Coleman's conjecture predicts that for g sufficiently large, there exists at most finitely many smooth curves of genus g (up to isomorphism) whose Jacobians are CM abelian varieties.  Based on the  recent work by  Tsimerman on the solution of André-Oort conjecture and my recent Joint works with  Xin Lu and with Ke Chen-Xin Lu on Oort conjectue for hyperelliptic curves and superelliptic curves we show Coleman's conjecture holds true for those two cases.