Let \(\Gamma\) be a discrete subgroup of \(\text{SL}_ 2(\mathbb R)\) such that the corresponding modular curve \(X=X(\Gamma)\) has finite volume. It is of interest to study the subgroup \(C(\Gamma)\) of \(J=\text{Jac}(X)\) generated by the divisors of degree \( 0\) supported on the cusps of \(X\). If \(\Gamma\) is a congruence subgroup, then Manin and Drinfeld used the theory of Hecke operators to prove that \(C(\Gamma)\) is finite. A second proof can be given by explicitly constructing modular functions with the appropriate zeros and poles [see \textit{D. S. Kubert} and \textit{S. Lang}, ``Modular units'' (1981; Zbl 0492.12002)]. In this paper the authors give a third proof, based on ideas of \textit{B. Schoeneberg} [``Elliptic modular functions'' (1974; Zbl 0285.10016)], \textit{G. Stevens} [``Arithmetic on modular curves'', Prog. Math. 20 (1982; Zbl 0529.10028)], and \textit{A. J. Scholl} [Math. Proc. Camb. Philos. Soc. 99, 11--17 (1986; Zbl 0564.10023)]. Associated to a cuspidal divisor is a differential of the third kind, which in turn is given by an Eisenstein series of weight 2; and the divisor has finite order in \(J\) if and only if the Eisenstein series has algebraic Fourier coefficients. The authors review this material, and then use Ramanujan sums to obtain an explicit expression for the Fourier coefficients. Since this expression is visibly algebraic, they conclude that \(C(\Gamma)\) is finite. In case \(\Gamma\) is not a congruence subgroup, it is possible for \(C(\Gamma)\) to be infinite. The authors next consider the well-known realization of the Fermat curve \(F_ N: X^ N+Y^ N=1\) as \(X(\Gamma)\) for a non-congruence subgroup, where the cusps are the \(N\) points ``at infinity''. They find that the finiteness of \(C(\Gamma)\) is equivalent to the algebraicity of a complicated (but very explicit) expression involving generalized Ramanujan sums. Since \textit{D. E. Rohrlich} [Invent. Math. 39, 95--127 (1977; Zbl 0357.14010)] has shown in this case that \(C(\Gamma)\) is finite, the authors conclude that their expression is algebraic. In the final section the authors look at the (unramified) correspondence between \(F_ N\) and \(X(2N)\) over \(X(2)\) considered by Kubert and Lang (op. cit.). This gives a divisor on the surface \(F_ N\times X(2N)\), and they show that this divisor has finite order in the relative Néron-Severi group \(\text{NS}(F_ N\times X(2N))/(\text{NS}(F_ N)\oplus \text{NS}(X(2N)))\).