Basic structures of function field arithmetic
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Bibliographic Information
Basic structures of function field arithmetic
（Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge,
Springer, c1996
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Includes bibliographical references (p. [401]-417) and index
Description and Table of Contents
Description
The arithmetic of function fields over finite fields has been an area of extensive research since the mid-1970s with interesting results being obtained. This text offers an introduction to basic concepts such as Drinfeld modules, T-modules, shtuka, exponentiation of ideals and characteristic p L-functions and Gamma-functions in all their various manifestations. Insight from classical number theory, differential equations/algebra and algebraic geometry is presented when needed. Unsolved problems are posed to the reader and a comprehensive list of references is included.
Table of Contents
1. Additive Polynomials.- 1.1. Basic Properties.- 1.2. Classification of Additive Polynomials.- 1.3. The Moore Determinant.- 1.4. The Relationship Between k[x] and k{?}.- 1.5. The p-resultant.- 1.6. The Left and Right Division Algorithms.- 1.7. The ?-adjoint of an Additive Polynomial.- 1.8. Dividing A1 by Finite Additive Groups.- 1.9. Analogs in Differential Equations/Algebra.- 1.10. Divisibility Theory.- 1.11. The Semi-invariants of Additive Polynomials.- 2. Review of Non-Archimedean Analysis.- 3. The Carlitz Module.- 3.1. Background.- 3.2. The Carlitz Exponential.- 3.3. The Carlitz Module.- 3.4. The Carlitz Logarithm.- 3.5. The Polynomials Ed(x).- 3.6. The Carlitz Module over Arbitrary A-fields.- 3.7. The Adjoint of the Carlitz Module.- 4. Drinfeld Modules.- 4.1. Introduction.- 4.2. Lattices and Their Exponential Functions.- 4.3. The Drinfeld Module Associated to a Lattice.- 4.4. The General Definition of a Drinfeld Module.- 4.5. The Height and Rank of a Drinfeld Module.- 4.6. Lattices and Drinfeld Modules over C?.- 4.7. Morphisms of Drinfeld Modules.- 4.8. Primality in F{?} and A.- 4.9. The Action of Ideals on Drinfeld Modules.- 4.10. The Reduction Theory of Drinfeld Modules.- 4.11. Review of Central Simple Algebra.- 4.12. Drinfeld Modules over Finite Fields.- 4.13. Rigidity of Drinfeld Modules.- 4.14. The Adjoint of a General Drinfeld Module.- 5. T-Modules.- 5.1. Vector Bundles.- 5.2. Sheaves and Differential Equations.- 5.3. ?-sheaves.- 5.4. Basic Concepts of T-modules.- 5.5. Pure T-modules.- 5.6. Torsion Points.- 5.7. Tensor Products.- 5.8. The Tensor Powers of the Carlitz Module.- 5.9. Uniformization.- 5.10. The Tensor Powers of the Carlitz Module redux.- 5.11. Scattering Matrices.- 6. Shtukas.- 6.1. Review of Some Algebraic Geometry.- 6.2. The Shtuka Correspondence.- 7. Sign Normalized Rank 1 Drinfeld Modules.- 7.1. Class-fields as Moduli.- 7.2. Sign Normalization.- 7.3. Fields of Definition of Drinfeld Modules.- 7.4. The Normalizing Field.- 7.5. Division Fields.- 7.6. Principal Ideal Theorems.- 7.7. A Rank One Version of Serre's Theorem.- 7.8. Classical Partial Zeta Functions.- 7.9. Unit Calculations.- 7.10. Period Computations.- 7.11. The Connection with Shtukas and Examples.- 8. L-series.- 8.1. The "Complex Plane" S?.- 8.2. Exponentiation of Ideals.- 8.3. ?-adic Exponentiation of Ideals.- 8.4. Continuous Functions on ? p.- 8.5. Entire Functions on S?.- 8.6. L-series of Characteristic p Arithmetic.- 8.7. Formal Dirichlet Series.- 8.8. Estimates.- 8.9. L-series of Finite Characters.- 8.10. The Question of Local Factors.- 8.11. The Generalized Teichmuller Character.- 8.12. Special-values at Negative Integers.- 8.13. Trivial Zeroes.- 8.14. Applications to Class Groups.- 8.15. "Geometric" Versus "Arithmetic" Notions.- 8.16. The Arithmetic Criterion for Cyclicity.- 8.17. The "Geometric Artin Conjecture".- 8.18. Special-values at Positive Integers.- 8.19. The Functional Equation of the Special-values.- 8.20. Applications to Class Groups.- 8.21. The Geometric Criterion for Cyclicity.- 8.22. Magic Numbers.- 8.23. Finiteness in Local and Global Fields.- 8.24. Towards a Theory of the Zeroes.- 8.25. Kapranov's Higher Dimensional Theory.- 9. ?-functions.- 9.1. Basic Properties of the Carlitz Factorial.- 9.2. Bernoulli-Carlitz Numbers.- 9.3. The ?-ideal.- 9.4. The Arithmetic ?-function.- 9.5. Functional Equations.- 9.6. Finite Interpolations.- 9.7. Another ?-adic ?-function.- 9.8. Gauss Sums.- 9.9. The Geometric ?-function.- 10. Additional Topics.- 10.1. The Geometric Fermat Equation.- 10.2. Geometric Deligne Reciprocity and Solitons.- 10.3. The Tate Conjecture for Drinfeld Modules.- 10.4. Meromorphic Continuations of L-functions.- 10.5. The Structure of the A-module of Rational Points.- 10.6. Log-algebraicity and Special Points.- References.
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