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}.
Reduction modulo p of two-dimensional crystalline representations of G_{Q_p} of slope less than three
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abstract
We use the p-adic local Langlands correspondence for GL_2(Q_p) to find the reduction modulo p of certain two-dimensional crystalline Galois representations. In particular, we resolve a conjecture of Breuil, Buzzard, and Emerton in the case when the slope is strictly between one and three, and prove partial results towards this conjecture for arbitrary slopes. Moreover, we partially classify the reduction modulo p of these representations when the slope is equal to one.
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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}.