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1.

図書

図書
Daniel A. Marcus
出版情報: Cham : Springer, c2018  xviii, 203 p. ; 24 cm
シリーズ名: Universitext
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2.

図書

図書
I. Reiner
出版情報: London ; New York : Academic Press, 1975  xii, 395 p. ; 24 cm
シリーズ名: L.M.S. monographs ; 5
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目次情報: 続きを見る
Foreword
Preface
Acknowledgements
Permanent notation
Algebraic preliminaries / 1:
Integral closure
Homological algebra / 2:
Localization / 3:
Dedekind domains / 4:
Completions and valuations / 5:
Radicals of rings / 6:
Semisimple rings and simple algebras / 7:
Orders
Definitions and examples / 8:
Reduced norms and traces / 9:
Existence of maximal orders; discriminants / 10:
Localization of orders / 11:
Maximal orders in skewfields (local case)
Uniqueness of maximal orders / 12:
Ramification index; inertial degree / 13:
Finite residue class field case / 14:
Morita equivalence
Progenerators / 15:
Morita correspondence / 16:
Maximal orders over discrete valuation rings
Maximal orders (complete local case) / 17:
Maximal orders (local case) / 18:
Ideals / 19:
Different, discriminant / 20:
Maximal orders over Dedekind domains
Basic results / 21:
Ideal theory / 22:
Alternate approach to global ideal theory / 23:
Norms of ideals / 24:
Ideal classes; Jordan-Zassenhaus theorem / 25:
Genus / 27:
Crossed-product algebras
Brauer groups / 28:
Cyclic algebras / 29:
Cyclic algebras over local fields / 31:
Simple algebras over global fields
Splitting of simple algebras / 32:
Reduced norms / 33:
Eichler's Theorem / 34:
Ideal class groups / 35:
K[subscript 0] of maximal orders / 36:
Picard groups / 37:
Non-maximal orders / 38:
Hereditary orders
Local theory of hereditary orders / 39:
Global theory of hereditary orders / 40:
Group rings / 41:
References
Index
Foreword
Preface
Acknowledgements
3.

図書

図書
Jean-Pierre Serre
出版情報: Paris : Hermann, 1962  243 p. ; 24 cm
シリーズ名: Publications de l'Institut de mathématique de l'Université de Nancago ; 8
Actualités scientifiques et industrielles ; 1296
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4.

図書

図書
Meyer, Curt
出版情報: Berlin : Akademie-Verlag, 1957  ix, 132 p ; 25 cm
シリーズ名: Mathematische Lehrbücher und Monographien ; II. Abteilung, Mathematische Monographien ; Bd. 5
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5.

図書

図書
David Winter
出版情報: New York : Springer-Verlag, c1974  xii, 205 p. ; 25 cm
シリーズ名: Graduate texts in mathematics ; 16
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6.

図書

図書
by Robert P. Langlands
出版情報: Princeton, N.J. : Princeton University Press , Tokyo : University of Tokyo Press, 1980  vii, 236 p. ; 25 cm
シリーズ名: Annals of mathematics studies ; no. 96
Tokyo University international edition ; no. 172
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7.

図書

図書
Paul J. McCarthy
出版情報: Waltham, Mass. : Blaisdell, c1966  ix, 166 p. ; 24 cm
シリーズ名: A Blaisdell book in pure and applied mathematics
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8.

図書

図書
J.S. Milne
出版情報: Boston ; Tokyo : Academic Press, c1986  x, 419 p. ; 24 cm
シリーズ名: Perspectives in mathematics ; v. 1
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目次情報: 続きを見る
Preface
Notations and Conventions
Galois Cohomology / Chapter I:
Preliminaries / 0:
Duality relative to a class formation / 1:
Local fields / 2:
Abelian varieties over local fields / 3:
Global fields / 4:
Global Euler-Poincare characteristics / 5:
Abelian varieties over global fields / 6:
An application to the conjecture of Birch and Swinnerton-Dyer / 7:
Abelian class field theory, in the sense of Langlands / 8:
Other applications / 9:
Class field theory for function fields / Appendix A:
Etale Cohomology / Chapter II:
Local results
Global results: preliminary calculations
Global results: the main theorem
Global results: complements
Global results: abelian schemes
Global results: singular schemes
Global results: higher dimensions
Flat Cohomology / Chapter III:
Local results: mixed characteristic, finite group schemes
Local results: mixed characteristic, abelian varieties
Global results: number field case
Local results: mixed characteristic, perfect residue field
Two exact sequences
Local fields of characteristic p
Local results: equicharacteristic
Global results: curves over finite fields, finite sheaves
Global results: curves over finite fields, Neron models
Local results: equicharacteristic, perfect residue field / 10:
Global results: curves over perfect fields / 11:
Embedding finite group schemes
Extending finite group schemes / Appendix B:
Biextensions and Neron models / Appendix C:
Bibliography
Index
Preface
Notations and Conventions
Galois Cohomology / Chapter I:
9.

