Adele ring

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

The ring of adeles is the restricted product

$A = {\prod_{p}}' \mathbb{Q}_p \times \mathbb{R}$

of all the p-adic completions $\mathbb{Q}_p$ and the real numbers. In this case the restricted product means that for an adele $(a_2, a_3, a_5, ...., a_\infty)$ all but a finite number of the $a p$ are p-adic integers.

Topologically, the adeles $A$ are a locally compact group with the rational numbers $\mathbb{Q}$ contained as a discrete subgroup. The use of adele rings in connection with Fourier transforms was exploited in Tate's thesis.

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 matrices over the adeles — a non-trivial construction in this case, as far as topology goes.

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(Q) with G(A), saying how G(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 case-by-case and took decades to complete.

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(Q) might be brought into relation with the E(Q_p) was one motivation and signpost, on the way from numerical evidence to the conjecture.

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