Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Artin $L$-function; character; Galois Gauss sum; special value
Artin $L$-function; character; Galois Gauss sum; special value
Summary:
We give a simple proof that critical values of any Artin $L$-function attached to a representation $\rho $ with character $\chi _{\rho }$ are stable under twisting by a totally even character $\chi $, up to the $\dim \rho $-th power of the Gauss sum related to $\chi $ and an element in the field generated by the values of $\chi _{\rho }$ and $\chi $ over $\mathbb {Q}$. This extends a result of Coates and Lichtenbaum as well as the previous work of Ward.
References:
[1] Coates, J., Lichtenbaum, S.: On $l$-adic zeta functions. Ann. Math. (2) 98 (1973), 498-550. DOI 10.2307/1970916 | MR 0330107 | Zbl 0279.12005
[2] Klingen, H.: Über die Werte der Dedekindschen Zetafunktion. Math. Ann. 145 (1962), 265-272 German. DOI 10.1007/BF01451369 | MR 0133304 | Zbl 0101.03002
[3] Martinet, J.: Character theory and Artin $L$-functions. Algebraic Number Fields Proc. Symp. London math. Soc., Univ. Durham 1975, Academic Press, London (1977), 1-87. MR 0447187 | Zbl 0359.12015
[4] Neukirch, J.: Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften 322, Springer, Berlin (1999). DOI 10.1007/978-3-662-03983-0 | MR 1697859 | Zbl 0956.11021
[5] Siegel, C. L.: Über die Fourierschen Koeffizienten von Modulformen. Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 3 (1970), 15-56 German. MR 0285488 | Zbl 0225.10031
[6] Ward, K.: Values of twisted Artin $L$-functions. Arch. Math. 103 (2014), 285-290. DOI 10.1007/s00013-014-0692-7 | MR 3266371 | Zbl 1314.11035
Search
Partner of
