Function Field Arithmetic -- Contents -- 1. Number fields and Function fields -- 1.1 Global fields: Basic analogies and contrasts -- 1.2 Genus and Riemann-Roch theorem -- 1.3 Zeta function and class group -- 1.4 Class field theory and Galois group -- Class field theory in terms of ideal (or divisor) groups: -- Class field theory in terms of idele groups: -- Class field theory via generalized Jacobians: -- Class field theory via cohomological approach: -- The Galois group: -- 2. Drinfeld modules -- 2.1 Carlitz module and related arithmetical objects -- 2.2 Drinfeld modules: Basic definitions -- 2.3 Torsion points -- 2.4 Analytic theory -- 2.5 Explicit calculations for Carlitz module -- 2.6 Reductions -- 2.7 Endomorphisms -- 2.8 Field of definition -- 2.9 Points on Drinfeld modules -- 2.10 Adjoints and duality -- 2.11 Useful tools in non-archimedean or finite field setting -- (a) Properties of k{} -- (b) Moore determinant -- (c) q-resultants -- (d) Non- archimedean calculus -- (e) Dwork's trace formula -- 3. Explicit class field theory -- 3.1 Torsion of rank one Drinfeld modules -- 3.2 Sign normalization of the top coefficient -- 3.3 Normalizing Field as a class field -- 3.4 Smallest field of definition as a class field -- 3.5 Ring of definition -- 3.6 Cyclotomic fields -- 3.7 Moduli approach -- 3.8 Summary -- 3.9 Maximal abelian extension -- 3.10 Cyclotomic theory of Fq[t] -- 3.11 Cyclotomic units and conjectures of Brumer and Stark -- 3.12 Some contrasts and open questions -- 4. Gauss sums and Gamma functions -- 4.1 Gauss and Jacobi sums: Definitions -- 4.2 Gauss and Jacobi sums: Fq[t] case -- 4.3 Gauss and Jacobi sums: General A -- 4.4 Sign of Gauss sums for Fq[t] -- 4.5 Arthimetic Factorial and Gamma: Definitions -- (a) Fq[t] case -- (b) General A -- 4.6 Functional equations in arithmetic case -- 4.7 Special values for arthimetic.

(a) Periods: Fq[t] case -- (b) Periods: General A -- 4.8 Special values of arithmetic -- (a) Fq[t] case: Analog of Gross-Koblitz -- (b) General A -- 4.9 Geometric Factorial and Gamma: Definitions -- 4.10 Functional equations in geometric case -- Reflection formula for II: -- Multiplication formula for II: -- Multiplication formula for II: -- Reflection formula for II: -- 4.11 Special values of geometric and: Fq[t] case -- 4.12 Comparisons and uniform framework -- 4.13 More analogies for Fq[t]: Divisibilities -- 4.14 Binomial coefficients -- (a) Binomial coefficients as nice basis -- (b) Difference and differentiation operators -- 4.15 Relations between the two notions of binomials -- 4.16 Bernoulli numbers and polynomials -- 4.17 Note on finite differences and q-analogs -- 5. Zeta functions -- 5.1 Zeta values at integers: Definitions -- 5.2 Values at positive integers -- 5.3 Values at non-positive integers -- 5.4 Mutiplicities of trivial zeros -- 5.5 Zeta function interpolation on character space -- (a) adic interpolation -- (b) adic interpolation -- 5.6 Power sums -- 5.7 Zeta measure -- 5.8 Zero distribution -- 5.9 Low values and multi-logarithms -- 5.10 Multizeta values -- (a) Complex valued multizeta -- (b) Finite characteristic variants -- (c) Interpolations -- 5.11 Analytic properties of zeta and Fredholm determinant -- 5.12 Note on classical interpolations -- 6. Higher rank theory -- 6.1 Elliptic modules -- 6.2 Modular forms -- 6.3 Galois representations -- 6.4 DeRham Cohomology -- (a) Elliptic curves case: Motivation -- (b) Drinfeld modules case -- 6.5 Hypergeometric functions -- (a) The first analog -- (b) The second analog -- 7. Higher dimensions and geometric tools -- 7.1 t-modules and t-motives -- 7.2 Torsion -- 7.3 Purity -- 7.4 Exponential, period lattice and uniformizability -- 7.5 Cohomology realizations.

7.6 Example: Carlitz-Tate twist Cn -- 7.7 Drinfeld dictionary in the simplest case -- 7.8 Krichever/Drinfeld dictionary in more generality -- 8. Applications to Gauss sums, Gamma and Zeta values -- 8.1 Cn and (n) -- 8.2 Shtuka and Jacobi sums -- (a) Gauss sums and Theta divisor -- (b) Examples and applications -- (c) The case g = d = 1 -- 8.3 Another Gamma function -- (a) Analog of Gross-Koblitz -- (b) Interpolation at for new Gamma -- 8.4 Fermat motives and solitons -- 8.5 Another approach to solitons -- 8.6 Analog of Gross-Koblitz for Geometric Gamma: Fq[t] case -- 8.7 What is known or expected in general case? -- 8.8 Gamma values to Periods connection via solitons: Sketch -- 8.9 Log-algebraicity, Cyclotomic module and Vandiver conjecture -- 8.10 Explicit Log-Algebraicity formulas -- 9. Diophantine approximation -- 9.1 Approximation exponents -- 9.2 Good approximations: Continued fractions -- 9.3 Range of exponents : Frobenius -- 9.4 Range of exponents: Differentiation -- 9.5 Connection with deformation theory -- (a) Height inequalities for algebraic points -- (b) Exponent hierarchy -- (c) Approximation by algebraic functions -- 9.6 Note on connection with Diophantine equations -- 10. Transcendence results -- 10.1 Approximation techniques and irrationality -- 10.2 Transcendence results on Drinfeld modules -- 10.3 Application to Zeta and Gamma values -- 10.4 Transcendence results in higher dimensions -- 10.5 Application to Zeta and Gamma values -- 11. Automata and algebraicity: Applications -- 11.1 Automata and algebraicity -- 11.2 Some useful automata tools -- 11.3 Applications to transcendence of gamma values and monomials -- 11.4 Applications to transcendence: periods and modular functions -- 11.5 Classifying finite characteristic numbers -- 11.6 Computational classes and basic tools -- 11.7 Algebraic properties of computational classes.

11.8 Applications to refined transcendence -- Note on the Notation -- Bibliography.