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The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.
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Differential structures on unit-root Frobenius modules
The fact that the Frobenius produces a differential equation is classical. Dwork was the first to find it out. As an reference in your exact context you have the paper of Fontaine in the Grothendieck …