Bounds for $G{L}_3$ $L$-functions in depth aspect

Let $f$ be a Hecke-Maass cusp form for $S{L}_3(\mathbb{Z})$ and $\chi$ a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^{\kappa }$ with $p$ prime and $\kappa \ge 10$. We prove a subconvexity bound $$L(\frac{1}{2},\pi \otimes \chi ){\ll }_{p,\pi ,\epsilon }{\mathfrak{q}}^{3\mathrm{/}4-3\mathrm{/}40+\epsilon }$$ for any $\epsilon >0$, where the dependence of the implied constant on $p$ is explicit and polynomial. We obtain this result by applying the circle method of Kloosterman’s version, summation formulas of Poisson and Voronoi’s type and a conductor lowering mechanism introduced by Munshi [14]. The main new technical estimates are the essentially square root bounds for some twisted multi-dimensional character sums, which are proved by an elementary method.