Arithmetic Geometry Over Global Function Fields PDF Download
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Author | : Gebhard Böckle |
Publisher | : Springer |
Total Pages | : 350 |
Release | : 2014-11-13 |
Genre | : Mathematics |
ISBN | : 3034808534 |
Download Arithmetic Geometry over Global Function Fields Book in PDF, ePub and Kindle
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 | : X. W. C. Faber |
Publisher | : |
Total Pages | : 254 |
Release | : 2008 |
Genre | : |
ISBN | : |
Download Topics in Arithmetic Geometry Over Function Fields Book in PDF, ePub and Kindle
Author | : Bruno Anglès |
Publisher | : Springer Nature |
Total Pages | : 337 |
Release | : 2021-03-03 |
Genre | : Mathematics |
ISBN | : 3030662497 |
Download Arithmetic and Geometry over Local Fields Book in PDF, ePub and Kindle
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 | : Lin Weng |
Publisher | : World Scientific |
Total Pages | : 411 |
Release | : 2006-06-02 |
Genre | : Mathematics |
ISBN | : 9814477931 |
Download Arithmetic Geometry And Number Theory Book in PDF, ePub and Kindle
Mathematics is very much a part of our culture; and this invaluable collection serves the purpose of developing the branches involved, popularizing the existing theories and guiding our future explorations.More precisely, the goal is to bring the reader to the frontier of current developments in arithmetic geometry and number theory through the works of Deninger-Werner in vector bundles on curves over p-adic fields; of Jiang on local gamma factors in automorphic representations; of Weng on Deligne pairings and Takhtajan-Zograf metrics; of Yoshida on CM-periods; of Yu on transcendence of special values of zetas over finite fields. In addition, the lecture notes presented by Weng at the University of Toronto from October to November 2005 explain basic ideas and the reasons (not just the language and conclusions) behind Langlands' fundamental, yet notably difficult, works on the Eisenstein series and spectral decompositions.And finally, a brand new concept by Weng called the Geometric Arithmetic program that uses algebraic and/or analytic methods, based on geometric considerations, to develop the promising and yet to be cultivated land of global arithmetic that includes non-abelian Class Field Theory, Riemann Hypothesis and non-abelian Zeta and L Functions, etc.
Author | : Lisa Berger |
Publisher | : American Mathematical Soc. |
Total Pages | : 131 |
Release | : 2020-09-28 |
Genre | : Mathematics |
ISBN | : 1470442191 |
Download Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields Book in PDF, ePub and Kindle
The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $mathbb F_p(t)$, when $p$ is prime and $rge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $mathbb F_q(t^1/d)$.
Author | : Michael Rosen |
Publisher | : Springer Science & Business Media |
Total Pages | : 355 |
Release | : 2013-04-18 |
Genre | : Mathematics |
ISBN | : 1475760469 |
Download Number Theory in Function Fields Book in PDF, ePub and Kindle
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 | : Dinesh S. Thakur |
Publisher | : World Scientific |
Total Pages | : 405 |
Release | : 2004 |
Genre | : Mathematics |
ISBN | : 9812388397 |
Download Function Field Arithmetic Book in PDF, ePub and Kindle
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 | : Jean-Louis Colliot-Thélène |
Publisher | : Springer |
Total Pages | : 251 |
Release | : 2010-10-27 |
Genre | : Mathematics |
ISBN | : 3642159451 |
Download Arithmetic Geometry Book in PDF, ePub and Kindle
Arithmetic Geometry can be defined as the part of Algebraic Geometry connected with the study of algebraic varieties through arbitrary rings, in particular through non-algebraically closed fields. It lies at the intersection between classical algebraic geometry and number theory. A C.I.M.E. Summer School devoted to arithmetic geometry was held in Cetraro, Italy in September 2007, and presented some of the most interesting new developments in arithmetic geometry. This book collects the lecture notes which were written up by the speakers. The main topics concern diophantine equations, local-global principles, diophantine approximation and its relations to Nevanlinna theory, and rationally connected varieties. The book is divided into three parts, corresponding to the courses given by J-L Colliot-Thelene, Peter Swinnerton Dyer and Paul Vojta.
Author | : Chih-Yun Chuang |
Publisher | : American Mathematical Soc. |
Total Pages | : 76 |
Release | : 2015-08-21 |
Genre | : Mathematics |
ISBN | : 1470414198 |
Download Brandt Matrices and Theta Series over Global Function Fields Book in PDF, ePub and Kindle
The aim of this article is to give a complete account of the Eichler-Brandt theory over function fields and the basis problem for Drinfeld type automorphic forms. Given arbitrary function field k together with a fixed place ∞, the authors construct a family of theta series from the norm forms of "definite" quaternion algebras, and establish an explicit Hecke-module homomorphism from the Picard group of an associated definite Shimura curve to a space of Drinfeld type automorphic forms. The "compatibility" of these homomorphisms with different square-free levels is also examined. These Hecke-equivariant maps lead to a nice description of the subspace generated by the authors' theta series, and thereby contributes to the so-called basis problem. Restricting the norm forms to pure quaternions, the authors obtain another family of theta series which are automorphic functions on the metaplectic group, and this results in a Shintani-type correspondence between Drinfeld type forms and metaplectic forms.
Author | : David Goss |
Publisher | : Springer Science & Business Media |
Total Pages | : 433 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3642614809 |
Download Basic Structures of Function Field Arithmetic Book in PDF, ePub and Kindle
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