Reductions of $2$-dimensional semi-stable representations with large $\mathcal{L}$-invariant

Abstract

We determine reductions of 2-dimensional, irreducible, semi-stable, and non-crystalline representations of $\mathrm{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$ with Hodge--Tate weights $0<k-1$ and with $\mathcal{L}$-invariant whose $p$-adic norm is sufficiently large, depending on $k$. Our main result provides the first systematic examples of the reductions for $k\ge p$.

23 pages. Minor revision, now final version (with updated numerical data). Accepted to appear in J. Inst. Math. Jussieu

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