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Reductions of $2$-dimensional semi-stable representations with large $\mathcal L$-invariant
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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 $k \geq p$.
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Reductions of some two-dimensional crystalline representations via Kisin modules
For v_p(a_p) > floor((k-1)/p), the semisimple mod p reduction of the crystalline representation V_{k,a_p} is V_{k,0}.
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