Documenta Math. 943 p-Jets of Finite Algebras, I: p-Divisible Groups

p-jets of finite flat maps of schemes are generally neither finite nor flat. However, for p-isogenies, and in particular for p-divisible groups, this pathology tends to disappear “in the limit”. We illustrate this in the case of Gm, elliptic curves, and formal groups of finite height. 2010 Mathematics Subject Classification: 14 L 05, 13 K 05


Introduction
This paper is the first in a series of papers where we investigate p-jet spaces (in the sense of [3]) of finite flat schemes/algebras.The understanding of such p-jet spaces seems to hold the key to a number of central questions about arithmetic differential equations [4].The present paper deals with p-isogenies, and in particular with p-divisible groups; a sequel to this paper [8] will deal with algebras of Witt vectors of finite length.Let p be an odd prime and let R = Z ur p be the p-adic completion of the maximum unramified extension of the ring Z p of p-adic numbers.(Throughout the paper the symbol means p-adic completion.)Let k = R/pR be the residue field of R. Then for each integer n ≥ 0 a functor J n was introduced in [3] that attaches to any scheme of finite type X/R a (Noetherian) p-adic formal scheme J n (X) over R called the p-jet space of X of order n.For each X there are morphisms J n (X) → J n−1 (X), n ≥ 1, functorial in X, and J 0 (X) = X.We refer to [3,4] for an exposition of the theory and for some of the applications of these spaces; see also [6,2] for a several prime version of the theory.The functors J n behave nicely on smooth schemes and étale morphisms: in particular if X is smooth over R then J n (X) are p-adic completions of smooth schemes over R; and if X → Y is an étale morphism then J n (X) ≃ J n (Y )× Y X.So, in particular, if X → Y is finite and étale then the map J n (X) → J n (Y ) is again finite and étale.However if X → Y is, say, finite and flat then J n (X) → J n (Y ) is generally neither finite nor flat.This basic pathology can be seen, in its simplest form, for X a finite flat group scheme of degree a p-power over Y = Spec R; or for p-isogenies X → Y (i.e.isogenies of degree a p-power) between smooth group schemes over R. The present paper offers an analysis of p-jets of p-isogenies [p ν ] : X → X and of p-divisible groups (X ν ; ν ≥ 1), X ν := Ker [p ν ], where X is either the multiplicative group, or an elliptic curve, or a (one dimensional) formal group of finite height.(For the latter case one still has at one's disposal a p-jet space theory.)The main moral of the story will be that although the p-jets of order n of the individual X ν 's are generally highly pathological order tends to be restored "in the limit", when either n → ∞ or ν → ∞.One of our main motivations for trying to understand the pjets of p-isogenies is their apparent link with the problem of understanding the U -operator (and the Hecke operator T (p)) on differential modular forms.Discussing this link here would lead us too far afield; the interested reader can see a hint of this in [5,7].In order to state some of our main results let us recall/introduce some basic notation.For any scheme of finite type X/R the rings of global functions O n (X) := O(J n (X)) form an inductive system; its direct limit is denoted by O ∞ (X).There are canonical (non-linear) operators δ : O n (X) → O n+1 (X) that can be viewed as arithmetic analogues of the total derivative operator in differential geometry/mechanics.Here is a basic example that we are going to be interested in.Let x, x ′ , x ′′ , ... be variables and consider the rings A n := R[x, x ′ , ..., x (n) ] ⊂ A := R{x} := R[x, x ′ , x ′′ , ...] whose elements are referred to as δ-polynomials.Let φ : R → R be the unique ring automorphism that lifts the p-power Frobenius on k and let φ : A → A be the unique ring homomorphism which is the φ above on R and sends x, x ′ , x ′′ , ... into x p + px ′ , (x ′ ) p + px ′′ , (x ′′ ) p + px ′′′ , ... respectively.Then one defines the following map of sets (the Fermat quotient operator): This map induces maps δ : A n → A n+1 , and by continuity, maps δ : (A n )ˆ→ (A n+1 )ˆ(where ˆalways denotes in this paper the p-adic completion).Note that if A 1 = Spec R[x] = Spec A 0 is the affine line over R then J n (A 1 ) = Spf (A n )ˆand the arithmetic analogues of the total derivatives δ : O n (A 1 ) → O n+1 (A 1 ) identify with the Fermat quotient operators δ : Documenta Mathematica 18 (2013) 943-969 So it becomes crucial to compute the δ-polynomials δ n (x p ν ).Consider the filtration of A by the subrings: and consider the ideal I = (x ′ , x ′′ , ...) ⊂ A. Also consider the ideals I [p ν ] of A generated by all δ-polynomials of the form p i (x (s) ) p j , with s ≥ 1, i, j ≥ 0, i + j = ν.By abuse of notation, we will often denote by [S] an element of a set S. The starting point of this paper will be the following "leading term computation".Let n, ν ≥ 1.
