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Arithmetic of Quadratic Forms 2010 ed. [Kietas viršelis]

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  • Formatas: Hardback, 238 pages, aukštis x plotis: 235x155 mm, weight: 1170 g, XII, 238 p., 1 Hardback
  • Serija: Springer Monographs in Mathematics
  • Išleidimo metai: 30-Jun-2010
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1441917314
  • ISBN-13: 9781441917317
Kitos knygos pagal šią temą:
Arithmetic of Quadratic Forms 2010 ed.Didesnis paveikslėlis
  • Formatas: Hardback, 238 pages, aukštis x plotis: 235x155 mm, weight: 1170 g, XII, 238 p., 1 Hardback
  • Serija: Springer Monographs in Mathematics
  • Išleidimo metai: 30-Jun-2010
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1441917314
  • ISBN-13: 9781441917317
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781441917317.html
Raktažodžiai:
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 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 Eichlers, 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.

Recenzijos

From the reviews:

This impressive monograph is devoted to some aspects of the arithmetic theory of quadratic forms to which the author has made important contributions in recent years. Anyone with an interest in the books topic will find this volume well worth his or her time. (Ch. Baxa, Monatshefte für Mathematik, Vol. 169 (1), January, 2013)

In this book Goro Shimura presents some very interesting material in a clear and readable style. There is plenty to learn from an introduction to a highly technical subject by one of its leaders. The book will serve well as a basis for a couple of postgraduate courses. the material is essentially self-contained the exposition is often very technical . (Peter Shiu, The Mathematical Gazette, Vol. 96 (536), July, 2012)

Beginners and experts have much to gain from the authors insights and perspectives presented in this book on the arithmetic theory of quadratic forms. In order to make the treatment of these topics accessible to readers with a general background in abstract algebra, the author includes preliminary material on algebraic number theory and the theory of semisimple algebras. Throughout the book, the presentation is concise and elegant. (A. G. Earnest, Mathematical Reviews, Issue 2011 m)

Goro Shimura is one of the worlds premier arithmeticians, with his name attached to a number of marvelous things. it presents a lot more than its title suggests and does so magnificently. Shimuras Arithmetic of Quadratic Forms is another very important monograph by this fine scholar and, to use a hackneyed but apt phrase, will richly repay the reader who invests his time in a careful study of its pages. (Michael Berg, The Mathematical Association of America, August, 2010)

Preface v
Notation and terminology xi
Chapter I The quadratic reciprocity law
1(14)
1 Elementary facts
1(3)
2 Structure of (Z/mZ)x
4(1)
3 The quadratic reciprocity law
5(6)
4 Lattices in a vector space
11(1)
5 Modules over a principal ideal domain
12(3)
Chapter II Arithmetic in an algebraic number field
15(32)
6 Valuations and p-adic numbers
15(7)
7 Hensel's lemma and its applications
22(3)
8 Integral elements in algebraic extensions
25(2)
9 Order functions in algebraic extensions
27(8)
10 Ideal theory in an algebraic number field
35(12)
Chapter III Various basic theorems
47(32)
11 The tensor product of fields
47(3)
12 Units and the class number of a number field
50(7)
13 Ideals in an extension of a number field
57(2)
14 The discriminant and different
59(7)
15 Adeles and ideles
66(5)
16 Galois extensions
71(4)
17 Cyclotomic fields
75(4)
Chapter IV Algebras over a field
79(36)
18 Semisimple and simple algebras
79(7)
19 Central simple algebras
86(9)
20 Quaternion algebras
95(5)
21 Arithmetic of semisimple algebras
100(15)
Chapter V Quadratic forms over a field
115(38)
22 Algebraic theory of quadratic forms
115(5)
23 Clifford algebras
120(7)
24 Clifford groups and spin groups
127(6)
25 Lower-dimensional cases
133(7)
26 The Hilbert reciprocity law
140(3)
27 The Hasse principle
143(10)
Chapter VI Deeper arithmetic of quadratic forms
153(50)
28 Classification of quadratic forms over local and global fields
153(8)
29 Lattices in a quadratic space
161(10)
30 The genus and class of a lattice and a matrix
171(8)
31 Integer-valued quadratic forms
179(7)
32 Strong approximation in the indefinite case
186(11)
33 Integer-valued symmetric forms
197(6)
Chapter VII Quadratic Diophantine equations
203(30)
34 A historical perspective
203(3)
35 Basic theorems of quadratic Diophantine equations
206(7)
36 Classification of binary forms
213(11)
37 New mass formulas
224(4)
38 The theory of genera
228(5)
References 233(2)
Index 235