SPECIAL L-VALUES OF GEOMETRIC MOTIVES

This paper proposes a conceptual unification of Beilinson's conjecture about special L-values for motives over Q, the Tate conjecture over F-p and Soule's conjecture about pole orders of zeta-functions of schemes over Z. We conjecture the following: the order of L(M, s) at s = 0 is given by the negative Euler characteristic of motivic cohomology of M-V(-1). Up to a nonzero rational factor, the L-value at s = 0 is given by the determinant of the pairing of Arakelov motivic cohomology of M with the motivic homology of M: L*(M, 0) = Pi(i is an element of Z) det(Hi-2(M, -1)circle times(H) over cap (i) (M) -> R)((-1) i+1) (mod Q(x)). Under standard assumptions concerning mixed motives over Q, F-p, and Z, this conjecture is equivalent to the conjunction of the above-mentioned conjectures of Beilinson, Tate, and Soul'e. We use this to unconditionally prove the Beilinson conjecture for all Tate motives and, up to an n-th root of a rational number, for all Artin-Tate motives.