Let `$F$`

be a non-archimedean local field of residue characteristic `$p$`

. The classical local Langlands correspondence is a 1-1 correspondence between 2-dimensional irreducible complex representations of the Weil group of `$F$`

and certain smooth irreducible complex representations of `$GL_2(F)$`

. The number-theoretic applications made it necessary to seek such correspondence of representations on vector spaces over a field of characteristic `$p$`

. In this talk, however, I will show that for `$F$`

of residue degree `$> 1$`

, unfortunately, there is no such 1-1 mod `$p$`

correspondence. This result is an elaboration of the arguments of Breuil and Paskunas to an arbitrary local field of residue degree `$> 1$`

.

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Last updated: 18 May 2024