Drinfeld Modular Curves
Author: Ernst-Ulrich Gekeler
Publisher: Springer
Published: 2006-11-14
Total Pages: 122
ISBN-13: 3540473866
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The Arithmetic of Drinfeld Modules
Author: Dragos Florin Ghioca
Publisher:
Published: 2005
Total Pages: 340
ISBN-13:
DOWNLOAD EBOOKFunction Field Arithmetic
Author: Dinesh S. Thakur
Publisher: World Scientific
Published: 2004
Total Pages: 405
ISBN-13: 9812388397
DOWNLOAD EBOOKThis 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.
Arithmetic and Geometry over Local Fields
Author: Bruno Anglès
Publisher: Springer Nature
Published: 2021-03-03
Total Pages: 337
ISBN-13: 3030662497
DOWNLOAD EBOOKThis 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.
Cohomology of Drinfeld Modular Varieties, Part 1, Geometry, Counting of Points and Local Harmonic Analysis
Author: Gérard Laumon
Publisher: Cambridge University Press
Published: 1996
Total Pages: 362
ISBN-13: 0521470609
DOWNLOAD EBOOKOriginally published in 1995, Cohomology of Drinfeld Modular Varieties aimed to provide an introduction, in two volumes, both to this subject and to the Langlands correspondence for function fields. These varieties are the analogues for function fields of the Shimura varieties over number fields. The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field. By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case. The present volume is devoted to the geometry of these varieties, and to the local harmonic analysis needed to compute their cohomology. Though the author considers only the simpler case of function rather than number fields, many important features of the number field case can be illustrated.
Drinfeld Modules
Author: Mihran Papikian
Publisher: Springer Nature
Published: 2023-03-31
Total Pages: 541
ISBN-13: 3031197070
DOWNLOAD EBOOKThis textbook offers an introduction to the theory of Drinfeld modules, mathematical objects that are fundamental to modern number theory. After the first two chapters conveniently recalling prerequisites from abstract algebra and non-Archimedean analysis, Chapter 3 introduces Drinfeld modules and the key notions of isogenies and torsion points. Over the next four chapters, Drinfeld modules are studied in settings of various fields of arithmetic importance, culminating in the case of global fields. Throughout, numerous number-theoretic applications are discussed, and the analogies between classical and function field arithmetic are emphasized. Drinfeld Modules guides readers from the basics to research topics in function field arithmetic, assuming only familiarity with graduate-level abstract algebra as prerequisite. With exercises of varying difficulty included in each section, the book is designed to be used as the primary textbook for a graduate course on the topic, and may also provide a supplementary reference for courses in algebraic number theory, elliptic curves, and related fields. Furthermore, researchers in algebra and number theory will appreciate it as a self-contained reference on the topic.
The Arithmetic of Function Fields
Author: David Goss
Publisher: Walter de Gruyter
Published: 2011-06-24
Total Pages: 493
ISBN-13: 3110886154
DOWNLOAD EBOOKThisseries 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.
Basic Structures of Function Field Arithmetic
Author: David Goss
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 433
ISBN-13: 3642614809
DOWNLOAD EBOOKFrom 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
Arithmetic Differential Equations
Author: Alexandru Buium
Publisher: American Mathematical Soc.
Published: 2005
Total Pages: 346
ISBN-13: 0821838628
DOWNLOAD EBOOKFor most of the book the only prerequisites are the basic facts of algebraic geometry and number theory."--BOOK JACKET.
Drinfeld Moduli Schemes and Automorphic Forms
Author: Yuval Z Flicker
Publisher: Springer Science & Business Media
Published: 2013-01-04
Total Pages: 150
ISBN-13: 1461458889
DOWNLOAD EBOOKDrinfeld Moduli Schemes and Automorphic Forms: The Theory of Elliptic Modules with Applications is based on the author’s original work establishing the correspondence between ell-adic rank r Galois representations and automorphic representations of GL(r) over a function field, in the local case, and, in the global case, under a restriction at a single place. It develops Drinfeld’s theory of elliptic modules, their moduli schemes and covering schemes, the simple trace formula, the fixed point formula, as well as the congruence relations and a "simple" converse theorem, not yet published anywhere. This version, based on a recent course taught by the author at The Ohio State University, is updated with references to research that has extended and developed the original work. The use of the theory of elliptic modules in the present work makes it accessible to graduate students, and it will serve as a valuable resource to facilitate an entrance to this fascinating area of mathematics.
