Given a Riemann surface it is an classical problem in analysis to bound
Laplace eigenfunctions,
either in terms of the eigenvalue or in terms of
the surface (such as its volume) or in terms of both.
We are interested in surfaces that can be realized in an arithmetic way.
A prime example are unions
of ellipsoids, realized as a suitable quotient of a positive
definite quaternion algebra over a totally real
number field.
If we restrict our attention to Laplace eigenfunctions that are
"arithmetically interesting" (that is, Hecke
eigenfunctions), we can use an amplified pre-trace formula to provide
strong sup-norm bounds. The
heart of proof are uniform bounds for representation numbers of quadratic
forms of large discriminant
of integral vectors that are almost parallel or almost orthogonal to a given vector.
We also discuss connections to subconvexity, supersingular elliptic curves
and automorphic forms on SO(4).