Hybrid subconvexity bounds for twists of $\rm GL(3)$ $L$-functions

TL;DR

This paper establishes a new subconvexity bound for twisted $L$-functions associated with $SL(3,Z)$ cusp forms and primitive Dirichlet characters of prime power conductor, improving understanding of their size in critical strips.

Contribution

It proves a novel subconvexity bound for $L$-functions of $SL(3)$ cusp forms twisted by prime power conductor characters, extending previous bounds in the field.

Findings

Established a subconvexity bound with explicit dependence on prime power conductor and spectral parameter.

Improved the exponent in the convexity bound for these $L$-functions.

Provides tools potentially applicable to other higher-rank $L$-functions.

Abstract

Let $\pi$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form and $\chi$ a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^k$ with $p$ prime. In this paper we will prove the following subconvexity bound $$ L\left(\frac{1}{2}+it,\pi\times \chi\right)\ll_{\pi,\varepsilon} p^{3/4}\big(\mathfrak{q}(1+|t|)\big)^{3/4-3/40+\varepsilon}, $$ for any $\varepsilon >0$ and $t \in \mathbb{R}$.