Objects
There are a number of related Langlands conjectures. There are many different groups over many different fields for which they can be stated, and for each field there are several different versions of the conjectures. Some versions of the Langlands conjectures are vague, or depend on objects such as the Langlands groups, whose existence in unproven, or on the L-group that has several inequivalent definitions. Moreover, the Langlands conjectures have evolved since Langlands first stated them in 1967.
There are different types of objects for which the Langlands conjectures can be stated:
- Representations of reductive groups over local fields (with different subcases corresponding to archimedean local fields, p-adic local fields, and completions of function fields)
- Automorphic forms on reductive groups over global fields (with subcases corresponding to number fields or function fields).
- Finite fields. Langlands did not originally consider this case, but his conjectures have analogues for it.
- More general fields, such as function fields over the complex numbers.
Read more about this topic: Langlands Program
Famous quotes containing the word objects:
“in the mind of man,
A motion and a spirit, that impels
All thinking things, all objects of all thought,
And rolls through all things.”
—William Wordsworth (1770–1850)
“Let the maiden, with erect soul, walk serenely on her way, accept the hint of each new experience, search in turn all the objects that solicit her eye, that she may learn the power and charm of her new-born being, which is the kindling of a new dawn in the recesses of space.”
—Ralph Waldo Emerson (1803–1882)
“Little minds mistake little objects for great ones, and lavish away upon the former that time and attention which only the latter deserve. To such mistakes we owe the numerous and frivolous tribe of insect-mongers, shell-mongers, and pursuers and driers of butterflies, etc. The strong mind distinguishes, not only between the useful and the useless, but likewise between the useful and the curious.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)