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

Abstract

Let $p = 4q + 1$ be a prime integer, where $q$ is an odd prime. Let $L = \mathbb{Q}(ζ_p)^+$ denote the maximal totally real subfield of the $p$-th cyclotomic field $\mathbb{Q}(ζ_p)$. 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 $\varepsilon \in U_K^+$, the absolute trace satisfies the sharp lower bound $\Tr_{K/\mathbb{Q}}(\varepsilon) \ge p - 4$. As an application, we show that a spherical fusion category $\C$ 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 $p\neq 29$.