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

Nicholas M. Katz

Published by Princeton University Press 2002

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

Nicholas M. Katz

Volume 150 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9781400824885

Overview
Contents

About this book

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.

Author / Editor 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.

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	v
Introduction Requires Authentication Unlicensed Licensed Download PDF	3
Part I: Background Material
Chapter 1: "Abstract" Theorems of Big Monodromy Requires Authentication Unlicensed Licensed Download PDF	23
Appendix to Chapter 1: A Result of Zalesskii Requires Authentication Unlicensed Licensed Download PDF	43
Chapter 2: Lefschetz Pencils, Especially on Curves Requires Authentication Unlicensed Licensed Download PDF	51
Chapter 3: Induction Requires Authentication Unlicensed Licensed Download PDF	71
Chapter 4: Middle Convolution Requires Authentication Unlicensed Licensed Download PDF	79
Part II: Twist Sheaves, over an Algebraically Closed Field
Chapter 5: Twist Sheaves and Their Monodromy Requires Authentication Unlicensed Licensed Download PDF	85
Part III: Twist Sheaves, over a Finite Field
Chapter 6: Dependence on Parameters Requires Authentication Unlicensed Licensed Download PDF	117
Chapter 7: Diophantine Applications over a Finite Field Requires Authentication Unlicensed Licensed Download PDF	125
Chapter 8: Average Order of Zero in Twist Families Requires Authentication Unlicensed Licensed Download PDF	147
Part IV: Twist Sheaves, over Schemes of Finite Type over ℤ
Chapter 9: Twisting by "Primes", and Working over ℤ Requires Authentication Unlicensed Licensed Download PDF	179
Chapter 10: Horizontal Results Requires Authentication Unlicensed Licensed Download PDF	207
References Requires Authentication Unlicensed Licensed Download PDF	235
Index Requires Authentication Unlicensed Licensed Download PDF	241

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