Weil's conjecture on Tamagawa numbers

In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number τ(G) of a simply connected simple algebraic group defined over a number field is 1. Weil (1959) did not explicitly conjecture this, but calculated the Tamagawa number in many cases and observed that in the cases he calculated it was an integer, and equal to 1 when the group is simply connected. The first observation does not hold for all groups: Ono (1963) found some examples whose 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. Several authors checked this in many cases, and finally Kottwitz proved it for all groups in 1988.

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

Tamagawa numbers were introduced by Tamagawa (1966), and named after him by Weil (1959).

Here simply connected is in the algebraic group theory sense of not having a proper algebraic covering, which is not always the topologists' meaning.

Tamagawa measure and Tamagawa numbers

Let k be a global field, A its ring of adeles, and G a semisimple algebraic group defined over k.

Choose Haar measures on the completions k_v of k such that O_v has volume 1 for all but finitely many places v. These then induce a Haar measure on A, which we further assume is normalised so that A/k has volume 1 with respect to the induced quotient measure.

The Tamagawa measure on the adelic algebraic group G(A) is now defined as follows. Take a left-invariant n-form ω on G(k) defined over k, where n is the dimension of G. This, together with the above choices of Haar measure on the k_v , induces Haar measures on G(k_v) for all places of v. As G is semisimple, the product of these measures yields a Haar measure on G(A), called the Tamagawa measure. The Tamagawa measure does not depend on the choice of ω, nor on our choice of measures on the k_v, because multiplying ω by an element of k* multiplies the Haar measure on G(A) by 1, using the product formula for valuations.

The Tamagawa number τ(G) is defined to be the Tamagawa measure of G(A)/G(k).

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.

In 2011, Jacob Lurie and Dennis Gaitsgory announced a proof of the conjecture for algebraic groups over function fields over finite fields.Lurie (2011) Lurie (2014)

See also

• Ran space

References

• Hazewinkel, Michiel, ed. (2001), "Tamagawa number", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Chernousov, V. I. (1989), "The Hasse principle for groups of type E8", Soviet Math. Dokl., 39: 592–596, MR 1014762
• Kottwitz, Robert E. (1988), "Tamagawa numbers", Ann. of Math. (2), Annals of Mathematics, 127 (3): 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 581580
• 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
• Ono, Takashi (1963), "On the Tamagawa number of algebraic tori", Annals of Mathematics. Second Series, 78: 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: 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., 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, 5, pp. 249–257
• Weil, André (1982) [1961], Adeles and algebraic groups, Progress in Mathematics, 23, Boston, MA: Birkhäuser Boston, ISBN 978-3-7643-3092-7, MR 670072
• Lurie, Jacob (2011), Tamagawa Numbers via Nonabelian Poincaré Duality
• Lurie, Jacob (2014), Tamagawa Numbers via Nonabelian Poincaré Duality

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.