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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.

MSC:
11F80 Galois representations
11S15 Ramification and extension theory
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