The Voronoi formula on GL(3) with ramification

TL;DR

This paper extends the Voronoi formula to automorphic forms on GL(3) with ramification, providing new tools for analyzing L-functions and their functional equations in cases with specific conductor conditions.

Contribution

It proves the Voronoi formula applies to GL(3) automorphic forms with ramification for certain conductor conditions, and explores implications for L-functions.

Findings

Voronoi formula applies to GL(3) with ramification when conductor divides N.

Derived functional equations for twisted L-functions on GL(3).

Extended the applicability of Voronoi summation in automorphic forms analysis.

Abstract

Firstly we prove that the Voronoi formula of Miller-Schmid type applies to automorphic forms on GL(3) for the congruence subgroup $\Gamma_0(N)$, when the conductor of the additive character in the formula is a multiple of $N$. As an application, we produce a result about the functional equation of $L$-function of the automorphic form on GL(3) twisted by Dirichlet characters. Secondly we prove that a similar formula applies to automorphic forms on GL(3) for the congruence subgroup $\Gamma_0(N)$, when the conductor of the additive character in the formula is coprime with $N$.