An algorithm for computing the reduction of $2$-dimensional crystalline representations of $\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$

TL;DR

This paper introduces an algorithm to compute the modulo p reduction of 2-dimensional crystalline Galois representations over p-adic fields, utilizing the semi-simple modulo p Langlands correspondence, with practical examples implemented in SAGE.

Contribution

The paper presents a novel algorithm for calculating reductions of 2D crystalline Galois representations using the semi-simple modulo p Langlands correspondence.

Findings

Successfully computed several examples of reductions.

Algorithm implemented in SAGE for practical computations.

Provides new computational tools for p-adic Galois representations.

Abstract

We describe an algorithm to compute the reduction modulo $p$ of a crystalline Galois representation of dimension $2$ of $\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$ with distinct Hodge-Tate weights via the semi-simple modulo $p$ Langlands correspondence. We give some examples computed with an implementation of this algorithm in SAGE.