## On some modular $$p$$-adic representations of $$\text{GL}_2(\mathbb Q_p)$$. II. (Sur quelques représentations modulaires et $$p$$-adiques de $$\text{GL}_2(\mathbb Q_p)$$. II.)(French)Zbl 1165.11319

In Part I [Compos. Math. 138, No. 2, 165–188 (2003; Zbl 1044.11041)], the author showed that there exists a correspondence between a particular family of $$\bar{{\mathbb F}}_ p$$-representations of $$\text{GL}_2(\mathbb Q_p)$$, the supersingular representations, and the $$2$$-dimensional continuous irreducible $$\bar{{\mathbb F}}_p$$-representations of $$\text{Gal}(\bar{\mathbb Q}_p/\mathbb Q_p)$$. In this second part he conjectures that the reduction modulo $$p$$ of irreducible crystalline two-dimensional representations over $$\bar{\mathbb Q}_p$$ of $$\text{Gal}(\bar{\mathbb Q}_p/\mathbb Q_p)$$ can be obtained from the reduction modulo $$p$$ of locally algebraic $$p$$-adic representations of $$\text{GL}_2(\mathbb Q_p)$$. He confirms this conjecture by giving some explicit calculations of these reductions. Moreover this suggests a nontrivial arithmetic connection between the two types of representations.

### MSC:

 11F80 Galois representations 11F33 Congruences for modular and $$p$$-adic modular forms 11F85 $$p$$-adic theory, local fields

Zbl 1044.11041
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