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We show that C_2-cofiniteness is enough to prove a modular invarianceproperty of vertex operator algebras without assuming the semisimplicity of Zhualgebra. For example, if a VOA V=\oplus_{m=0}^{\infty}V_m is C_2-cofinite, thenthe space spanned by generalized characters of V-modules is invariant under theaction of SL_2(\Z). In this case, the central charge and conformal weights areall rational numbers. Namely, a VOA satisfying C_2-cofiniteness is a rationalconformal field theory in a sense. We also show that C_2-cofiniteness isequivalent to the condition that every weak module is an \N-graded weak modulewhich is a direct sum of generalized eigenspaces of L(0).

Modular invariance of vertex operator algebras satisfying C 2 -cofiniteness