#### Vol. 14, No. 3, 2020

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

### Sandra Rozensztajn

Vol. 14 (2020), No. 3, 643–700
##### Abstract

We consider the family of irreducible crystalline representations of dimension $2$ of $Gal\left({\stackrel{̄}{ℚ}}_{p}∕{ℚ}_{p}\right)$ given by the ${V}_{k,{a}_{p}}$ for a fixed weight $k\ge 2$. 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.

Primary: 11F80
Secondary: 14G22
##### Milestones
Revised: 26 August 2019
Accepted: 30 September 2019
Published: 1 June 2020
##### Authors
 Sandra Rozensztajn UMPA, ÉNS de Lyon UMR 5669 du CNRS Lyon France