Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces

Abstract

Let $D$ be an indefinite quaternion division algebra over $\mathbb{Q}$. We approach the problem of bounding the sup-norms of automorphic forms $\phi$ on $D^\times(\mathbb{A})$ that belong to irreducible automorphic representations and transform via characters of unit groups of orders of $D$. We obtain a non-trivial upper bound for $\|\phi\|_\infty$ in the level aspect that is valid for arbitrary orders. This generalizes and strengthens previously known upper bounds for $\|\phi\|_\infty$ in the setting of newforms for Eichler orders. In the special case when the index of the order in a maximal order is a squarefull integer $N$, our result specializes to $\|\phi\|_\infty \ll_{\pi_\infty, \epsilon} N^{1/3 + \epsilon} \|\phi\|_2$. A key application of our result is to automorphic forms $\phi$ which correspond at the ramified primes to either minimal vectors (in the sense of Hu-Nelson-Saha), or $p$-adic microlocal lifts (in the sense of Nelson). For such forms, our bound specializes to $\| \phi\|_\infty \ll_{\epsilon} C^{\frac16 + \epsilon}\|\phi\|_2$ where $C$ is the conductor of the representation $\pi$ generated by $\phi$. This improves upon the previously known local bound $\|\phi\|_\infty \ll_{\lambda, \epsilon} C^{\frac14 + \epsilon}\|\phi\|_2$ in these cases.