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
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