A proof that quasi-semisimple mod p G-valued Galois representations admit crystalline lifts with arbitrary fixed crystalline abelianization, for split reductive groups and tame quasi-split groups.
Conrad, Lifting global representations with local properties , preprint, available at https://math.stanford.edu/~conrad/papers/locchar.pdf
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Crystalline lifts of semisimple $G$-valued Galois representations with fixed determinant
A proof that quasi-semisimple mod p G-valued Galois representations admit crystalline lifts with arbitrary fixed crystalline abelianization, for split reductive groups and tame quasi-split groups.