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

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 $a_p$.
Keywords :
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-01519169
Contributor : Sandra Rozensztajn <>
Submitted on : Friday, May 5, 2017 - 11:05:31 PM
Last modification on : Thursday, January 11, 2018 - 6:12:31 AM

### Identifiers

• HAL Id : hal-01519169, version 1
• ARXIV : 1705.01060

### Citation

Sandra Rozensztajn. On the locus of 2-dimensional crystalline representations with a given reduction modulo p. 2017. ⟨hal-01519169⟩

Record views