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Let $K$ be a number field, $\mathcal O _K^\times$ its group of integral units and $\mathcal O _{K,+} ^\times$ its group of totally positive units. Denote further by $\widehat{\mathcal O }_K^\times$ the units of the finite, integral adeles with its idelic topology.

Is it possible to describe the closures of $\mathcal O _K^\times$ and $\mathcal O _{K,+} ^\times$, respectively, in $\widehat{\mathcal O}_K^\times$ (explicitly)? Is there a natural/useful characterization? I think for the rational field and imaginary quadratic fields the integral units are discrete using a theorem saying that the rational points of the Serre torus (attached to these number fields) are discrete in its adelic points. What about, for example, real quadratic fields? In my applications I sometimes encounter these closures but don't have a good intuition about their size, unfortunately.

Thank you very much in advance for your help.

Edit: One (abstract) characterization of the closure $\overline{\mathcal O _{K,+}^\times}$ of the totally positive units is given in terms of class field theory, namely the closure is equal to the kernel of the (restricted) Artin homomorphism $rec : \widehat{\mathcal O}_K^\times \to Gal(K^{ab}/K)$.

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This sounds like it should depend on Leopoldt's conjecture. – KConrad May 27 '11 at 16:20
@KConrad: Really? The absolute value on the finite ideles, restricted to the image of $K^{\times}$ is the product of the inverses of the archimedian absolute values... – David Hansen May 27 '11 at 21:16
The question is sufficiently ambiguous (for me) to decide whether it depends on Leopoldt ;-) For example, one way of "describing" the closure in the finite adeles is describing the closure in each of the finite local completions (Leopoldt) and then "describing" the amount the global closure differs from the local closures. David's comment just indicates that there are other approaches. I think that already in the real quadratic case though you get subtleties... – Kevin Buzzard May 28 '11 at 7:06
...I think the question is very analogous to… where it seemed to me to be tricky to "describe" the global closure in the product of the local closures (even in the rank 1 case, where Leopoldt issues do not exist). – Kevin Buzzard May 28 '11 at 7:08
PS my intuition about the size is that the closure is finite if the unit group is finite, and uncountably infinite otherwise... – Kevin Buzzard May 28 '11 at 7:10

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