Weil conjecture on Tamagawa numbers
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In mathematics, the Weil conjecture on Tamagawa numbers was formulated by André Weil in the late 1950s. It states that the Tamagawa number τ(G), where G is any connected and simply connected semisimple algebraic group G, defined over a number field K, satisfies
- τ(G) = 1.
This is with an understanding on normalization (cf. Voskresenskii book Ch. 5); in any case the conjecture was of the value in this case. (Here simply connected has the usual meaning for algebraic group theory, of not having a proper algebraic covering, which is not exactly the topologists' meaning in all cases.) Weil checked this in enough classical group cases to propose the conjecture.
It remained not completely proved for around 30 years. Robert Langlands (1966) introduced harmonic analysis methods to show it for Chevalley groups. J. G. M. Mars gave further results during the 1960s. K. F. Lai (1980) extended the class of known cases to quasisplit reductive groups. Kottwitz (1988) proved it in the absence of factors E8. V. I. Chernousov (1989) removed the restriction, by means of an approximation theorem in the resistant E8 case (see strong approximation in algebraic groups); thus making Weil's conjecture a theorem.
[edit] Reference
V. E. Voskresenskii, Algebraic Groups and their Birational Invariants, AMS translation 1991
