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 $\mathbb{F}_p$ and Soulé’s conjecture about pole orders of $\zeta$-functions of schemes over $\mathbb{Z}$. 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 $M$:\[L^{*} (M, 0) \equiv \prod_{i \in Z} \mathrm{det} (H_{i-2} (M,-1) \otimes \hat{H}^i (M) \to \mathbb{R})^{(-1)^{i+1}} \: (\mathrm{mod} \: \mathbb{Q}^{\times}) \: \textrm{.}\]Under standard assumptions concerning mixed motives over $\mathbb{Q}$, $\mathbb{F}_p$, and $\mathbb{Z}$, this conjecture is equivalent to the conjunction of the above-mentioned conjectures of Beilinson, Tate, and Soulé. We use this to unconditionally prove the Beilinson conjecture for all Tate motives and, up to an $n \textrm{-th}$ root of a rational number, for all Artin–Tate motives.

Keywords

$L$-functions, Beilinson conjecture, motives, $K$-theory, Deligne cohomology, Arakelov motivic cohomology

2010 Mathematics Subject Classification

Primary 11G40, 14F42, 19E15. Secondary 14G40.

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Received 13 July 2015

Published 5 July 2017