Bowman-Bradley type theorem for finite multiple zeta values in ${\mathcal{A}}_2$

Bowman and Bradley obtained a remarkable formula among multiple zeta values. The formula states that the sum of multiple zeta values for indices which consist of the shuffle of two kinds of the strings $\{1,3,\dots ,1,3\}$ and $\{2,\dots ,2\}$ is a rational multiple of a power of ${\pi }^2$. Recently, Saito and Wakabayashi proved that analogous but more general sums of finite multiple zeta values in an adelic ring ${\mathcal{A}}_1$ vanish. In this paper, we partially lift Saito-Wakabayashi’s theorem from ${\mathcal{A}}_1$ to ${\mathcal{A}}_2$. Our result states that a Bowman-Bradley type sum of finite multiple zeta values in ${\mathcal{A}}_2$ is a rational multiple of a special element and this is closer to the original Bowman-Bradley theorem.