On critical values of twisted Artin $L$-functions

Peng-Jie Wong

Czechoslovak Mathematical Journal (2017)

• Volume: 67, Issue: 2, page 551-555
• ISSN: 0011-4642

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Abstract

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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.

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Wong, Peng-Jie. "On critical values of twisted Artin $L$-functions." Czechoslovak Mathematical Journal 67.2 (2017): 551-555. <http://eudml.org/doc/288218>.

@article{Wong2017,
abstract = {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.},
author = {Wong, Peng-Jie},
journal = {Czechoslovak Mathematical Journal},
keywords = {Artin $L$-function; character; Galois Gauss sum; special value},
language = {eng},
number = {2},
pages = {551-555},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On critical values of twisted Artin $L$-functions},
url = {http://eudml.org/doc/288218},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Wong, Peng-Jie
TI - On critical values of twisted Artin $L$-functions
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 551
EP - 555
AB - 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.
LA - eng
KW - Artin $L$-function; character; Galois Gauss sum; special value
UR - http://eudml.org/doc/288218
ER -

References

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1. Coates, J., Lichtenbaum, S., 10.2307/1970916, Ann. Math. (2) 98 (1973), 498-550. (1973) Zbl0279.12005MR0330107DOI10.2307/1970916
2. Klingen, H., 10.1007/BF01451369, Math. Ann. 145 (1962), 265-272 German. (1962) Zbl0101.03002MR0133304DOI10.1007/BF01451369
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. (1977) Zbl0359.12015MR0447187
4. Neukirch, J., 10.1007/978-3-662-03983-0, Grundlehren der Mathematischen Wissenschaften 322, Springer, Berlin (1999). (1999) Zbl0956.11021MR1697859DOI10.1007/978-3-662-03983-0
5. Siegel, C. L., Über die Fourierschen Koeffizienten von Modulformen, Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 3 (1970), 15-56 German. (1970) Zbl0225.10031MR0285488
6. Ward, K., 10.1007/s00013-014-0692-7, Arch. Math. 103 (2014), 285-290. (2014) Zbl1314.11035MR3266371DOI10.1007/s00013-014-0692-7

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