The local converse theorem for generic representations of p-adic general linear groups

Let $F$ be a non-archimedean local field, let $G_n={\mathrm{GL}}_n(F)$, and fix an additive character $\psi$ of $F$. Let $\pi_1,\pi_2$ be irreducible generic representations of $G_n$. For an irreducible generic representation $\tau$ of $G_r$, write $\gamma(s,\pi_i\times\tau,\psi)$ for the associated local gamma factor.

Local converse conjecture. If $\pi_1$ and $\pi_2$ have the same central character and

$$\gamma(s,\pi_1\times\tau,\psi)=\gamma(s,\pi_2\times\tau,\psi)$$

as functions of the complex variable $s$ for every irreducible generic representation $\tau$ of $G_r$ and every $r$ with $1\leq r\leq \left[\frac n2\right]$, then $\pi_1\cong\pi_2$.

This is the standard local converse conjecture, asserting that a generic representation is determined by its Rankin–Selberg gamma factors against generic representations of sufficiently small rank. The paper’s abstract states that the conjecture is completely proved; the source also records earlier proofs in several cases, including $n=3$, so its status is solved.