Arithmetic of Quadratic Forms

verfasst von: Goro Shimura

Verlag: Springer New York

Buchreihe : Springer Monographs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book can be divided into two parts. The ?rst part is preliminary and consists of algebraic number theory and the theory of semisimple algebras. The raison d’ˆ etre of the book is in the second part, and so let us ?rst explain the contents of the second part. There are two principal topics: (A) Classi?cation of quadratic forms; (B) Quadratic Diophantine equations. Topic (A) can be further divided into two types of theories: (a1) Classi?cation over an algebraic number ?eld; (a2) Classi?cation over the ring of algebraic integers. To classify a quadratic form ? over an algebraic number ?eld F, almost all previous authors followed the methods of Helmut Hasse. Namely, one ?rst takes ? in the diagonal form and associates an invariant to it at each prime spot of F, using the diagonal entries. A superior method was introduced by Martin Eichler in 1952, but strangely it was almost completely ignored, until I resurrected it in one of my recent papers. We associate an invariant to ? at each prime spot, which is the same as Eichler’s, but we de?ne it in a di?erent and more direct way, using Cli?ord algebras. In Sections 27 and 28 we give an exposition of this theory. At some point we need the Hasse norm theorem for a quadratic extension of a number ?eld, which is included in class ?eld theory. We prove it when the base ?eld is the rational number ?eld to make the book self-contained in that case.

Inhaltsverzeichnis

Frontmatter

Chapter I. The Quadratic Reciprocity Law

Abstract

In this section we recall several well-known elementary facts, mostly without proof. We give the proof for some of them. An ideal I of a commutative ring R is called a prime ideal if R/I has no zero divisors; I is called principal if I = αR with some \(\,{\alpha}\in R.\) An integral domain (that is, a commutative ring with identity element that has no zero divisors) R is called a principal ideal domain if every ideal of R is principal. It is known that for a field F and an indeterminate x the polynomial ring F[x] is a principal ideal domain. Also, the ring Z. is a principal ideal domain. An integral domain is called a unique factorization domain if every principal ideal I of R different from {0} can be written uniquely in the form \(I=P_1^{e_1}\cdots P_r^{e_r}\) with prime ideals P _i that are principal and \(\,0 < e_i\in{\bold{Z}}.\)

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Chapter II. Arithmetic in an Algebraic Number Field

Abstract

6.1. Let F be a field. A map \(\,\nu : F\to {\bold{R}}\cup\{\infty\}\) is called an order function of F if it satisfies the following conditions:

(i)

\(\,\nu(x)=\infty\,\Longleftrightarrow\, x=0;\)

(ii)

\(\,\nu(xy)=\nu(x) +\nu(y);\)

(iii)

\(\,\nu(x+y)\ge\text{Min}\big\{\nu(x),\,\nu(y)\big\};\)

(iv)

There exists an element \(\,z\in F,\,\ne0,\,\) such that \(\,\nu(z)\ne 0.\)

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Chapter III. Various Basic Theorems

Abstract

Let A be a vector space over F that has a law of multiplication which, together with the existing law of addition, makes A an associative ring. We call A an algebra over F, or simply an F -algebra, if \(\,c({\alpha}{\beta}) =(c{\alpha}){\beta}={\alpha}(c{\beta})\,\) for every \(\,c\in F\) and \(\,{\alpha},\,{\beta}\in A.\) If A has an identity element \(1_A,\) then identifying \(\,c\,\) with \(\,c1_A,\) we can view F as a subfield of A. Notice that \(\,c{\alpha}=(c1_A){\alpha}={\alpha}(c1_A) ,\) and so two laws of multiplication for the elements of F (one in the vector space and the other in the ring) are the same. Every field extension of F can naturally be viewed as an F-algebra.

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Chapter IV. Algebras Over a Field

Abstract

18.1. Let us first recall the notion of an algebra over a field that we introduced in §11.1. By an algebra over a field F, or simply by an F -algebra, we understand an associative ring A which is also a vector space over F such that \((ax)(by)=abxy\) for \(\,a,\,b\in F\) and \(\,x,\,y\in A.\) If A has an identity element, we denote it by \(1_A,\) or simply by \(1.\) Identifying \(\,a\,\) with \(\,a1_A\) for every \(\,a\in F,\) we can view F as a subring of A.

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Chapter V. Quadratic Forms

Abstract

We take a base field F and consider a finite-dimensional vector space V over F and an F-valued F-bilinear form \(\,\varphi :V\times V\to F.\) We call, as usual, \(\,\varphi\,\) symmetric if \(\,\varphi(x,\,y)=\varphi(y,x)\) for every \(\,x,\,y \in V.\)

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Chapter VI. Deeper Arithmetic of Quadratic Forms

Abstract

Let F be either an algebraic number field or its completion at a prime \(\,v\in{\bold{v}},\) and \((V,\,\varphi)\) a nondegenerate quadratic space over F of dimension \(\,n.\) We then define the characteristic (quaternion) algebra of \((V,\,\varphi)\) as follows.

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Chapter VII. Quadratic Diophantine Equations

Abstract

34.1. We take a nondegenerate quadratic space \((V,\,{\varphi})\) of dimension \(\,n\,\) over a local or global field F in the sense of §21.1.

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Backmatter

Titel: Arithmetic of Quadratic Forms
verfasst von: Goro Shimura
Verlag: Springer New York
Electronic ISBN: 978-1-4419-1732-4
Print ISBN: 978-1-4419-1731-7
DOI: https://doi.org/10.1007/978-1-4419-1732-4