図書

図書
Wang Yuan
出版情報: Berlin ; New York : Springer-Verlag, c1991  xvi, 168 p. ; 25 cm
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10.

図書

図書
Carlos Moreno
出版情報: Cambridge [England] ; New York : Cambridge University Press, 1991  ix, 246 p. ; 24 cm
シリーズ名: Cambridge tracts in mathematics ; 97
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目次情報: 続きを見る
Preface
Algebraic curves and function fields / 1:
Geometric aspects / 1.1:
Introduction / 1.1.1:
Affine varieties / 1.1.2:
Projective varieties / 1.1.3:
Morphisms / 1.1.4:
Rational maps / 1.1.5:
Non-singular varieties / 1.1.6:
Smooth models of algebraic curves / 1.1.7:
Algebraic aspects / 1.2:
Points on the projective line P[superscript 1] / 1.2.1:
Extensions of valuation rings / 1.2.3:
Points on a smooth curve / 1.2.4:
Independence of valuations / 1.2.5:
Exercises
Notes
The Riemann-Roch theorem / 2:
Divisors / 2.1:
The vector space L(D) / 2.2:
Principal divisors and the group of divisor classes / 2.3:
The Riemann theorem / 2.4:
Pre-adeles (repartitions) / 2.5:
Pseudo-differentials (the Riemann-Roch theorem) / 2.6:
Zeta functions / 3:
The zeta functions of curves / 3.1:
The functional equation / 3.3:
Consequences of the functional equation / 3.3.1:
The Riemann hypothesis / 3.4:
The L-functions of curves and their functional equations / 3.5:
Preliminary remarks and notation / 3.5.1:
Exponential sums / 3.5.2:
The zeta function of the projective line / 4.1:
Gauss sums: first example of an L-function for the projective line / 4.2:
Properties of Gauss sums / 4.3:
Cyclotomic extensions: basic facts / 4.3.0:
Elementary properties / 4.3.1:
The Hasse-Davenport relation / 4.3.2:
Stickelberger's theorem / 4.3.3:
Kloosterman sums / 4.4:
Second example of an L-function for the projective line / 4.4.1:
A Hasse-Davenport relation for Kloosterman sums / 4.4.2:
Third example of an L-function for the projective line / 4.5:
Basic arithmetic theory of exponential sums / 4.6:
Part I: L-functions for the projective line / 4.6.1:
Part II: Artin-Schreier coverings / 4.6.2:
The Hurwitz-Zeuthen formula for the covering [pi]: C [right arrow] C / 4.6.3:
Goppa codes and modular curves / 5:
Elementary Goppa codes / 5.1:
The affine and projective lines / 5.2:
Affine line A[superscript 1](k) / 5.2.1:
Projective line P[superscript 1] / 5.2.2:
Goppa codes on the projective line / 5.3:
Algebraic curves / 5.4:
Separable extensions / 5.4.1:
Closed points and their neighborhoods / 5.4.2:
Differentials / 5.4.3:
The theorems of Riemann-Roch, of Hurwitz and of the Residue / 5.4.4:
Linear series / 5.4.6:
Algebraic geometric codes / 5.5:
Algebraic Goppa codes / 5.5.1:
Codes with better rates than the Varshamov-Gilbert bound / 5.5.2:
The theorem of Tsfasman, Vladut and Zink / 5.6:
Modular curves / 5.6.1:
Elliptic curves over C / 5.6.2:
Elliptic curves over the fields F[subscript p], Q / 5.6.3:
Torsion points on elliptic curves / 5.6.4:
Igusa's theorem / 5.6.5:
The modular equation / 5.6.6:
The congruence formula / 5.6.7:
The Eichler-Selberg trace formula / 5.6.8:
Proof of the theorem of Tsfasman, Vladut and Zink / 5.6.9:
Examples of algebraic Goppa codes / 5.7:
The Hamming (7,4) code / 5.7.1:
BCH codes / 5.7.2:
The Fermat cubic (Hermite form) / 5.7.3:
Elliptic codes (according to Driencourt-Michon) / 5.7.4:
The Klein quartic / 5.7.5:
Simplification of the singularities of algebraic curves / Appendix:
Homogeneous coordinates in the plane / A.1:
Basic lemmas / A.2:
Dual curves / A.3:
Plucker formulas / A.3.1:
Quadratic transformations / A.4:
Quadratic transform of a plane curve / A.4.1:
Quadratic transform of a singularity / A.4.2:
Singularities off the exceptional lines / A.4.3:
Reduction of singularities / A.4.4:
Bibliography
Index
Preface
Algebraic curves and function fields / 1:
Geometric aspects / 1.1:
11.

