A new proof of a theorem of Hida
for all r > 0. (Here Γ denotes the group of units in Zp congruent to one modulo p, Γr denotes the kernel of the reduction of Γ modulo p, and Λ denotes the completed group ring Zp[[Γ]] = lim←− r Zp[Γ/Γr].) Hida proves this through a series of group cohomological calculations combined with his theory of the ordinary part of the p-adic Hecke algebra. In this note we present a simple proof of the same result (Theorem 5.3 below) using only the elementary algebraic topology of the Riemann surfaces Y1(p). As with Hida, we also consider the case of auxiliary Γ1(N)-level structure, for some N prime to p.