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Author: Emmanuel Peyre
Publisher: Springer Nature
ISBN: 3030575594
Category : Mathematics
Languages : en
Pages : 469
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Book Description
Bridging the gap between novice and expert, the aim of this book is to present in a self-contained way a number of striking examples of current diophantine problems to which Arakelov geometry has been or may be applied. Arakelov geometry can be seen as a link between algebraic geometry and diophantine geometry. Based on lectures from a summer school for graduate students, this volume consists of 12 different chapters, each written by a different author. The first chapters provide some background and introduction to the subject. These are followed by a presentation of different applications to arithmetic geometry. The final part describes the recent application of Arakelov geometry to Shimura varieties and the proof of an averaged version of Colmez's conjecture. This book thus blends initiation to fundamental tools of Arakelov geometry with original material corresponding to current research. This book will be particularly useful for graduate students and researchers interested in the connections between algebraic geometry and number theory. The prerequisites are some knowledge of number theory and algebraic geometry.
Author: Emmanuel Peyre
Publisher: Springer Nature
ISBN: 3030575594
Category : Mathematics
Languages : en
Pages : 469
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Book Description
Bridging the gap between novice and expert, the aim of this book is to present in a self-contained way a number of striking examples of current diophantine problems to which Arakelov geometry has been or may be applied. Arakelov geometry can be seen as a link between algebraic geometry and diophantine geometry. Based on lectures from a summer school for graduate students, this volume consists of 12 different chapters, each written by a different author. The first chapters provide some background and introduction to the subject. These are followed by a presentation of different applications to arithmetic geometry. The final part describes the recent application of Arakelov geometry to Shimura varieties and the proof of an averaged version of Colmez's conjecture. This book thus blends initiation to fundamental tools of Arakelov geometry with original material corresponding to current research. This book will be particularly useful for graduate students and researchers interested in the connections between algebraic geometry and number theory. The prerequisites are some knowledge of number theory and algebraic geometry.
Author: C. Soulé
Publisher: Cambridge University Press
ISBN: 9780521477093
Category : Mathematics
Languages : en
Pages : 190
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Book Description
An account for graduate students of this new technique in diophantine geometry; includes account of higher dimensional theory.
Author: Marc Hindry
Publisher: Springer Science & Business Media
ISBN: 1461212103
Category : Mathematics
Languages : en
Pages : 574
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Book Description
This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.
Author: Huayi Chen
Publisher: Springer Nature
ISBN: 9811517282
Category : Mathematics
Languages : en
Pages : 452
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Book Description
The purpose of this book is to build the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for research on arithmetic geometry in several directions. By adelic curve is meant a field equipped with a family of absolute values parametrized by a measure space, such that the logarithmic absolute value of each non-zero element of the field is an integrable function on the measure space. In the literature, such construction has been discussed in various settings which are apparently transversal to each other. The authors first formalize the notion of adelic curves and discuss in a systematic way its algebraic covers, which are important in the study of height theory of algebraic points beyond Weil–Lang’s height theory. They then establish a theory of adelic vector bundles on adelic curves, which considerably generalizes the classic geometry of vector bundles or that of Hermitian vector bundles over an arithmetic curve. They focus on an analogue of the slope theory in the setting of adelic curves and in particular estimate the minimal slope of tensor product adelic vector bundles. Finally, by using the adelic vector bundles as a tool, a birational Arakelov geometry for projective variety over an adelic curve is developed. As an application, a vast generalization of Nakai–Moishezon’s criterion of positivity is proven in clarifying the arguments of geometric nature from several fundamental results in the classic geometry of numbers. Assuming basic knowledge of algebraic geometry and algebraic number theory, the book is almost self-contained. It is suitable for researchers in arithmetic geometry as well as graduate students focusing on these topics for their doctoral theses.
Author: G., van der Geer
Publisher: Springer Science & Business Media
ISBN: 1461204577
Category : Mathematics
Languages : en
Pages : 450
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Book Description
Arithmetic algebraic geometry is in a fascinating stage of growth, providing a rich variety of applications of new tools to both old and new problems. Representative of these recent developments is the notion of Arakelov geometry, a way of "completing" a variety over the ring of integers of a number field by adding fibres over the Archimedean places. Another is the appearance of the relations between arithmetic geometry and Nevanlinna theory, or more precisely between diophantine approximation theory and the value distribution theory of holomorphic maps. Research mathematicians and graduate students in algebraic geometry and number theory will find a valuable and lively view of the field in this state-of-the-art selection.
Author: Jean-Benoît Bost
Publisher: Springer Nature
ISBN: 3030443299
Category : Mathematics
Languages : en
Pages : 365
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Book Description
This book presents the most up-to-date and sophisticated account of the theory of Euclidean lattices and sequences of Euclidean lattices, in the framework of Arakelov geometry, where Euclidean lattices are considered as vector bundles over arithmetic curves. It contains a complete description of the theta invariants which give rise to a closer parallel with the geometric case. The author then unfolds his theory of infinite Hermitian vector bundles over arithmetic curves and their theta invariants, which provides a conceptual framework to deal with the sequences of lattices occurring in many diophantine constructions. The book contains many interesting original insights and ties to other theories. It is written with extreme care, with a clear and pleasant style, and never sacrifices accessibility to sophistication.
Author: Pieter Belmans
Publisher: Cambridge University Press
ISBN: 1009063286
Category : Mathematics
Languages : en
Pages : 308
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Book Description
The Stacks Project Expository Collection (SPEC) compiles expository articles in advanced algebraic geometry, intended to bring graduate students and researchers up to speed on recent developments in the geometry of algebraic spaces and algebraic stacks. The articles in the text make explicit in modern language many results, proofs, and examples that were previously only implicit, incomplete, or expressed in classical terms in the literature. Where applicable this is done by explicitly referring to the Stacks project for preliminary results. Topics include the construction and properties of important moduli problems in algebraic geometry (such as the Deligne–Mumford compactification of the moduli of curves, the Picard functor, or moduli of semistable vector bundles and sheaves), and arithmetic questions for fields and algebraic spaces.
Author: Yu. I. Manin
Publisher: Springer Science & Business Media
ISBN: 3540276920
Category : Mathematics
Languages : en
Pages : 519
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Book Description
This edition has been called ‘startlingly up-to-date’, and in this corrected second printing you can be sure that it’s even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories.
Author: Hideaki Ikoma
Publisher: Cambridge University Press
ISBN: 1108845959
Category : Mathematics
Languages : en
Pages : 179
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Book Description
This book provides a self-contained proof of the Mordell conjecture (Faltings's theorem) and a concise introduction to Diophantine geometry.
Author: Yujiro Kawamata
Publisher: American Mathematical Soc.
ISBN: 0821807692
Category : Mathematics
Languages : en
Pages : 152
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Book Description
This work, intended for research mathematicians, presents proceedings from the Japan-US Mathematics Institute (JAMI) Conference on Birational Algebraic Geometry in memory of Wei-Liang Chow, held at the John's Hopkins University in Baltimore in April 1996. These proceedings bring to light the many directions in which birational algebraic geometry is headed. Featured are: problems on special models, such as Fanos and their fibrations; adjunctions and subadjunction formulae; projectivity and projective embeddings; and more. Some papers reflect the frontiers of this rapidly-developing area of mathematics. Therefore, in these cases, only directions are given without complete explanations or proofs.
