We contribute to the classification of modular categories $\mathcal{ C}$ with $\operatorname{ FPdim}(\mathcal{ C})\equiv 2 \pmod 4$. We prove that such categories have group of invertibles of even order, and that they factorize as $\mathcal C\cong \widetilde{ \mathcal C} \boxtimes \operatorname{ sem}$, where $\widetilde{ \mathcal C}$ is an odd-dimensional modular category and $\operatorname{ sem}$ is the rank 2 pointed modular category. This reduces the classification of these categories to the classification of odd-dimensional modular categories. It follows that modular categories $\mathcal C$ with $\operatorname{ FPdim}(\mathcal{ C})\equiv 2 \pmod 4$ of rank up to 46 are pointed. More generally, we prove that if $\mathcal C$ is a weakly integral MTC and $p$ is an odd prime dividing the order of the group of invertibles that has multiplicity one in $\operatorname{ FPdim}(\mathcal C)$, then we have a factorization $\mathcal C \cong \widetilde{ \mathcal C} \boxtimes \operatorname{ Vec}_{ \mathbb Z_p}^{ \chi},$ for $\widetilde{ \mathcal C}$ an MTC with dimension not divisible by $p$.
Categories: math.QA