This computation will have a number of consequences (both in characteristic zero and in characteristic p).Here is a consequence in characteristic zero.Let J n (µ p ν ) 1 be the kernel of the projection J n (µ p ν ) → J 0 (µ p ν ) and write . Then for n ≥ 1 we have: Let us mention some consequences in characteristic p.Before doing so we introduce some notation.For any ring B we denote B = B/(p); and for any b ∈ B we let b ∈ B be the image of B. In particular for any scheme of finite type X/R we set Note that the morphisms O n (X) → O ∞ (X) are generally not injective !(They are injective, however, if X/R is smooth [3].)It turns out that for non-smooth X/R a special role is then played by the rings: According to our notation above we may consider the ring A = k[x, x ′ , x ′′ , ...] and its filtration with subrings n) ].Also we may consider the reduction mod p, I = (x ′ , x ′′ , ...) ⊂ A, of the ideal where, by Theorem 1.1: The "smallest" interesting case is ν = 1, n = 3, Indeed finiteness is clear; injectivity follows by looking at dimensions; and flatness follows from the general fact that finite surjective maps of non-singular (irreducible) varieties are automatically flat (cf., say, [9], Theorem 18.16.).
Corollary 1.3 trivially implies the following determination of O n (µ p ν ).
The statement of the corollary above should be contrasted with the fact that, as we shall see, for all n, ν ≥ 1 the the rings O n (µ p ν ) have positive Krull dimension (actually they are polynomial rings in min{n, ν} variables over some explicit local Artin rings).
Remark 1.6.As we saw the mod p Corollary 1.5 follows trivially from our characteristic zero Theorem 1.1.We will also present an alternative proof of this mod p result using a Witt vector computation; we are indebted to the referee for this alternative proof.We included both approaches because each has its own advantage: the Witt vector computation yields a shorter argument (but apparently working only mod p) whereas the computation in Theorem 1.1 is valid in characteristic zero and has other consequences as well.
The above theory has an analogue for formal groups of height ≥ 2 which we now explain.We consider the rings Consider the filtration of A by the subrings: and consider the ideals I [p ν ] of A generated by all δ-polynomials of the form p i (x (s) ) p j , with s ≥ 0, i, j ≥ 0, i + j = ν.(Note that, unlike in the case of the ideals I [p ν ] , the superscript s here is allowed to be 0! So, for instance .) The lift of Frobenius φ : A → A on A = R{x} and the Fermat quotient operator δ : A → A induce obvious maps φ : A → A and δ : ] be a formal group law (in one variable) of finite height and let F[p ν ] be the kernel of the multiplication by p ν viewed as a finite flat group scheme over R. As we shall see it turns out that , where ] is the series giving the multiplication by p in F and F •ν is its ν-th iterate.So we shall be interested in computing δ n (F •ν ).More generally start with any series ], F (0) = 0; any F of the form [p] F (x) (F a formal group) has this shape.Then one easily sees that where ) boils down to computing the quantities δ m (p i G j (x p j )) for i + j = ν and m ≤ n.Here is our main characteristic zero "leading term computation" of such quantities.
