On Reductions of Families of Crystalline Galois Representations

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.

2010 Mathematics Subject Classification: 11F80, 11F85.

Keywords and Phrases: Wach modules, $(\varphi ,\Gamma )$-modules, reductions of crystalline Galois representations.

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