Overview

"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."

Full Product Details

Author:   Nicholas M. Katz
Publisher:   Princeton University Press
Imprint:   Princeton University Press
Edition:   New edition
Volume:   164
Dimensions:   Width: 15.20cm , Height: 1.30cm , Length: 23.50cm
Weight:   0.340kg
ISBN:  

9780691091518

ISBN 10:   069109151
Pages:   264
Publication Date:   24 February 2002
Audience:   Professional and scholarly ,  College/higher education ,  Professional & Vocational ,  Tertiary & Higher Education
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.
Language:   English

Table of Contents

"*FrontMatter, pg. i*Contents, pg. v*Introduction, pg. 3*Chapter 1: ""Abstract"" Theorems of Big Monodromy, pg. 23*Appendix to Chapter 1: A Result of Zalesskii, pg. 43*Chapter 2: Lefschetz Pencils, Especially on Curves, pg. 51*Chapter 3: Induction, pg. 71*Chapter 4: Middle Convolution, pg. 79*Chapter 5: Twist Sheaves and Their Monodromy, pg. 85*Chapter 6: Dependence on Parameters, pg. 117*Chapter 7: Diophantine Applications over a Finite Field, pg. 125*Chapter 8: Average Order of Zero in Twist Families, pg. 147*Chapter 9: Twisting by ""Primes"", and Working over Z, pg. 179*Chapter 10: Horizontal Results, pg. 207*References, pg. 235*Index, pg. 241"

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Author Information

Nicholas M. Katz is Professor of Mathematics at Princeton University. He is the author of four other books in this series: Arithmetic Moduli of Elliptic Curves (with Barry Mazur); Gauss Sums, Kloosterman Sums, and Monodromy Groups; Exponential Sums and Differential Equations; and Rigid Local Systems.

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