The arithmetic of function fields over finite fields is a relatively new area of mathematics. The author has been in the forefront of research in this field since it started and this book is the first systematic treatment of this field. Many applications and open problems are presented and the last section of the book introduces the reader to some of the most vital areas of current research.

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 ShtukaCorrespondence.- 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 Teichmüller 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.