Basic Structures of Function Field Arithmetic

by Goss, David

• ISBN13: 9783540635413
• ISBN10: 3540635416
• Format: Paperback
• Copyright: 1997-11-01
• Publisher: Springer Verlag

Summary

From the reviews:"The book...is a thorough and very readable introduction to the arithmetic of function fields of one variable over a finite field, by an author who has made fundamental contributions to the field. It serves as a definitive reference volume, as well as offering graduate students with a solid understanding of algebraic number theory the opportunity to quickly reach the frontiers of knowledge in an important area of mathematics...The arithmetic of function fields is a universe filled with beautiful surprises, in which familiar objects from classical number theory reappear in new guises, and in which entirely new objects play important roles. Goss'clear exposition and lively style make this book an excellent introduction to this fascinating field." MR 97i:11062

Table of Contents

1. Additive Polynomials	1	(34)
1.1. Basic Properties	1	(2)
1.2. Classification of Additive Polynomials	3	(2)
1.3. The Moore Determinant	5	(4)
1.4. The Relationship Between k[x] and k{T}	9	(1)
1.5. The p-resultant	10	(2)
1.6. The Left and Right Division Algorithms	12	(3)
1.7. The T-adjoint of an Additive Polynomial	15	(3)
1.8. Dividing A(1) by Finite Additive Groups	18	(2)
1.9. Analogs in Differential Equations/Algebra	20	(6)
1.10. Divisibility Theory	26	(5)
1.11. The Semi-invariants of Additive Polynomials	31	(4)
2. Review of Non-Archimedean Analysis	35	(8)
3. The Carlitz Module	43	(20)
3.1. Background	43	(4)
3.2. The Carlitz Exponential	47	(5)
3.3. The Carlitz Module	52	(4)
3.4. The Carlitz Logarithm	56	(1)
3.5. The Polynomials E(d)(x)	57	(2)
3.6. The Carlitz Module over Arbitrary A-fields	59	(2)
3.7. The Adjoint of the Carlitz Module	61	(2)
4. Drinfeld Modules	63	(74)
4.1. Introduction	63	(2)
4.2. Lattices and Their Exponential Functions	65	(1)
4.3. The Drinfeld Module Associated to a Lattice	66	(3)
4.4. The General Definition of a Drinfeld Module	69	(1)
4.5. The Height and Rank of a Drinfeld Module	70	(3)
4.6. Lattices and Drinfeld Modules over C INFINITY	73	(6)
4.7. Morphisms of Drinfeld Modules	79	(7)
4.8. Primality in F{T} and A	86	(1)
4.9. The Action of Ideals on Drinfeld Modules	86	(2)
4.10. The Reduction Theory of Drinfeld Modules	88	(5)
4.11. Review of Central Simple Algebra	93	(9)
4.12. Drinfeld Modules over Finite Fields	102	(13)
4.13. Rigidity of Drinfeld Modules	115	(7)
4.14. The Adjoint of a General Drinfeld Module	122	(15)
5. T-Modules	137	(42)
5.1. Vector Bundles	137	(2)
5.2. Sheaves and Differential Equations	139	(1)
5.3. XXX-sheaves	140	(2)
5.4. Basic Concepts of T-modules	142	(6)
5.5. Pure T-modules	148	(3)
5.6. Torsion Points	151	(5)
5.7. Tensor Products	156	(1)
5.8. The Tensor Powers of the Carlitz Module	157	(2)
5.9. Uniformization	159	(13)
5.10. The Tensor Powers of the Carlitz Module redux	172	(3)
5.11. Scattering Matrices	175	(4)
6. Shtukas	179	(14)
6.1. Review of Some Algebraic Geometry	179	(3)
6.2. The Shtuka Correspondence	182	(11)
7. Sign Normalized Rank 1 Drinfeld Modules	193	(42)
7.1. Class-fields as Moduli	193	(3)
7.2. Sign Normalization	196	(5)
7.3. Fields of Definition of Drinfeld Modules	201	(1)
7.4. The Normalizing Field	202	(3)
7.5. Division Fields	205	(6)
7.6. Principal Ideal Theorems	211	(3)
7.7. A Rank One Version of Serre's Theorem	214	(4)
7.8. Classical Partial Zeta Functions	218	(3)
7.9. Unit Calculations	221	(2)
7.10. Period Computations	223	(4)
7.11. The Connection with Shtukas and Examples	227	(8)
8. L-series	235	(114)
8.1. The "Complex Plane" S INFINITY	236	(2)
8.2. Exponentiation of Ideals	238	(5)
8.3. v-adic Exponentiation of Ideals	243	(2)
8.4. Continuous Functions on Z(p)	245	(3)
8.5. Entire Functions on S INFINITY	248	(6)
8.6. L-series of Characteristic p Arithmetic	254	(5)
8.7. Formal Dirichlet Series	259	(2)
8.8. Estimates	261	(3)
8.9. L-series of Finite Characters	264	(5)
8.10. The Question of Local Factors	269	(1)
8.11. The Generalized Teichmuller Character	270	(2)
8.12. Special-values at Negative Integers	272	(4)
8.13. Trivial Zeroes	276	(3)
8.14. Applications to Class Groups	279	(4)
8.15. "Geometric" Versus "Arithmetic" Notions	283	(1)
8.16. The Arithmetic Criterion for Cyclicity	284	(1)
8.17. The "Geometric Artin Conjecture"	285	(5)
8.18. Special-values at Positive Integers	290	(7)
8.19. The Functional Equation of the Special-values	297	(4)
8.20. Applications to Class Groups	301	(6)
8.21. The Geometric Criterion for Cyclicity	307	(2)
8.22. Magic Numbers	309	(10)
8.23. Finiteness in Local and Global Fields	319	(3)
8.24. Towards a Theory of the Zeroes	322	(19)
8.25. Kapranov's Higher Dimensional Theory	341	(8)
9. XXX-functions	349	(40)
9.1. Basic Properties of the Carlitz Factorial	350	(3)
9.2. Bernoulli-Carlitz Numbers	353	(6)
9.3. The XXX-ideal	359	(2)
9.4. The Arithmetic XXX-function	361	(7)
9.5. Functional Equations	368	(2)
9.6. Finite Interpolations	370	(4)
9.7. Another v-adic XXX-function	374	(2)
9.8. Gauss Sums	376	(5)
9.9. The Geometric XXX-function	381	(8)
10. Additional Topics	389	(12)
10.1. The Geometric Fermat Equation	389	(1)
10.2. Geometric Deligne Reciprocity and Solitons	390	(3)
10.3. The Tate Conjecture for Drinfeld Modules	393	(1)
10.4. Meromorphic Continuations of L-functions	394	(2)
10.5. The Structure of the A-module of Rational Points	396	(1)
10.6. Log-algebraicity and Special Points	397	(4)
References	401	(20)
Index	421