Twisted L-Functions and Monodromy

Twisted L-Functions and Monodromy
Author: Nicholas M. Katz
Publisher: Princeton University Press
Total Pages: 258
Release: 2009-01-10
Genre: Mathematics
ISBN: 1400824885
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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.

Annals of Mathematics Studies

Annals of Mathematics Studies
Author: Nicholas M. Katz
Publisher:
Total Pages: 249
Release: 1940
Genre: L-functions
ISBN: 9780691091501
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Arithmetic of L-functions

Arithmetic of L-functions
Author: Cristian Popescu
Publisher: American Mathematical Soc.
Total Pages: 517
Release:
Genre: Mathematics
ISBN: 0821886983
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Number Theory, Analysis and Geometry

Number Theory, Analysis and Geometry
Author: Dorian Goldfeld
Publisher: Springer Science & Business Media
Total Pages: 715
Release: 2011-12-21
Genre: Mathematics
ISBN: 1461412609
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Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry, and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas of the field, namely Number Theory, Analysis, and Geometry, representing Lang’s own breadth of interest and impact. A special introduction by John Tate includes a brief and fascinating account of the Serge Lang’s life. This volume's group of 6 editors are also highly prominent mathematicians and were close to Serge Lang, both academically and personally. The volume is suitable to research mathematicians in the areas of Number Theory, Analysis, and Geometry.

Ranks of Elliptic Curves and Random Matrix Theory

Ranks of Elliptic Curves and Random Matrix Theory
Author: J. B. Conrey
Publisher: Cambridge University Press
Total Pages: 5
Release: 2007-02-08
Genre: Mathematics
ISBN: 0521699649
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This comprehensive volume introduces elliptic curves and the fundamentals of modeling by a family of random matrices.

Contributions to Automorphic Forms, Geometry, and Number Theory

Contributions to Automorphic Forms, Geometry, and Number Theory
Author: Haruzo Hida
Publisher: JHU Press
Total Pages: 946
Release: 2004-03-11
Genre: Mathematics
ISBN: 9780801878602
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In Contributions to Automorphic Forms, Geometry, and Number Theory, Haruzo Hida, Dinakar Ramakrishnan, and Freydoon Shahidi bring together a distinguished group of experts to explore automorphic forms, principally via the associated L-functions, representation theory, and geometry. Because these themes are at the cutting edge of a central area of modern mathematics, and are related to the philosophical base of Wiles' proof of Fermat's last theorem, this book will be of interest to working mathematicians and students alike. Never previously published, the contributions to this volume expose the reader to a host of difficult and thought-provoking problems. Each of the extraordinary and noteworthy mathematicians in this volume makes a unique contribution to a field that is currently seeing explosive growth. New and powerful results are being proved, radically and continually changing the field's make up. Contributions to Automorphic Forms, Geometry, and Number Theory will likely lead to vital interaction among researchers and also help prepare students and other young mathematicians to enter this exciting area of pure mathematics. Contributors: Jeffrey Adams, Jeffrey D. Adler, James Arthur, Don Blasius, Siegfried Boecherer, Daniel Bump, William Casselmann, Laurent Clozel, James Cogdell, Laurence Corwin, Solomon Friedberg, Masaaki Furusawa, Benedict Gross, Thomas Hales, Joseph Harris, Michael Harris, Jeffrey Hoffstein, Hervé Jacquet, Dihua Jiang, Nicholas Katz, Henry Kim, Victor Kreiman, Stephen Kudla, Philip Kutzko, V. Lakshmibai, Robert Langlands, Erez Lapid, Ilya Piatetski-Shapiro, Dipendra Prasad, Stephen Rallis, Dinakar Ramakrishnan, Paul Sally, Freydoon Shahidi, Peter Sarnak, Rainer Schulze-Pillot, Joseph Shalika, David Soudry, Ramin Takloo-Bigash, Yuri Tschinkel, Emmanuel Ullmo, Marie-France Vignéras, Jean-Loup Waldspurger.

Recent Perspectives in Random Matrix Theory and Number Theory

Recent Perspectives in Random Matrix Theory and Number Theory
Author: F. Mezzadri
Publisher: Cambridge University Press
Total Pages: 530
Release: 2005-06-21
Genre: Mathematics
ISBN: 0521620589
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Provides a grounding in random matrix techniques applied to analytic number theory.

Moments, Monodromy, and Perversity

Moments, Monodromy, and Perversity
Author: Nicholas M. Katz
Publisher: Princeton University Press
Total Pages: 492
Release: 2005-10-02
Genre: Mathematics
ISBN: 9780691123301
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It is now some thirty years since Deligne first proved his general equidistribution theorem, thus establishing the fundamental result governing the statistical properties of suitably "pure" algebro-geometric families of character sums over finite fields (and of their associated L-functions). Roughly speaking, Deligne showed that any such family obeys a "generalized Sato-Tate law," and that figuring out which generalized Sato-Tate law applies to a given family amounts essentially to computing a certain complex semisimple (not necessarily connected) algebraic group, the "geometric monodromy group" attached to that family. Up to now, nearly all techniques for determining geometric monodromy groups have relied, at least in part, on local information. In Moments, Monodromy, and Perversity, Nicholas Katz develops new techniques, which are resolutely global in nature. They are based on two vital ingredients, neither of which existed at the time of Deligne's original work on the subject. The first is the theory of perverse sheaves, pioneered by Goresky and MacPherson in the topological setting and then brilliantly transposed to algebraic geometry by Beilinson, Bernstein, Deligne, and Gabber. The second is Larsen's Alternative, which very nearly characterizes classical groups by their fourth moments. These new techniques, which are of great interest in their own right, are first developed and then used to calculate the geometric monodromy groups attached to some quite specific universal families of (L-functions attached to) character sums over finite fields.

Current Trends in Number Theory

Current Trends in Number Theory
Author: S.D. Adhikari
Publisher: Hindustan Book Agency and Indian National Science Academy
Total Pages: 288
Release: 2002
Genre: Mathematics
ISBN:
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The book gives a glimpse of current research in combinatorial, algebraic, and analytic aspects of number theory. The articles are refereed and expanded versions of talks given at the International Conference on Number Theory held at the Harish-Chandra Research Institute (Allahaba, India). Included also are some articles on arithmetic algebraic geometry.