Optimal lifting for the projective action of SL3(ℤ)

Kamber, Amitay; Lavner, Hagai

doi:10.2140/ant.2023.17.749

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Optimal lifting for the projective action of $\operatorname{SL}_3(\mathbb{Z})$

Amitay Kamber and Hagai Lavner

Vol. 17 (2023), No. 3, 749–774

DOI: 10.2140/ant.2023.17.749

Abstract

Let $𝜖 > 0$ and let $q \to \infty$ be a prime. We prove that with high probability, given $x$ , $y$ in the projective plane over $𝔽_{q}$ , there exists $γ \in {SL}_{3} (ℤ)$ , with coordinates bounded by $q^{1 ∕ 3 + 𝜖}$ , whose projection to ${SL}_{3} (𝔽_{q})$ sends $x$ to $y$ . The exponent $\frac{1}{3}$ is optimal and the result is a high rank generalization of Sarnak’s optimal strong approximation theorem for ${SL}_{2} (ℤ)$ .

Keywords

optimal lifting, congruence subgroups

Mathematical Subject Classification

Primary: 11F06

Milestones

Received: 9 August 2021

Revised: 14 March 2022

Accepted: 10 May 2022

Published: 12 April 2023

Authors

Amitay Kamber
Centre for Mathematical Sciences University of Cambridge Cambridge United Kingdom

Hagai Lavner
Einstein Institute of Mathematics The Hebrew University of Jerusalem Jerusalem Israel

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Vol. 17, No. 3, 2023