Near Counterexamples to Weil's Converse Theorem

TL;DR

This paper demonstrates that in Weil's converse theorem, a certain number of functional equations for multiplicative twists are necessary to establish modularity for (p), highlighting limitations in the theorem's conditions.

Contribution

It identifies the minimum number of functional equations required in Weil's converse theorem to prove modularity for (p), providing new insights into the theorem's constraints.

Findings

Functional equations needed grow with p

At least ((p))/3 equations are necessary

Limits the applicability of Weil's converse theorem

Abstract

We show that in Weil's converse theorem the functional equations of multiplicative twists for at least the first $\sqrt{\frac{p-24}{3}}$ moduli are needed in order to prove the modularity for $\Gamma_0(p)$.