Reductions of $\mathrm{GL}_2(\mathbb Q_{p^f})$-Banach spaces of slopes in $(0,1)$

TL;DR

This paper investigates conditions under which certain mod p reductions of p-adic Banach space representations of GL2 over Q_{p^f} are supercuspidal, focusing on slopes in (0,1).

Contribution

It provides new criteria for the irreducible quotients of these reductions to be supercuspidal, especially for small weights and degrees.

Findings

Identifies conditions for supercuspidality of irreducible quotients.

Confirms non-zero mod p reductions for small weights and degrees.

Analyzes the structure of reductions of p-adic representations.

Abstract

Let $p$ be an odd prime and $f \geq 1$. We consider a $p$-adic locally algebraic $\text{GL}_2(\mathbb Q_{p^f})$-representation attached to a tuple of $f$ weights $k=(k_i)$ for $0 \leq i \leq f-1$ and a $p$-adic integer $a_p$ with valuation in $(0,1)$. We give conditions under which the irreducible quotients of the subquotients in a filtration on the reduction mod $p$ of the natural integral structure on this space are supercuspidal. We also check that for small $k$ and $f$ the integral structure is a lattice so that the mod $p$ reduction is nonzero.