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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
##### Keywords:
mod $$p$$ Langlands correspondence; Serre weights; socle
Full Text:
##### References:
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