## Number Theory – Condition on a Fontaine Laffaille module that specifies the image of the associated Galois representation

The establishment:

Let $$mathbb {F}$$ to be a finite feature body $$p$$ and $$W ( mathbb {F})$$ the Witt vector ring with residue field $$mathbb {F}$$.
Recall that filtered Dieudonné & # 39; $$W ( mathbb {F})$$-module also known as Fontaine-Laffaille module is a $$W ( mathbb {F})$$-module equipped with a separate filtration, decreasing, exhaustive sub-modules $${F ^ i M }$$ and for each integer $$i$$ a $$sigma$$semilinear $$varphi ^ i = varphi_M ^ i: F ^ i M rightarrow M$$. These cards are necessary to satisfy two conditions

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

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

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

The functor of Fontaine-Laffaille $$text {U}: text {MF} _ {tor} ^ {f,[0,p]} rightarrow text {Rep} _ {W ( mathbb {F})} ( text {G} _ { mathbb {Q} _p})$$
or $$text {Rep} _ {W ( mathbb {F})} ( text {G} _ { mathbb {Q} _p})$$ is the category of continuous $$W ( mathbb {F})[text{G}_{mathbb{Q}_p}]$$ finite length modules $$W ( mathbb {F})$$-modules. It is a fundamental fact that if $$M$$ has the structure of a free $$W ( mathbb {F})$$-module of rank $$n$$ (more in detail, $$F ^ j M$$ are all free $$W ( mathbb {F})$$-modules and maps $$varphi ^ j$$ and $$W ( mathbb {F})$$-modules and can be seen as matrices with entries in $$W ( mathbb {F})$$) then $$rho_M: = text {U} (M)$$ is a representation of Galois $$rho_M: text {G} _ { mathbb {Q} _p} rightarrow text {GL} _n (W ( mathbb {F}))$$.

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

Comment: For classical groups like $$text {GSp} _ {2n}$$ I know how to do that. It involves making use of the alternating form and the functoriality of $$U$$.

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