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Conductors of \(\ell\)-adic representations. (English) Zbl 1383.11076

Summary: We give a new formula for the Artin conductor of an \( \ell \)-adic representation of the Weil group of a local field of residue characteristic \( p\neq \ell \).

MSC:

11F80 Galois representations
11S37 Langlands-Weil conjectures, nonabelian class field theory
11S20 Galois theory

References:

[1] Artin, E., Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlk\"orper, J. Reine Angew. Math., 164, 1-11 (1931) · JFM 57.0200.02 · doi:10.1515/crll.1931.164.1
[2] Clifford, A. H., Representations induced in an invariant subgroup, Ann. of Math. (2), 38, 3, 533-550 (1937) · Zbl 0017.29705 · doi:10.2307/1968599
[3] Darmon, Henri; Diamond, Fred; Taylor, Richard, Fermat’s last theorem. Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993), 2-140 (1997), Int. Press, Cambridge, MA
[4] Deligne, P., Les constantes des \'equations fonctionnelles des fonctions \(L\). Modular functions of one variable, II, Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972, 501-597. Lecture Notes in Math., Vol. 349 (1973), Springer, Berlin · Zbl 0264.00002
[5] Dokchitser, T.; V. Dokchitser, Growth of III in towers for isogenous curves, Compositio Mathematica, FirstView, 1-25 (2015) · Zbl 1351.11035 · doi:10.1112/S0010437X15007423
[6] Rohrlich, David E., Elliptic curves and the Weil-Deligne group. Elliptic curves and related topics, CRM Proc. Lecture Notes 4, 125-157 (1994), Amer. Math. Soc., Providence, RI · Zbl 0852.14008
[7] [Serre70] J.-P. Serre, \newblockFacteurs locaux des fonctions z\^eta des vari\'et\'es alg\'ebriques (d\'efinitions et conjectures), \newblockS\'eminaire Delange-Pisot-Poitou: 1969/70, Th\'eorie des Nombres, Fasc. 2, Exp. 19, page 12, Secr\'etariat math\'ematique, Paris, 1970. · Zbl 0214.48403
[8] Serre, Jean-Pierre, Local fields, Graduate Texts in Mathematics 67, viii+241 pp. (1979), Springer-Verlag, New York-Berlin · Zbl 0423.12016
[9] [SerreLALG] J.-P. Serre. \newblockLie algebras and Lie groups, volume 1500 of Lecture Notes in Mathematics. \newblock Springer-Verlag, Berlin, 2006. \newblock1964 lectures given at Harvard University, Corrected fifth printing of the second (1992) edition.
[10] Serre, Jean-Pierre; Tate, John, Good reduction of abelian varieties, Ann. of Math. (2), 88, 492-517 (1968) · Zbl 0172.46101
[11] Tate, J., Number theoretic background. Automorphic forms, representations and \(L\)-functions, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977, Proc. Sympos. Pure Math., XXXIII, 3-26 (1979), Amer. Math. Soc., Providence, R.I. · Zbl 0422.12007
[12] [WieseGR] G. Wiese, \newblockGalois representations, \newblock Version dated 13 February 2012, downloaded from http://math.uni.lu/\textasciitildewiese/notes/GalRep.pdf, 2012.
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