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On some modular and \(p\)-adic representations of \(\text{GL}_{2}(\mathbb Q_{p})\). I. (Sur quelques représentations modulaires et \(p\)-adiques de \(\text{GL}_{2}(\mathbb Q_{p})\). I.) (French) Zbl 1044.11041
Let \(p\) be a prime number and \(G:= \text{GL}(2,{\mathbb Q}_p) \supset K:= \text{GL}(2,{\mathbb Z}_p).\) Let \(T\) be the standard generator for the Hecke algebra of \(G\) relative to \(K\). If \(\pi\) is an irreducible smooth and unramified representation of \(G\) on an \(\overline {{\mathbb F}_p}\)-vector space \(V\), the space of \(K\)-invariants \(V^K\) is one-dimensional, and \(T\) acts on it by multiplication by some scalar \(\lambda\). Following L. Barthel and R. Livné [J. Number Theory 55, No. 1, 1–27 (1995; Zbl 0841.11026)], the representation \(\pi\) is called supersingular if \(\lambda = 0\). These are representations which do not have good analogs in characteristic zero.
In the paper under consideration, the author constructs supersingular representations and gives a natural bijection between the sets of (isomorphism classes of) supersingular representations and of (isomorphism classes of) irreducible two-dimensional representations of \(\text{Gal}(\overline {{\mathbb Q}}_p / {\mathbb Q}_p)\) on \(\overline {{\mathbb F}}_p\)-vector spaces.

MSC:
11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
11F85 \(p\)-adic theory, local fields
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