Here for a an integer we set a + = max{a, 0}.The above (and indeed a much weaker statement) implies in particular that the natural homomorphism is injective; this is in the spirit of Corollary 1.2.A more precise consequence of the above can be obtained by combining Theorems 1.1 and 1.7 to give a "leading term computation" for δ m (F •ν ) in characterististic zero: all one has to do is to replace the expression δ m−i (x p j ) in the formula of Theorem 1.7 by its value given in Theorem 1.1.Rather than stating this consequence in characteristic zero we look at some of its effects in characteristic p.
According to our conventions we may consider the ring ..] and its filtration with subrings n) ].Also we may consider the reduction mod p of the ideal I, I = (x, x ′ , x ′′ , ...) ⊂ A. with F (0) = 0 and to write its ν-th iterate as in Equation 1.3.Note that ] so we may consider the series Then our Theorem 1.7 will imply the following.Let ν ≥ 1 and n ≥ 0.
Corollary 1.8.The element δ n (F •ν ) ∈ A n is given by Remark 1.9.The case ν = 1 of Corollary 1.8 above can be interpreted as follows.Let us consider A 1 = Spec R[x], the affine line over R, and its reduction mod p, , is the most general Rmorphism lifting the relative (k-linear) Frobenius A 1 → A 1 and sending 0 into 0; Corollary 1.8 provides then, in particular, a description of the reduction mod p of the induced map (which sends x, x ′ , ..., x (n) into F, δF, ..., δ n F ).Note that the map J n (Φ) is generally neither finite nor flat and its behavior depends in an essential way on the series f (x).
Another immediate consequence of Corollary 1.8 is the following structure theorem for O n (F[p ν ]).We actually prove a slightly more general result covering cases that do not come from formal groups.Let n, ν ≥ 1.
Again Corollary 1.10 should be contrasted to the fact that O n (X ν ) have positive Krull dimension (they are, again, polynomial rings in min{n, ν} variables over some explicit local Artin rings).
Remark 1.11.We already mentioned that Corollary 1.5 can be proved independently of Theorem 1.1 via a Witt vector computation argument.It would Documenta Mathematica 18 (2013) 943-969 be very interesting to find a similar Witt vector argument for Corollary 1.10 which is independent of Theorem 1.7, at least in the case when F (x) is of the form [p] F (x) for some formal group F.
Remark 1.12.Results similar to Corollaries 1.5 and 1.10 above are obtained in the body of the paper for the p-divisible groups of elliptic curves.The case of ordinary elliptic curves is deduced from (a twisted version of) the results for G m while the case of supersingular elliptic curves is deduced from the results on formal groups.In the ordinary case the shape of the results depends on the value of the Serre-Tate parameter.
Remark 1.13.It would be interesting to have a generalization of our computations (in characteristic zero or at least in characteristic p) to the case of arbitrary p-divisible groups.
Remark 1.14.It is interesting to note the following phenomenon.Let X 0 , ..., X ν be closed subschemes of the affine line A 1 over R which are, say, finite and flat over R, and let (scheme theoretic union inside A 1 , defined by the intersection of the defining ideals).Then, in general, as closed subschemes of J n (A 1 ).An example is provided by the case when where ζ p i is a p i -th root of unity.In this case J n (X 0 ) = Spec R and, as we shall see later in the paper, J n (X i ) = ∅ for i ≥ 1 and n ≥ 1; on the other hand X = µ p ν (kernel of multiplication by p ν on G m over R) and hence J n (µ p ν ) has a non-reduced reduction mod p by Theorem 1.10.It would be interesting to understand this phenomenon more generally when, for instance, X i = Spec R[α i ] with α i integers in a finite ramified extension of the fraction field of R.
Remark 1.15.In a sequel to this paper [8] we shall investigate the p-jet spaces of another remarkable example of finite flat schemes over R namely schemes of the form Spec W m (R) where W m (R) are the rings of Witt vectors of finite length on R.
