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

Abstract

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 $O(T^{3+2/3+\epsilon })$ of Balkanova and Frolenkov to $O(T^{3+1/2+\epsilon })$ 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 ${\hat{I}}_c^0(T)$ of Maass cusp forms on ${\mathrm{PSL}}_2(\mathbb{Z}[i])\backslash {\mathbb{H}}^3$ to the family ${\hat{I}}_c(T,\sqrt{T})$ of cuspidal representations on ${\mathrm{PSL}}_2(\mathbb{Z}[i])\backslash {\mathrm{PSL}}_2(\mathbb{C})$.

20 pages; further refinement and remarks added; changed title; accepted in Sci. China Math