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

##### 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
Zbl 0265.00008