Idèle

2020 Mathematics Subject Classification: Primary: 11-02 Secondary: 11Rxx11Sxx [MSN][ZBL]

An idele (also: idèle) is an invertible element of the ring of adeles (adèles) of a global field (cf. Adele). The set of all ideles forms a group under multiplication, called the idele group. The elements of the idele group of the field of rational numbers are sequences of the form $$a = (a_\infty,a_2,\dots,a_p,\dots),$$ where $a_\infty$ is a non-zero real number, $a_p$ is a non-zero $p$-adic number, $p=2,3,5,7,\dots,$ and $|a_p|=1$ for all but finitely many $p$ (here $|x|_p$ is the $p$-adic norm). A sequence of ideles $$a^{(n)} = (a_\infty^{(n)},a_2^{(n)},\dots,a_p^{(n)},\dots),$$ is said to converge to an idele $a$ if it converges to $a$ componentwise and if there exists an $N$ such that $|a_p^{-1}a_p^{(n)}|_p = 1$ for $n>N$ and all $p$. The idele group is a locally compact topological group in this topology. The idele group of an arbitrary number field is constructed in an analogous way.

The multiplicative group of the field of rational numbers is isomorphically imbedded in the idele group of this field. Every rational number $r\ne 0$ is associated with the sequence $$(r,r,\dots,r,\dots),$$ which is an idele. Such an idele is said to be a principal idele. The subgroup consisting of all principal ideles is a discrete subgroup of the idele group.

The concepts of an idele and an adele were introduced by C. Chevalley in 1936 for the purposes of algebraic number theory. The new language proved useful in the study of arithmetic aspects of the theory of algebraic groups. To those ends, A. Weil generalized the definitions of an adele and an idele to the case of an arbitrary linear algebraic group defined over a number field.

References

[1]	A. Weil, Basic number theory, Springer (1973) | MR1344916 | Zbl 0823.11001
[2]	J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1967) | MR0215665 | Zbl 0153.07403

Comments

Let $I$ be an index set and for each $i\in I$ let there be given a locally compact topological ring or group $G_i$ and an open compact subring or subgroup $B_i$. The restricted direct product $G=\Pi' G_i$ of the $G_i$ with respect to the $B_i$ consists of all families $(g_i)_{i\in I}$ such that $g_i\in B_i$ for all but finitely many $i$. $G$ becomes a locally compact group (ring) by taking as a basis of open neighbourhoods of the identity (zero) the sets $\prod_i U_i$ with $U_i$ open in $G_i$ for all $i$ and $U_i = B_i$ for all but finitely many $i$. For each finite set $S\subset I$ let $G_S = \prod_{i\in S} \times \prod_{i\notin S} B_i$. Then $G$ is the union (direct limit) of the $G_S$.

Now let $k$ be a number field (or, more generally, a global field). Let $I$ be the set of all prime divisors of $k$ (both finite and infinite ones). For each $\def\fp{\mathfrak{p}} \fp\in I$ let $k_\fp$ be the completion of $k$ with respect to the norm of $\fp$, and let $A_\fp$ be the ring of integers of $k_\fp$. (Set $A_\fp = k_\fp$ if $\fp$ is infinite.) Then the restricted product of the $k_\fp$ with respect to the $A_\fp$ is the ring of adeles $A_k$ of $k$.

Now for each $\fp\in I$ let $k_\fp^*$ be the group of non-zero elements of $k_\fp$ and let $U_\fp$ be the group of units of $k_\fp^*$ (if $\fp$ is infinite take $U_\fp = k_\fp^*$). The restricted product of the $k_\fp^*$ with respect to the $U_\fp$ is the group of ideles of $k$. As a set the group of ideles $I_k$ is the set of invertible elements of $A_k$. But the topology on $I_k$ is stronger than that induced by $A_k$.

The quotient of $I_k$ by the diagonal subgroup $k^* = \{(\alpha)_{i\in I}\}$ of principal ideles is called the idele class group; it is important in class field theory.

The name idele derives from "ideal element". This got abbreviated "id.el.", which, pronounced in French, gave rise to "idèle".