On certain cusp forms on a definite quaternion algebra

TL;DR

This paper computes the number of specific cusp forms on a definite quaternion algebra over the rationals, linking their properties to supersingular elliptic curves via a Deuring-type correspondence.

Contribution

It provides explicit formulas for counting cusp forms with given ramification and conductor on a definite quaternion algebra, expanding understanding of their structure.

Findings

Derived formulas for the number of cusp forms with specified properties

Established a Deuring-type correspondence relating elliptic curves and quaternion algebra cosets

Connected the theory of cusp forms to supersingular elliptic curves in characteristic p

Abstract

If $D$ is the definite quaternion algebra over $\qu$ of discriminant $p$, we compute, for any prime $p>3$, the number of infinite dimensional cusp forms on $D^*$ 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 $D^*$.