Lectures on the Arithmetic Riemann-Roch Theorem. (AM-127)

Author: Gerd Faltings

Publisher: Princeton University Press

ISBN: 1400882478

Category: Mathematics

Page: 118

View: 4630

The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof. It should enable mathematicians with a background in arithmetic algebraic geometry to understand some basic techniques in the rapidly evolving field of Arakelov-theory.
Release

Lectures on Arakelov Geometry

Author: C. Soulé,D. Abramovich,J. F. Burnol

Publisher: Cambridge University Press

ISBN: 9780521477093

Category: Mathematics

Page: 177

View: 2134

Arakelov theory is a new geometric approach to diophantine equations. It combines algebraic geometry, in the sense of Grothendieck, with refined analytic tools such as currents on complex manifolds and the spectrum of Laplace operators. It has been used by Faltings and Vojta in their proofs of outstanding conjectures in diophantine geometry. This account presents the work of Gillet and Soulé, extending Arakelov geometry to higher dimensions. It includes a proof of Serre's conjecture on intersection multiplicities and an arithmetic Riemann-Roch theorem. To aid number theorists, background material on differential geometry is described, but techniques from algebra and analysis are covered as well. Several open problems and research themes are also mentioned.
Release

Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties

Author: Jorg Jahnel

Publisher: American Mathematical Soc.

ISBN: 1470418827

Category: Mathematics

Page: 267

View: 6371

The central theme of this book is the study of rational points on algebraic varieties of Fano and intermediate type--both in terms of when such points exist and, if they do, their quantitative density. The book consists of three parts. In the first part, the author discusses the concept of a height and formulates Manin's conjecture on the asymptotics of rational points on Fano varieties. The second part introduces the various versions of the Brauer group. The author explains why a Brauer class may serve as an obstruction to weak approximation or even to the Hasse principle. This part includes two sections devoted to explicit computations of the Brauer-Manin obstruction for particular types of cubic surfaces. The final part describes numerical experiments related to the Manin conjecture that were carried out by the author together with Andreas-Stephan Elsenhans. The book presents the state of the art in computational arithmetic geometry for higher-dimensional algebraic varieties and will be a valuable reference for researchers and graduate students interested in that area.
Release

Computational Aspects of Modular Forms and Galois Representations

How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Author: Bas Edixhoven,Jean-Marc Couveignes

Publisher: Princeton University Press

ISBN: 9780691142029

Category: Mathematics

Page: 440

View: 4991

Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.
Release

Books in Print

Author: N.A

Publisher: N.A

ISBN: N.A

Category: American literature

Page: N.A

View: 2090

Release

Rational Points

Seminar Bonn/Wuppertal 1983/84 A Publication of the Max-Planck-Institut für Mathematik, Bonn

Author: Gerd Faltings

Publisher: Springer-Verlag

ISBN: 3322839184

Category: Mathematics

Page: 268

View: 965

Release

Books in Print

Author: R.R. Bowker Company

Publisher: N.A

ISBN: N.A

Category: American literature

Page: N.A

View: 2723

Books in print is the major source of information on books currently published and in print in the United States. The database provides the record of forthcoming books, books in-print, and books out-of-print.
Release

Forthcoming Books

Author: Rose Arny

Publisher: N.A

ISBN: N.A

Category: American literature

Page: N.A

View: 4297

Release

Séminaire Bourbaki

volume 2005/2006 : exposés 952-966

Author: Société mathématique de France

Publisher: N.A

ISBN: N.A

Category: Algebra

Page: 401

View: 7849

As in the preceding volumes of this seminar, one finds here fifteen survey lectures on topics of current interest: two lectures on algebraic geometry, three on partial differential equations, one on Arakelov geometry, one on $p$-adic analytic geometry, one on Galois representations, one on number theory, one on $p$-divisible groups, one on graph theory, one on operator algebras, one on representation theory, one on singular integral operators, and one on dynamical systems.
Release

Endliche Gruppen

Eine Einführung in die Theorie der endlichen Gruppen

Author: H. Kurzweil

Publisher: Springer-Verlag

ISBN: 3642953131

Category: Mathematics

Page: 190

View: 3656

Release

Helmut Hasse und Emmy Noether

Author: Helmut Hasse,Emmy Noether,Franz Lemmermeyer,Peter Roquette

Publisher: Universitätsverlag Göttingen

ISBN: 3938616350

Category: Class field theory

Page: 301

View: 1882

Release