exponential map (Riemannian geometry). --- Fiber bundle. --- Floor and ceiling functions. --- Formal group. --- Formal power series. --- Formal scheme. --- Fundamental group. --- Geometric Langlands correspondence. --- Geometry. --- Heegner point. --- Hodge structure. --- Hodge theory. --- Homomorphism. --- I0. --- Integer. --- Intersection number. --- Irreducible component. --- Isogeny. --- Isomorphism class. --- Jacobian variety. --- L-function. --- Langlands dual group. --- Laurent series. --- Linear combination. --- Local system. --- Logarithmic derivative. --- Logarithmic form. --- Mathematics. --- Modular form. --- Moduli space. --- Monotonic function. --- Natural topology. --- P-adic analysis. --- P-adic number. --- Pell's equation. --- Perverse sheaf. --- Polylogarithm. --- Polynomial. --- Power series. --- Presheaf (category theory). --- Prime number. --- Projective space. --- Quaternion algebra. --- Rational point. --- Real number. --- Reductive group. --- Rigid analytic space. --- Roth's theorem. --- Series expansion. --- Shafarevich conjecture. --- Sheaf (mathematics). --- Shimura variety. --- Siegel zero. --- Special case. --- Stack (mathematics). --- Subset. --- Summation. --- Szpiro's conjecture. --- Tate conjecture. --- Tate module. --- Taylor series. --- Theorem. --- Theta function. --- Topological ring. --- Topology. --- Torsor (algebraic geometry). --- Upper and lower bounds. --- Vector bundle. --- Weil group. --- Witt vector. --- Zariski topology.