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Abstract I will discuss a proof of the conjecture of almost twenty years on the modular invariance of (logarithmic) intertwining operators.
Let V be a C_2-cofinite vertex operator algebra of positive energy.The conjecture states that the vector space spanned by pseudo-q-traces
shifted by -c/24 of products of (logarithmic) intertwining operators among grading-restricted generalized V-modules is a module for
the modular group SL(2, Z). In 2015, Fiordalisi proved that such pseudo-q-traces are absolutely convergent and have the genus-one
associativity property and some other properties. Recently, I have proved this conjecture completely. This modular invariace gives
a construction of C_2-cofinite genus-one logarithmic conformal field theories. It might play an important role in the future proofs of the
conjectures on the corresponding braided tensor categories, especially their rigidity and modularity.