A few words about the structure of the paper.We begin by recalling from [3,4] some of the basic concepts we shall be dealing with.Then we will study the filtrations A {n} and I [p ν ] in a general setting and we will prove Theorem 1.1.Then, in subsequent sections, we will investigate the p-jets of the divisible groups of G m , ordinary elliptic curves, formal groups of finite height, and supersingular elliptic curves respectively.The G m case will be used as a step in the analysis of all the other cases.
Acknowledgment.The author would like to thank the referee for communicating to him the Witt vector computation argument yielding an alternative proof of Corollary 1.5.The material in this paper is based upon work supported by the National Science Foundation under Grant No. 0852591 and by the IHES, France and MPI, Bonn.Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation, IHES, or MPI.
2. Review of some basic concepts [3,4] Rings in this paper will always be assumed commutative with unity.A pderivation δ : A → A on a ring A is a set theoretic map satisfying where C p is the polynomial: If δ is as above then φ : A → A, φ(x) = x p + pδx, is a ring homomorphism.Note that δ(xy) = x p δy + φ(y)δx = y p δx + φ(x)δy.Also δ and φ commute.If A is p-torsion free then δ is, of course, uniquely determined by φ; also where Now the ring R = Z ur p has a unique p-derivation defined by δx = (φ(x) − x p )/p where φ : R → R is the unique ring automorphism lifting the p-power Frobenius on R/pR.Let x be a variable (or more generally an N -tuple of variables x 1 , ..., x N .)We consider the δ-polynomial ring R{x} = R[x, x ′ , x ′′ , ...]; this is the polynomial ring in variables x, x ′ , x ′′ , ..., x (n) , ..., where x ′ , x ′′ , ... are variables (or N -tuples of variables), equipped with the unique p-derivation δ : R{x} → R{x} such that δx = x ′ , δx ′ = x ′′ , etc.For X a scheme of finite type over R one defines the p-jet spaces J n (X), n ≥ 0 [3].The latter are p-adic formal schemes over R fitting into a projective system Note that X → J n (X) are functors commuting with open immersions and more generally with étale maps in the sense that if for a tuple of variables x and a tuple of polynomials f and then J n (Z).The rings O n (X) := O(J n (X)) form an inductive system, the pderivation δ on R{x} induces operators δ : O n (X) → O n+1 (X), and the direct limit of these δ's induces a p-derivation δ on the direct limit O ∞ (X) of the rings O n (X).The following universality property holds.Assume for simplicity X is affine.Then any R-algebra homomorphism of O(X) into a p-adically complete ring B equipped with a p-derivation δ is induced by a unique R-algebra homomorphism O ∞ (X) → B that commutes with δ.

Filtrations
In this section we will introduce and study some basic filtrations, especially on rings equipped with p-derivations.In the next section we will specialize to the case when the ring in question is the ring R{x} of δ-polynomials.Let A be a ring which for simplicity we assume p-torsion free and let I be an ideal in A. For any integer ν ≥ 0 we denote by I [p ν ] the ideal generated by all the elements of the form p i f p j with f ∈ I, i, j ≥ 0, i + j = ν.(N.B.Sometimes the superscript [ ] is used to denote divided powers of ideals; our use of this superscript here has nothing to do with divided powers but rather it generalizes the notation used for Frobenius powers of ideals in characteristic p.)In particular I [1] = I.Also note that I [p ν ] ⊂ I(pA + I) ν , where J ν is, as usual, the ν-th power of an ideal J.
4) If I is generated by a family {f s ; s ∈ S} then I [p ν ] is generated by the family {p i f p j s ; s ∈ S, i, j ≥ 0, i + j = ν}.
Proof.Assertions 1 and 2 are clear.For assertion 3 if f = N s=1 p is f p js s g s with f s ∈ I, g s ∈ A, and i s + j s = ν for all s then (3.1) To check assertion 4 it is sufficient to prove that if g ∈ I then g p t is in the ideal generated by the family {p i f p j s ; s ∈ S, i, j ≥ 0, i + j = t}.One proves this by induction on t ≥ 0. The case t = 0 is clear.Now if the statement is true for t = ν and we set f = g p ν then we are done by equation (3.1).
