Abstract view
$L$functions for Quadratic Characters and Annihilation of Motivic Cohomology Groups


Published:20150423
Printed: Sep 2015
Jonathan W. Sands,
Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05401, USA
Abstract
Let $n$ be a positive even integer, and let $F$ be a totally real
number field and $L$ be an abelian Galois extension which is totally
real or CM.
Fix a finite set $S$ of primes of $F$ containing the infinite primes
and all those which ramify in
$L$, and let $S_L$ denote the primes of $L$ lying above those in
$S$. Then $\mathcal{O}_L^S$ denotes the ring of $S_L$integers of $L$.
Suppose that $\psi$ is a quadratic character of the Galois group of
$L$ over $F$. Under the assumption of the motivic Lichtenbaum
conjecture, we obtain a nontrivial annihilator of the motivic
cohomology group
$H_\mathcal{M}^2(\mathcal{O}_L^S,\mathbb{Z}(n))$ from the lead term of the Taylor series for the
$S$modified Artin $L$function $L_{L/F}^S(s,\psi)$ at $s=1n$.
MSC Classifications: 
11R42, 11R70, 14F42, 19F27 show english descriptions
Zeta functions and $L$functions of number fields [See also 11M41, 19F27] $K$theory of global fields [See also 19Fxx] Motivic cohomology; motivic homotopy theory [See also 19E15] Etale cohomology, higher regulators, zeta and $L$functions [See also 11G40, 11R42, 11S40, 14F20, 14G10]
11R42  Zeta functions and $L$functions of number fields [See also 11M41, 19F27] 11R70  $K$theory of global fields [See also 19Fxx] 14F42  Motivic cohomology; motivic homotopy theory [See also 19E15] 19F27  Etale cohomology, higher regulators, zeta and $L$functions [See also 11G40, 11R42, 11S40, 14F20, 14G10]
