Number Theory and Algebraic Geometry
Title | Number Theory and Algebraic Geometry PDF eBook |
Author | Miles Reid |
Publisher | Cambridge University Press |
Pages | 312 |
Release | 2003 |
Genre | Mathematics |
ISBN | 9780521545181 |
This volume honors Sir Peter Swinnerton-Dyer's mathematical career spanning more than 60 years' of amazing creativity in number theory and algebraic geometry.
Algebraic Geometry and Number Theory
Title | Algebraic Geometry and Number Theory PDF eBook |
Author | victor ginzburg |
Publisher | Springer Science & Business Media |
Pages | 656 |
Release | 2007-12-31 |
Genre | Mathematics |
ISBN | 0817645322 |
This book represents a collection of invited papers by outstanding mathematicians in algebra, algebraic geometry, and number theory dedicated to Vladimir Drinfeld. Original research articles reflect the range of Drinfeld's work, and his profound contributions to the Langlands program, quantum groups, and mathematical physics are paid particular attention. These ten original articles by prominent mathematicians, dedicated to Drinfeld on the occasion of his 50th birthday, broadly reflect the range of Drinfeld's own interests in algebra, algebraic geometry, and number theory.
Algebraic Geometry and Number Theory
Title | Algebraic Geometry and Number Theory PDF eBook |
Author | Hussein Mourtada |
Publisher | Birkhäuser |
Pages | 232 |
Release | 2017-05-16 |
Genre | Mathematics |
ISBN | 9783319477787 |
This lecture notes volume presents significant contributions from the “Algebraic Geometry and Number Theory” Summer School, held at Galatasaray University, Istanbul, June 2-13, 2014. It addresses subjects ranging from Arakelov geometry and Iwasawa theory to classical projective geometry, birational geometry and equivariant cohomology. Its main aim is to introduce these contemporary research topics to graduate students who plan to specialize in the area of algebraic geometry and/or number theory. All contributions combine main concepts and techniques with motivating examples and illustrative problems for the covered subjects. Naturally, the book will also be of interest to researchers working in algebraic geometry, number theory and related fields.
Geometric Methods in Algebra and Number Theory
Title | Geometric Methods in Algebra and Number Theory PDF eBook |
Author | Fedor Bogomolov |
Publisher | Springer Science & Business Media |
Pages | 365 |
Release | 2006-06-22 |
Genre | Mathematics |
ISBN | 0817644172 |
* Contains a selection of articles exploring geometric approaches to problems in algebra, algebraic geometry and number theory * The collection gives a representative sample of problems and most recent results in algebraic and arithmetic geometry * Text can serve as an intense introduction for graduate students and those wishing to pursue research in algebraic and arithmetic geometry
Number Theory and Geometry: An Introduction to Arithmetic Geometry
Title | Number Theory and Geometry: An Introduction to Arithmetic Geometry PDF eBook |
Author | Álvaro Lozano-Robledo |
Publisher | American Mathematical Soc. |
Pages | 488 |
Release | 2019-03-21 |
Genre | Arithmetical algebraic geometry |
ISBN | 147045016X |
Geometry and the theory of numbers are as old as some of the oldest historical records of humanity. Ever since antiquity, mathematicians have discovered many beautiful interactions between the two subjects and recorded them in such classical texts as Euclid's Elements and Diophantus's Arithmetica. Nowadays, the field of mathematics that studies the interactions between number theory and algebraic geometry is known as arithmetic geometry. This book is an introduction to number theory and arithmetic geometry, and the goal of the text is to use geometry as the motivation to prove the main theorems in the book. For example, the fundamental theorem of arithmetic is a consequence of the tools we develop in order to find all the integral points on a line in the plane. Similarly, Gauss's law of quadratic reciprocity and the theory of continued fractions naturally arise when we attempt to determine the integral points on a curve in the plane given by a quadratic polynomial equation. After an introduction to the theory of diophantine equations, the rest of the book is structured in three acts that correspond to the study of the integral and rational solutions of linear, quadratic, and cubic curves, respectively. This book describes many applications including modern applications in cryptography; it also presents some recent results in arithmetic geometry. With many exercises, this book can be used as a text for a first course in number theory or for a subsequent course on arithmetic (or diophantine) geometry at the junior-senior level.
Noncommutative Geometry and Number Theory
Title | Noncommutative Geometry and Number Theory PDF eBook |
Author | Caterina Consani |
Publisher | Springer Science & Business Media |
Pages | 374 |
Release | 2007-12-18 |
Genre | Mathematics |
ISBN | 3834803529 |
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.
Algebraic Geometry and Arithmetic Curves
Title | Algebraic Geometry and Arithmetic Curves PDF eBook |
Author | Qing Liu |
Publisher | Oxford University Press |
Pages | 593 |
Release | 2006-06-29 |
Genre | Mathematics |
ISBN | 0191547808 |
This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the funadmental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises.