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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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