Special L-values of geometric motives

TL;DR

This paper proposes a conjecture relating special L-values of geometric motives over Z to motivic cohomology and Arakelov theory, connecting several major conjectures in number theory and algebraic geometry.

Contribution

It introduces a unified conjecture linking L-values of motives over Z to motivic cohomology and Arakelov theory, extending and connecting existing conjectures.

Findings

Conjecture relates L-values at s=0 to motivic cohomology and pairings.

Shows equivalence of the conjecture to several major existing conjectures.

Provides a framework unifying various conjectures in number theory.

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 Soul\'e'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 $F_p$.