  Let $V$ be a $C_2$-cofinite vertex operator algebra without nonzero elements
of negative weights. We prove the conjecture that the spaces spanned by
analytic extensions of pseudo-$q$-traces ($q=e^{2\pi i\tau}$) shifted by
$-\frac{c}{24}$ of products of geometrically-modified (logarithmic)
intertwining operators among grading-restricted generalized $V$-modules are
invariant under modular transformations. The convergence and analytic extension
result needed to formulate this conjecture and some consequences on such
shifted pseudo-$q$-traces were proved by Fiordalisi in [F1] and [F2] using the
method developed in [H2]. The method that we use to prove this conjecture is
based on the theory of the associative algebras $A^{N}(V)$ for $N\in
\mathbb{N}$, their graded modules and their bimodules introduced and studied by
the author in [H8] and [H9]. This modular invariance result gives a
construction of $C_2$-cofinite genus-one logarithmic conformal field theories
from the corresponding genus-zero logarithmic conformal field theories.
