Tamagawa number

Template:Short description In mathematics, the Tamagawa number <math>\tau(G)</math> of a semisimple algebraic group defined over a global field Template:Math is the measure of <math>G(\mathbb{A})/G(k)</math>, where <math>\mathbb{A}</math> is the adele ring of Template:Math. Tamagawa numbers were introduced by Template:Harvs, and named after him by Template:Harvs.

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

DefinitionEdit

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

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

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

The Tamagawa number Template:Math is defined to be the Tamagawa measure of Template:Math.

Weil's conjecture on Tamagawa numbersEdit

Template:See also Weil's conjecture on Tamagawa numbers states that the Tamagawa number Template:Math of a simply connected (i.e. not having a proper algebraic covering) simple algebraic group defined over a number field is 1. Template:Harvs 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. Template:Harvtxt 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 Template:Harvs and for the analogue over function fields over finite fields by Template:Harvtxt.

Tamagawa number

Contents

DefinitionEdit

Weil's conjecture on Tamagawa numbersEdit

See alsoEdit

ReferencesEdit

Further readingEdit