• DOI:10.4310/AJM.2017.V21.N2.A2
• Corpus ID: 115172058

Special L-values of geometric motives

@article{Scholbach2010SpecialLO,
  title={Special L-values of geometric motives},
  author={Jakob Scholbach},
  journal={Asian Journal of Mathematics},
  year={2010},
  volume={21},
  pages={225-264}
}

• Jakob Scholbach
• Published 5 March 2010
• Mathematics
• Asian Journal of Mathematics

This paper proposes a conjecture about special values of L-functions L(M,s) := Q p det(Id Fr 1 p s |iM`) 1 of geometric motives M over Z. This includes L-functions of mixed motives over Q and Hasse-Weil �-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 mo- tivic cohomology of D(M) := M _ (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): L � (M,0) � Y i det(H i(M)H i (D(M)) ! R) ( 1) i+1 (mod Q × ). Under standard assumptions concerning mixed motives over Q, Fp, and Z, this conjecture is essentially equivalent to the conjunction of Soule's conjecture about pole orders of �-functions of schemes over Z, Beilinson's conjecture about special L-values for motives over Q and the Tate conjec- ture over Fp.