mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Tamagawa number

\tau(G)

of a

semisimple algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...

defined over a global field is the measure of

G(\mathbb)/G(k)

, where

\mathbb

is the

adele ring Adele Laurie Blue Adkins (, ; born 5 May 1988), professionally known by the mononym Adele, is an English singer and songwriter. After graduating in arts from the BRIT School in 2006, Adele signed a reco ...

of . Tamagawa numbers were introduced by , and named after him by .

Tsuneo Tamagawa Tsuneo Tamagawa (Japanese: 玉河恒夫, ''Tamagawa Tsuneo'', 11 December 1925 in Tokyo – 30 December 2017 in New Haven, Connecticut) was a mathematician. He worked on the arithmetic of classical groups. Tamagawa received his PhD in 1954 at t ...

's observation was that, starting from an invariant

differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...

ω on , defined over , the measure involved was

well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A funct ...

: 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

semisimple group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...

s contains important parts of classical

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

theory.

Definition

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

Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though ...

s 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 In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

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 .

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 In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

defined over a number field is 1. calculated the Tamagawa number in many cases of

classical group In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or ske ...

s 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

Lurie Lurie is often a Jewish surname, but also an Irish and English surname. The name is sometimes transliterated from/to other languages as Lurye, Luriye (from Russian), Lourié (in French). Other variants include: Lurey (surname), Loria, Luria, Lur ...

and Gaitsgory in 2011.

References

* *. * * * * * *{{Citation , last=Lurie , first=Jacob , author-link=Jacob Lurie , title=Tamagawa Numbers via Nonabelian Poincaré Duality , year=2014 , url=http://www.math.harvard.edu/~lurie/282y.html

Definition

Weil's conjecture on Tamagawa numbers

See also

References

Further reading