Integral almost square-free modular categories

Abstract

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius–Perron dimension [Formula: see text], where [Formula: see text] is a prime number, [Formula: see text] is a square-free natural number and [Formula: see text]. We prove that, if [Formula: see text] or [Formula: see text] is prime with [Formula: see text], then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and Tucker. As an application, we prove that an integral modular category whose Frobenius–Perron dimension is odd and less than [Formula: see text] is group-theoretical.