In what follows we assume we are given a p-derivation δ : Proof.First, for any f ∈ A, we have the following computation: where P is a polynomial with Z-coefficients; indeed this is because for 2 ≤ m ≤ p ν (and since p is odd) we have In particular if Let's prove assertion 1.In view of the equation (2.1) it is enough to note that for i by Lemma 3.1 and because δ(p i ) is either 0 or in p i−1 A according as i = 0 or i ≥ 1.
To prove assertion 2 note that if In what follows we assume we are given, in addition, a filtration on by subrings A n such that δA n ⊂ A n+1 for all n ≥ 0. Then we define a new filtration by subrings Proof.A trivial exercise.
We will also need the following general: Lemma 3.4.For any f, g ∈ A 0 we have the following equality of ideals in A n : Documenta Mathematica 18 (2013) 943-969 Proof.Induction on n.The induction step follows from the congruence In this section we specialize the discussion of the previous section to the case when (4.1) So in this case, explicitly, is the ideal of R{x} generated by all δ-polynomials of the form p i (x (s) ) p j , with s ≥ 1, i, j ≥ 0, i + j = ν; cf.assertion 4 in Lemma 3.1.
We start by proving Theorem 1.1 in the Introduction.
Proof of Theorem 1.1.First note that for n ≤ ν + 1 we have by Lemma 3.2 and hence (4.2) by Lemma 3.1.Similarly, for n ≥ ν + 2, we have by Lemma 3.2.We also claim that (4.4) To check (4.4) it is enough, by Lemma 3.2, to check that δ(x p ν ) ∈ I [p ν ] ; this however follows from equation (3.2).In view of (4.2), (4.3), (4.4), in order to prove our theorem it is enough to prove that We fix ν and proceed by induction on n ≥ 1.For n = 1 we are done by (3.2).
Next assume the theorem is true for n and we prove it for n + 1. Assume first n ≤ ν + 1.By Lemma 3.3, φ n−1 (x ′ ) ∈ A {1} .So we have Clearly the last term in the last equation is in p ν−n+1 A {1} .Also, by Lemma 3.3: So for n ≤ ν we get which ends the induction step in case n ≤ ν.
For n = ν + 1 we get which ends the induction step in case n = ν + 1. Assume now n ≥ ν + 2; then, by Lemma 3.3, This ends the induction step in case n ≥ ν + 2.
The following will also be useful later.
. Proof.Trivial induction on n.

p-jets of µ p ν
Start again with the multiplicative group G m and the isogeny [ be the kernel of [p ν ] Gm .More generally, (for the purpose of looking later at p-divisible groups of elliptic curves) we consider, for any a ∈ R × , the finite flat scheme µ a p ν := Spec R[x]/(x p ν − a).Its functor of points is given by µ a p ν (S) = {s ∈ S; s p ν = a} for any R-algebra S. Then µ a p ν has a natural structure of µ p ν -torsor.More generally, for any a, b ∈ R × we have a natural morphism given on S-points by (s, t) → st.We also have a natural isomorphism given on points by s → s/a.

Consider the group U
so division by b gives an isomorphism µ a p ν ≃ µ p ν .Finally the system (µ p ν ; ν ≥ 1) is a p-divisible group with embeddings µ p ν ⊂ µ p ν+1 given by the inclusions on points.More generally for any a ∈ R × and any ν 0 ≥ 1, the schemes (µ a p ν−ν 0 p ν ; ν ≥ ν 0 ) form an inductive system with embeddings (5.1) given by the inclusions on points.Note that if a ∈ U ν0+1 then we can write a = b p ν 0 and hence division by b gives an isomorphism between the inductive system (5.1) and the inductive system (5.2) ) is given at the level of rings by x → x p ν , x ′ → δ(x p ν ), etc.By the commutation of J n with fiber products it follows that More generally, if a ∈ R × , and if we still denote by a : Spec R → G m the point defined by x → a then It follows that: .
In particular .
