Hodge Cycles, Motives, and Shimura Varieties
Hodge Cycles, Motives, and Shimura Varieties

Author: Pierre Deligne

Publisher: Springer Science & Business Media

Published: 1982

Total Pages: 423

ISBN-13: 3540111743

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This volume collects six related articles. The first is the notes (written by J.S. Milne) of a major part of the seminar "Periodes des Int grales Abeliennes" given by P. Deligne at I'.B.E.S., 1978-79. The second article was written for this volume (by P. Deligne and J.S. Milne) and is largely based on: N Saavedra Rivano, Categories tannakiennes, Lecture Notes in Math. 265, Springer, Heidelberg 1972. The third article is a slight expansion of part of: J.S. Milne and Kuang-yen Shih, Sh ura varieties: conjugates and the action of complex conjugation 154 pp. (Unpublished manuscript, October 1979). The fourth article is based on a letter from P. De1igne to R. Langlands, dated 10th April, 1979, and was revised and completed (by De1igne) in July, 1981. The fifth article is a slight revision of another section of the manuscript of Milne and Shih referred to above. The sixth article, by A. Ogus, dates from July, 1980.

Motives
Motives

Author: Uwe Jannsen

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 766

ISBN-13: 0821827979

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'Motives' were introduced in the mid-1960s by Grothendieck to explain the analogies among the various cohomology theories for algebraic varieties, and to play the role of the missing rational cohomology. This work contains the texts of the lectures presented at the AMS-IMS-SIAM Joint Summer Research Conference on Motives, held in Seattle, in 1991.

Contributions to Automorphic Forms, Geometry, and Number Theory
Contributions to Automorphic Forms, Geometry, and Number Theory

Author: Haruzo Hida

Publisher: JHU Press

Published: 2004-03-11

Total Pages: 946

ISBN-13: 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.

Motives
Motives

Author:

Publisher: American Mathematical Soc.

Published: 1994-02-28

Total Pages: 694

ISBN-13: 0821827987

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'Motives' were introduced in the mid-1960s by Grothendieck to explain the analogies among the various cohomology theories for algebraic varieties, and to play the role of the missing rational cohomology. This work contains the texts of the lectures presented at the AMS-IMS-SIAM Joint Summer Research Conference on Motives, held in Seattle, in 1991.

The Arithmetic and Geometry of Algebraic Cycles
The Arithmetic and Geometry of Algebraic Cycles

Author: B. Brent Gordon

Publisher: American Mathematical Soc.

Published: 2000-01-01

Total Pages: 468

ISBN-13: 9780821870204

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From the June 1998 Summer School come 20 contributions that explore algebraic cycles (a subfield of algebraic geometry) from a variety of perspectives. The papers have been organized into sections on cohomological methods, Chow groups and motives, and arithmetic methods. Some specific topics include logarithmic Hodge structures and classifying spaces; Bloch's conjecture and the K-theory of projective surfaces; and torsion zero-cycles and the Abel-Jacobi map over the real numbers.

Periods and Nori Motives
Periods and Nori Motives

Author: Annette Huber

Publisher: Springer

Published: 2017-03-08

Total Pages: 381

ISBN-13: 3319509268

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This book casts the theory of periods of algebraic varieties in the natural setting of Madhav Nori’s abelian category of mixed motives. It develops Nori’s approach to mixed motives from scratch, thereby filling an important gap in the literature, and then explains the connection of mixed motives to periods, including a detailed account of the theory of period numbers in the sense of Kontsevich-Zagier and their structural properties. Period numbers are central to number theory and algebraic geometry, and also play an important role in other fields such as mathematical physics. There are long-standing conjectures about their transcendence properties, best understood in the language of cohomology of algebraic varieties or, more generally, motives. Readers of this book will discover that Nori’s unconditional construction of an abelian category of motives (over fields embeddable into the complex numbers) is particularly well suited for this purpose. Notably, Kontsevich's formal period algebra represents a torsor under the motivic Galois group in Nori's sense, and the period conjecture of Kontsevich and Zagier can be recast in this setting. Periods and Nori Motives is highly informative and will appeal to graduate students interested in algebraic geometry and number theory as well as researchers working in related fields. Containing relevant background material on topics such as singular cohomology, algebraic de Rham cohomology, diagram categories and rigid tensor categories, as well as many interesting examples, the overall presentation of this book is self-contained.

Shimura Varieties
Shimura Varieties

Author: Thomas Haines

Publisher: Cambridge University Press

Published: 2020-02-20

Total Pages: 341

ISBN-13: 1108632068

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This is the second volume of a series of mainly expository articles on the arithmetic theory of automorphic forms. It forms a sequel to On the Stabilization of the Trace Formula published in 2011. The books are intended primarily for two groups of readers: those interested in the structure of automorphic forms on reductive groups over number fields, and specifically in qualitative information on multiplicities of automorphic representations; and those interested in the classification of I-adic representations of Galois groups of number fields. Langlands' conjectures elaborate on the notion that these two problems overlap considerably. These volumes present convincing evidence supporting this, clearly and succinctly enough that readers can pass with minimal effort between the two points of view. Over a decade's worth of progress toward the stabilization of the Arthur-Selberg trace formula, culminating in Ngo Bau Chau's proof of the Fundamental Lemma, makes this series timely.

Noncommutative Geometry and Number Theory
Noncommutative Geometry and Number Theory

Author: Caterina Consani

Publisher: Springer Science & Business Media

Published: 2007-12-18

Total Pages: 374

ISBN-13: 3834803529

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In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new connections between the fields of number theory, algebraic geometry and noncommutative geometry. The articles collected in this volume present new noncommutative geometry perspectives on classical topics of number theory and arithmetic such as modular forms, class field theory, the theory of reductive p-adic groups, Shimura varieties, the local L-factors of arithmetic varieties. They also show how arithmetic appears naturally in noncommutative geometry and in physics, in the residues of Feynman graphs, in the properties of noncommutative tori, and in the quantum Hall effect.