On Potentially Semi - Stable Representations of Hodge - Tate Type ( 0 , 1 )

The purpose of this note is to complement part of a theorem from the remarkable paper of Fontaine and Mazur on geometric Galois representations [5]. Fix a prime p, and let K be a finite extension of the p-adic numbers Qp. Fix an algebraic closure K̄ of K and let G be the Galois group of K̄ over K. Fontaine’s theory [3] classifies various types of representations ρ : G→Aut(V) on finite-dimensional Qp-vector spaces V , and we refer to op. cit. for terminology. The theorem (C2. (ii) ⇔ (iii)) in question from [5] says the following: If p ≥ 5 and (V, ρ) is a two-dimensional irreducible Hodge-Tate representation of Hodge-Tate type (0,1), then ρ is potentially semi-stable if and only if it is potentially crystalline. Of course, one should emphasize that the rest of the theorem gives much more detail, namely, a complete list of possibilities, and the theorem to follow is by no means a substitute for the refined statements. However, it might be worth remarking that at least this part admits an entirely simple proof in greater generality. We note also that [5] C2. (i) ⇔(ii), the equivalence between crystalline representations of Hodge-Tate type (0, 1) and BarsottiTate representations, has been proved for p 6= 2 and arbitrary dimension by [2], [6] in the small ramification case and [1] in general. We are grateful to the referee for suggesting improvements.