$\Phi$-modules and coefficient spaces

We define and study certain moduli stacks of modules endowed with a Frobenius semi-linear endomorphism. These stacks can be thought of as parametrizing the coefficients of a variable Galois representation and are global variants of the spaces of Kisin–Breuil $\Phi$-modules used by Kisin in his study of deformation spaces of local Galois representations. A version of a rigid analytic period map is defined for these spaces, and it is shown how their local structure can be described in terms of “local models”. We also show how Bruhat–Tits buildings can be used to study their special fibers.