LAUSR

Let D be an indefinite quaternion division algebra over $${{\mathbb {Q}}}$$ Q . We approach the problem of bounding the sup-norms of automorphic forms $$\phi $$ ϕ on $$D^\times ({{\mathbb {A}}})$$ D × ( 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 $$\Vert \phi \Vert _\infty $$ ‖ ϕ ‖ ∞ in the level aspect that is valid for arbitrary orders. This generalizes and strengthens previously known upper bounds for $$\Vert \phi \Vert _\infty $$ ‖ ϕ ‖ ∞ 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 $$\Vert \phi \Vert _\infty \ll _{\pi _\infty , \epsilon } N^{1/3 + \epsilon } \Vert \phi \Vert _2$$ ‖ ϕ ‖ ∞ ≪ π ∞ , ϵ N 1 / 3 + ϵ ‖ ϕ ‖ 2 . A key application of our result is to automorphic forms $$\phi $$ ϕ which correspond at the ramified primes to either minimal vectors, in the sense of Hu et al. (Commun Math Helv, to appear) or p -adic microlocal lifts, in the sense of Nelson in “Microlocal lifts and and quantum unique ergodicity on $$\mathrm{GL}_2({{\mathbb {Q}}}_{p})$$ GL 2 ( Q p ) ” (Algebra Number Theory 12(9):2033–2064, 2018 ). For such forms, our bound specializes to $$\Vert \phi \Vert _\infty \ll _{ \epsilon } C^{\frac{1}{6} + \epsilon }\Vert \phi \Vert _2$$ ‖ ϕ ‖ ∞ ≪ ϵ C 1 6 + ϵ ‖ ϕ ‖ 2 where C is the conductor of the representation $$\pi $$ π generated by $$\phi $$ ϕ . This improves upon the previously known local bound $$\Vert \phi \Vert _\infty \ll _{\lambda , \epsilon } C^{\frac{1}{4} + \epsilon }\Vert \phi \Vert _2$$ ‖ ϕ ‖ ∞ ≪ λ , ϵ C 1 4 + ϵ ‖ ϕ ‖ 2 in these cases.