The Prime Geodesic Theorem for the Picard Orbifold

Abstract

We establish the prime geodesic theorem for the Picard orbifold $\mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathbb{H}^{3}$, wherein the error term shrinks proportionally to improvements in the subconvex exponent for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. Our result sheds light on a venerable conjecture by attaining an unconditional exponent of $1.483$ and a conditionally superior exponent of $1.425$ under the generalised Lindel\"{o}f hypothesis. The argument synthesises, among other elements, the complete resolution of Koyama's (2001) mean Lindel\"{o}f hypothesis over $\mathbb{Q}(i)$, an improved Brun-Titchmarsh-type theorem over short intervals, a bootstrapped multiplicative exponent pair in the limiting regime, and a zero density theorem for the symplectic family of quadratic characters. Notably, despite the theoretical strength of our manifestations towards the mean Lindel\"{o}f hypothesis, the fundamental toolbox relies exclusively on the optimal mean square asymptotics for the Fourier coefficients of Maass cusp forms via the pre-Kuznetsov formula.