An extension of Venkatesh's converse theorem to the Selberg class

TL;DR

This paper extends Venkatesh's converse theorem to the broader context of the Selberg class, utilizing the Petersson trace formula to handle more general gamma factors and levels.

Contribution

It generalizes the converse theorem for modular forms to the Selberg class, accommodating arbitrary level, character, and degree 2 gamma factors.

Findings

Extended the converse theorem to the Selberg class.

Applied the Petersson trace formula to broader gamma factors.

Handled arbitrary level and character in the proof.

Abstract

We extend Venkatesh's proof of the converse theorem for classical holomorphic modular forms to arbitrary level and character. The method of proof, via the Petersson trace formula, allows us to treat arbitrary degree 2 gamma factors of Selberg class type.