Crystalline Representations. By a p-adic Galois representation we mean a finite dimensional Qp-vector space V equipped with a continuous Qp-linear action of the Galois group GK. In the following we shall make a number of assertions. These are all well- known when the ring Bmax is replaced by the usual ring of crystalline periods Bcrys (see for instance [11, Section 5]). From these standard results one easily deduces the statements we make here using that there are inclusions ϕ(Bmax) ⊂ Bcrys ⊂ Bmax (for this see [6, III.2]). Construction 2.2.1. If V is a p-adic Galois representation then GK acts on the tensor product V ⊗Qp Bmax via the respective actions on each max component. Since BGK = K0 the GK-invariants Dcrys(V ) = (V ⊗Qp Bmax)GK form a K0-vector space. It can be shown that multiplication Dcrys(V ) ⊗K0 Bmax → V ⊗Qp Bmax is injective which implies Dcrys(V ) is finite-dimensional over K0. The injective frobenius endomorphism ϕ on Bmax induces an injective (and therefore bijective) frobenius semilinear endomorphism ϕ of Dcrys(V ).
Crystalline Representations. Let K be a p-adic local field. In the 1980s Xxxxxxxx introduced the period ring BdR, which is a filtered K-algebra endowed with a GK-action, and its subring Bcris on which one can define a natural Frobenius map (see [Ber04, §II.2,§II.3] for the exact definitions). If V is an n-dimensional Qp-vector space and ρ : GK → GL(V ) a representation. Then we will say ρ is de Rham (resp. crystalline) if (B bQp V ) is a free module over K bQ Qp (resp. K0 bQp Qp) of rank GK p n for B = BdR (resp. B = Bcris), where GK acts diagonally on the tensor product. It turns out that any crystalline representation is also de Rham. ÐÝ Some intuition for this definition can be given as follows. Suppose E is an elliptic curve defined over Qp. For any prime l we can look at its l-adic Xxxx module TlE := limn Erlns(Qp) on which GQp := Gal(Qp/Qp) acts naturally. We may wonder whether it is possible to determine whether E has good reduction at p simply from studying the properties of the repre- sentation ρE : GQp → GL2(Ql) obtained from this action. Indeed, if p ‰ l the criterion of N´xxxx, Ogg and Xxxxxxxxxxx says that E has good reduction at p if and only if this representation is unramified. When l = p, however, the condition of unramifiedness does not work any longer and needs to be replaced by the condition that the representation is crystalline. Therefore, one (vague) way of thinking about crystalline representations is as repre- sentations GQp → GLn(Qp) coming from geometric objects over Qp with good reduction at p. p
