THE CANONICAL SUBGROUP OF E IS SpecR[x]/(x p +

Let p be a prime. In this note we make explicit some results on the canonical subgroup of an elliptic curve E over the ring of integers Rp of Cp implicit in [K-pPMF]. In particular, if ω generates Ω1E/Rp and E has a canonical subgroup CE , knowlege of the Hasse invariant of the reduction of (E,ω) modulo p is equivalent to knowledge of the pair (CE , ω|CE). 1. Group Schemes of order p. Let μ denote the group of (p− 1)-st roots of unity in Zp and A the subring of Qp {r ∈ Zp: ∃n ∈ N, pnr ∈ Z[μ, 1/(p− 1)]}. Suppose R is an A-algebra, e.g., a p-adically complete ring with identity. For a ∈ R, let Ra = R[x]/(xp + ax) and Ba = SpecRa and for ǫ ∈ μp−1(R), [ǫ]a the automorphism of Ba corresponding to x 7→ ǫx. If a 6= 0, the automorphisms α of Ba such that α ◦ [ǫ]a = [ǫ]a ◦ α for ǫ ∈ μ are the [γ]a for γ ∈ μp−1(R). Suppose ∃b ∈ R such that ab = p. Because then, d(xp+ax) = a(1+bxp−1)dx and (1+bxp−1)(1−bxp−1/(1−p)) = 1, Ω1Ba/R ∼= Ba/aBa. Proposition 1.1. Suppose G is a group scheme of order p over R. Then the R-module of invariant differentials ΩG/R on G over R is cyclic and if ω is a generator, there are a, b ∈ R such that ab = p and a unique isomorphism of schemes h:Ba → G such that h ◦ [ǫ]a = [ǫ]G ◦ h, for ǫ ∈ μ, and h∗ω = (1 + bxp−1)−1dx. Moreover, ΩG/R ∼= R/aR, a is determined modulo a2R and b is determined modulo pR. In particular, if R is integrally closed and G is self-dual both a and b are determined modulo pR. Proof. We know from [O-T, pp.13-14] that there are universal constants wi ∈ A, i ≥ 1, such that w1 = 1, wj ∈ Z∗p, j < p, wp = pwp−1 and there are u, v ∈ R such that uv = wp, an isomorphism g:B−u → G over R for which the pullback of the group law on G to B−u is