History[edit]

Weil (1959) calculated the Tamagawa number in many cases of classical groups and observed that it is an integer in all considered cases and that it was equal to 1 in the cases when the group is simply connected. The first observation does not hold for all groups: Ono (1963) found examples where the Tamagawa numbers are not integers. The second observation, that the Tamagawa numbers of simply connected semisimple groups seem to be 1, became known as the Weil conjecture.


Robert Langlands (1966) introduced harmonic analysis methods to show it for Chevalley groups. 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. In 2011, Jacob Lurie and Dennis Gaitsgory announced a proof of the conjecture for algebraic groups over function fields over finite fields,[1] formally published in Gaitsgory & Lurie (2019), and a future proof using a version of the Grothendieck-Lefschetz trace formula will be published in a second volume.

Applications[edit]

Ono (1965) used the Weil conjecture to calculate the Tamagawa numbers of all semisimple algebraic groups.


For spin groups, the conjecture implies the known Smith–Minkowski–Siegel mass formula.[1]

Tamagawa number

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Tamagawa number"

Chernousov, V. I. (1989), "The Hasse principle for groups of type E8", Soviet Math. Dokl., 39: 592–596,  1014762

MR

; Lurie, Jacob (2019), Weil's Conjecture for Function Fields (Volume I), Annals of Mathematics Studies, vol. 199, Princeton: Princeton University Press, pp. viii, 311, ISBN 978-0-691-18213-1, MR 3887650, Zbl 1439.14006

Gaitsgory, Dennis

Kottwitz, Robert E. (1988), "Tamagawa numbers", Ann. of Math., 2, 127 (3), Annals of Mathematics: 629–646, :10.2307/2007007, JSTOR 2007007, MR 0942522.

doi

Lai, K. F. (1980), , Compositio Mathematica, 41 (2): 153–188, MR 0581580

"Tamagawa number of reductive algebraic groups"

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,  0213362

MR

(2014), Tamagawa Numbers via Nonabelian Poincaré Duality

Lurie, Jacob

(1963), "On the Tamagawa number of algebraic tori", Annals of Mathematics, Second Series, 78 (1): 47–73, doi:10.2307/1970502, ISSN 0003-486X, JSTOR 1970502, MR 0156851

Ono, Takashi

Ono, Takashi (1965), , Annals of Mathematics, Second Series, 82 (1): 88–111, doi:10.2307/1970563, ISSN 0003-486X, JSTOR 1970563, MR 0177991

"On the relative theory of Tamagawa numbers"

Tamagawa, Tsuneo (1966), "Adèles", Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., vol. IX, Providence, R.I.: , pp. 113–121, MR 0212025

American Mathematical Society

Voskresenskii, V. E. (1991), Algebraic Groups and their Birational Invariants, AMS translation

(1959), Exp. No. 186, Adèles et groupes algébriques, Séminaire Bourbaki, vol. 5, pp. 249–257

Weil, André

(1982) [1961], Adeles and algebraic groups, Progress in Mathematics, vol. 23, Boston, MA: Birkhäuser Boston, ISBN 978-3-7643-3092-7, MR 0670072

Weil, André

Aravind Asok, Brent Doran and Frances Kirwan, , February 22, 2013

"Yang-Mills theory and Tamagawa Numbers: the fascination of unexpected links in mathematics"

J. Lurie, posted June 8, 2012.

The Siegel Mass Formula, Tamagawa Numbers, and Nonabelian Poincaré Duality