Converse theorems assuming a partial Euler product

TL;DR

This paper explores converse theorems for automorphic forms, assuming a partial Euler product, and proposes an approach under specific additional assumptions related to second order modular forms.

Contribution

It extends previous converse theorems by assuming a partial Euler product and introduces a new approach involving additional assumptions, possibly linked to second order modular forms.

Findings

Proposes a new approach to converse theorems under partial Euler product assumptions.

Connects assumptions to properties of second order modular forms.

Provides a framework for future research on automorphic form characterization.

Abstract

Associated to a newform $f(z)$ is a Dirichlet series $L_f(s)$ with functional equation and Euler product. Hecke showed that if the Dirichlet series $F(s)$ has a functional equation of a particular form, then $F(s)=L_f(s)$ for some holomorphic newform $f(z)$ on $\Gamma(1)$. Weil extended this result to $\Gamma_0(N)$ under an assumption on the twists of $F(s)$ by Dirichlet characters. Conrey and Farmer extended Hecke's result for certain small $N$, assuming that the local factors in the Euler product of $F(s)$ were of a special form. We make the same assumption on the Euler product and describe an approach to the converse theorem using certain additional assumptions. Some of the assumptions may be related to second order modular forms.