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Filtered modules with coefficients. (English) Zbl 1251.11044
Let $$E$$ be a finite extension of $$\mathbb Q_p$$. Arithmetic geometry, and in particular elliptic curves and modular forms, is a source of continuous representations $$\rho : \mathrm{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p) \to \mathrm{GL}_2(E)$$. Some of these representations are potentially semi-stable in the sense of Fontaine and are then classified by $$E$$-linear filtered $$(\varphi,N,G_{K/\mathbb Q_p})$$-modules (see J.-M. Fontaine [Representations $$p$$-adiques semi-stables, Asterisque 223, 113–184 (1994; Zbl 0865.14009)]). These objects are much simpler than the original representation $$\rho$$, but even in dimension $$2$$ it can be tricky to classify them all. The purpose of the article under review is to write down a list of these objects and hence to “essentially classify [them]”. The list is given in terms of the type of the representation. This type can be special, ramified principal series, unramified supercuspidal or ramified supercuspidal. In each case, the corresponding modules are given explicitly.

##### MSC:
 11F80 Galois representations
Full Text:
##### References:
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