Behaviour of Certain Crystalline Representations modulo $2$

Abstract

We compute the explicit form of the semisimplified reduction modulo $2$ of the $2$-adic crystalline Galois representations $V_{k,a_2}$ at small slopes in $(0,1]$, using the compatibility of $2$-adic and mod-$2$ local Langlands correspondence. We find parameters $\alpha'(k,a_{2})$ and $\alpha(k,a_{2})$, which play a crucial role in determining the reduction of $V_{k,a_{2}}$ for slopes in the range $(0,1)$ and slope $1$ respectively.