Remarks on symmetric fusion categories of low rank in positive characteristic

Abstract: We give lower bounds for the rank of a symmetric fusion category in characteristic $p\geq 5$ in terms of $p$. We prove that the second Adams operation $\psi_2$ is not the identity for any non-trivial symmetric fusion category, and that symmetric fusion categories satisfying $\psi_2^a=\psi_2^{a-1}$ for some positive integer $a$ are super Tannakian. As an application, we classify all symmetric fusion categories of rank 3 and those of rank 4 with exactly two self dual simple objects.