词条 | Tamagawa number |
释义 |
In mathematics, the Tamagawa number of a semisimple algebraic group defined over a global field {{math|k}} is the measure of , where is the adele ring of {{math|k}}. Tamagawa numbers were introduced by {{harvs|txt|authorlink=Tsuneo Tamagawa|last=Tamagawa|year=1966}}, and named after him by {{harvs|txt|authorlink=André Weil|last=Weil|year=1959}}. Tsuneo Tamagawa's observation was that, starting from an invariant differential form ω on {{math|G}}, defined over {{math|k}}, the measure involved was well-defined: while {{math|ω}} could be replaced by {{math|cω}} with {{math|c}} a non-zero element of , the product formula for valuations in {{math|k}} is reflected by the independence from {{math|c}} of the measure of the quotient, for the product measure constructed from {{math|ω}} on each effective factor. The computation of Tamagawa numbers for semisimple groups contains important parts of classical quadratic form theory. DefinitionLet {{math|k}} be a global field, {{math|A}} its ring of adeles, and {{math|G}} a semisimple algebraic group defined over {{math|k}}. Choose Haar measures on the completions {{math|kv of k}} such that {{math|Ov}} has volume 1 for all but finitely many places {{math|v}}. These then induce a Haar measure on {{math|A}}, which we further assume is normalized so that {{math|A/k}} has volume 1 with respect to the induced quotient measure. The Tamagawa measure on the adelic algebraic group {{math|G(A)}} is now defined as follows. Take a left-invariant {{math|n}}-form {{math|ω}} on {{math|G(k)}} defined over {{math|k}}, where {{math|n}} is the dimension of {{math|G}}. This, together with the above choices of Haar measure on the {{math|kv}}, induces Haar measures on {{math|G(kv)}} for all places of {{math|v}}. As {{math|G}} is semisimple, the product of these measures yields a Haar measure on {{math|G(A)}}, called the Tamagawa measure. The Tamagawa measure does not depend on the choice of ω, nor on the choice of measures on the {{math|kv}}, because multiplying {{math|ω}} by an element of {{math|k*}} multiplies the Haar measure on {{math|G(A)}} by 1, using the product formula for valuations. The Tamagawa number {{math|τ(G)}} is defined to be the Tamagawa measure of {{math|G(A)/G(k)}}. Weil's conjecture on Tamagawa numbers{{see also|Weil conjecture on Tamagawa numbers}}Weil's conjecture on Tamagawa numbers states that the Tamagawa number {{math|τ(G)}} of a simply connected (i.e. not having a proper algebraic covering) simple algebraic group defined over a number field is 1. {{harvs|txt|authorlink=André Weil|last=Weil|year=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. {{harvtxt|Ono|1963}} found examples where the Tamagawa numbers are not integers, but the conjecture about the Tamagawa number of simply connected groups was proven in general by several works culminating in a paper by {{harvs|txt|authorlink=Robert Kottwitz|last=Kottwitz|year=1988}} and for the analogue over function fields over finite fields by Lurie and Gaitsgory in 2011.{{sfn|Lurie|2014}} See also
References
Further reading
2 : Algebraic groups|Algebraic number theory |
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