Cohomological Theory Of Crystals Over Function Fields

Author : Gebhard Böckle
Publisher : European Mathematical Society
ISBN 13 : 9783037190746
Total Pages : 200 pages
Book Rating : 4.44/5 ( download)

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Download or read book Cohomological Theory of Crystals Over Function Fields written by Gebhard Böckle and published by European Mathematical Society. This book was released on 2009 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book develops a new cohomological theory for schemes in positive characteristic $p$ and it applies this theory to give a purely algebraic proof of a conjecture of Goss on the rationality of certain $L$-functions arising in the arithmetic of function fields. These $L$-functions are power series over a certain ring $A$, associated to any family of Drinfeld $A$-modules or, more generally, of $A$-motives on a variety of finite type over the finite field $\mathbb{F}_p$. By analogy to the Weil conjecture, Goss conjectured that these $L$-functions are in fact rational functions. In 1996 Taguchi and Wan gave a first proof of Goss's conjecture by analytic methods a la Dwork. The present text introduces $A$-crystals, which can be viewed as generalizations of families of $A$-motives, and studies their cohomology. While $A$-crystals are defined in terms of coherent sheaves together with a Frobenius map, in many ways they actually behave like constructible etale sheaves. A central result is a Lefschetz trace formula for $L$-functions of $A$-crystals, from which the rationality of these $L$-functions is immediate. Beyond its application to Goss's $L$-functions, the theory of $A$-crystals is closely related to the work of Emerton and Kisin on unit root $F$-crystals, and it is essential in an Eichler - Shimura type isomorphism for Drinfeld modular forms as constructed by the first author. The book is intended for researchers and advanced graduate students interested in the arithmetic of function fields and/or cohomology theories for varieties in positive characteristic. It assumes a good working knowledge in algebraic geometry as well as familiarity with homological algebra and derived categories, as provided by standard textbooks. Beyond that the presentation is largely self contained.

Author : Gebhard Böckle
Publisher : Springer
ISBN 13 : 3034808534
Total Pages : 350 pages
Book Rating : 4.38/5 ( download)

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Download or read book Arithmetic Geometry over Global Function Fields written by Gebhard Böckle and published by Springer. This book was released on 2014-11-13 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume collects the texts of five courses given in the Arithmetic Geometry Research Programme 2009-2010 at the CRM Barcelona. All of them deal with characteristic p global fields; the common theme around which they are centered is the arithmetic of L-functions (and other special functions), investigated in various aspects. Three courses examine some of the most important recent ideas in the positive characteristic theory discovered by Goss (a field in tumultuous development, which is seeing a number of spectacular advances): they cover respectively crystals over function fields (with a number of applications to L-functions of t-motives), gamma and zeta functions in characteristic p, and the binomial theorem. The other two are focused on topics closer to the classical theory of abelian varieties over number fields: they give respectively a thorough introduction to the arithmetic of Jacobians over function fields (including the current status of the BSD conjecture and its geometric analogues, and the construction of Mordell-Weil groups of high rank) and a state of the art survey of Geometric Iwasawa Theory explaining the recent proofs of various versions of the Main Conjecture, in the commutative and non-commutative settings.

Author : Gerard B. M. van der Geer
Publisher : Springer Science & Business Media
ISBN 13 : 0817644474
Total Pages : 323 pages
Book Rating : 4.75/5 ( download)

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Download or read book Number Fields and Function Fields – Two Parallel Worlds written by Gerard B. M. van der Geer and published by Springer Science & Business Media. This book was released on 2006-11-24 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: Invited articles by leading researchers explore various aspects of the parallel worlds of function fields and number fields Topics range from Arakelov geometry, the search for a theory of varieties over the field with one element, via Eisenstein series to Drinfeld modules, and t-motives Aimed at graduate students, mathematicians, and researchers interested in geometry and arithmetic and their connections

Author : Fabrizio Catanese
Publisher : Springer Science & Business Media
ISBN 13 : 3540354808
Total Pages : 508 pages
Book Rating : 4.02/5 ( download)

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Download or read book Global Aspects of Complex Geometry written by Fabrizio Catanese and published by Springer Science & Business Media. This book was released on 2006-09-29 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of surveys present an overview of recent developments in Complex Geometry. Topics range from curve and surface theory through special varieties in higher dimensions, moduli theory, Kähler geometry, and group actions to Hodge theory and characteristic p-geometry. Written by established experts this book will be a must for mathematicians working in Complex Geometry

Author : Chris Christensen
Publisher : Springer Science & Business Media
ISBN 13 : 9783540004752
Total Pages : 810 pages
Book Rating : 4.50/5 ( download)

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Download or read book Algebra, Arithmetic and Geometry with Applications written by Chris Christensen and published by Springer Science & Business Media. This book was released on 2004 with total page 810 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is the proceedings of the Conference on Algebra and Algebraic Geometry with Applications which was held July 19 – 26, 2000, at Purdue University to honor Professor Shreeram S. Abhyankar on the occasion of his seventieth birthday. Eighty-five of Professor Abhyankar's students, collaborators, and colleagues were invited participants. Sixty participants presented papers related to Professor Abhyankar's broad areas of mathematical interest. There were sessions on algebraic geometry, singularities, group theory, Galois theory, combinatorics, Drinfield modules, affine geometry, and the Jacobian problem. This volume offers an outstanding collection of papers by authors who are among the experts in their areas.

