Reductions Of Crystalline Representations Of Fractional Slope $<p-1$

Abstract

Let $p$ be an odd prime and let $V_{k,a_p}$ be the two-dimensional crystalline representation of the Galois group of ${\mathbb Q}_p$ of weight $k \geq 2$ and parameter $a_p \in \bar{\mathbb{Q}}_p$. We study the semi-simplification $\bar{V}_{k,a_p}$ of the mod $p$ reduction of $V_{k,a_p}$ when the slope (valuation of $a_p$) is a positive fraction $< p-1$ using the mod $p$ local Langlands correspondence. We describe the $\textit{exact shape}$ of $\bar{V}_{k,a_p}$ for all such slopes and all (sufficiently large, depending on the slope) weights $k$, as long as certain Jordan-H\"older factors of dimension $p-1$ do not intervene in the computation (when $k$ is odd), though we also provide some criteria which further determine the shape of $\bar{V}_{k,a_p}$ in some of these exceptional cases. To keep this paper a reasonable length, we assume that for certain bad congruence classes of $k$ mod $p$, the slope is less than the representative - taken in the range $[1,p-1]$ - of the congruence class of $k-2$ mod $(p-1)$, which is generically the case if the slope is small. Finally, a folklore conjecture predicts that the reduction $\bar{V}_{k,a_p}$ is $\textit{irreducible}$ for fractional slopes if $k$ is even. We deduce this conjecture for all fractional slopes $< p-2$ and all (sufficiently large, even) weights $k$ under the aforementioned slope assumption.