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Monodromy groups.

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Functions, Zeta. --- L-functions. --- Random matrices. --- Limit theorems (Probability theory) --- Monodromy groups. --- Group theory --- Probabilities --- Matrices, Random --- Matrices --- Functions, L --- -Number theory --- Zeta functions --- Functions, Zeta --- L-functions --- Monodromy groups --- Random matrices

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Differential geometry. Global analysis --- 51 <082.1> --- Mathematics--Series --- Permutation groups. --- Curves. --- Monodromy groups. --- Riemann surfaces. --- Symmetry groups. --- Groupes de permutations. --- Courbes. --- Groupes de monodromie. --- Riemann, Surfaces de. --- Groupes de symétrie. --- Curves --- Monodromy groups --- Permutation groups --- Riemann surfaces --- Symmetry groups --- Groups, Symmetry --- Symmetric groups --- Crystallography, Mathematical --- Quantum theory --- Representations of groups --- Surfaces, Riemann --- Functions --- Substitution groups --- Group theory --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Differential --- Geometry, Enumerative --- Mathematics --- Shapes

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In singularity theory and algebraic geometry the monodromy group is embodied in the Picard-Lefschetz formula and the Picard-Fuchs equations. It has applications in the weakened 16th Hilbert problem and in mixed Hodge structures. In the theory of systems of linear differential equations one has the Riemann-Hilbert problem, the Stokes phenomena and the hypergeometric functions with their multidimensional generalizations. In the theory of homomorphic foliations there appear the Ecalle-Voronin-Martinet-Ramis moduli. On the other hand, there is a deep connection of monodromy theory with Galois theory of differential equations and algebraic functions. All this is presented in this book, underlining the unifying role of the monodromy group. The material is addressed to a wide audience, ranging from specialists in the theory of ordinary differential equations to algebraic geometers. The book contains a lot of results which are usually spread in many sources. Readers can quickly get introduced to modern and vital mathematical theories, such as singularity theory, analytic theory of ordinary differential equations, holomorphic foliations, Galois theory, and parts of algebraic geometry, without searching in vast literature.

Monodromy groups. --- Riemann-Hilbert problems. --- Hilbert-Riemann problems --- Riemann problems --- Boundary value problems --- Group theory --- Algebra. --- Algebraic topology. --- Functions of complex variables. --- Differential Equations. --- Functions, special. --- Algebraic Topology. --- Functions of a Complex Variable. --- Ordinary Differential Equations. --- Special Functions. --- Special functions --- Mathematical analysis --- 517.91 Differential equations --- Differential equations --- Complex variables --- Elliptic functions --- Functions of real variables --- Topology --- Mathematics --- Differential equations. --- Special functions.

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For hundreds of years, the study of elliptic curves has played a central role in mathematics. The past century in particular has seen huge progress in this study, from Mordell's theorem in 1922 to the work of Wiles and Taylor-Wiles in 1994. Nonetheless, there remain many fundamental questions where we do not even know what sort of answers to expect. This book explores two of them: What is the average rank of elliptic curves, and how does the rank vary in various kinds of families of elliptic curves? Nicholas Katz answers these questions for families of ''big'' twists of elliptic curves in the function field case (with a growing constant field). The monodromy-theoretic methods he develops turn out to apply, still in the function field case, equally well to families of big twists of objects of all sorts, not just to elliptic curves. The leisurely, lucid introduction gives the reader a clear picture of what is known and what is unknown at present, and situates the problems solved in this book within the broader context of the overall study of elliptic curves. The book's technical core makes use of, and explains, various advanced topics ranging from recent results in finite group theory to the machinery of l-adic cohomology and monodromy. Twisted L-Functions and Monodromy is essential reading for anyone interested in number theory and algebraic geometry.

L-functions. --- Monodromy groups. --- Functions, L --- -L-functions. --- Group theory --- -Number theory --- L-functions --- Monodromy groups --- Abelian variety. --- Absolute continuity. --- Addition. --- Affine space. --- Algebraically closed field. --- Ambient space. --- Average. --- Betti number. --- Birch and Swinnerton-Dyer conjecture. --- Blowing up. --- Codimension. --- Coefficient. --- Computation. --- Conjecture. --- Conjugacy class. --- Convolution. --- Critical value. --- Differential geometry of surfaces. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Divisor (algebraic geometry). --- Divisor. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Equation. --- Equidistribution theorem. --- Existential quantification. --- Factorization. --- Finite field. --- Finite group. --- Finite set. --- Flat map. --- Fourier transform. --- Function field. --- Functional equation. --- Goursat's lemma. --- Ground field. --- Group representation. --- Hyperplane. --- Hypersurface. --- Integer matrix. --- Integer. --- Irreducible component. --- Irreducible polynomial. --- Irreducible representation. --- J-invariant. --- K3 surface. --- L-function. --- Lebesgue measure. --- Lefschetz pencil. --- Level of measurement. --- Lie algebra. --- Limit superior and limit inferior. --- Minimal polynomial (field theory). --- Modular form. --- Monodromy. --- Morphism. --- Numerical analysis. --- Orthogonal group. --- Percentage. --- Polynomial. --- Prime number. --- Probability measure. --- Quadratic function. --- Quantity. --- Quotient space (topology). --- Representation theory. --- Residue field. --- Riemann hypothesis. --- Root of unity. --- Scalar (physics). --- Set (mathematics). --- Sheaf (mathematics). --- Subgroup. --- Summation. --- Symmetric group. --- System of imprimitivity. --- Theorem. --- Trivial representation. --- Zariski topology.