We study special values of Artin $L$-functions for dihedral extensions at negative integers. We give a relation between these values and orders of the $\chi$-parts of certain étale cohomology groups.
References
M. Kolster, T. Nguyen Quang Do, V. Fleckinger, Twisted $S$-Units, $p$-adic class number formulas, and the Lichtenbaum conjectures, Duke Mathematical Journal 84 (1996), no.3, 679–717. MR1408541 10.1215/S0012-7094-96-08421-5 euclid.dmj/1077244040
M. Kolster, T. Nguyen Quang Do, V. Fleckinger, Twisted $S$-Units, $p$-adic class number formulas, and the Lichtenbaum conjectures, Duke Mathematical Journal 84 (1996), no.3, 679–717. MR1408541 10.1215/S0012-7094-96-08421-5 euclid.dmj/1077244040
Y. Konomi, The ideal class groups of dihedral extensions over imaginary quadratic fields and the special values of the Artin $L$-function, J. Number Theory 131 (2011), no.6, 1062–1069. MR2772488 10.1016/j.jnt.2010.11.011 Y. Konomi, The ideal class groups of dihedral extensions over imaginary quadratic fields and the special values of the Artin $L$-function, J. Number Theory 131 (2011), no.6, 1062–1069. MR2772488 10.1016/j.jnt.2010.11.011
V. Voevodsky, On motivic cohomology with $\mathbb{Z}/l$-coefficients, Annals of Mathematics 174 (2011), 401–438. MR2811603 10.4007/annals.2011.174.1.11 V. Voevodsky, On motivic cohomology with $\mathbb{Z}/l$-coefficients, Annals of Mathematics 174 (2011), 401–438. MR2811603 10.4007/annals.2011.174.1.11
C. Weibel, 2007 Trieste lectures on the proof of the Bloch-Kato conjecture, some recent developments in algebraic $K$-theory, ICTP Lecture Notes 23 (Abdus Salam International Centre for Theoretical Physics, Trieste, 2008), 277–305. MR2509183 C. Weibel, 2007 Trieste lectures on the proof of the Bloch-Kato conjecture, some recent developments in algebraic $K$-theory, ICTP Lecture Notes 23 (Abdus Salam International Centre for Theoretical Physics, Trieste, 2008), 277–305. MR2509183
A. Wiles, On a conjecture of Brumer, Annals of Mathematics 131 (1990), 555–565. MR1053490 10.2307/1971470 A. Wiles, On a conjecture of Brumer, Annals of Mathematics 131 (1990), 555–565. MR1053490 10.2307/1971470