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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.

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
Full Text: DOI
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