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By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary. This book explains in detail the following topics about PEL-type Shimura varieties and their compactifications: A construction of smooth integral models of PEL-type Shimura varieties by defining and representing moduli problems of abelian schemes with PEL structures An analysis of the degeneration of abelian varieties with PEL structures into semiabelian schemes, over noetherian normal complete adic base rings A construction of toroidal and minimal compactifications of smooth integral models of PEL-type Shimura varieties, with detailed descriptions of their structure near the boundary Through these topics, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai).

Shimura varieties. --- Arithmetical algebraic geometry. --- FourierЊacobi expansions. --- Hecke actions. --- Hermitian symmetric spaces. --- KodairaГpencer morphisms. --- Koecher's principle. --- Langlands program. --- Lie algebra conditions. --- PEL structures. --- PEL-type Shimura varieties. --- PEL-type Shimura. --- PEL-type structures. --- Raynaud extensions. --- Siegel moduli schemes. --- Weil-pairing calculation. --- abelian schemes. --- abelian varieties. --- algebraic stacks. --- analysis. --- arithmetic minimal compactifications. --- arithmetic toroidal compactifications. --- automorphic forms. --- biextensions. --- codimension counting. --- compactifications. --- complex abelian varieties. --- cubical structures. --- cusp labels. --- deformation theory. --- degeneration data. --- degeneration theory. --- degeneration. --- dual abelian varieties. --- dual objects. --- endomorphism structures. --- functoriality. --- geometry. --- good algebraic models. --- isogeny classes. --- isomorphism classes. --- isomorphism. --- level structures. --- linear algebraic assumptions. --- local moduli functors. --- minimal compactifications. --- modular curves. --- moduli problems. --- multiplicative type. --- number theory. --- polarized abelian schemes. --- polarized abelian varieties. --- prorepresentability. --- reductive groups. --- representability. --- semi-abelian schemes. --- tale topology. --- toroidal compactifications. --- toroidal embeddings. --- torsors.

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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.

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An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjects--elliptic curves and modular forms--come together in Eichler-Shimura theory, which constructs elliptic curves out of modular forms of a special kind. The converse, that all rational elliptic curves arise this way, is called the Taniyama-Weil Conjecture and is known to imply Fermat's Last Theorem. Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match. The theory covered by Anthony Knapp in this book is, therefore, a window into a broad expanse of mathematics--including class field theory, arithmetic algebraic geometry, and group representations--in which the concidence of L functions relates analysis and algebra in the most fundamental ways. Developing, with many examples, the elementary theory of elliptic curves, the book goes on to the subject of modular forms and the first connections with elliptic curves. The last two chapters concern Eichler-Shimura theory, which establishes a much deeper relationship between the two subjects. No other book in print treats the basic theory of elliptic curves with only undergraduate mathematics, and no other explains Eichler-Shimura theory in such an accessible manner.

Curves, Elliptic. --- Affine plane (incidence geometry). --- Affine space. --- Affine variety. --- Algebra homomorphism. --- Algebraic extension. --- Algebraic geometry. --- Algebraic integer. --- Algebraic number theory. --- Algebraic number. --- Analytic continuation. --- Analytic function. --- Associative algebra. --- Automorphism. --- Big O notation. --- Binary quadratic form. --- Birch and Swinnerton-Dyer conjecture. --- Bounded set (topological vector space). --- Change of variables. --- Characteristic polynomial. --- Coefficient. --- Compactification (mathematics). --- Complex conjugate. --- Complex manifold. --- Complex number. --- Conjecture. --- Coprime integers. --- Cusp form. --- Cyclic group. --- Degeneracy (mathematics). --- Dimension (vector space). --- Dirichlet character. --- Dirichlet series. --- Division algebra. --- Divisor. --- Eigenform. --- Eigenvalues and eigenvectors. --- Elementary symmetric polynomial. --- Elliptic curve. --- Elliptic function. --- Elliptic integral. --- Equation. --- Euler product. --- Finitely generated abelian group. --- Fourier analysis. --- Function (mathematics). --- Functional equation. --- General linear group. --- Group homomorphism. --- Group isomorphism. --- Hecke operator. --- Holomorphic function. --- Homomorphism. --- Ideal (ring theory). --- Integer matrix. --- Integer. --- Integral domain. --- Intersection (set theory). --- Inverse function theorem. --- Invertible matrix. --- Irreducible polynomial. --- Isogeny. --- J-invariant. --- Linear fractional transformation. --- Linear map. --- Liouville's theorem (complex analysis). --- Mathematical induction. --- Meromorphic function. --- Minimal polynomial (field theory). --- Modular form. --- Monic polynomial. --- Möbius transformation. --- Number theory. --- P-adic number. --- Polynomial ring. --- Power series. --- Prime factor. --- Prime number theorem. --- Prime number. --- Principal axis theorem. --- Principal ideal domain. --- Principal ideal. --- Projective line. --- Projective variety. --- Quadratic equation. --- Quadratic function. --- Quadratic reciprocity. --- Riemann surface. --- Riemann zeta function. --- Simultaneous equations. --- Special case. --- Summation. --- Taylor series. --- Theorem. --- Torsion subgroup. --- Transcendence degree. --- Uniformization theorem. --- Unique factorization domain. --- Variable (mathematics). --- Weierstrass's elliptic functions. --- Weil conjecture.