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

• 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
• Physical Description: 1 online resource (264 p.)

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.

Table of Contents

Frontmatter -- Contents -- Introduction -- Part I: Background Material -- Chapter 1: "Abstract" Theorems of Big Monodromy -- Appendix to Chapter 1: A Result of Zalesskii -- Chapter 2: Lefschetz Pencils, Especially on Curves -- Chapter 3: Induction -- Chapter 4: Middle Convolution -- Part II: Twist Sheaves, over an Algebraically Closed Field -- Chapter 5: Twist Sheaves and Their Monodromy -- Part III: Twist Sheaves, over a Finite Field -- Chapter 6: Dependence on Parameters -- Chapter 7: Diophantine Applications over a Finite Field -- Chapter 8: Average Order of Zero in Twist Families -- Part IV: Twist Sheaves, over Schemes of Finite Type over ℤ -- Chapter 9: Twisting by "Primes", and Working over ℤ -- Chapter 10: Horizontal Results -- References -- Index