Function field arithmetic

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Function field arithmetic / by Dinesh S. Thakur

資料種別:: 図書
出版情報:: Singapore : World Scientific, c2004
形態:: xv, 388 p. ; 24 cm
著者名:: Thakur, Dinesh S., 1961- <DA14499332>
ISBN:: 9789812388391 [9812388397]
書誌ID:: BA6795124X

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注記:

Includes bibliographical references (p. 373-388) and index

主題:

Arithmetic functions; Drinfeld modules; Fields, Algebraic

言語:

英語

目次情報:

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：

Zeta function and class group / 1.3：

Class field theory and Galois group / 1.4：

Drinfeld modules / 2：

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