Holmstrom Scholbach conjecture

arXiv:1003.1215 Special L-values of geometric motives from arXiv Front: math.NT by Jakob Scholbach This paper proposes a conjecture about special values of L-functions of geometric motives over Z. We conjecture the following: the pole order of the L-function L(M, s) of M at s=0 is given by the negative Euler characteristic of motivic cohomology of $D(M) := M\dual(1)[2]$. Up to a nonzero rational factor, the L-value at s=0 is given by the determinant of a pairing coupling an Arakelov-like variant of motivic cohomology of M with the motivic cohomology of D(M). Under standard assumptions concerning mixed motives over Q, finite fields 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.

nLab page on Scholbach conjecture