Values of zeta-functions, étale cohomology, and algebraic \(K\)-theory. (English) Zbl 0284.12005

Lichtenbaum, Stephen

Algebraic \(K\)-Theory II, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 342, 489-501 (1973).

Let \(F\) be an algebraic number field of finite degree over \(\mathbb Q\) and let \(\zeta(F,s)\) be the associated zeta-function. The author makes a number of conjectures connecting the absolute value of \(\zeta(F,-m)\), \(m\) a positive odd integer, with the orders of various higher \(K\)-groups and étale cohomology groups. The most spectacular is that \[ | \zeta(F,-m)| = | K_{2m}(\mathcal O)|/| K_{2m+1}(\mathcal O)| \] with 2-torsion, where \(\mathcal O\) is the integers of \(F\). The conjectures are based on various relations (also mostly conjectural) between the \(K_i(\mathcal O)\)’s and cohomology groups, together with the theory of \(p\)-adic \(L\)-functions.
For the entire collection see Zbl 0265.00008.

Reviewer: M. E. Keating (London)

MSC:

11R42	Zeta functions and \(L\)-functions of number fields
11R34	Galois cohomology
11R70	\(K\)-theory of global fields
19F99	\(K\)-theory in number theory

Citations:

Zbl 0265.00008