On the Twisted Spectral Large Sieve Inequality for $\mathrm{PGL}_2 (\mathbb{Z}[i]) \backslash \mathrm{PGL}_2 (\mathbb{C})$

TL;DR

This paper extends the twisted spectral large sieve inequalities from classical modular groups to the setting of $ ext{PGL}_2( ext{Z}[i])$, broadening the scope of these inequalities in automorphic forms.

Contribution

It generalizes existing spectral large sieve inequalities to the group $ ext{PGL}_2( ext{Z}[i])$, a significant step in automorphic form theory.

Findings

Extended large sieve inequalities to $ ext{PGL}_2( ext{Z}[i])$

Established bounds analogous to classical cases

Enhanced tools for automorphic form analysis

Abstract

In this paper, we extend the twisted spectral large sieve inequalities of Deshouillers, Iwaniec, Luo, and Young from $\mathrm{SL}_2 (\mathbb{Z})$ onto $\mathrm{PGL}_2 (\mathbb{Z}[i])$.