Number theory in function fields

✍ Scribed by Michael I Rosen

• Publisher: Springer
• Year: 2002
• Tongue: English
• Leaves: 374
• Series: Graduate texts in mathematics, 210
• Category: Library

✦ Synopsis

Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting analogues of various theorems. The later chapters probe the analogy between global function fields and algebraic number fields. Topics include the ABC-conjecture, Brumer-Stark conjecture, and Drinfeld modules.

✦ Table of Contents

Content: Preface --
1. Polynomials over finite fields --
2. Primes, Arithmetic functions, and the zeta function --
3. The reciprocity law --
4. Dirichlet L-series and primes in an arithmetic progression --
5. Algebraic function fields and global function fields --
6. Weil differentials and the canonical class --
7. Extensions of function fields, Riemann-Hurwitz, and the ABC theorem --
8. Constant field extensions --
9. Galois extensions : Hecke and Artin L-series --
10. Artin's primitive root conjecture --
11. The behavior of the class group in constant field extensions --
12. Cyclotomic function fields --
13. Drinfeld modules : an introduction --
14. S-units, S-class group, and the corresponding L-functions --
15. The Brumer-Stark conjecture --
16. The class number formulas in quadratic and cyclotomic function fields --
17. Average value theorems in function fields --
Appendix. A proof of the function field Riemann hypothesis.