In 1987, Coleman submitted a certain conjecture for curves of genus greater than one over complete discrete valuation fields of mixed characteristics. Roughly speaking, this conjecture asserts that the residue fields of the torsion points of the Jacobian lying on the curve are unramified over the base field. As an application, (the already proven part of) this conjecture gives another proof of the Manin-Mumford conjecture (Raynaud's theorem) on the finiteness of torsion points on curves. In this talk, after overviewing some known results on the Coleman conjecture by Coleman, Tamagawa, Hoshi, et al., I explain my recent approach to the conjecture using Raynaud's classification of vector space schemes and discuss “quasi-supersingular group schemes'', which I introduced in another possible approach to the conjecture.