# What is the Pontryagin dual of the rationals?

Endow the rational numbers (or any global field) with the discrete topology, what will be the (compact) Pontryagin dual of the additive group and of the multiplicative group?

I am suprised nobody mentioned this: but the part of the question of the additive group of the rational is answered here already: Representation theory of the additive group of the rationals?

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late_learner: I had mentioned the link in your 2nd paragraph as a comment to my answer below. –  KCd Jul 2 '11 at 3:15

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Nice. Is it too much to hope for that the answer for the multiplicative group is $A^\times /Q^\times$ is the answer for the multiplicative group? –  plusepsilon.de Jun 28 '11 at 10:19
If you add a http:// in front of a link, it becomes clickable. –  t.b. Jun 28 '11 at 10:19
Okay, I see you use that $Q$ is a lattice in $A$. So the same argument works for the finite adeles $A_f^\times / Q^{times}$ for the multiplicative group, nice. –  plusepsilon.de Jun 28 '11 at 10:25
Nice answser, so it seems I was a bit too quick with mine (partly ashamed, partly amused). Now I wonder if there's a way to fix my earlier reasoning : To choose a character, you need to chose the image of $1$ in $\mathbb{R}/\mathbb{Z}$ (that's the part I forgot earlier), then for all $n \in \mathbb{N}^*$ choose an $n^{\textrm{th}}$ root of it (the image of $\frac{1}{n}$) in a compatible way yielding an element of $\hat{\mathbb{Z}}$. In the end you'd get $\hat{\mathbb{Z}} \times \mathbb{R}/\mathbb{Z}$, right ? –  Joel Cohen Jun 28 '11 at 10:44
@Joel: Dear Joel, $\mathbb A/\mathbb Q$ is not the same as $\hat{\mathbb Z}\times \mathbb R/\mathbb Z$; e.g. the former is a $\mathbb Q$-vector space, while the latter contains torsion elements. There is a surjection from $\mathbb A/\mathbb Q$ to $\mathbb R/\mathbb Z$, with kernel equal to $\hat{\mathbb Z}$, but this surjection does not split. Regards, –  Matt E Jun 29 '11 at 21:08

For those who don't know what is $A_Q$...

Hewitt and Ross, Abstract Harmonic Analysis, p. 404. The dual of the discrete rationals is described as an $\mathbf{a}$-adic solenoid. An inverse limit of a sequence of circles, $T_n$, say, where the map of $T_{n+1}$ onto $T_n$ wraps around $n$ times.

Their notes say this is due to Makoto Abe (1940) and independently to Anzai and Kakutani (1943).

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