Special L-values of geometric motives

Abstract

This paper proposes a conjecture on special values of L-functions of geometric motives over Z. This includes L-functions of mixed motives over Q and Hasse-Weil 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 $M^{\vee }(-1)$. 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. Under standard assumptions concerning mixed motives over Q, $F_p$, and Z, this conjecture is essentially equivalent to the conjunction of Soulé's conjecture about pole orders of zeta-functions of schemes over Z, Beilinson's conjecture about special L-values for motives over Q and the Tate conjecture over $F_p$.

Final version. To appear in Asian Journal of Mathematics. Most recent changes: Corollary 0.3. (L-values of Artin-Tate motives), Theorem 4.5 (relation of Beilinson-Soulé vanishing vs. perfectness of Arakelov intersection pairing)

References in corpus (1)

• Mixed Artin-Tate motives over number rings

Cited by in corpus (3)

• Arakelov motivic cohomology I
• Overconvergent global analytic geometry
• f-cohomology and motives over number rings