Tamagawa number

 ( Algebraic Groups ) 

In mathematics, the Tamagawa number $\backslash tau(G)$ of a semisimple algebraic group defined over a global field is the measure of $G(\backslash mathbb\{A\})/G(k)$, where $\backslash mathbb\{A\}$ is the adele ring of . Tamagawa numbers were introduced by , and named after him by .

Tsuneo Tamagawa's observation was that, starting from an invariant differential form ω on , defined over , the measure involved was well-defined: while could be replaced by with a non-zero element of $k$, the product formula for valuations in is reflected by the independence from of the measure of the quotient, for the product measure constructed from on each effective factor. The computation of Tamagawa numbers for contains important parts of classical quadratic form theory.

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

Choose on the completions such that has volume 1 for all but finitely many places . These then induce a Haar measure on , which we further assume is normalized so that has volume 1 with respect to the induced quotient measure.

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

The Tamagawa number is defined to be the Tamagawa measure of .

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Weil's conjecture on Tamagawa numbers Weil's conjecture on Tamagawa numbers states that the Tamagawa number of a simply connected (i.e. not having a proper algebraic covering) simple algebraic group defined over a number field is 1. calculated the Tamagawa number in many cases of 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. 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 and for the analogue over function fields over finite fields by .

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See also

• Adelic algebraic group

• .

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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 video, posted June 8, 2012.

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