# Condition on a Fontaine Laffaille module which prescribes the image of the associated Galois representation

The Setup:

Let $$m\geq 1$$ be an integer, $$\mathbb{F}$$ be a finite field of characteristic $$p$$ and $$W(\mathbb{F})$$ the ring of Witt-vectors with residue field $$\mathbb{F}$$ and $$\sigma:W(\mathbb{F})\xrightarrow{\sim} W(\mathbb{F})$$ a lift of the Frobenius $$\bar{\sigma}:\mathbb{F}\xrightarrow{\sim}\mathbb{F}$$. Recall that filtered Dieudonne' $$W(\mathbb{F})$$-module also known as a Fontaine-Laffaille module is a $$W(\mathbb{F})$$-module furnished with a decreasing, exhaustive, separated filtration of submodules $$\{F^i M\}$$ and for each integer $$i$$ a $$\sigma$$-semilinear map $$\varphi^i=\varphi_M^i:F^i M\rightarrow M$$. These maps are required to satisfy two conditions

1. the following compatibility relation is satisfied $$\varphi^{i+1}=p \varphi^i$$,

2. $$\sum_i \varphi^i(F^i M)=M$$.

Let $$\text{MF}_{tor}^{f}$$ denote the category of Fontaine-Laffaille modules $$M$$ with morphisms satisfying the conditions alluded to above. For $$a let $$\text{MF}_{tor}^{f,[a,b]}$$ let the full subcategory of $$\text{MF}_{tor}^{f}$$ whose underlying modules $$M$$ satisfy $$F^0 M=M$$ and $$F^p M=0$$.

The Fontaine-Laffaille functor $$\text{U}:\text{MF}_{tor}^{f,[0,p]}\rightarrow \text{Rep}_{W(\mathbb{F})}(\text{G}_{\mathbb{Q}_p})$$ where $$\text{Rep}_{W(\mathbb{F})}(\text{G}_{\mathbb{Q}_p})$$ is the category of continuous $$W(\mathbb{F})[\text{G}_{\mathbb{Q}_p}]$$ modules that are finite-length $$W(\mathbb{F})$$-modules. It is a basic fact that if $$M$$ has the structure of a free $$W(\mathbb{F})/p^m$$-module of rank $$n$$ (in greater detail, $$F^j M$$ are all free $$W(\mathbb{F})/p^m$$-modules and the maps $$\varphi^j$$ and semilinear $$W(\mathbb{F})$$-module maps) then $$\rho_M:=\text{U}(M)$$ is a Galois representation $$\rho_M:\text{G}_{\mathbb{Q}_p}\rightarrow \text{GL}_n(W(\mathbb{F})/p^m)$$.

Question: Let $$G\subset \text{GL}_n$$ be an algebraic subgroup of $$\text{GL}_n$$ defined over $$\mathbb{Q}$$ (I'm mainly interested in the exceptional groups, like for instance $$G_2$$). What condition on the maps $$\varphi^j_M$$ ensures that the image of $$\rho_M:=\text{U}(M)$$ lies in $$G(W(\mathbb{F})/p^m)$$ so that it is a Galois representation $$\rho_M:\text{G}_{\mathbb{Q}_p}\rightarrow G(W(\mathbb{F})/p^m )$$.

Comment: For the classical groups like $$\text{GSp}_{2n}$$ this is done in section 2 of the paper of Clozel, Harris and Taylor "Automorphy of some $$l$$-adic lifts of Automorphic Mod $$l$$ Galois Representations". It involves making use of the alternating form and the functoriality of $$U$$.