A Brauer-Siegel theorem for Fermat surfaces over finite fields

Abstract: We prove an analogue of the Brauer-Siegel theorem for Fermat surfaces over a finite field. Namely, letting $F_d$ be the Fermat surface of degree $d$ over $\mathbb{F}_q$ and $p_g(F_d)$ be its geometric genus, we consider the product of the order of the Brauer group $\mathrm{Br}(F_d)$ of $F_d$ times the absolute value of a Gram determinant of the N\'eron-Severi group of $F_d$ with respect to the intersection form (the regulator $\mathrm{Reg}(F_d)$ of $F_d$). We show that this product grows like $q^{p_g(F_d)}$ when $d$ tends to infinity: $$ \log\left( |\mathrm{Br}(F_d)|\cdot \mathrm{Reg}(F_d)\right) \sim \log q^{p_g(F_d)}.$$