  Integer counting processes increment of an integer value at transitions
between states of an underlying Markov process. The generator of a counting
process, which depends on a parameter conjugate to the increments, defines a
complex algebraic curve through its characteristic equation, and thus a compact
Riemann surface. We show that the probability of a counting process can then be
written as a contour integral on that Riemann surface. Several examples are
discussed in details.
