# On the locus of $2$-dimensional crystalline representations with a given reduction modulo $p$

## TL;DR

This paper investigates the set of parameters for 2-dimensional crystalline Galois representations with fixed weight where the reduction modulo p is constant, providing methods to determine this locus through finite computations and extending results to other types.

## Contribution

It introduces a way to characterize the locus of parameters with a fixed reduction modulo p for 2-dimensional crystalline representations, including a finite computational approach and generalizations.

## Key findings

- The locus can be computed by finite reduction calculations.
- Qualitative descriptions of the locus are provided.
- Results are extended to other Galois types.

## Abstract

We consider the family of irreducible crystalline representations of dimension $2$ of ${\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)$ given by the $V_{k,a_p}$ for a fixed weight integer $k\geq 2$. We study the locus of the parameter $a_p$ where these representations have a given reduction modulo $p$. We give qualitative results on this locus and show that for a fixed $p$ and $k$ it can be computed by determining the reduction modulo $p$ of $V_{k,a_p}$ for a finite number of values of the parameter $a_p$. We also generalize these results to other Galois types.