Subconvexity for twisted $L$-functions on $\mathrm{GL}_3$ over the Gaussian number field

Abstract

Let $q \in \mathbb{Z} [i]$ be prime and $\chi $ be the primitive quadratic Hecke character modulo $q$. Let $\pi$ be a self-dual Hecke automorphic cusp form for $\mathrm{SL}_3 (\mathbb{Z} [i] )$ and $f$ be a Hecke cusp form for $\Gamma_0 (q) \subset \mathrm{SL}_2 (\mathbb{Z} [i])$. Consider the twisted $L$-functions $ L (s, \pi \otimes f \otimes \chi) $ and $L (s, \pi \otimes \chi)$ on $\mathrm{GL}_3 \times \mathrm{GL}_2$ and $\mathrm{GL}_3$. We prove the subconvexity bounds \begin{equation*} L \big(\tfrac 1 2, \pi \otimes f \otimes \chi \big) \ll_{\, \varepsilon, \pi, f } \mathrm{N} (q)^{5/4 + \varepsilon}, L \big(\tfrac 1 2 + it, \pi \otimes \chi \big) \ll_{\, \varepsilon, \pi, t } \mathrm{N} (q)^{5/8 + \varepsilon}, \end{equation*} for any $\varepsilon > 0$.