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

Abstract

Let $p$ be an odd prime and $f\ge 1$. 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 $0\le i\le f-1$ 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.

51 pages, minor changes, appendix added