Adele ring

In number theory, the adele ring is a topological ring containing the field of rational numbers (or, more generally, to an algebraic number field). It involves all the completions of the field.

The word "adele" is short for "additive idele", and is also a French girl's name. Adeles were called valuation vectors or repartitions before about 1950.

Definitions

The profinite completion of the integers ${\hat {\mathbb {Z} }}$ is the inverse limit of the rings $\mathbb {Z} /n\mathbb {Z}$:

${\hat {\mathbb {Z} }}=\lim _{\leftarrow }\mathbb {Z} /n\mathbb {Z}$

By the Chinese remainder theorem it is isomorphic to the product of all the rings of p-adic integers:

${\hat {\mathbb {Z} }}=\prod _{p}\mathbb {Z} _{p}$

The ring of integral adeles A_Z is the product

$\mathbb {A} _{\mathbb {Z} }=\mathbb {R} \times {\hat {\mathbb {Z} }}$

The ring of (rational) adeles A_Q is the tensor product

$\mathbb {A} _{\mathbb {Q} }=\mathbb {Q} \otimes \mathbb {A} _{\mathbb {Z} }$

(topologized so that A_Z is an open subring). More generally the ring of adeles A_K of any algebraic number field K is the tensor product

$\mathbb {A} _{\mathbb {K} }=\mathbb {K} \otimes \mathbb {A} _{\mathbb {Z} }$

(topologized as the product of deg(K) copies of A_Q).

The ring of (rational) adeles can also be defined as the restricted product

$\mathbb {A} _{\mathbb {Q} }=\mathbb {R} \times {\prod _{p}}'\mathbb {Q} _{p}$

of all the p-adic completions $\mathbb {Q} _{p}$ and the real numbers (or in other words as the restricted product of all completions of the rationals). In this case the restricted product means that for an adele $(a_{\infty },a_{2},a_{3},a_{5},....)$ all but a finite number of the $a_{p}$ are p-adic integers.

The adeles of a function field over a finite field can be defined in a similar way, as the restricted product of all completions.

Properties

The rational adeles $A$ are a locally compact group with the rational numbers $\mathbb {Q}$ contained as a discrete co-compact subgroup. The use of adele rings in connection with Fourier transforms was exploited in Tate's thesis. One key property of the additive group of adeles is that it is isomorphic to its Pontryagin dual.

Applications

The ring A is much used in advanced parts of number theory, often as the coefficients in matrix groups: that is, combined with the theory of algebraic groups to construct adelic algebraic groups. The idele group of class field theory appears as the group of 1×1 invertible matrices over the adeles. (It is not given the subset topology, as the inverse is not continuous in this topology. Instead the ideles are identified with the closed subset of all pairs (x,y) of A×A with xy=1, with the subset topology.)

An important stage in the development of the theory was the definition of the Tamagawa number for an adelic linear algebraic group. This is a volume measure relating $G(\mathbb {Q} )$ with G(A), saying how $G(\mathbb {Q} )$, which is a discrete group in G(A), lies in the latter. A conjecture of André Weil was that the Tamagawa number was always 1 for a simply-connected G. This arose out of Weil's modern treatment of results in the theory of quadratic forms; the proof was finally completed by Kottwitz.

Meanwhile the influence of the Tamagawa number idea was felt in the theory of abelian varieties. There the application by no means works, in any straightforward way. But during the formulation of the Birch and Swinnerton-Dyer conjecture, the consideration that for an elliptic curve E the group of rational points $E(\mathbb {Q} )$ might be brought into relation with the $E(\mathbb {Q} _{p})$ was one motivation and signpost, on the way from numerical evidence to the conjecture.

See also

• Adelic algebraic group.
• Schwartz-Bruhat function

References

Almost any book on modern algebraic number theory, such as:

• Algebraic Number Theory by J. W. S. Cassels, A. Frohlich ISBN 0-12-163251-2