Katz Twisted L-Functions and Monodromy

Core Textbook
ISBN: 978-1-4008-2488-5
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 264 Seiten

Reihe: Annals of Mathematics Studies

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.

Autoren/Hrsg.

Katz, Nicholas M.

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie
• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

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

Über Autor(innen)

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.