We investigate trace functions of modules for vertex operator
algebras (VOA) satisfying $C_2$ -cofiniteness. For the modular
invariance property, Zhu assumed two conditions in [Z]:
(1) $A(V)$
is semisimple and (2) $C_2$ -cofiniteness.
We show that $C_2$ -cofiniteness is enough to prove a modular
invariance property. For example, if a VOA
$V=\bigoplus_{m=0}^{\infty}V_m$
is $C_2$ -cofinite, then the space
spanned by generalized characters (pseudotrace functions of the
vacuum element) of $V$ -modules is a finite-dimensional
$\SL_2(\mathbb{Z})$ $\SL_2(\mathbb{Z})$ -invariant space and the central charge and
conformal weights are all rational numbers. Namely, we show that
$C_2$ -cofiniteness implies "rational conformal field theory" in
a sense as expected in Gaberdiel and Neitzke [GN]. Viewing a
trace map as a symmetric linear map and using a result of
symmetric algebras, we introduce "pseudotraces" and pseudotrace
functions and then show that the space spanned by such pseudotrace
functions has a modular invariance property. We also show that
$C_2$ -cofiniteness is equivalent to the condition that every weak
module is an $\mathbb{N}$ -graded weak module that is a direct sum
of generalized eigenspaces of $L(0)$ .