Remark 5.2.Let J n (µ p ν ) 1 be the kernel of the projection J n (µ p ν ) → J 0 (µ p ν ) = µ p ν and write O n (µ p ν ) 1 := O(J n (µ p ν ) 1 ).Since µ p ν has a lift of Frobenius x → x p in the category of group schemes we get an isomorphism compatible with the group laws.Equivalently we have as groups in the category of formal p-adic schemes over R.
Since, by Theorem 1.1 the denominator in the last equation is in (p ν−n+1 ) we are done by the following well known fact (whose proof we recall).Proof.The map from A to the projective limit is clearly injective.It is surjective because if (by [13], Theorems 8.2 and 8.14 According to our general notation (Equations 1.1 and 1.2) we next recall the rings Proof.To prove assertion 1 let a ∈ U m \U m+1 , 1 ≤ m ≤ ν.Then a simple induction shows that δ m a ∈ R × hence, by Corollary 1.3, the reduction mod p δ m (x p ν ) − δ m a is in k × .By Proposition 5.1 O N (µ a p ν ) = 0 for all N ≥ m and hence O n (µ a p ν ) = 0 for all n ≥ 1.To prove assertion 2 note that the equalities (5.4) and (5.5) follow from Corollary 1.3 and Proposition 5.1; in particular we have lim .

Now (5.3) follows from the fact that the intersection
in the ring k[x, x ′ , x ′′ , ...] equals the ideal By the above Proposition we get: We also remark the following: ) is a finite k-algebra which is not the case, cf.equations (5.4) and (5.5) hence in O n (µ p ν ).Hence x ′ is in the denominator of the ring in equation ( 5.3), a contradiction.
Remark 5.8.The author is indebted to A. Saha for pointing out assertion 2 in Proposition 5.7 above.
Remark 5.9.We end this section by providing an alternative argument for Corollary 1.5, hence of Proposition 5.5 in case a = 1; the author is indebted to the referee for this argument.We start by recalling that for finitely generated R-algebras B and k-algebras C there are isomorphisms functorial in both B and C and compatible with varying n. (Here W n (C) = (R n+1 , +, ×) is the ring of p-typical Witt vectors of length n + 1.)This follows from the theory in [1]; cf.Section 3.4 of that paper.(The indexing of rings of Witt vectors is also taken from [1] and is not the classical one: the above W n are usually denoted by W n+1 .)Using the isomorphism above Corollary 1.5 is easily seen to be equivalent to the following statement.
Proposition 5.10.For any k-algebra C and any Witt vector x = (x 0 , ..., x n ) ∈ W n (C) we have Proof.By multiplying x with the inverse of the Teichmüller lift (x 0 , 0, ..., 0) of x 0 one may reduce to the case when x 0 = 1.Assume this from now on.Then by induction it is enough to show that for all 0 ≤ i ≤ n − ν − 1 and under the assumption x p ν 1 = ... = x p ν i = 0 we have To show the latter we may further assume that i = n − ν − 1.Now write x = 1 + V (z) where V is the Verschiebung and z = (x 1 , ..., x n ).Then we have (5.7) x Since C has characteristic p, if F is the Frobenius, we have F V = V F = p hence (5.8) . By Equations 5.7 and 5.8 we are left to prove that for all j ≥ 2, (5.9) Let r = ord p p j .Then Equation 5.9 is equivalent (due to the general It is therefore enough to show that for j ≥ 2, which is true (for p ≥ 3) for elementary reasons.

p-jets of the irreducible components of µ p ν
Next note that µ p ν is connected and has ν + 1 irreducible components: where The scheme theoretic intersection of these components is In deep contrast with Proposition 5.5 the p-jets of these components are completely uninteresting: Proof.The first equality is clear.The second follows from the fact that Φ p i (x+ 1) is an Eisenstein polynomial plus the following general fact: Proposition 6.2.Let f (x) = x e + a 1 x e−1 + ... + a e−1 x + a e ∈ R[x] be an Eisentein polynomial (i.e. e ≥ 2, a 1 , ..., a e ∈ pR, a e ∈ p 2 R) and let Proof.It is enough to show that δf (x) is invertible in O 1 (X) ⊗ k.Now we have: δf (x) = δ(x e ) + δ(a 1 x e−1 ) + ... + δ(a e−1 x) + δa e + C p (x e , a 1 x e−1 , ..., a e ) ≡ ex p(e−1) x ′ + x p(e−1) (δa 1 ) + ... + x p (δa e−1 ) + δa e mod p.
