Abstract

It is a classical problem to understand the value of the Riemann zeta-function at n=2, 3, 4, ..... For n even this value is πn times a non-zero rational number, but for n odd much less is known. In the 70s Borel proved that, for a number field k, and n≥2, up to some simple factors and a non-zero rational number, ζk(n) equals the regulator Rn(k) associated to the K-group K2n−1(k) defined earlier by Quillen. Because ζQ(s)=ζ(s), and K3(Q), K7(Q), ... are finite whereas K5(Q), K9(Q), ... have rank one, this explains the behaviour of ζ(n). Using the functional equation of ζk(n), Borel’s result gives the main ingredient of the first non-zero coefficient in the Taylor expansion at s=1−n of ζk(s), but the Lichtenbaum conjecture is more precise: up to sign and a power of 2, this coefficient should equal |K2n−1(Ok)tor|-1|K2n−2(Ok)| Rn(k), where Ok is the ring of algebraic integers in k. Nowadays there are various conjectures on special values of zeta-functions and their generalisations. Techniques to prove results have involved Iwasawa theory, as well as more explicit descriptions of the K-groups involved in terms of complexes of algebraic cycles or formal generators and relations. One of those conjectures (predating Borel’s result and Lichtenbaum’s conjecture, and, in fact, one of the Millenium Prize Problems of the Clay Mathematical Institute) is the conjecture by Birch and Swinnerton-Dyer, on the behaviour at s=1 of L(E, s) for an elliptic curve E over a number field, where L(E, s) is the analogue of the zeta-function in this context. Its statement involves another K-group, K0(E), as well as periods of 1-forms on the associated Riemann surface, a height pairing, and some more refined arithmetic invariants. Such periods tie up with Hodge theory, one of the other areas covered by this workshop.