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.
Read more about Weil Conjecture On Tamagawa Numbers: Tamagawa Measure and Tamagawa Numbers, History
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