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… 
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