# local langlands conjecturesの例文

- Proved the local Langlands conjectures for the symplectic similitude group GSp ( 4 ) and used that in to deduce it for the symplectic group Sp ( 4 ).
- The local Langlands conjectures state ( very roughly ) that representations of an algebraic group over a local field should be closely related to conjugacy classes in the Langlands dual group.
- Proved the local Langlands conjectures for the general linear group GL " n " ( " K " ) for positive characteristic local fields " K " . gave an exposition of their work.
- Proved the local Langlands conjectures for the general linear group GL " n " ( " K " ) for characteristic 0 local fields " K " . gave another proof . and gave expositions of their work.
- Taylor received the 2014 Breakthrough Prize in Mathematics " for numerous breakthrough results in the theory of automorphic forms, including the Taniyama Weil conjecture, the local Langlands conjecture for general linear groups, and the Sato Tate conjecture . " He also received the 2007 Shaw Prize in Mathematical Sciences for his work on the Langlands program with Robert Langlands.
- The local Langlands conjecture for GL 2 of a local field says that there is a ( unique ) bijection ? from 2-dimensional semisimple Deligne representations of the Weil group to irreducible smooth representations of GL 2 ( " F " ) that preserves " L "-functions, ?-factors, and commutes with twisting by characters of " F " *.
- The local Langlands conjectures state ( roughly ) that the L-packets of a reductive group " G " over a local field " F " are conjecturally parameterized by certain homomorphisms of the Langlands group of " F " to the L-group of " G ", and Arthur has given a conjectural description of the representations in a given L-packet.