The prime geodesic theorem for PSL2(ℤ[i]) and spectral exponential sums

Kaneko, Ikuya

doi:10.2140/ant.2022.16.1845

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The prime geodesic theorem for $\operatorname{PSL}_{2}(\mathbb{Z}[i])$ and spectral exponential sums

Ikuya Kaneko

Vol. 16 (2022), No. 8, 1845–1887

DOI: 10.2140/ant.2022.16.1845

Abstract

This work addresses the prime geodesic theorem for the Picard manifold $ℳ = {PSL}_{2} (ℤ [i]) ∖ 𝔥^{3}$ , which asks for the asymptotic evaluation of a counting function for the closed geodesics on $ℳ$ . Let $E_{Γ} (X)$ be the error term in the prime geodesic theorem. We establish that $E_{Γ} (X) = O_{𝜖} (X^{3 ∕ 2 + 𝜖})$ on average as well as many pointwise bounds. The second moment bound parallels an analogous result for $Γ = {PSL}_{2} (ℤ)$ due to Balog et al. and our innovation features the delicate analysis of sums of Kloosterman sums with an explicit manipulation of oscillatory integrals. The proof of the pointwise bounds requires Weyl-strength subconvexity for quadratic Dirichlet $L$ -functions over $ℚ (i)$ . Moreover, an asymptotic formula for a spectral exponential sum in the spectral aspect for a cofinite Kleinian group $Γ$ is given. Our numerical experiments visualise in particular that $E_{Γ} (X)$ obeys a conjectural bound of size $O_{𝜖} (X^{1 + 𝜖})$ .

Keywords

prime geodesic theorem, Picard manifold, second moment, $L$-functions, Selberg trace formula, Kuznetsov formula, Kloosterman sums, spectral exponential sums, subconvexity

Mathematical Subject Classification

Primary: 11M36

Secondary: 11F72, 11L05, 11M26

Milestones

Received: 28 March 2020

Revised: 17 February 2021

Accepted: 1 April 2021

Published: 29 November 2022

Authors

Ikuya Kaneko
The Division of Physics, Mathematics and Astronomy California Institute of Technology Pasadena, CA United States

Vol. 16, No. 8, 2022