\(K\)-theory and values of zeta functions. (English) Zbl 1096.11503
Bass, H. (ed.) et al., Algebraic \(K\)-theory and its applications. Proceedings of the workshop and symposium, ICTP, Trieste, Italy, September 1–19, 1997. Singapore: World Scientific (ISBN 981-02-3491-0/hbk). 255-283 (1999).
From the text: The author surveys various conjectures and results relating algebraic \(K\)-groups and values of zeta-functions, putting the emphasis on concrete examples in the one-dimensional case. First a short section on varieties over finite fields is given. Then he treats the case of rings of integers in number fields. The following topics are considered: (1) Computations of \(K_i(\mathbb Z)\) using homology of the space of lattices and their conjectural relation to Bernoulli numbers. (2) The Borel regulator and its \(p\)-adic variant. (3) Relation to the Main Conjecture in Iwasawa theory. (4) Polylogarithms and Zagier’s conjecture.
For the entire collection see [Zbl 0949.00018].
For the entire collection see [Zbl 0949.00018].
MSC:
| 11G55 | Polylogarithms and relations with \(K\)-theory |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 11R42 | Zeta functions and \(L\)-functions of number fields |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |
