Suppose I have a prime $p$ and a finite group $G$ together with representations $\sigma: G \to GL\_n(\mathbb{Q}\_p)$ and $\pi: G \to GL\_n(\mathbb{F}\_p)$. My question is:



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> When does there exist a representation $\rho: G \to GL\_n(\mathbb{Z}\_p)$ such that $\sigma$ is isomorphic to the base change of $\rho$ to $\mathbb{Q}\_p$, and $\pi$ is isomorphic to the base change of $\rho$ to $\mathbb{F}\_p$?
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One necessary condition is that for any $p$-regular element $g \in G\_{reg}$, the trace of $\sigma(g)$ is equal to the Brauer character of $\pi(g)$. However, I can't tell if this is sufficient or not - the results in Serre's Linear Representations book only give me existence theorems up to semisimplification. I tried to cook up a suitable $G$-stable $\mathbb{Z}\_p$-lattice using affine buildings, but I got hopelessly lost.



Can we glue characteristic 0 and characteristic p representations of a finite group given equality of (Brauer) characters?
The condition on Brauer characters is not sufficient.


Let $G$ be a $p$-group, $\pi$ any nontrivial representation over $\mathbb{F}\_p$, and $\sigma$ the trivial representation over $\mathbb{Q}\_p$ of the same degree as $\pi$. Then a representation over $\mathbb{Z}\_p$ whose base change to $\mathbb{Q}\_p$ is isomorphic to $\sigma$ must also be trivial, so its base change to $\mathbb{F}\_p$ can’t be isomorphic to $\pi$.


But the condition on Brauer characters is trivially satisfied.


