Crystalline Lifts of Semisimple $G$-Valued Galois Representations with Fixed Determinant
Vol. 33 (2026), No. 3, Page 359-379.
Aoki, K.
Crystalline Lifts of Semisimple $G$-Valued Galois Representations with Fixed Determinant
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Abstract:
For a finite extension $K/\mathbb{Q}_p$ and a split reductive group $G$ over $\mathcal{O}_L$ for a finite extension $L/K$, let $\overline{\rho} \colon \rm{Gal}$$_K$ $\to G(\overline{k}_L)$ be a continuous quasi-semisimple mod $p$ $G$-valued representation of the absolute Galois group $\rm{Gal}$$_K$. Let $\overline{\rho}^\rm{ab}$ be the abelianization of $\overline{\rho}$ and fix a crystalline lift $\psi$ of $\overline{\rho}^\rm{ab}$. We show the existence of a crystalline lift $\rho$ of $\overline{\rho}$ with regular Hodge-Tate weights such that the abelianization of $\rho$ coincides with $\psi$. We also show analogous results in the case that $G$ is a quasi-split tame group and $\overline{\rho} \colon \rm{Gal}$$_K$ $\to {^L}G(\overline{k}_L)$ is a semisimple mod $p$ $L$-parameter. These theorems are generalizations of those of Lin and Böckle-Iyengar-Paškūnas.
Keywords: Galois representation, crystalline representation.
Mathematics Subject Classification (2020): 11F80.
Received: 2025-02-25
