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L-functions for Quadratic Characters and Annihilation of Motivic Cohomology Groups
Published online by Cambridge University Press: 20 November 2018
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 $\text{CM}$. 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 $s=1-n$.
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- Copyright © Canadian Mathematical Society 2015