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Towards a modulo \(p\) Langlands correspondence for \(\mathrm{GL}_{2}\). (English) Zbl 1245.22010
Mem. Am. Math. Soc. 1016, 114 p. (2012).
The authors discuss the possibility to extend the conjectural Langlands correspondence to a mod-\(p\)-situation.
More specifically, given the quotient field \(F\) of the ring of Witt vectors \({\mathcal O}_F\) of a finite field \(k\) in characteristic \(p,\) let \(\rho:\text{Gal}(\overline F / F) \to \text{GL}_2(\overline k)\) be a continuous representation. The authors describe supersingular representations \(\pi\) of \(\text{GL}_2(F)\) which can be associated to \(\rho\) by means of their maximal semisimple \(\text{GL}_2({\mathcal O}_F)\)-submodule. They, in particular, discuss that \(\pi\) will not be uniquely characterized by the desired properties and propose a means to deal with this situation.
The results are very technical and hard to describe in detail in this review.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
11F80 Galois representations
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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