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

Abstract

Let $f$ be a Hecke-Maass cusp form for $S{L}_3(\mathbb{Z})$ and $\ddot{I}‡$ a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^{\hat{I}}º$ with $p$ prime and $\hat{I}º\ge 10$. We prove a subconvexity bound $$L\left(\frac{1}{2},\ddot{I}€\otimes \ddot{I}‡\right){\ll }_{p,\ddot{I}€,\epsilon }{\mathfrak{q}}^{3/4-3/40+\epsilon }$$ 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.

20 pages. Comments welcome!