図書

図書
by Cyrus Colton MacDuffee
出版情報: New York : J. Wiley & sons, inc. , London : Chapman & Hall, limited, c1940  vii, 303 p ; 24 cm
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12.

図書

図書
by T.G. Room and P.B. Kirkpatrick
出版情報: London : Cambridge Univ. Press, 1971  viii, 176 p. ; 22 cm
シリーズ名: Cambridge tracts in mathematics and mathematical physics ; no. 60
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目次情報: 続きを見る
Algebraic Background / Part I:
Two algebraic systems with nine elements / 1:
Field-Planes / Part II:
Projective planes / 2:
Galois planes of orders 3 and 9 / 3:
Miniquaternion Planes / Part III:
The planes ÃŽ© and ÃŽ©D / 4:
The plane ο / 5:
Algebraic Background / Part I:
Two algebraic systems with nine elements / 1:
Field-Planes / Part II:
13.

図書

図書
by George Bachman
出版情報: New York : Academic Press, c1964  ix, 173 p. ; 21 cm
シリーズ名: Academic paperbacks ; Mathematics
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14.

図書

図書
Irving Kaplansky
出版情報: Chicago : University of Chicago Press, [1969]  ix, 198 p. ; 21 cm
シリーズ名: Chicago lectures in mathematics
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15.

図書

図書
[by] Marvin J. Greenberg
出版情報: New York : W. A. Benjamin, 1969  167 p ; 23 cm
シリーズ名: Mathematics lecture note series
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16.

図書

図書
Serge Lang
出版情報: Reading, Mass. : Addison-Wesley Pub., c1964  ix, 163 p. ; 24 cm
シリーズ名: Addison-Wesley series in mathematics / Eric Reissner, consulting editor
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17.

図書

図書
Masayoshi Nagata
出版情報: New York : M. Dekker, c1977  vii, 268 p. ; 24 cm
シリーズ名: Monographs and textbooks in pure and applied mathematics ; 40
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18.

図書

図書
Max Deuring
出版情報: Berlin ; New York : Springer-Verlag, 1973  vi, 151 p. ; 26 cm
シリーズ名: Lecture notes in mathematics ; 314
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19.

図書

図書
Paul Gérardin
出版情報: Berlin ; New York : Springer-Verlag, 1975  180 p. ; 25 cm
シリーズ名: Lecture notes in mathematics ; 462
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20.

図書

図書
Gregory Karpilovsky
出版情報: Amsterdam ; Tokyo : North Holland, 1989 , New York, N.Y., U.S.A. : Sole distributors for the U.S.A. and Canada, Elsevier Science Pub. Co.  xi, 546 p. ; 25 cm
シリーズ名: North-Holland mathematics studies ; 155
Notas de matemática ; 124
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21.

図書

図書
Serge Lang
出版情報: New York ; Berlin : Springer-Verlag, c1978-1980  2 v. ; 25 cm
シリーズ名: Graduate texts in mathematics ; 59,69
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22.