Author : Anders Björn
Publisher : European Mathematical Society
ISBN 13 : 9783037190999
Total Pages : 422 pages
Book Rating : 4.9X/5 ( download)

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Download or read book Nonlinear Potential Theory on Metric Spaces written by Anders Björn and published by European Mathematical Society. This book was released on 2011 with total page 422 pages. Available in PDF, EPUB and Kindle. Book excerpt: The $p$-Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory, and image processing. Its solutions, called p-harmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs, and Heisenberg groups. Nonlinear potential theory of p-harmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theories. This monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers. The aim is to serve both as an introduction to the area for interested readers and as a reference text for active researchers. The presentation is rather self contained, but it is assumed that readers know measure theory and functional analysis. The first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study p-harmonic functions on metric spaces, and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric space. Each chapter contains historical notes with relevant references, and an extensive index is provided at the end of the book.

Author : Irena Peeva
Publisher : Springer Science & Business Media
ISBN 13 : 1461452929
Total Pages : 705 pages
Book Rating : 4.28/5 ( download)

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Download or read book Commutative Algebra written by Irena Peeva and published by Springer Science & Business Media. This book was released on 2013-02-01 with total page 705 pages. Available in PDF, EPUB and Kindle. Book excerpt: This contributed volume brings together the highest quality expository papers written by leaders and talented junior mathematicians in the field of Commutative Algebra. Contributions cover a very wide range of topics, including core areas in Commutative Algebra and also relations to Algebraic Geometry, Algebraic Combinatorics, Hyperplane Arrangements, Homological Algebra, and String Theory. The book aims to showcase the area, especially for the benefit of junior mathematicians and researchers who are new to the field; it will aid them in broadening their background and to gain a deeper understanding of the current research in this area. Exciting developments are surveyed and many open problems are discussed with the aspiration to inspire the readers and foster further research.

Author : Lenny Taelman
Publisher : Cambridge University Press
ISBN 13 : 1316502597
Total Pages : 132 pages
Book Rating : 4.94/5 ( download)

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Download or read book Sheaves and Functions Modulo p written by Lenny Taelman and published by Cambridge University Press. This book was released on 2016 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt: Describes how to use coherent sheaves and cohomology to prove combinatorial and number theoretical identities over finite fields.

Author : Hans Triebel
Publisher : European Mathematical Society
ISBN 13 : 9783037190852
Total Pages : 314 pages
Book Rating : 4.5X/5 ( download)

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Download or read book Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration written by Hans Triebel and published by European Mathematical Society. This book was released on 2010 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first chapters of this book deal with Haar bases, Faber bases and some spline bases for function spaces in Euclidean $n$-space and $n$-cubes. These are used in the subsequent chapters to study sampling and numerical integration preferably in spaces with dominating mixed smoothness. The subject of the last chapter is the symbiotic relationship between numerical integration and discrepancy, measuring the deviation of sets of points from uniformity. This book is addressed to graduate students and mathematicians who have a working knowledge of basic elements of function spaces and approximation theory and who are interested in the subtle interplay between function spaces, complexity theory and number theory (discrepancy).

Author : Erich Novak
Publisher : European Mathematical Society
ISBN 13 : 9783037190845
Total Pages : 684 pages
Book Rating : 4.41/5 ( download)

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Download or read book Tractability of Multivariate Problems: Standard information for functionals written by Erich Novak and published by European Mathematical Society. This book was released on 2008 with total page 684 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the second volume of a three-volume set comprising a comprehensive study of the tractability of multivariate problems. The second volume deals with algorithms using standard information consisting of function values for the approximation of linear and selected nonlinear functionals. An important example is numerical multivariate integration. The proof techniques used in volumes I and II are quite different. It is especially hard to establish meaningful lower error bounds for the approximation of functionals by using finitely many function values. Here, the concept of decomposable reproducing kernels is helpful, allowing it to find matching lower and upper error bounds for some linear functionals. It is then possible to conclude tractability results from such error bounds. Tractability results, even for linear functionals, are very rich in variety. There are infinite-dimensional Hilbert spaces for which the approximation with an arbitrarily small error of all linear functionals requires only one function value. There are Hilbert spaces for which all nontrivial linear functionals suffer from the curse of dimensionality. This holds for unweighted spaces, where the role of all variables and groups of variables is the same. For weighted spaces one can monitor the role of all variables and groups of variables. Necessary and sufficient conditions on the decay of the weights are given to obtain various notions of tractability. The text contains extensive chapters on discrepancy and integration, decomposable kernels and lower bounds, the Smolyak/sparse grid algorithms, lattice rules and the CBC (component-by-component) algorithms. This is done in various settings. Path integration and quantum computation are also discussed. This volume is of interest to researchers working in computational mathematics, especially in approximation of high-dimensional problems. It is also well suited for graduate courses and seminars. There are 61 open problems listed to stimulate future research in tractability.