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: 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.
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 , and planned to be completed in a second volume using the
Grothendieck-Lefschetz trace formula and the Ran space. ==Applications==