Twisted L-functions and monodromy

Includes index.

Includes bibliographical references (p. 235-239).

Introduction; Part I: Background Material; 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
Chapter 9: Twisting by 'Primes', & Working over Z.
Chapter 10: Horizontal Results.

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 f.