on the one hand, we use an adelic setup to treat the number field situation appropriately. On the other hand, at the heart of the amplification is Iwaniec’s idea of playing off various subgroups against each other, and so we need to keep track carefully of the various levels occurring in the course of the argument.

Perhaps the most appealing application of Theorem \[theorem1\] is to combine it with the formula of Waldspurger [@Wa] and its extensions by Shimura [@Sh2], Khuri-Makdisi [@KM], Kojima [@Ko], Baruch–Mao [@BM] and others in order to bound the Fourier coefficients of half-integral weight Hilbert modular forms. For $K ={\mathbb{Q}}$, the original breakthrough was achieved by Iwaniec [@Iw1], and the currently strongest bounds are given in [@BH1]. For a totally real number field $K$ other than ${\mathbb{Q}}$, there does not seem to be an explicit reference in the literature.

\[cor5\] Let $(\tilde{\pi},V_{\tilde\pi})$ be an irreducible cuspidal representation of $\,\widetilde{\operatorname{SL}}_2(K)\backslash
\widetilde{\operatorname{SL}}_2(\mathbb{A}_K)$, orthogonal to one-dimensional theta series, and let $r \in {\mathfrak{o}}$ be a nonzero squarefree integer. Define the $r$-th normalized Fourier coefficient $\rho_{\tilde{\phi}}(r)$ of a pure tensor $\tilde{\phi}=\otimes_v\tilde\phi_v \in V_{\tilde{\pi}}$ by the left hand side of below. Then $$\sqrt{|{\mathcal{N}}r|} \rho_{\tilde{\phi}}(r) \ll_{\tilde{\phi}, K, {\varepsilon}} |{\mathcal{N}}r|^{\frac{1}{4}- \frac{1}{16}(1-2\theta) + {\varepsilon}}.$$