Since the image of x in O 1 (X) ⊗ k is nilpotent and the image of δa e in the same ring is invertible it follows that the image of δf (x) in this ring is invertible which ends the proof.

p-jets of E[p ν ] for ordinary elliptic curves
We start with a review of extensions of p −ν Z/Z by µ p ν .For any group (respectively group scheme) G we denote by G[N ] the kernel of the multiplication by N map.For a finite group Γ we continue to denote by Γ the étale group scheme over R attached to Γ; so for any connected R-algebra S, Γ(S) = Γ.In particular we have the connected R-group scheme We also view, when appropriate, µ p ν and p −ν Z/Z as R m schemes via base change.Then, by Kummer theory, (7.1) We will need to recall the following explicit description of the above isomorphism.Let q ∈ 1 + pR.Consider the finite flat R-scheme This is a group scheme with multiplication given by , where 0 ≤ l < p ν , i + j ≡ l mod p ν .The functor of points of Γ q p ν is given by Γ q p ν (S) = {(s, i); s ∈ S × , 0 ≤ i < p ν , s p ν = q i } for any R-algebra S with connected spectrum; the multiplication on points is given by (s, i) • (t, j) = (st, i + j) if i + j < p ν and (s, i) • (t, j) = (st/q, i + j − p ν ) if i + j ≥ p ν .We have an extension (7.2) 0 → µ p ν → Γ q p ν → p −ν Z/Z → 0 (the second map being given on points by (s, i) → i p ν + Z).Then Kummer theory gives: Lemma 7.1.The isomorphism ( 7.1) is given by attaching to the class of q ∈ 1 + pR in (1 + pR m )/(1 + p ν+1 R m ) the class of the extension ( 7.2) in Ext 1 Rm (p −ν Z/Z, µ p ν ).Note that the system (Γ q p ν ; ν ≥ 1) is a p-divisible group via the morphisms Γ q p ν → Γ q p ν+1 given on points by (s, i) → (s, pi) and given on schemes by the inclusions µ q i p ν ⊂ µ q pi p ν+1 .The p-divisible group (Γ q p ν ; ν ≥ 1) is an extension of the p-divisible group (p −ν Z/Z; ν ≥ 1) by the p-divisible group (µ p ν ; ν ≥ 1), where the latter are viewed as p-divisible groups with respect to the natural inclusions.
Next we consider an elliptic curve E/R.References for this are [14,10].Let E/k be its reduction mod p and let E f or be the formal group attached to E. (We use the superscript for rather than because the latter is used in the present paper to denote p-adic completion).Let E f or [p ν ] be the kernel of the multiplication by p ν on E f or , viewed as a finite flat group scheme over R.
Assume in what follows that E is ordinary.Then (7.3) E f or ≃ G f or m ; we fix such an isomorphism.So we have induced isomorphisms E f or [p ν ] ≃ µ p ν .Moreover we fix isomorphisms (7.4) With the isomorphisms (7.3) and (7.4) fixed one defines the Serre-Tate parameter q = q(E) ∈ 1 + pR of E as follows.The isomorphisms (7.4) define a basis (α ν ) of the Tate module lifts α ν then one defines the Serre-Tate parameter q(E) ∈ 1 + pR as the image of lim p ν A ν ∈ E f or (R) via the isomorphism E f or (R) ≃ 1 + pR induced by (7.3); cf.[10], section 2. On the other hand with the isomorphisms (7.3) and (7.4) fixed there are induced exact sequences of finite flat group schemes over R: Cf., say, [10].Also by loc.cit.we have Lemma 7.2.The class of the extension ( 7.5) is the image of the Serre-Tate parameter q(E) ∈ 1 + pR under the isomorphism ( 7.1).
