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:
“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)
“I think of consciousness as a bottomless lake, whose waters seem transparent, yet into which we can clearly see but a little way. But in this water there are countless objects at different depths; and certain influences will give certain kinds of those objects an upward influence which may be intense enough and continue long enough to bring them into the upper visible layer. After the impulse ceases they commence to sink downwards.”
—Charles Sanders Peirce (1839–1914)
“We have good reason to believe that memories of early childhood do not persist in consciousness because of the absence or fragmentary character of language covering this period. Words serve as fixatives for mental images. . . . Even at the end of the second year of life when word tags exist for a number of objects in the child’s life, these words are discrete and do not yet bind together the parts of an experience or organize them in a way that can produce a coherent memory.”
—Selma H. Fraiberg (20th century)