# Towards the Jacquet conjecture on the Local Converse Problem for $p$-adic $\mathrm {GL}_n$

### Dihua Jiang

University of Minnesota, Minneapolis, USA### Chufeng Nien

National Cheng Kung University, Tainan, Taiwan### Shaun Stevens

University of East Anglia, Norwich, UK

## Abstract

The Local Converse Problem is to determine how the family of the local gamma factors $\gamma(s,\pi\times\tau,\psi)$ characterizes the isomorphism class of an irreducible admissible generic representation $\pi$ of $\mathrm {GL}_n(F)$, with $F$ a non-archimedean local field, where $\tau$ runs through all irreducible supercuspidal representations of $\mathrm {GL}_r(F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,\ldots,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, we formulate a general approach to prove the Jacquet conjecture. With this approach, the Jacquet conjecture is proved under an assumption which is then verified in several cases, including the case of level zero representations.

## Cite this article

Dihua Jiang, Chufeng Nien, Shaun Stevens, Towards the Jacquet conjecture on the Local Converse Problem for $p$-adic $\mathrm {GL}_n$. J. Eur. Math. Soc. 17 (2015), no. 4, pp. 991–1007

DOI 10.4171/JEMS/524