I will consider Maass forms transforming with respect to (unitary
characters of) unit groups of orders of an indefinite quaternion
division algebra over Q. I will assume that these Maass forms are
eigenfunctions of the Hecke operators at almost all primes not
dividing the level (or equivalently, that they generate an irreducible
automorphic representation). However, I will not make any assumptions
about the local components of these Maass forms at the primes dividing
the level (in particular they do not need to be newforms). I will
present an upper bound for the sup-norm of such Maass forms in the
level aspect that is valid for general orders and improves upon the
trivial bound. The key element in the proof is a uniform counting
result for points in quaternionic lattices.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/7866