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 Lindelöf hypothesis. The argument synthesises, among other elements, the complete resolution of Koyama's (2001) mean Lindelöf 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 Lindelöf 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.