The Arithmetic Of Function Fields

Author : David Goss
Publisher : Springer Science & Business Media
Page : 433 pages
File Size : 55,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642614804

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Basic Structures of Function Field Arithmetic by David Goss Pdf

From the reviews:"The book...is a thorough and very readable introduction to the arithmetic of function fields of one variable over a finite field, by an author who has made fundamental contributions to the field. It serves as a definitive reference volume, as well as offering graduate students with a solid understanding of algebraic number theory the opportunity to quickly reach the frontiers of knowledge in an important area of mathematics...The arithmetic of function fields is a universe filled with beautiful surprises, in which familiar objects from classical number theory reappear in new guises, and in which entirely new objects play important roles. Goss'clear exposition and lively style make this book an excellent introduction to this fascinating field." MR 97i:11062

Author : Dinesh S. Thakur
Publisher : World Scientific
Page : 405 pages
File Size : 42,6 Mb
Release : 2004
Category : Mathematics
ISBN : 9789812388391

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Function Field Arithmetic by Dinesh S. Thakur Pdf

This book provides an exposition of function field arithmetic with emphasis on recent developments concerning Drinfeld modules, the arithmetic of special values of transcendental functions (such as zeta and gamma functions and their interpolations), diophantine approximation and related interesting open problems. While it covers many topics treated in 'Basic Structures of Function Field Arithmetic' by David Goss, it complements that book with the inclusion of recent developments as well as the treatment of new topics such as diophantine approximation, hypergeometric functions, modular forms, transcendence, automata and solitons. There is also new work on multizeta values and log-algebraicity. The author has included numerous worked-out examples. Many open problems, which can serve as good thesis problems, are discussed.

Author : David Goss,David R. Hayes,Michael Rosen
Publisher : Walter de Gruyter
Page : 493 pages
File Size : 44,9 Mb
Release : 2011-06-24
Category : Mathematics
ISBN : 9783110886153

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The Arithmetic of Function Fields by David Goss,David R. Hayes,Michael Rosen Pdf

Thisseries is devoted to the publication of monographs, lecture resp. seminar notes, and other materials arising from programs of the OSU Mathemaical Research Institute. This includes proceedings of conferences or workshops held at the Institute, and other mathematical writings.

Author : Michael Rosen
Publisher : Springer Science & Business Media
Page : 355 pages
File Size : 40,6 Mb
Release : 2013-04-18
Category : Mathematics
ISBN : 9781475760460

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Number Theory in Function Fields by Michael Rosen Pdf

Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting analogues of various theorems. The later chapters probe the analogy between global function fields and algebraic number fields. Topics include the ABC-conjecture, Brumer-Stark conjecture, and Drinfeld modules.

Author : Henning Stichtenoth
Publisher : Springer Science & Business Media
Page : 360 pages
File Size : 47,9 Mb
Release : 2009-02-11
Category : Mathematics
ISBN : 9783540768784

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Algebraic Function Fields and Codes by Henning Stichtenoth Pdf

This book links two subjects: algebraic geometry and coding theory. It uses a novel approach based on the theory of algebraic function fields. Coverage includes the Riemann-Rock theorem, zeta functions and Hasse-Weil's theorem as well as Goppa' s algebraic-geometric codes and other traditional codes. It will be useful to researchers in algebraic geometry and coding theory and computer scientists and engineers in information transmission.

Author : Michael D. Fried,Moshe Jarden
Publisher : Springer Science & Business Media
Page : 812 pages
File Size : 44,6 Mb
Release : 2005
Category : Algebraic fields
ISBN : 354022811X

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Field Arithmetic by Michael D. Fried,Moshe Jarden Pdf

Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements. Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)?

Author : Gebhard Böckle,David Burns,David Goss,Dinesh Thakur,Fabien Trihan,Douglas Ulmer
Publisher : Springer
Page : 350 pages
File Size : 55,8 Mb
Release : 2014-11-13
Category : Mathematics
ISBN : 9783034808538

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Arithmetic Geometry over Global Function Fields by Gebhard Böckle,David Burns,David Goss,Dinesh Thakur,Fabien Trihan,Douglas Ulmer Pdf

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 : Gabriel Daniel Villa Salvador
Publisher : Springer Science & Business Media
Page : 658 pages
File Size : 51,6 Mb
Release : 2007-10-10
Category : Mathematics
ISBN : 9780817645151

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Topics in the Theory of Algebraic Function Fields by Gabriel Daniel Villa Salvador Pdf

The fields of algebraic functions of one variable appear in several areas of mathematics: complex analysis, algebraic geometry, and number theory. This text adopts the latter perspective by applying an arithmetic-algebraic viewpoint to the study of function fields as part of the algebraic theory of numbers. The examination explains both the similarities and fundamental differences between function fields and number fields, including many exercises and examples to enhance understanding and motivate further study. The only prerequisites are a basic knowledge of field theory, complex analysis, and some commutative algebra.

