On the Symmetric Square Large Sieve for $\mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathrm{PSL}_2 (\mathbb{C}) $ and the Prime Geodesic Theorem for $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3 $

In this paper, we improve the error term in the prime geodesic theorem for the Picard manifold $\mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3$. Instead of $\mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3$, we establish a spectral large sieve inequality for symmetric squares over $\mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathrm{PSL}_2 (\mathbb{C})$. This enables us to improve the bound $O (T^{3+2/3+\varepsilon})$ of Balkanova and Frolenkov into $O (T^{3+1/2+\varepsilon})$ for the second moment of symmetric square $L$-functions over $\mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3$. The basic idea is to enlarge the spherical family $\Pi_c^{0} (T)$ of Maass cusp forms on $\mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3$ into the family $\Pi_c (T, \sqrt{T})$ of cuspidal representations on $\mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathrm{PSL}_2 (\mathbb{C})$.