Open Access
March 2015 On the special values of Artin $L$-functions for dihedral extensions
Yutaka Konomi
Funct. Approx. Comment. Math. 52(1): 109-116 (March 2015). DOI: 10.7169/facm/2015.52.1.8
Abstract

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

1.

A. Grothendieck et al, Théorie des Topos et Cohomologie Etale des Schémas, Lecture Notes in Mathmatics, vol. 270, 1972.  MR354653 A. Grothendieck et al, Théorie des Topos et Cohomologie Etale des Schémas, Lecture Notes in Mathmatics, vol. 270, 1972.  MR354653

2.

I. Martin Isaacs, Character Theory of Finite Groups, Pure and Applied Mathematics 69, Academic Press, 1976.  MR460423 I. Martin Isaacs, Character Theory of Finite Groups, Pure and Applied Mathematics 69, Academic Press, 1976.  MR460423

3.

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

4.

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

5.

J. Neukirch, Algebraic Number Theory, GdmW., vol. 322, Springer, 1999.  MR1697859 J. Neukirch, Algebraic Number Theory, GdmW., vol. 322, Springer, 1999.  MR1697859

6.

V.P. Snaith, Algebraic $K$-groups as Galois Modules, Progress in Mathematics 206, Birkhäuser, 2002.  MR1897817 V.P. Snaith, Algebraic $K$-groups as Galois Modules, Progress in Mathematics 206, Birkhäuser, 2002.  MR1897817

7.

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

8.

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

9.

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
Copyright © 2015 Adam Mickiewicz University
Yutaka Konomi "On the special values of Artin $L$-functions for dihedral extensions," Functiones et Approximatio Commentarii Mathematici 52(1), 109-116, (March 2015). https://doi.org/10.7169/facm/2015.52.1.8
Published: March 2015
Vol.52 • No. 1 • March 2015
Back to Top