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

## TL;DR

This paper establishes new bounds on the maximum size of automorphic forms on compact arithmetic surfaces, improving previous results and applying to forms with specific local properties, with implications for understanding their growth in the level aspect.

## Contribution

It provides a generalized upper bound for automorphic forms' sup-norms in the level aspect for arbitrary orders, extending prior bounds for Eichler orders and special cases.

## Key findings

- Bound $
orm{}{}_ \, ext{in terms of level } N^{1/3 + psilon}
- Improved local bounds for forms with minimal vectors or $p$-adic microlocal lifts
- Enhanced understanding of automorphic forms' growth on 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.