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 and proved in 1989. 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.)

[edit] History

Weil checked this in enough classical group cases to propose the conjecture. In particular for spin groups it implies the known Smith-Minkowski-Siegel mass formula.

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 for all groups satisfying the Hasse principle, which at the time was known for all groups without E₈ factors. V. I. Chernousov (1989) removed this restriction, by proving the Hasse principle for the resistant E₈ case (see strong approximation in algebraic groups), thus completing the proof of Weil's conjecture.

[edit] References

• Chernousov, V. I. (1989), “The Hasse principle for groups of type E8”, Soviet Math. Dokl. 39: 592--596, MR1014762 
• Kottwitz, Robert E. (1988), “Tamagawa numbers”, Ann. of Math. (2) 127 (3): 629--646, MR0942522, <http://links.jstor.org/sici?sici=0003-486X%28198805%292%3A127%3A3%3C629%3ATN%3E2.0.CO%3B2-7> .
• Lai, K. F. (1980), “Tamagawa number of reductive algebraic groups”, Compositio Mathematica 41 (2): 153-188, MR581580, <http://www.numdam.org/item?id=CM_1980__41_2_153_0> 
• Langlands, R. P. (1966), “The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups”, Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., Providence, R.I.: Amer. Math. Soc., pp. 143--148, MR0213362 
• Voskresenskii, V. E. (1991), Algebraic Groups and their Birational Invariants, AMS translation