By M. Tsfasman, S.G. Vladut
1. Codes.- 1.1. Codes and their parameters.- 1.2. Examples and constructions.- 1.3. Asymptotic problems.- 2. Curves.- 2.1. Algebraic curves.- 2.2. Riemann-Roch theorem.- 2.3. Rational points.- 2.4. Elliptic curves.- 2.5. Singular curves.- 2.6. mark downs and schemes.- three. AG-Codes.- 3.1. structures and properties.- 3.2. Examples.- 3.3. Decoding.- 3.4. Asymptotic results.- four. Modular Codes.- 4.1. Codes on classical modular curves.- 4.2. Codes on Drinfeld curves.- 4.3. Polynomiality.- five. Sphere Packings.- 5.1. Definitions and examples.- 5.2. Asymptotically dense packings.- 5.3. quantity fields.- 5.4. Analogues of AG-codes.- Appendix. precis of effects and tables.- A.1. Codes of finite length.- A.1.1. Bounds.- A.1.2. Parameters of yes codes.- A.1.3. Parameters of yes constructions.- A.1.4. Binary codes from AG-codes.- A.2. Asymptotic bounds.- A.2.1. record of bounds.- A.2.2. Diagrams of comparison.- A.2.3. Behaviour on the ends.- A.2.4. Numerical values.- A.3. extra bounds.- A.3.1. consistent weight codes.- A.3.2. Self-dual codes.- A.4. Sphere packings.- A.4.1. Small dimensions.- A.4.2. definite families.- A.4.3. Asymptotic results.- writer index.- record of symbols.
By Herwig Hauser, Joseph Lipman, Frans Oort, Adolfo Quiros
In September 1997, the operating Week on answer of Singularities used to be held at Obergurgi within the Tyrolean Alps. Its target was once to occur the cutting-edge within the box and to formulate significant questions for destiny study. The 4 classes given in this week have been written up by means of the audio system and make up half I of this quantity. they're complemented partly II by means of fifteen chosen contributions on particular themes and backbone theories.
The quantity is meant to supply a huge and available advent to answer of singularities best the reader on to concrete study difficulties.
Series: growth in arithmetic, Vol. 181
By Michal Krizek, Florian Luca, Lawrence Somer, A. Solcova
The pioneering paintings of French mathematician Pierre de Fermat has attracted the eye of mathematicians for over 350 years. This publication used to be written in honor of the four-hundredth anniversary of his delivery, offering readers with an summary of the numerous homes of Fermat numbers and demonstrating their purposes in parts comparable to quantity conception, chance concept, geometry, and sign processing. This booklet introduces a basic mathematical viewers to uncomplicated mathematical rules and algebraic tools hooked up with the Fermat numbers.
By Serge Lang (auth.)
Diophantine difficulties symbolize the various most powerful aesthetic points of interest to algebraic geometry. They consist in giving standards for the lifestyles of suggestions of algebraic equations in earrings and fields, and at last for the variety of such strategies. the basic ring of curiosity is the hoop of normal integers Z, and the basic box of curiosity is the sector Q of rational numbers. One discovers quickly that to have all of the technical freedom wanted in dealing with normal difficulties, one needs to think about earrings and fields of finite sort over the integers and rationals. additionally, one is ended in reflect on additionally finite fields, p-adic fields (including the genuine and complicated numbers) as representing a localization of the issues into consideration. we will care for worldwide difficulties, all of that allows you to be of a qualitative nature. at the one hand we've got curves outlined over say the rational numbers. Ifthe curve is affine one may well ask for its issues in Z, and due to Siegel, you possibly can classify all curves that have infinitely many imperative issues. This challenge is taken care of in bankruptcy VII. One could ask additionally for these that have infinitely many rational issues, and for this, there's simply Mordell's conjecture that if the genus is :;;; 2, then there's just a finite variety of rational points.
By John W. Rutter
This survey covers teams of homotopy self-equivalence sessions of topological areas, and the homotopy form of areas of homotopy self-equivalences. For manifolds, the complete crew of equivalences and the mapping classification staff are in comparison, as are the corresponding areas. incorporated are tools of calculation, quite a few calculations, finite new release effects, Whitehead torsion and different components. a few 330 references are given. The e-book assumes familiarity with phone complexes, homology and homotopy. Graduate scholars and proven researchers can use it for studying, for reference, and to figure out the present country of knowledge.
By Harry Reimann
This monograph is worried with the Shimura style connected to a quaternion algebra over a wholly actual quantity box. For anywhere of fine (or reasonably undesirable) relief, the corresponding (semi-simple) neighborhood zeta functionality is expressed by way of (semi-simple) neighborhood L-functions hooked up to automorphic representations. In an appendix a conjecture of Langlands and Rapoport at the relief of a Shimura sort in a really normal case is restated in a marginally better shape. The reader is anticipated to be accustomed to the elemental techniques of algebraic geometry, algebraic quantity idea and the idea of automorphic representation.
By Michael Harris
This ebook goals first to turn out the neighborhood Langlands conjecture for GLn over a p-adic box and, moment, to spot the motion of the decomposition crew at a primary of undesirable aid at the l-adic cohomology of the "simple" Shimura kinds. those difficulties pass hand in hand. the implications characterize an important develop in algebraic quantity idea, eventually proving the conjecture first proposed in Langlands's 1969 Washington lecture as a non-abelian generalization of neighborhood category box theory.The neighborhood Langlands conjecture for GLn(K), the place okay is a p-adic box, asserts the life of a correspondence, with yes formal homes, concerning n-dimensional representations of the Galois workforce of ok with the illustration idea of the in the community compact workforce GLn(K). This booklet constructs a candidate for this type of neighborhood Langlands correspondence at the vanishing cycles hooked up to the undesirable aid over the integer ring of okay of a undeniable kinfolk of Shimura types. And it proves that this is often approximately suitable with the worldwide Galois correspondence discovered at the cohomology of a similar Shimura forms. The neighborhood Langlands conjecture is received as a corollary.Certain options constructed during this booklet may still expand to extra common Shimura forms, delivering new cases of the neighborhood Langlands conjecture. in addition, the geometry of the distinct fibers is exactly analogous to that of Shimura curves and will be anticipated to have purposes to a number of questions in quantity conception.