In the book "Introduction to the arithmetic theory of automorphic functions", section 2.6, Shimura gives general formulas to compute the dimension of the space of integral and cusp forms of a given weight, for any Fuchsian group of the first kind. His proof uses the theory of divisors of automorphic forms and Riemann-Roch theorem.
Does anybody know if there is a general proof of these results which does not rely on the Riemann-Roch theorem (for every Fuchsian group of the first kind, not only $\Gamma(1)$)?
Thanks!