on each occurrence. We embed $K$ as a ${\mathbb{Q}}$-algebra into $K_{\infty}:= {\mathbb{R}}^d$ using the $d$ real field embeddings $r \mapsto (r^{\sigma_1}, \ldots, r^{\sigma_d})$. We denote by ${K_{\infty,+}^{\times}}:={\mathbb{R}}^d_{>0}$ the set of totally positive elements of $K_{\infty}$, and we put $${K_{\infty,+}^{\text{diag}}}:= \{(x, \ldots, x) \mid x
\in {\mathbb{R}}_{>0} \} \subseteq {K_{\infty,+}^{\times}}.$$ For $r \in K$ we write $$\operatorname{sgn}(r):=(\operatorname{sgn}(r^{\sigma_1}), \ldots, \operatorname{sgn}(r^{\sigma_d})) \in
\{\pm 1\}^d,$$ and we write $r >> 0$ for a totally positive integer $r \in {\mathfrak{o}}$. We denote by $U^{+} \subseteq U$ the group of totally positive units and the group of units of ${\mathfrak{o}}$, respectively.

Let $\mathbb{A}$ be the adele ring of $K$, with $K$ being embedded diagonally (this defines in particular a multiplication $K \times
\mathbb{A} \to \mathbb{A}$). We shall often write $\mathbb{A} =
K_{\infty} \times \mathbb{A}_{\text{fin}}$. We shall label the archimedean places with elements of $\{1,\dots,d\}$ and the non-archimedean places with prime ideals ${\mathfrak{p}}$ of $K$ in an obvious way. As usual, we shall denote the module of an idele $x \in
\mathbb{A}^\times$ by $|x|:=|x_\infty||x_\text{fin}|$, where $|x_\infty|:=\prod_{j=1}^d|x_j|$ and $|x_\text{fin}|:=\prod_{\mathfrak{p}}|x_{\mathfrak{p}}|$. We denote by $\psi:\mathbb{A}
\to S^1$ the unique continuous additive character which is trivial on $K$, agrees with $x\mapsto e(x_1+\cdots+x_d)$ on $K_\infty$, and on $K_{\mathfrak{p}}$ it is trivial on 