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Twisted L-Functions and Monodromy. (AM-150), Volume 150 : Annals of Mathematics Studies - Nicholas M. Katz

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

Annals of Mathematics Studies


Published: 4th February 2002
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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.

Introductionp. 3
Background Material
"Abstract" Theorems of Big Monodromyp. 23
A Result of Zalesskiip. 43
Lefschetz Pencils, Especially on Curvesp. 51
Inductionp. 71
Middle Convolutionp. 79
Twist Sheaves, over an Algebraically Closed Field
Twist Sheaves and Their Monodromyp. 85
Twist Sheaves, over a Finite Field
Dependence on Parametersp. 117
Diophantine Applications over a Finite Fieldp. 125
Average Order of Zero in Twist Familiesp. 147
Twist Sheaves, over Schemes of Finite Type over Z
Twisting by "Primes", and Working over Zp. 179
Horizontal Resultsp. 207
Referencesp. 235
Indexp. 241
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780691091518
ISBN-10: 069109151X
Series: Annals of Mathematics Studies
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 264
Published: 4th February 2002
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 1.32
Weight (kg): 0.34