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On some modular \(p\)-adic representations of \(\text{GL}_2(\mathbb Q_p)\). II. (Sur quelques repr√©sentations 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.

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