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

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.