Néron Models

Néron Models PDF

Author: Siegfried Bosch

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 336

ISBN-13: 3642514383

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Néron models were invented by A. Néron in the early 1960s in order to study the integral structure of abelian varieties over number fields. Since then, arithmeticians and algebraic geometers have applied the theory of Néron models with great success. Quite recently, new developments in arithmetic algebraic geometry have prompted a desire to understand more about Néron models, and even to go back to the basics of their construction. The authors have taken this as their incentive to present a comprehensive treatment of Néron models. This volume of the renowned "Ergebnisse" series provides a detailed demonstration of the construction of Néron models from the point of view of Grothendieck's algebraic geometry. In the second part of the book the relationship between Néron models and the relative Picard functor in the case of Jacobian varieties is explained. The authors helpfully remind the reader of some important standard techniques of algebraic geometry. A special chapter surveys the theory of the Picard functor.

Néron Models and Base Change

Néron Models and Base Change PDF

Author: Lars Halvard Halle

Publisher: Springer

Published: 2016-03-02

Total Pages: 154

ISBN-13: 3319266381

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Presenting the first systematic treatment of the behavior of Néron models under ramified base change, this book can be read as an introduction to various subtle invariants and constructions related to Néron models of semi-abelian varieties, motivated by concrete research problems and complemented with explicit examples. Néron models of abelian and semi-abelian varieties have become an indispensable tool in algebraic and arithmetic geometry since Néron introduced them in his seminal 1964 paper. Applications range from the theory of heights in Diophantine geometry to Hodge theory. We focus specifically on Néron component groups, Edixhoven’s filtration and the base change conductor of Chai and Yu, and we study these invariants using various techniques such as models of curves, sheaves on Grothendieck sites and non-archimedean uniformization. We then apply our results to the study of motivic zeta functions of abelian varieties. The final chapter contains a list of challenging open questions. This book is aimed towards researchers with a background in algebraic and arithmetic geometry.

A Celebration of Algebraic Geometry

A Celebration of Algebraic Geometry PDF

Author: Brendan Hassett

Publisher: American Mathematical Soc.

Published: 2013-09-11

Total Pages: 614

ISBN-13: 0821889834

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This volume resulted from the conference A Celebration of Algebraic Geometry, which was held at Harvard University from August 25-28, 2011, in honor of Joe Harris' 60th birthday. Harris is famous around the world for his lively textbooks and enthusiastic teaching, as well as for his seminal research contributions. The articles are written in this spirit: clear, original, engaging, enlivened by examples, and accessible to young mathematicians. The articles in this volume focus on the moduli space of curves and more general varieties, commutative algebra, invariant theory, enumerative geometry both classical and modern, rationally connected and Fano varieties, Hodge theory and abelian varieties, and Calabi-Yau and hyperkähler manifolds. Taken together, they present a comprehensive view of the long frontier of current knowledge in algebraic geometry. Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

The Arithmetic and Geometry of Algebraic Cycles

The Arithmetic and Geometry of Algebraic Cycles PDF

Author: B. Brent Gordon

Publisher: Springer Science & Business Media

Published: 2000-02-29

Total Pages: 652

ISBN-13: 9780792361947

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The subject of algebraic cycles has thrived through its interaction with algebraic K-theory, Hodge theory, arithmetic algebraic geometry, number theory, and topology. These interactions have led to such developments as a description of Chow groups in terms of algebraic K-theory, the arithmetic Abel-Jacobi mapping, progress on the celebrated conjectures of Hodge and Tate, and the conjectures of Bloch and Beilinson. The immense recent progress in algebraic cycles, based on so many interactions with so many other areas of mathematics, has contributed to a considerable degree of inaccessibility, especially for graduate students. Even specialists in one approach to algebraic cycles may not understand other approaches well. This book offers students and specialists alike a broad perspective of algebraic cycles, presented from several viewpoints, including arithmetic, transcendental, topological, motives and K-theory methods. Topics include a discussion of the arithmetic Abel-Jacobi mapping, higher Abel-Jacobi regulator maps, polylogarithms and L-series, candidate Bloch-Beilinson filtrations, applications of Chern-Simons invariants to algebraic cycles via the study of algebraic vector bundles with algebraic connection, motivic cohomology, Chow groups of singular varieties, and recent progress on the Hodge and Tate conjectures for Abelian varieties.

F-O

F-O PDF

Author: Library of Congress. Office for Subject Cataloging Policy

Publisher:

Published: 1990

Total Pages: 1636

ISBN-13:

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Motivic Integration

Motivic Integration PDF

Author: Antoine Chambert-Loir

Publisher: Springer

Published: 2018-09-15

Total Pages: 526

ISBN-13: 149397887X

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This monograph focuses on the geometric theory of motivic integration, which takes its values in the Grothendieck ring of varieties. This theory is rooted in a groundbreaking idea of Kontsevich and was further developed by Denef & Loeser and Sebag. It is presented in the context of formal schemes over a discrete valuation ring, without any restriction on the residue characteristic. The text first discusses the main features of the Grothendieck ring of varieties, arc schemes, and Greenberg schemes. It then moves on to motivic integration and its applications to birational geometry and non-Archimedean geometry. Also included in the work is a prologue on p-adic analytic manifolds, which served as a model for motivic integration. With its extensive discussion of preliminaries and applications, this book is an ideal resource for graduate students of algebraic geometry and researchers of motivic integration. It will also serve as a motivation for more recent and sophisticated theories that have been developed since.

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