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

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 $\text{CM}$ . 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 $H_{\mathcal{M}}^{2}\left( \mathcal{O}_{L}^{S},\mathbb{Z}\left( n \right) \right)$ from the lead term of the Taylor series for the S-modified Artin $L$ -function $L_{L/F}^{S}\left( s,\psi \right)$ at $s=1-n$ .