In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number τ ( G ) {\displaystyle \tau (G)} of a simply connected simple algebraic group defined over a number field is 1. In this case, simply connected means "not having a proper algebraic covering" in the algebraic group theory sense, which is not always the topologists' meaning.
History 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, with part of the argument published in Gaitsgory & Lurie (2019), and planned to be completed in a second volume using the Grothendieck-Lefschetz trace formula and the Ran space.
Applications 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.
See also Tamagawa number
References
"Tamagawa number", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Chernousov, V. I. (1989), "The Hasse principle for groups of type E8", Soviet Math. Dokl., 39: 592–596, MR 1014762 Gaitsgory, Dennis; 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 Kottwitz, Robert E. (1988), "Tamagawa numbers", Ann. of Math., 2, 127 (3), Annals of Mathematics: 629–646, doi:10.2307/2007007, JSTOR 2007007, MR 0942522. Lai, K. F. (1980), "Tamagawa number of reductive algebraic groups", Compositio Mathematica, 41 (2): 153–188, MR 0581580 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, MR 0213362 Lurie, Jacob (2014), Tamagawa Numbers via Nonabelian Poincaré Duality Ono, Takashi (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 (1965), "On the relative theory of Tamagawa numbers", Annals of Mathematics, Second Series, 82 (1): 88–111, doi:10.2307/1970563, ISSN 0003-486X, JSTOR 1970563, MR 0177991 Tamagawa, Tsuneo (1966), "Adèles", Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., vol. IX, Providence, R.I.: American Mathematical Society, pp. 113–121, MR 0212025 Voskresenskii, V. E. (1991), Algebraic Groups and their Birational Invariants, AMS translation Weil, André (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
Further reading Aravind Asok, Brent Doran and Frances Kirwan, "Yang-Mills theory and Tamagawa Numbers: the fascination of unexpected links in mathematics", February 22, 2013 J. Lurie, The Siegel Mass Formula, Tamagawa Numbers, and Nonabelian Poincaré Duality posted June 8, 2012.