図書

図書
by Dinesh S. Thakur
出版情報: Singapore : World Scientific, c2004  xv, 388 p. ; 24 cm
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目次情報: 続きを見る
Number fields and Function fields / 1:
Global fields: Basic analogies and contrasts / 1.1:
Genus and Riemann-Roch theorem / 1.2:
Zeta function and class group / 1.3:
Class field theory and Galois group / 1.4:
Drinfeld modules / 2:
Carlitz module and related arithmetical objects / 2.1:
Drinfeld modules: Basic definitions / 2.2:
Torsion points / 2.3:
Analytic theory / 2.4:
Explicit calculations for Carlitz module / 2.5:
Reductions / 2.6:
Endomorphisms / 2.7:
Field of definition / 2.8:
Points on Drinfeld modules / 2.9:
Adjoints and duality / 2.10:
Useful tools in non-archimedean or finite field setting / 2.11:
Properties of k{[tau]} / (a):
Moore determinant / (b):
q-resultants / (c):
Non-archimedean calculus / (d):
Dwork's trace formula / (e):
Explicit class field theory / 3:
Torsion of rank one Drinfeld modules / 3.1:
Sign normalization of the top coefficient / 3.2:
Normalizing Field as a class field / 3.3:
Smallest field of definition as a class field / 3.4:
Ring of definition / 3.5:
Cyclotomic fields / 3.6:
Moduli approach / 3.7:
Summary / 3.8:
Maximal abelian extension / 3.9:
Cyclotomic theory of F[subscript q t] / 3.10:
Cyclotomic units and conjectures of Brumer and Stark / 3.11:
Some contrasts and open questions / 3.12:
Gauss sums and Gamma functions / 4:
Gauss and Jacobi sums: Definitions / 4.1:
Gauss and Jacobi sums: F[subscript q t] case / 4.2:
Gauss and Jacobi sums: General A / 4.3:
Sign of Gauss sums for F[subscript q t] / 4.4:
Arithmetic Factorial and Gamma: Definitions / 4.5:
F[subscript q t] case
General A
Functional equations in arithmetic case / 4.6:
Special values for arithmetic [Gamma subscript infinity] / 4.7:
Periods: F[subscript q t] case
Periods: General A
Special values of arithmetic [Gamma subscript v] / 4.8:
F[subscript q t] case: Analog of Gross-Koblitz
Geometric Factorial and Gamma: Definitions / 4.9:
Functional equations in geometric case / 4.10:
Special values of geometric [Gamma] and [Gamma subscript v]: F[subscript q t] case / 4.11:
Comparisons and uniform framework / 4.12:
More analogies for F[subscript q t]: Divisibilities / 4.13:
Binomial coefficients / 4.14:
Binomial coefficients as nice basis
Difference and differentiation operators
Relations between the two notions of binomials / 4.15:
Bernoulli numbers and polynomials / 4.16:
Note on finite differences and q-analogs / 4.17:
Zeta functions / 5:
Zeta values at integers: Definitions / 5.1:
Values at positive integers / 5.2:
Values at non-positive integers / 5.3:
Multiplicities of trivial zeros / 5.4:
Zeta function interpolation on character space / 5.5:
[infinity]-adic interpolation
p adic interpolation
Power sums / 5.6:
Zeta measure / 5.7:
Zero distribution / 5.8:
Low values and multi-logarithms / 5.9:
Multizeta values / 5.10:
Complex valued multizeta
Finite characteristic variants
Interpolations
Analytic properties of zeta and Fredholm determinant / 5.11:
Note on classical interpolations / 5.12:
Higher rank theory / 6:
Elliptic modules / 6.1:
Modular forms / 6.2:
Galois representations / 6.3:
DeRham Cohomology / 6.4:
Elliptic curves case: Motivation
Drinfeld modules case
Hypergeometric functions / 6.5:
The first analog
The second analog
Higher dimensions and geometric tools / 7:
t-modules and t-motives / 7.1:
Torsion / 7.2:
Purity / 7.3:
Exponential, period lattice and uniformizability / 7.4:
Cohomology realizations / 7.5:
Example: Carlitz-Tate twist C[superscript multiply sign in circle n] / 7.6:
Drinfeld dictionary in the simplest case / 7.7:
Krichever/Drinfeld dictionary in more generality / 7.8:
Applications to Gauss sums, Gamma and Zeta values / 8:
C[superscript multiply sign in circle n] and [xi](n) / 8.1:
Shtuka and Jacobi sums / 8.2:
Gauss sums and Theta divisor
Examples and applications
The case g = d[subscript infinity] = 1
Another Gamma function / 8.3:
Analog of Gross-Koblitz
Interpolation at [infinity] for new Gamma
Fermat motives and Solitons / 8.4:
Another approach to solitons / 8.5:
Analog of Gross-Koblitz for Geometric Gamma: F[subscript q t] case / 8.6:
What is known or expected in general case? / 8.7:
Gamma values to Periods connection via solitons: Sketch / 8.8:
Log-algebraicity, Cyclotomic module and Vandiver conjecture / 8.9:
Explicit Log-Algebraicity formulas / 8.10:
Diophantine approximation / 9:
Approximation exponents / 9.1:
Good approximations: Continued fractions / 9.2:
Range of exponents: Frobenius / 9.3:
Range of exponents: Differentiation / 9.4:
Connection with deformation theory / 9.5:
Height inequalities for algebraic points
Exponent hierarchy
Approximation by algebraic functions
Note on connection with Diophantine equations / 9.6:
Transcendence results / 10:
Approximation techniques and irrationality / 10.1:
Transcendence results on Drinfeld modules / 10.2:
Application to Zeta and Gamma values / 10.3:
Transcendence results in higher dimensions / 10.4:
Automata and algebraicity: Applications / 10.5:
Automata and algebraicity / 11.1:
Some useful automata tools / 11.2:
Applications to transcendence of gamma values and monomials / 11.3:
Applications to transcendence: periods and modular functions / 11.4:
Classifying finite characteristic numbers / 11.5:
Computational classes and basic tools / 11.6:
Algebraic properties of computational classes / 11.7:
Applications to refined transcendence / 11.8:
Note on the Notation
Bibliography
Number fields and Function fields / 1:
Global fields: Basic analogies and contrasts / 1.1:
Genus and Riemann-Roch theorem / 1.2:
23.

