| Number fields and Function fields | p. 1 |
| Global fields: Basic analogies and contrasts | p. 1 |
| Genus and Riemann-Roch theorem | p. 9 |
| Zeta function and class group | p. 13 |
| Class field theory and Galois group | p. 20 |
| Drinfeld modules | p. 31 |
| Carlitz module and related arithmetical objects | p. 31 |
| Drinfeld modules: Basic definitions | p. 34 |
| Torsion points | p. 39 |
| Analytic theory | p. 41 |
| Explicit calculations for Carlitz module | p. 44 |
| Reductions | p. 48 |
| Endomorphisms | p. 50 |
| Field of definition | p. 52 |
| Points on Drinfeld modules | p. 53 |
| Adjoints and duality | p. 54 |
| Useful tools in non-archimedean or finite field setting | p. 56 |
| Properties of k{[tau]} | p. 57 |
| Moore determinant | p. 58 |
| q-resultants | p. 58 |
| Non-archimedean calculus | p. 59 |
| Dwork's trace formula | p. 61 |
| Explicit class field theory | p. 63 |
| Torsion of rank one Drinfeld modules | p. 64 |
| Sign normalization of the top coefficient | p. 68 |
| Normalizing Field as a class field | p. 70 |
| Smallest field of definition as a class field | p. 73 |
| Ring of definition | p. 74 |
| Cyclotomic fields | p. 76 |
| Moduli approach | p. 77 |
| Summary | p. 79 |
| Maximal abelian extension | p. 80 |
| Cyclotomic theory of F[subscript q t] | p. 82 |
| Cyclotomic units and conjectures of Brumer and Stark | p. 83 |
| Some contrasts and open questions | p. 85 |
| Gauss sums and Gamma functions | p. 87 |
| Gauss and Jacobi sums: Definitions | p. 88 |
| Gauss and Jacobi sums: F[subscript q t] case | p. 91 |
| Gauss and Jacobi sums: General A | p. 94 |
| Sign of Gauss sums for F[subscript q t] | p. 99 |
| Arithmetic Factorial and Gamma: Definitions | p. 101 |
| F[subscript q t] case | p. 101 |
| General A | p. 103 |
| Functional equations in arithmetic case | p. 105 |
| Special values for arithmetic [Gamma subscript infinity] | p. 109 |
| Periods: F[subscript q t] case | p. 109 |
| Periods: General A | p. 110 |
| Special values of arithmetic [Gamma subscript v] | p. 113 |
| F[subscript q t] case: Analog of Gross-Koblitz | p. 113 |
| General A | p. 114 |
| Geometric Factorial and Gamma: Definitions | p. 116 |
| Functional equations in geometric case | p. 118 |
| Special values of geometric [Gamma] and [Gamma subscript v]: F[subscript q t] case | p. 122 |
| Comparisons and uniform framework | p. 124 |
| More analogies for F[subscript q t]: Divisibilities | p. 130 |
| Binomial coefficients | p. 132 |
| Binomial coefficients as nice basis | p. 134 |
| Difference and differentiation operators | p. 136 |
| Relations between the two notions of binomials | p. 138 |
| Bernoulli numbers and polynomials | p. 142 |
| Note on finite differences and q-analogs | p. 149 |
| Zeta functions | p. 153 |
| Zeta values at integers: Definitions | p. 154 |
| Values at positive integers | p. 158 |
| Values at non-positive integers | p. 162 |
| Multiplicities of trivial zeros | p. 166 |
| Zeta function interpolation on character space | p. 170 |
| [infinity]-adic interpolation | p. 170 |
| p adic interpolation | p. 173 |
| Power sums | p. 174 |
| Zeta measure | p. 176 |
| Zero distribution | p. 178 |
| Low values and multi-logarithms | p. 182 |
| Multizeta values | p. 185 |
| Complex valued multizeta | p. 186 |
| Finite characteristic variants | p. 187 |
| Interpolations | p. 195 |
| Analytic properties of zeta and Fredholm determinant | p. 197 |
| Note on classical interpolations | p. 199 |
| Higher rank theory | p. 205 |
| Elliptic modules | p. 205 |
| Modular forms | p. 209 |
| Galois representations | p. 213 |
| DeRham Cohomology | p. 218 |
| Elliptic curves case: Motivation | p. 219 |
| Drinfeld modules case | p. 221 |
| Hypergeometric functions | p. 226 |
| The first analog | p. 227 |
| The second analog | p. 233 |
| Higher dimensions and geometric tools | p. 237 |
| t-modules and t-motives | p. 239 |
| Torsion | p. 244 |
| Purity | p. 244 |
| Exponential, period lattice and uniformizability | p. 247 |
| Cohomology realizations | p. 253 |
| Example: Carlitz-Tate twist C[superscript multiply sign in circle n] | p. 254 |
| Drinfeld dictionary in the simplest case | p. 256 |
| Krichever/Drinfeld dictionary in more generality | p. 260 |
| Applications to Gauss sums, Gamma and Zeta values | p. 265 |
| C[superscript multiply sign in circle n] and [xi](n) | p. 266 |
| Shtuka and Jacobi sums | p. 267 |
| Gauss sums and Theta divisor | p. 269 |
| Examples and applications | p. 269 |
| The case g = d[subscript infinity] = 1 | p. 271 |
| Another Gamma function | p. 273 |
| Analog of Gross-Koblitz | p. 276 |
| Interpolation at [infinity] for new Gamma | p. 276 |
| Fermat motives and Solitons | p. 280 |
| Another approach to solitons | p. 286 |
| Analog of Gross-Koblitz for Geometric Gamma: F[subscript q t] case | p. 290 |
| What is known or expected in general case? | p. 290 |
| Gamma values to Periods connection via solitons: Sketch | p. 293 |
| Log-algebraicity, Cyclotomic module and Vandiver conjecture | p. 295 |
| Explicit Log-Algebraicity formulas | p. 298 |
| Diophantine approximation | p. 303 |
| Approximation exponents | p. 304 |
| Good approximations: Continued fractions | p. 306 |
| Range of exponents: Frobenius | p. 312 |
| Range of exponents: Differentiation | p. 319 |
| Connection with deformation theory | p. 322 |
| Height inequalities for algebraic points | p. 323 |
| Exponent hierarchy | p. 324 |
| Approximation by algebraic functions | p. 325 |
| Note on connection with Diophantine equations | p. 326 |
| Transcendence results | p. 329 |
| Approximation techniques and irrationality | p. 330 |
| Transcendence results on Drinfeld modules | p. 331 |
| Application to Zeta and Gamma values | p. 334 |
| Transcendence results in higher dimensions | p. 335 |
| Application to Zeta and Gamma values | p. 337 |
| Automata and algebraicity: Applications | p. 341 |
| Automata and algebraicity | p. 341 |
| Some useful automata tools | p. 347 |
| Applications to transcendence of gamma values and monomials | p. 348 |
| Applications to transcendence: periods and modular functions | p. 353 |
| Classifying finite characteristic numbers | p. 355 |
| Computational classes and basic tools | p. 357 |
| Algebraic properties of computational classes | p. 360 |
| Applications to refined transcendence | p. 364 |
| Note on the Notation | p. 371 |
| Bibliography | p. 373 |
| Table of Contents provided by Rittenhouse. All Rights Reserved. |
