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}.
Semi-simplified modulo $p$ of semi-stable representations: an algorithmic approach
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abstract
The aim of this paper is to present an algorithm the complexity of which is polynomial to compute the semi-simplified modulo $p$ of a semi-stable $\Q_p$-representation of the absolute Galois group of a $p$-adic field (\emph{i.e.} a finite extension of $\Q_p$). In order to do so, we use abundantly the $p$-adic Hodge theory and, in particular, the Breuil-Kisin modules theory.
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2019 1verdicts
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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}.