It is a classical problem to determine the span
of the theta series of a given quadraic space over
a small ring. In such a way, Jacobi proved his
famous formula of the number of ways of expressing
integers as sums of four squares.
For the norm form of a definite quaternion algebra B,
we determine the span integrally over very small ring
(for example, if B only ramifies at one prime p,
we shall determine the span over Z[1/(p-1)]).