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On the finiteness of mod \(p\) Galois representations of a local field. (English) Zbl 1205.11059
The finiteness problem of mod \(p\) Galois representations with bounded Artin conductor has been studied for global fields, and also for algebraic function fields. In this paper, the author considers the finiteness of \(\bmod\;p\) Galois representations of a local field.
Let \(K\) be a local field, \(G_ K\) its absolute Galois group and \(k\) an algebraically closed field of characteristic \(p\geq 0\). Let \(\rho: G_ K\to \text{GL}_ d(k)\) denote a semisimple continuous representation. In Section 1 the author proves:
Theorem 1: (1) There are finitely many \(\rho\) with bounded Artin conductor and bounded residue degree. (2) If \(p>0\) and \([K:{\mathbb Q}_ p]<\infty\), there are finitely many \(\rho\) of bounded residue degree.
Theorem 1 is a consequence of
Theorem 2: (1) There exist finitely many finite Galois extensions \(L/K\) with bounded different valuation such that \(\text{Gal}(L/K)\) can be embedded in \(\text{GL}_ d(k)\). (2) If \(p>0\) and \([K: {\mathbb Q}_ p]<\infty\) then there exist finitely many finite Galois extensions of bounded residue degree such that \(\text{Gal}(L/K)\) can be embedded in \(\text{GL}_ d(k)\).
The author presents another proof of Theorem 1 that gives an effective upper bound for the number of finite Galois extensions of \(K\) corresponding to irreducible representations.

11F80 Galois representations
11S15 Ramification and extension theory
Full Text: DOI
[1] L. M. Butler, A unimodality result in the enumeration of subgroups of a finite abelian group, Proc. Amer. Math. Soc. 101 (1987), 771–775. JSTOR: · Zbl 0647.20053 · doi:10.2307/2046687 · links.jstor.org
[2] G. Böckle and C. Khare, Finiteness results for mod \(l\) Galois representations over function fields, · Zbl 1078.11036
[3] K. Iwasawa, Local class field theory, Oxford Math. Monogr., Oxford Sci. Publ., The Clarendon Press, Oxford Univ. Press, New York, 1986. · Zbl 0604.12014
[4] N. M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Ann. of Math. Stud. 116, Princeton Univ. Press, Princeton, N.J., 1988. · Zbl 0675.14004
[5] C. Khare, Conjectures on finiteness of mod \(p\) Galois representations, J. Ramanujan Math. Soc. 15 (2000), 23–42. · Zbl 1017.11028
[6] M. J. Larsen and R. Pink, Finite subgroups of algebraic groups, · Zbl 1241.20054
[7] A. Lubotzky and D. Segal, Subgroup growth, Progr. Math. 212, Birkhäuser Verlag, Basel, 2003. · Zbl 1071.20033
[8] I. G. Macdonald, Symmetric functions and Hall polynomials, second edition, Oxford Math. Monogr., Oxford Sci. Publ., The Clarendon Press, Oxford Univ. Press, New York, 1995. · Zbl 0824.05059
[9] H. Moon, Finiteness results on certain mod \(p\) Galois representations, J. Number Theory 84 (2000), 156–165. · Zbl 0967.11021 · doi:10.1006/jnth.2000.2534
[10] H. Moon, The number of monomial mod \(p\) Galois representations with bounded conductor, Tohoku Math. J. (2) 55 (2003), 89–98. · Zbl 1047.11050 · doi:10.2748/tmj/1113247447
[11] H. Moon and Y. Taguchi, Mod \(p\)-Galois representations of solvable image, Proc. Amer. Math. Soc. 129 (2001), 2529–2534 (electronic). JSTOR: · Zbl 1033.11024 · doi:10.1090/S0002-9939-01-05894-4 · links.jstor.org
[12] J.-P. Serre, Linear representations of finite groups, Grad. Texts in Math. 42, Springer-Verlag, New York, 1977. · Zbl 0355.20006
[13] J.-P. Serre, Une “formule de masse” pour les extensions totalement ramifiées de degré donné d’un corps local, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), A1031–A1036. · Zbl 0388.12005
[14] J.-P. Serre, Local fields, Grad. Texts in Math. 67, Springer-Verlag, New York, 1979. · Zbl 0423.12016
[15] J.-P. Serre, Sur les représentations modulaires de degré \(2\) de \(\mathrmGal(\bar\boldsymbolQ/\boldsymbolQ)\), Duke Math. J. 54 (1987), 179–230. · Zbl 0641.10026 · doi:10.1215/S0012-7094-87-05413-5
[16] D. A. Suprunenko, Matrix groups, Trans. Math. Monogr. 45, Amer. Math. Soc., Providence, R.I., 1976. · Zbl 0317.20028
[17] Y. Taguchi, On the finiteness of various Galois representations, To appear in : Proc. Symp. “Primes and Knots”, (T. Kohno and M. Morishita, eds.), Contemp. Math., Amer. Math. Soc. · Zbl 1173.11061
[18] Y. Taguchi, Induction formula for the Artin conductors of mod \(l\) Galois representations, Proc. Amer. Math. Soc. 130 (2002), 2865–2869 (electronic). JSTOR: · Zbl 1045.11036 · doi:10.1090/S0002-9939-02-06524-3 · links.jstor.org
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