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1.

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:
2.

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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3.

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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4.

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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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
5.

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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6.

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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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:
7.

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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8.

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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9.

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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10.

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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