図書

図書
by Shreeram Abhyankar
出版情報: Princeton, N.J. : Princeton University Press, 1959  vi, 96 p. ; 25 cm
シリーズ名: Annals of mathematics studies ; no. 43
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目次情報: 続きを見る
Preface
Introduction
General Ramification Theory / I:
Elementary lemmas from ideal theory / 1:
Primary decomposition in non-noetherian rings / 2:
Integral dependence / 3:
Linear algebra / 4:
Discriminant of an algebra / 5:
The discriminant ideal / 6:
Galois theory of local rings / 7:
Valuation Theory / II:
Ordered abelian groups / 8:
Valuations / 9:
Specialization and composition of valuations / 10:
Ramification theory of valuations / 11:
Valuations of algebraic function fields / 12:
Noetherian Local Rings / III:
The dimension of a noetherian local ring / 13:
Quadratic transforms / 14:
Two-Dimensional Local Domains / IV:
Limits of quadratic sequences / 15:
Uniformization in a two-dimensional regular local domain / 16:
Uniformization for a two-dimensional algebraic function field / 17:
Varieties and Transformations / V:
Projective varieties / 18:
Resolution of singularities of algebraic surfaces / 19:
Bibliography
Preface
Introduction
General Ramification Theory / I:
24.

図書

図書
Emil Artin
出版情報: New York : Gordon and Breach, c1967  xiii, 349 p. ; 24 cm
シリーズ名: Notes on mathematics and its applications
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25.

図書

図書
by Gr. C. Moisil
出版情報: Oxford ; New York : Pergamon Press, [1969]  719 p. ; 23 cm
シリーズ名: International series of monographs on pure and applied mathematics ; v. 41
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26.

図書

図書
Yu. V. Linnik ; translated by M.S. Keane
出版情報: Berlin ; New York : Springer-Verlag, 1968  ix, 192 p. ; 24 cm
シリーズ名: Ergebnisse der Mathematik und ihrer Grenzgebiete ; Bd. 45
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27.

図書

図書
Gerald J. Janusz
出版情報: New York : Academic Press, 1973  x, 220 p. ; 24 cm
シリーズ名: Pure and applied mathematics ; 55
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28.

図書

図書
Jean-Pierre Serre ; written with the collaboration of Willem Kuyk and John Labute
出版情報: New York : W.A. Benjamin, 1968  1 v. (various pagings) ; 24 cm
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29.

図書

図書
David Goss
出版情報: Berlin : Springer, c1996  xiii, 422 p. ; 25 cm
シリーズ名: Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 35
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30.

図書

図書
Jean-Pierre Serre
出版情報: Paris : Hermann, 1968  245 p. ; 24 cm
シリーズ名: Publications de l'Institut de mathématique de l'Université de Nancago ; 8
Actualités scientifiques et industrielles ; 1296
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