Editing Langlands program (section)

===Local Langlands conjectures===
{{main|local Langlands conjectures}}

{{harvs|txt|last=Kutzko|first=Philip |authorlink=Philip Kutzko|year=1980}} proved the [[local Langlands correspondence]] for the general linear group GL(2, ''K'') over local fields.

{{harvs|txt|last=Laumon|first=Gérard |last2=Rapoport |author1-link=Gérard Laumon |first2=Michael |author2-link=Michael Rapoport |last3=Stuhler |first3=Ulrich |author3-link=Ulrich Stuhler |year=1993}} proved the local Langlands correspondence for the general linear group GL(''n'', ''K'') for positive characteristic local fields ''K''. Their proof uses a global argument, realizing smooth admissible representations of interest as the local components of automorphic representations of the group of units of a division algebra over a curve, then using the point-counting formula to study the properties of the global Galois representations associated to these representations.

{{harvs |txt |author2-link=Richard Taylor (mathematician) |first2=Richard |last2=Taylor |first1=Michael |last1=Harris |author1-link=Michael Harris (mathematician) |year=2001}} proved the local Langlands conjectures for the general linear group GL(''n'', ''K'') for characteristic 0 local fields ''K''. {{harvs |txt |last=Henniart |first=Guy |author-link=Guy Henniart |year=2000}} gave another proof. Both proofs use a global argument of a similar flavor to the one mentioned in the previous paragraph. {{harvs |txt |author-link=Peter Scholze |last=Scholze |first=Peter |year=2013}} gave another proof.