On certain cusp forms on a definite quaternion algebra

If $D$ is the definite quaternion algebra over $\text{qu}$ of discriminant $p$, we compute, for any prime $p>3$, the number of infinite dimensional cusp forms on $D^{\ast }$ which are trivial at infinity, tamely ramified at $p$, and have given conductor $N$ away from $p$. We include a detail explanation of a Deuring–type correspondence between supersingular elliptic curves in characteristic $p$ and a certain double coset arising from the adelic points of $D^{\ast }$.