TY - BOOK ID - 12029768 TI - Twisted L-functions and monodromy PY - 2002 VL - 150 SN - 1282820893 9786612820892 1400824885 069109151X 0691091501 9781400824885 9780691091501 9780691091518 9781282820890 6612820896 PB - Princeton Princeton University Press DB - UniCat KW - L-functions. KW - Monodromy groups. KW - Functions, L KW - -L-functions. KW - Group theory KW - -Number theory KW - L-functions KW - Monodromy groups KW - Abelian variety. KW - Absolute continuity. KW - Addition. KW - Affine space. KW - Algebraically closed field. KW - Ambient space. KW - Average. KW - Betti number. KW - Birch and Swinnerton-Dyer conjecture. KW - Blowing up. KW - Codimension. KW - Coefficient. KW - Computation. KW - Conjecture. KW - Conjugacy class. KW - Convolution. KW - Critical value. KW - Differential geometry of surfaces. KW - Dimension (vector space). KW - Dimension. KW - Direct sum. KW - Divisor (algebraic geometry). KW - Divisor. KW - Eigenvalues and eigenvectors. KW - Elliptic curve. KW - Equation. KW - Equidistribution theorem. KW - Existential quantification. KW - Factorization. KW - Finite field. KW - Finite group. KW - Finite set. KW - Flat map. KW - Fourier transform. KW - Function field. KW - Functional equation. KW - Goursat's lemma. KW - Ground field. KW - Group representation. KW - Hyperplane. KW - Hypersurface. KW - Integer matrix. KW - Integer. KW - Irreducible component. KW - Irreducible polynomial. KW - Irreducible representation. KW - J-invariant. KW - K3 surface. KW - L-function. KW - Lebesgue measure. KW - Lefschetz pencil. KW - Level of measurement. KW - Lie algebra. KW - Limit superior and limit inferior. KW - Minimal polynomial (field theory). KW - Modular form. KW - Monodromy. KW - Morphism. KW - Numerical analysis. KW - Orthogonal group. KW - Percentage. KW - Polynomial. KW - Prime number. KW - Probability measure. KW - Quadratic function. KW - Quantity. KW - Quotient space (topology). KW - Representation theory. KW - Residue field. KW - Riemann hypothesis. KW - Root of unity. KW - Scalar (physics). KW - Set (mathematics). KW - Sheaf (mathematics). KW - Subgroup. KW - Summation. KW - Symmetric group. KW - System of imprimitivity. KW - Theorem. KW - Trivial representation. KW - Zariski topology. UR - https://www.unicat.be/uniCat?func=search&query=sysid:12029768 AB - 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. ER -