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

Rozensztajn, Sandra

doi:10.2140/ant.2020.14.655

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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

DOI: 10.2140/ant.2020.14.655

Abstract

We consider the family of irreducible crystalline representations of dimension $2$ of $Gal ({\bar{ℚ}}_{p} ∕ ℚ_{p})$ given by the $V_{k, a_{p}}$ for a fixed weight $k \geq 2$ . 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.

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Keywords

Galois representations, p-adic representations

Mathematical Subject Classification 2010

Primary: 11F80

Secondary: 14G22

Milestones

Received: 13 September 2018

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

Vol. 14, No. 3, 2020