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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.

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Convolution and Equidistribution explores an important aspect of number theory--the theory of exponential sums over finite fields and their Mellin transforms--from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject. The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book's main theorem, using a mixture of geometric, categorical, and group-theoretic methods. By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject.

Mellin transform. --- Convolutions (Mathematics) --- Sequences (Mathematics) --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Convolution transforms --- Transformations, Convolution --- Distribution (Probability theory) --- Functions --- Integrals --- Transformations (Mathematics) --- Transform, Mellin --- Integral transforms --- ArtinГchreier reduced polynomial. --- Emanuel Kowalski. --- EulerАoincar formula. --- Frobenius conjugacy class. --- Frobenius conjugacy. --- Frobenius tori. --- GoursatЋolchinВibet theorem. --- Kloosterman sheaf. --- Laurent polynomial. --- Legendre. --- Pierre Deligne. --- Ron Evans. --- Tannakian category. --- Tannakian groups. --- Zeeev Rudnick. --- algebro-geometric. --- autodual objects. --- autoduality. --- characteristic two. --- connectedness. --- dimensional objects. --- duality. --- equidistribution. --- exponential sums. --- fiber functor. --- finite field Mellin transform. --- finite field. --- finite fields. --- geometrical irreducibility. --- group scheme. --- hypergeometric sheaf. --- interger monic polynomials. --- isogenies. --- lie-irreducibility. --- lisse. --- middle convolution. --- middle extension sheaf. --- monic polynomial. --- monodromy groups. --- noetherian connected scheme. --- nonsplit form. --- nontrivial additive character. --- number theory. --- odd characteristic. --- odd prime. --- orthogonal case. --- perverse sheaves. --- polynomials. --- pure weight. --- semisimple object. --- semisimple. --- sheaves. --- signs. --- split form. --- supermorse. --- theorem. --- theorems.

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L-functions --- Monodromy groups --- Sheaf theory

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