5 problems

Jacquet's local converse conjecture. If

Let $n$ be prime and let $\omega$ be a character of $\mathbb F_{p^n}^{\times}$; write $S(\chi)$ for the corresponding Gauss sum. Let $\hat k=k(p^n-1)/(p-1)$. Prime-characteristic G…

Let $E/{\mathcal F}$ be the quadratic extension implicit in the distinction notation, with $n\ge3$. Let $\pi$ be a supercuspidal representation of $\operatorname{GL}_n(E)$, and for…

Let $r$ be a prime divisor of $n$. For a regular character $\chi$ of $\mathbb F_{q^n}^{\times}$, let $\pi_\chi$ be the associated primitive cuspidal representation of…

Let $n>m$. Let $\pi$ and $\tau$ be irreducible cuspidal representations of $\operatorname{GL}_n(\mathbb F_q)$ and $\operatorname{GL}_m(\mathbb F_q)$, corresponding to regular chara…