A trace lower bound for totally positive units in $\mathbb{Q}({\zeta }_p{)}^{+}$ and its application to fusion category

Let $p=4q+1$ be a prime integer, where $q$ is an odd prime. Let $L=\mathbb{Q}({\zeta }_p{)}^{+}$ denote the maximal totally real subfield of the $p$-th cyclotomic field $\mathbb{Q}({\zeta }_p)$. In this paper, we prove that if $U_L^{+}=U_L^2$ (i.e., every totally positive unit is a square in $L$), then for any non-trivial subfield $K\subseteq L$ and any non-trivial totally positive unit $\epsilon \in U_K^{+}$, the absolute trace satisfies the sharp lower bound ${\text{Tr}}_{K\mathrm{/}\mathbb{Q}}(\epsilon )\ge p-4$. As an application, we show that a spherical fusion category $\text{C}$ with global dimension $\dim (\mathcal{C})=p=4q+1$ is pointed, where $2$ is a primitive root modulo $q$, or $p=4q+1<2^{22}$ and $p\mathrm{\ne }29$.