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, π\otimes f \otimes χ\big) \ll_{\, \varepsilon, π, f } \mathrm{N} (q)^{5/4 + \varepsilon}, L \big(\tfrac 1 2 + it, π\otimes χ\big) \ll_{\, \varepsilon, π, t } \mathrm{N} (q)^{5/8 + \varepsilon}, \end{equation*} for any $\epsilon >0$.

To appear in Trans. Amer. Math. Soc.. 32 pages

References in corpus (7)

• On the subconvexity problem for L-functions on GL(3)
• Bounds for twists of $\mathrm{GL}(3)$ $L$-functions
• Subconvexity for symmetric square $L$-functions
• Hybrid subconvexity bounds for twisted $L$-functions on $GL(3)$
• Subconvexity for $\mathrm{GL}(3)$ L-functions
• Subconvexity for L-functions of non-spherical cusp forms on GL(3)
• Cancellation in the Additive Twists of Fourier Coefficients for ${\mathrm{GL}}_2$ and ${\mathrm{GL}}_3$ over Number Fields

Cited by in corpus (1)

• Low-lying zeros of $L$-functions for Maass forms over imaginary quadratic fields