In the previous papers, we determined the cuspidal class numbers of the modular curves X_1(p~m) for prime numbers p≠2, 3. The purpose of this paper is to determine the cuspidal class number of the modular curve X_1(3~m). Let h' be the number obtained by the substitution of 3 for p in the cuspidal class number formula for the case p≠2, 3 ([8, Theorem 7.1, Theorem 8.1]). Let h_1(3~m) be the cuspidal class number of the curve X_1(3~m). Then our main results (Theorem 3.1, Theorem 4.1) show h_1(3~m)=h'/3 if m≧2. (If m=1, then h_1(3)=h'/3~2=l.) As is well known, the cuspidal divisor class groups of the modular curves are finite (Manin, Drinfeld).
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