Norm bounds on Eisenstein series (Q6591608)

In this paper under review, the authors provide some sup-norm bounds for Eisenstein series on certain arithmetic hyperbolic orbifolds. More precisely, let \(\Gamma=\mathrm{SL}_2(\mathbb{Z})\), it is shown that for any compact set \(\Omega \subset \Gamma \backslash \mathbb{H}\) there is a constant \(c = c(\Omega)\) such that for all \(T \geq 1\), \N\[\N\int_{-T}^T \left(\sup_{z\in \Omega}\left|E_\Gamma \left(z, \frac{1}{2}+it\right)\right| \right)^2 dt\leq c T \log^4 (T), \N\]\Nwhere \(E_\Gamma\) is the Eisenstein series at the unique cusp at \(\infty\). Similarly, for \(\Gamma=\mathrm{SL}_2(\mathbb{Z}[i])\) any compact set \(\Omega \subset \Gamma \backslash \mathbb{H}^3\), and for any \(\varepsilon> 0\), there is a constant \(c = c(\Omega, \varepsilon)\) such that \N\[\N\sup_{z\in \Omega} E_\Gamma \left(z, 1+iT \right)\leq c T^{1/2 +\varepsilon}.\N\]\NA key ingredient of independent interest is obtaining strong bounds on the Epstein zeta function and counting restricted values of indefinite quadratic forms at integer points.