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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
##### Keywords:
Galois representations; Artin conductor; local fields
Full Text:
##### References:
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