Weil's conjecture on Tamagawa numbers

In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number $\tau (G)$ 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,^[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

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]

References

^ ^a ^b Lurie 2014.

"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

Weil's conjecture on Tamagawa numbers

History

Applications

See also

References

Further reading