Let $K$ be a number field and let $A\_K$ be the adele ring of $K$. Then $K$ sits in $A\_K$ via the diagonal embedding and the quotient $A\_K/K$ is compact. All this is well known. Many proofs of the above fact first reduce the case to that of $K=\mathbb{Q}$ and solves the problem in this case. The proof also provides a fundamental domain for the quotient in terms of a fundamental domain for $\mathbb{Z}$ in $\mathbb{R}$.


I am trying to find a description of the fundamental domain for $A\_K/K$ involving a fundamental domain for $O\_K$, the ring of integers of $K$, in $K\_\infty := \prod\_{v | \infty} K\_v $ (all the notations are standard and I hope that its ok that I don't explain them).


Apparently such a description is possible only if the class number of $K$ is $1$. See the final remarks in [Conrad's notes](http://math.stanford.edu/%7Econrad/676Page/handouts/adelelattice.pdf) for example. I don't understand where exactly the problem occurs if the class number is bigger than $1$. Any help would be greatly appreciated.



Fundamental domain for $A_K/K$
One does not need that the class number is $1$, the construction in Brian Conrad's notes works in general. See Propositions 6-7 in Ch.V-4 of Weil: Basic Number Theory. The class group enters for $A\_K^\times/K^\times$, not for $A\_K/K$.


