Behaviour of Certain Crystalline Representations modulo $2$

TL;DR

This paper explicitly computes the mod 2 reduction of certain 2-adic crystalline Galois representations at small slopes, revealing key parameters that determine their behavior.

Contribution

It provides explicit formulas for the reduction of crystalline Galois representations modulo 2 at small slopes, using compatibility of local Langlands correspondences.

Findings

Explicit form of the semisimplified reduction modulo 2 for small slopes.

Identification of parameters '(k,a_2) and (k,a_2) crucial for reductions.

Reduction behavior characterized for slopes in (0,1] and at slope 1.

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.