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# $L$-functions for Quadratic Characters and Annihilation of Motivic Cohomology Groups

Published:2015-04-23
Printed: Sep 2015
• Jonathan W. Sands,
Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05401, USA
 Format: LaTeX MathJax PDF

## 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 non-trivial 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 $s=1-n$.
 Keywords: motivic cohomology, regulator, Artin L-functions
 MSC Classifications: 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]

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