Iwahori–Hecke model for the universal supersingular representation

Anand Chitrao and Asfak Soneji

Vol. 344 (2026), No. 1, 27–46

DOI: 10.2140/pjm.2026.344.27

Abstract

Let $F$ be a nonarchimedean local field with residue field ${\mathbb{𝔽}}_q$. Every supersingular representation of ${GL}_2(F)$ is a quotient of a universal supersingular representation ${\pi }_r$ for some integer $0\le r\le q-1$. When $F$ is a finite extension of ${\mathbb{Q}}_p$,

Anandavardhanan–Borisagar and Anandavardhanan–Jana introduced an Iwahori–Hecke model for ${\pi }_r$ in the regular case $0<r<q-1$. When $F={\mathbb{Q}}_p$, Chitrao introduced an Iwahori–Hecke model for ${\pi }_r$ when $r=0$, $p-1$. This model was then used by Chitrao–Ghate to compute the reduction mod $p$ of two-dimensional irreducible semistable representations of $\mathrm{Gal}<!--FUNCTION\; APPLICATION-->({\overline{\mathbb{Q}}}_p∕{\mathbb{Q}}_p)$. We extend this model to an arbitrary local field $F$ for $r=0$, $q-1$.

We also write down an explicit nonsplit self-extension of ${\pi }_r$ which has a four-dimensional space of $I(1)$-invariants when $F={\mathbb{Q}}_p$ and $r=0$, $p-1$.

Keywords

modular representations, Iwahori–Hecke model, supersingular representations

Mathematical Subject Classification

Primary: 11F70, 20G05, 22E50

Milestones

Received: 26 August 2025

Revised: 11 May 2026

Accepted: 13 May 2026

Published: 10 August 2026