# On reductions of families of crystalline Galois representations

### Gerasimos Dousmanis

## Abstract

Let $K_f$ be the finite unramified extension of ${Q}_p$ of degree $f$ and $E$ any finite large enough coefficient field containing $K_f.$ We construct analytic families of étale $(\varphi ,\Gamma )$-modules which give rise to families of crystalline $E$-representations of the absolute Galois group $G_{K_f}$ of $K_f.$ For any irreducible effective two-dimensional crystalline $E$-representation of $G_{K_f}$ with labeled Hodge-Tate weights ${0,-k_i}_{\tau _i}$ induced from a crystalline character of $G_{K_{2f}},$ we construct an infinite family of crystalline $E$ -representations of $G_{K_f}$ of the same Hodge-Tate type which contains it. As an application, we compute the semisimplified mod $p$ reductions of the members of each such family.

## Cite this article

Gerasimos Dousmanis, On reductions of families of crystalline Galois representations. Doc. Math. 15 (2010), pp. 873–938

DOI 10.4171/DM/317