Langlands conjecture

• Langlands' conjecture

Named after Canadian mathematician Robert Langlands; interpretable as a reference to the Langlands programme.

Langlands conjecture (plural Langlands conjectures)

1. (mathematics) Any of the conjectures that comprise the Langlands programme.
   • 1998, Daniel Bump, Automorphic Forms and Representations, Cambridge University Press, page 90:

     It^{[the Langlands functoriality conjecture]} is closely related to the local Langlands conjectures, which amount to a (conjectural) classification of the representations of reductive groups over local fields.

   • 2009, Huai-Dong Cao, Shing-Tung Yau, editors, Surveys in Differential Geometry, International Press, page 86:

     This abelian case of the Geometric Langlands Conjecture amounts to the well known result that any rank of one local system (or: line bundle with flat connection) on the curve C extends uniquely to J, and this extension is natural with respect to the Abel-Jacobi map.

   • 2013, J. W. Cogdell, 10: Langlands Conjectures for GL_n, Joseph Bernstein, Stephen Gelbart (editors), An Introduction to the Langlands Program, Springer, page 229,

     The Langlands conjectures predict the existence of a correspondence between the ${\displaystyle n}$-dimensional representations of ${\displaystyle {\mathcal {G}}_{k}}$ and the automorphic representations of ${\displaystyle \mathrm {GL} _{n}(\mathbb {A} )}$ which preserves these analytic invariants.

• Langlands program on Wikipedia.Wikipedia