Fundamental domain for $A_K/K$

Question

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 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.

Answer 1

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 $A_K/K$.

Answer 2

There is also a description in Tate's thesis, see section 4.1, and specifically Definition 4.1.2 and Theorem 4.1.3(i) (note he denotes the ring of adeles by $V$). In section 4.3 (Definition 4.3.2, Theorem 4.3.2(i)) he also gives an explicit description of a fundamental domain for ideles. The construction does depend on choosing representatives of the class group.