Author : Michael D. Fried,Moshe Jarden
Publisher : Springer Science & Business Media
Page : 803 pages
File Size : 50,5 Mb
Release : 2005-08-29
Category : Mathematics
ISBN : 9783540269496

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Field Arithmetic by Michael D. Fried,Moshe Jarden Pdf

Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements. Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)?

Author : Dennis Gaitsgory,Jacob Lurie
Publisher : Princeton University Press
Page : 320 pages
File Size : 49,8 Mb
Release : 2019-02-19
Category : Mathematics
ISBN : 9780691184432

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Weil's Conjecture for Function Fields by Dennis Gaitsgory,Jacob Lurie Pdf

A central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field K in terms of the behavior of various completions of K. This book looks at a specific example of a local-to-global principle: Weil’s conjecture on the Tamagawa number of a semisimple algebraic group G over K. In the case where K is the function field of an algebraic curve X, this conjecture counts the number of G-bundles on X (global information) in terms of the reduction of G at the points of X (local information). The goal of this book is to give a conceptual proof of Weil’s conjecture, based on the geometry of the moduli stack of G-bundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting l-adic sheaves. Using this theory, Dennis Gaitsgory and Jacob Lurie articulate a different local-to-global principle: a product formula that expresses the cohomology of the moduli stack of G-bundles (a global object) as a tensor product of local factors. Using a version of the Grothendieck-Lefschetz trace formula, Gaitsgory and Lurie show that this product formula implies Weil’s conjecture. The proof of the product formula will appear in a sequel volume.

Author : Bruno Anglès,Tuan Ngo Dac
Publisher : Springer Nature
Page : 337 pages
File Size : 55,6 Mb
Release : 2021-03-03
Category : Mathematics
ISBN : 9783030662493

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Arithmetic and Geometry over Local Fields by Bruno Anglès,Tuan Ngo Dac Pdf

This volume introduces some recent developments in Arithmetic Geometry over local fields. Its seven chapters are centered around two common themes: the study of Drinfeld modules and non-Archimedean analytic geometry. The notes grew out of lectures held during the research program "Arithmetic and geometry of local and global fields" which took place at the Vietnam Institute of Advanced Study in Mathematics (VIASM) from June to August 2018. The authors, leading experts in the field, have put great effort into making the text as self-contained as possible, introducing the basic tools of the subject. The numerous concrete examples and suggested research problems will enable graduate students and young researchers to quickly reach the frontiers of this fascinating branch of mathematics.

Author : John Knopfmacher
Publisher : Unknown
Page : 156 pages
File Size : 42,7 Mb
Release : 1979
Category : Mathematics
ISBN : UCAL:B4562804

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Analytic Arithmetic of Algebraic Function Fields by John Knopfmacher Pdf

Author : John Knopfmacher
Publisher : Unknown
Page : 142 pages
File Size : 41,5 Mb
Release : 2024-07-03
Category : Electronic
ISBN : 0608089478

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Analytic Arithmetic of Algebraic Function Fields by John Knopfmacher Pdf

Author : Gebhard Böckle,Richard Pink
Publisher : European Mathematical Society
Page : 200 pages
File Size : 54,6 Mb
Release : 2009
Category : Mathematics
ISBN : 3037190744

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Cohomological Theory of Crystals Over Function Fields by Gebhard Böckle,Richard Pink Pdf

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 : Gerard B. M. van der Geer,BJJ Moonen,René Schoof
Publisher : Springer Science & Business Media
Page : 323 pages
File Size : 45,7 Mb
Release : 2006-11-24
Category : Mathematics
ISBN : 9780817644475

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Number Fields and Function Fields – Two Parallel Worlds by Gerard B. M. van der Geer,BJJ Moonen,René Schoof Pdf

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