an:02011912
Zbl 1165.11319
Breuil, Christophe
On some modular \(p\)-adic representations of \(\text{GL}_2(\mathbb Q_p)\). II
FR
J. Inst. Math. Jussieu 2, No. 1, 23-58 (2003).
1474-7480 1475-3030
2003
j
11F80 11F33 11F85
crystalline representation; locally algebraic representation; reduction modulo \(p\).
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.
Olaf Ninnemann (Berlin)
1044.11041