Local Langlands Conjectures For GL2
The local Langlands conjecture for GL2 of a local field says that there is a (unique) bijection π from 2-dimensional semisimple Deligne representations of the Weil group to irreducible smooth representations of GL2(F) that preserves L-functions, ε-factors, and commutes with twisting by characters of F*.
Jacquet & Langlands (1970) verified the local Langlands conjectures for GL2 in the case when the residue field does not have characteristic 2. In this case the representations of the Weil group are all of cyclic or dihedral type. Gelfand & Graev (1962) classified the smooth irreducible representations of GL2(F) when F has odd residue characteristic (see also (Gelfand, Graev & Pyatetskii-Shapiro 1969, chapter 2)), and claimed incorrectly that the classification for even residue characteristic differs only insignifictanly from the odd residue characteristic case. Weil (1974) pointed out that when the residue field has characteristic 2, there are some extra exceptional 2-dimensional representations of the Weil group whose image in PGL2(C) is of tetrahedral or octahedral type. (For global Langlands conjectures, 2-dimensional representations can also be of icosahedral type, but this cannot happen in the local case as the Galois groups are solvable.) Tunnell (1978) proved the local Langlands conjectures for the general linear group GL2(K) over the 2-adic numbers, and over local fields containing a cube root of unity. Kutzko (1980, 1980b) proved the local Langlands conjectures for the general linear group GL2(K) over all local fields.
Cartier (1981) and Bushnell & Henniart (2006) gave expositions of the proof.
Read more about this topic: Local Langlands Conjectures
Famous quotes containing the words local and/or conjectures:
“The poets eye, in a fine frenzy rolling,
Doth glance from heaven to earth, from earth to heaven;
And as imagination bodies forth
The forms of things unknown, the poets pen
Turns them to shapes, and gives to airy nothing
A local habitation and a name.”
—William Shakespeare (15641616)
“Our conjectures pass upon us for truths; we will know what we do not know, and often, what we cannot know: so mortifying to our pride is the base suspicion of ignorance.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (16941773)