Beilinson: Higher regulators and values of L-functions (in Russian). (1984)
Some work by Rob de Jeu on curves over number fields.
MR0406981 (53 #10765) 12A70 Lichtenbaum, Stephen Values of zeta-functions, ´etale cohomology, and algebraicK-theory. Algebraic K-theory, II: ‘‘Classical’’ algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 489–501. Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973. The author expounds the Birch-Tate conjecture about the value of the zeta-function ?(F, s) of an algebraic number field F at s = −1 in terms of cohomology, as well as in terms of K2 [see the author, Ann. of Math. (2) 96 (1972), 338–360; MR0360527 (50 #12975)], and then he offers a conjecture in connection with Quillen’s conjecture, extending the latter to a conjecture for any odd negative number s. In the last section he defines the mth regulator of F and again proposes a conjecture about this and ?(F, s). {For the entire collection see MR0325308 (48 #3656b).}
MR721399 (85d:14017) 14C35 (13D15 18F25) Coombes, K. R. (1-OK); Srinivas,V. Srinivas,Vasudevan Aremark onK1 of an algebraic surface. Math. Ann. 265 (1983), no. 3, 335–342. The authors prove two interesting theorems, which are part of a general program initiated by Bloch and Beilinson whose goal is to relate higher algebraic K-groups of algebraic varieties to their real cohomology groups via a regulator map. Let X be a nonsingular projective surface over an uncountable algebraically closed field k, such that CH2(X), the group of codimension 2 cycles modulo linear equivalence, is finite-dimensional in the sense of Mumford. They prove that the cokernel of the product map PicX Z k? ! H1(X,K2) isN-torsion for some integerN. If, in addition, AlbX = 0, then the same result holds for Kn(k)!H0(X,Kn), for any n. They give the following application to commutative algebra. LetR = kx, y, z/(zpn −f(x, y)) be a normal domain of characteristic p. Let G be the Grothendieck group of R-modules of finite length and finite projective dimension. The theorem is that G is a direct sum of Z and an abelian group which is N-torsion for some N.
nLab page on Regulator references [private]