It is a classical problem to bound the \(L^\infty\)-norm (or sup-norm) \(\|\varphi\|_\infty\) of Laplace eigenfunctions, which are \(L^2\)-normalized (\(\|\varphi\|_2=1)\). Here the authors establish similar bounds for automorphic forms over a number field \(\mathbb F\) with \(p\) real places and \(q\) complex ones. They prove the existence of a constant \(C_{\mathbb F,\varepsilon}\), depending only on \(\mathbb F\) and \(\varepsilon>0\), such that \[ \|\phi\|_{\infty}\le C_{\mathbb F,\varepsilon} \left[ |\lambda|_{\infty}^{5/24}(\mathcal{N}\mathfrak{n})^{1/3} + |\lambda|_{\mathbb R}^{1/8} |\lambda|_{\mathbb C}^{1/4} (\mathcal{N}\mathfrak{n})^{1/4} \right] \left(|\lambda|_{\infty}\mathcal{N}\mathfrak{n}\right)^\varepsilon \|\phi\|_2 \] for any spherical cuspidal Maaß-Hecke newform \(\phi\) of square-free level \(\mathfrak{n}\) with norm \(\mathcal{N}\mathfrak{n}\) for the group \(\mathrm{GL}_2\), with eigenvalues \(\lambda=(\lambda_r)_{r=1}^{p+q}\) at different real and complex places. Moreover, for \(\mathbb F\) not totally real (i.e., with maximal totally real subfield \(\mathbb F_0\) strictly included in \(\mathbb F\)) and under the same assumptions than supra, the authors prove, for an appropriate constant \( \widetilde{C}_{\mathbb F,\varepsilon}\), the upper bound \[ \|\phi\|_{\infty}\le \widetilde{C}_{\mathbb F,\varepsilon} \left[ |\lambda|_{\infty}^{1/2}\mathcal{N}\mathfrak{n} \right]^{\frac12-(8[\mathbb F:\mathbb F_0]-4)^{-1}+\varepsilon} \|\phi\|_2. \] These new bounds by the authors reproduce or improve previously known upper bounds for automorphic forms over the rationals. So, it is worthwhile to compare this hybrid bounds (aspects eigenvalue and level) with existing upper bounds (see H. Iwaniec and P. Sarnak [Ann. Math. (2) 141, No. 2, 301–320 (1995; Zbl 0833.11019)]). Let us quote the general bound for eigenforms on compact locally symmetric space \(X\) \[ \|\varphi\|_{\infty} \le C_X |\lambda|_{X}^{(\dim X-\mathrm{rank} X)/4}\|\varphi\|_2 \] the sup-norm for Hecke-Laplace eigenfunctions on arithmetic surfaces (sphere \(\mathbb S^2\) and modular surface \(\mathrm{SL_2(\mathbb{R})}\backslash\mathbb H_{\mathrm{hyp}}^2\)) \[ \|\varphi\|_{\infty} \le C_\varepsilon\lambda^{5/24+\varepsilon}\|\varphi\|_2 \] and the hybrid sup-norm bounds for Hecke-Maaß cusp forms over congruence cover of level \(N\) proved by N. Templier [J. Eur. Math. Soc. (JEMS) 17, No. 8, 2069–2082 (2015; Zbl 1376.11030)] \[ \|\varphi\|_{\infty} \le \widetilde C_\varepsilon \lambda^{5/24}N^{1/3} (\lambda N)^\varepsilon \|\varphi\|_2. \]
The authors follow a similar strategy (e.g., adelic framework, Atkin-Lehner operator, pretrace formula, amplification for counting) used for recent works on sup-norm upper bounds, solving technical and crucial difficulties due to the number field \(\mathbb F\). E. Assing [“On sup-norm bounds part I: ramified Maaß newforms over number fields”, Preprint, arXiv:1710.00362] extended the present work without conditions on the level or the central character.