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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. 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: 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. used the Weil conjecture to calculate the Tamagawa numbers of all semisimple algebraic groups. Tamagawa numbers were introduced by , and named after him by . 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'' an algebraic group defined over ''k''. The Tamagawa measure on the adelic algebraic group ''G''(''A'') is defined as follows. Take a left-invariant ''n''-form ω on ''G''(''k'') defined over ''k'', where ''n'' is the dimension of ''G''. This induces Haar measures on ''G''(''k''''s'') for all places of ''s'', and hence a Haar measure on ''G''(''A''), if the product over all places converges. This Haar measure on ''G''(''A'') does not depend on the choice of ω, 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 the Tamagawa measure of ''G''(''A'')/''G''(''k''). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Weil conjecture on Tamagawa numbers」の詳細全文を読む スポンサード リンク
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