Mod P Galois Representations# Introduction Motivation Definitions Classification of \(\varphi\)-modules Connection with Galois Representations \(\varphi\)-modules and Galois Representations in Magma \(\varphi\)-modules Category Creation Functions PhiModule(M): AlgMatElt → PhiMod ElementaryPhiModule(S,d,h): RngSerLaur, RngIntElt, RngIntElt → PhiMod PhiModuleElement(x,D): AlgMatElt, PhiMod → PhiModElt Attributes of \(\varphi\)-modules Dimension(D): PhiMod → RngIntElt CoefficientRing(D): PhiMod → RngSerLaur FrobeniusMatrix(D): PhiMod → AlgMatElt Basic Operations and Properties of \(\varphi\)-modules IsEtale(D): PhiMod → BoolElt ChangePrecision(~D, prec): ~PhiMod, RngIntElt DirectSum(D1, D2): PhiMod, PhiMod → PhiMod BaseChange(~D, P): ~PhiMod, AlgMatElt RandomBaseChange(~D): PhiMod Phi(D, x): PhiMod, PhiModElt → PhiModElt Reduction of \(\varphi\)-modules and Galois Representations SemisimpleDecomposition(D): PhiMod → AlgMatElt, AlgMatElt, SeqEnum, SeqEnum Slopes(D): PhiMod → SeqEnum SSGaloisRepresentation(D): PhiMod → SSGalRep Semisimple Galois Representations Category Creation Functions SSGaloisRepresentation(E,K,w,P): FldFin, RngSerLaur, SeqEnum, SeqEnum → SSGalRep Basic Operations CoefficientRing(V): SSGalRep → FldFin FixedField(V): SSGalRep → RngSerLaur Weights(V): SSGalRep → SeqEnum Representation Associated to a \(\varphi\)-Module SSGaloisRepresentation(D): PhiMod → SSGalRep Examples Example: Slopes of a Direct Sum of Phi-Modules