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

TL;DR

This paper establishes hybrid bounds for twists of $GL(3)$ $L$-functions, providing subconvexity estimates under specific prime modulus conditions for the twisting characters.

Contribution

The authors derive new hybrid subconvexity bounds for $GL(3)$ $L$-functions twisted by Dirichlet characters with prime moduli, extending previous results to more general settings.

Findings

Established bounds improve upon convexity estimates.

Derived explicit bounds depending on the moduli and spectral parameter.

Applicable to a range of prime moduli under specified conditions.

Abstract

Let $\pi$ be a Hecke-Maass cusp form for $SL(3,\mathbb{Z})$ and $\chi=\chi_1\chi_2$ a Dirichlet character with $\chi_i$ primitive modulo $M_i$. Suppose that $M_1$, $M_2$ are primes such that $\max\{(M|t|)^{1/3+2\delta/3},M^{2/5}|t|^{-9/20}, M^{1/2+2\delta}|t|^{-3/4+2\delta}\}(M|t|)^{\varepsilon}<M_1< \min\{ (M|t|)^{2/5},(M|t|)^{1/2-8\delta}\}(M|t|)^{-\varepsilon}$ for any $\varepsilon>0$, where $M=M_1M_2$, $|t|\geq 1$ and $0<\delta< 1/52$. Then we have $$ L\left(\frac{1}{2}+it,\pi\otimes \chi\right)\ll_{\pi,\varepsilon} (M|t|)^{3/4-\delta+\varepsilon}. $$