Local Langlands Conjectures - Local Langlands Conjectures For GLn

Local Langlands Conjectures For GLn

The local Langlands conjectures for general linear groups state that there are unique bijections π ↔ ρπ from equivalence classes of irreducible admissible representations π of GLn(F) to equivalence classes of continuous Frobenius semisimple complex n-dimensional Weil–Deligne representations ρπ of the Weil group of F, that preserve L-functions and ε-factors of pairs of representations, and coincide with the Artin map for 1-dimensional representations. In other words,

  • L(sπ⊗ρπ') = L(s,π×π')
  • ε(sπ⊗ρπ',ψ) = ε(s,π×π',ψ)

Laumon, Rapoport & Stuhler (1993) proved the local Langlands conjectures for the general linear group GLn(K) for positive characteristic local fields K. Carayol (1992) gave an exposition of their work.

Richard Taylor and Michael Harris (2001) proved the local Langlands conjectures for the general linear group GLn(K) for characteristic 0 local fields K. Henniart (2001) gave another proof. Carayol (2000) and Wedhorn (2008) gave expositions of their work.

Read more about this topic:  Local Langlands Conjectures

Famous quotes containing the words local and/or conjectures:

    Surely there must be some way to find a husband or, for that matter, merely an escort, without sacrificing one’s privacy, self-respect, and interior decorating scheme. For example, men could be imported from the developing countries, many parts of which are suffering from a man excess, at least in relation to local food supply.
    Barbara Ehrenreich (b. 1941)

    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 (1694–1773)