Twisted L-Functions and Monodromy. (AM-150), Volume 150 /

Twisted L-Functions and Monodromy. (AM-150), Volume 150 / / Nicholas M. Katz.

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 w...

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Superior document:Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020
VerfasserIn:
Katz, Nicholas M.,
Place / Publishing House:Princeton, NJ : : Princeton University Press, , [2009]
©2002
Year of Publication:2009
Edition:Core Textbook
Language:English
Series:Annals of Mathematics Studies ; 150
Online Access:
Physical Description:1 online resource (264 p.)
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245 1 0 |a Twisted L-Functions and Monodromy. (AM-150), Volume 150 /  |c Nicholas M. Katz. 
250 |a Core Textbook 
264 1 |a Princeton, NJ :   |b Princeton University Press,   |c [2009] 
264 4 |c ©2002 
300 |a 1 online resource (264 p.) 
336 |a text  |b txt  |2 rdacontent 
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490 0 |a Annals of Mathematics Studies ;  |v 150 
505 0 0 |t Frontmatter --   |t Contents --   |t Introduction --   |t Part I: Background Material --   |t Chapter 1: "Abstract" Theorems of Big Monodromy --   |t Appendix to Chapter 1: A Result of Zalesskii --   |t Chapter 2: Lefschetz Pencils, Especially on Curves --   |t Chapter 3: Induction --   |t Chapter 4: Middle Convolution --   |t Part II: Twist Sheaves, over an Algebraically Closed Field --   |t Chapter 5: Twist Sheaves and Their Monodromy --   |t Part III: Twist Sheaves, over a Finite Field --   |t Chapter 6: Dependence on Parameters --   |t Chapter 7: Diophantine Applications over a Finite Field --   |t Chapter 8: Average Order of Zero in Twist Families --   |t Part IV: Twist Sheaves, over Schemes of Finite Type over ℤ --   |t Chapter 9: Twisting by "Primes", and Working over ℤ --   |t Chapter 10: Horizontal Results --   |t References --   |t Index 
506 0 |a restricted access  |u http://purl.org/coar/access_right/c_16ec  |f online access with authorization  |2 star 
520 |a 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. 
530 |a Issued also in print. 
538 |a Mode of access: Internet via World Wide Web. 
546 |a In English. 
588 0 |a Description based on online resource; title from PDF title page (publisher's Web site, viewed 31. Jan 2022) 
650 0 |a L-functions. 
650 0 |a Monodromy groups. 
650 7 |a MATHEMATICS / Number Theory.  |2 bisacsh 
653 |a Abelian variety. 
653 |a Absolute continuity. 
653 |a Addition. 
653 |a Affine space. 
653 |a Algebraically closed field. 
653 |a Ambient space. 
653 |a Average. 
653 |a Betti number. 
653 |a Birch and Swinnerton-Dyer conjecture. 
653 |a Blowing up. 
653 |a Codimension. 
653 |a Coefficient. 
653 |a Computation. 
653 |a Conjecture. 
653 |a Conjugacy class. 
653 |a Convolution. 
653 |a Critical value. 
653 |a Differential geometry of surfaces. 
653 |a Dimension (vector space). 
653 |a Dimension. 
653 |a Direct sum. 
653 |a Divisor (algebraic geometry). 
653 |a Divisor. 
653 |a Eigenvalues and eigenvectors. 
653 |a Elliptic curve. 
653 |a Equation. 
653 |a Equidistribution theorem. 
653 |a Existential quantification. 
653 |a Factorization. 
653 |a Finite field. 
653 |a Finite group. 
653 |a Finite set. 
653 |a Flat map. 
653 |a Fourier transform. 
653 |a Function field. 
653 |a Functional equation. 
653 |a Goursat's lemma. 
653 |a Ground field. 
653 |a Group representation. 
653 |a Hyperplane. 
653 |a Hypersurface. 
653 |a Integer matrix. 
653 |a Integer. 
653 |a Irreducible component. 
653 |a Irreducible polynomial. 
653 |a Irreducible representation. 
653 |a J-invariant. 
653 |a K3 surface. 
653 |a L-function. 
653 |a Lebesgue measure. 
653 |a Lefschetz pencil. 
653 |a Level of measurement. 
653 |a Lie algebra. 
653 |a Limit superior and limit inferior. 
653 |a Minimal polynomial (field theory). 
653 |a Modular form. 
653 |a Monodromy. 
653 |a Morphism. 
653 |a Numerical analysis. 
653 |a Orthogonal group. 
653 |a Percentage. 
653 |a Polynomial. 
653 |a Prime number. 
653 |a Probability measure. 
653 |a Quadratic function. 
653 |a Quantity. 
653 |a Quotient space (topology). 
653 |a Representation theory. 
653 |a Residue field. 
653 |a Riemann hypothesis. 
653 |a Root of unity. 
653 |a Scalar (physics). 
653 |a Set (mathematics). 
653 |a Sheaf (mathematics). 
653 |a Subgroup. 
653 |a Summation. 
653 |a Symmetric group. 
653 |a System of imprimitivity. 
653 |a Theorem. 
653 |a Trivial representation. 
653 |a Zariski topology. 
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773 0 8 |i Title is part of eBook package:  |d De Gruyter  |t Princeton University Press eBook-Package Backlist 2000-2013  |z 9783110442502 
776 0 |c print  |z 9780691091518 
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856 4 2 |3 Cover  |u https://www.degruyter.com/document/cover/isbn/9781400824885/original 
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