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.