We conclude by Lemmas 7.1 and 7.2 that if q = q(E) then E[p ν ] and Γ q p ν are isomorphic as extensions over R m for any m; the isomorphisms are compatible as m varies so we get the following: Corollary 7.3.E[p ν ] and Γ q p ν are isomorphic as extensions over R for q = q(E).In particular if 0 ≤ i < p ν and θ we fix ν 0 and an index 0 ≤ i 0 < p ν0 and if θ = i0 p ν 0 + Z ∈ p −ν0 Z/Z then the inductive system identifies with the inductive system Putting together Proposition 5.5 and Corollary 7.3 (and making the change of variables x → x + 1) we get: Proposition 7.4.Let E/R be an elliptic curve with ordinary reduction and Serre-Tate parameter q = q(E) ∈ U 1 .Let n ≥ 1, let θ ∈ m≥1 p −m Z/Z, and let ν 0 ≥ 1 be minimal with the property that θ ∈ p −ν0 Z/Z.Let ν ≥ ν 0 and let E[p ν ] θ be the connected component of E[p ν ] lying over θ.Then: Also Proposition 5.7 and Corollary 7.3 imply Corollary 7.6.Let n, ν ≥ 1.Then the map An analogue of Remark 1.4 should hold nevertheless in the elliptic case as well.

p-jets of F[p ν ]
Recall that a series F (x) ∈ xR[[x]] without constant term is said to have finite height if F ≡ 0 mod p; if this is the case the height of F is defined as the largest integer Remark 8.1.The main example we have in mind here arises as follows.Consider a formal group law F ∈ R[[x 1 , x 2 ]].By [14] the multiplication by p in F is given by a series F (x) := [p] F (x) satisfying F (x) ≡ px mod x 2 .The height of F is defined to be the height of F (x); if the height is finite then it is ≥ 1.Not every series of height ≥ 1 which is ≡ px mod x 2 is the multiplication by p of a formal group law F; indeed if F (x) = [p] F has height h then one knows that the x-adic valuation of the reduction mod p of F in k[[x]] is exactly p h ; cf.[14], p.127.
] be a series of finite height h ≥ 1 and let F •ν = F • ... • F be the ν fold composition of F with itself for ν ≥ 1.Let ep h be the x-adic valuation of the reduction mod p of F .(So if F (x) = [p] F (x) for some formal group law F then e = 1.)By "Weierstrass preparation" (cf.[11], p. 130) is monic of degree e ν p hν , P ν ≡ x e ν p hν mod p.Consider the scheme: the latter isomorphism follows from "Euclid division" by P ν in R[[x]]; cf.[11], p. 129.So X ν is a finite flat scheme over R of degree e ν p hν and we have a natural sequence of closed immersions (8.1) Our aim in this section is to understand the rings O n (X ν ).Note that if F (x) = [p] F (x) is the multiplication by p on some formal group law F then where the latter is the kernel of [p ν ] F on F and indeed the inductive system (8.1)coincides with the p-divisible group of F; cf.[15].
Remark 8.2.Recall (cf.[12], p. 480) that any formal group law F over R of height h = 1 is isomorphic to the multiplicative formal group law hence in particular F[p ν ] ≃ µ p ν .Hence our analysis in Section 5 applies to O n (F[p ν ]) in the height one case.We will consider in what follows the case of formal groups of arbitrary height ≥ 1.More generally we will treat the case of iterates of series of height ≥ 1 which are not necessarily coming from formal groups; so even for height 1 our analysis below will not be covered by Section 5.
To understand the rings O n (X ν ) we will first perform some computations in characteristic zero culminating with a proof of Theorem 1.7.Then we will reduce mod p the outcome of these computations.
Proof.Induction on n.The case n = 1 is clear.Now assume the above is true for some n ≥ 1.If n ≤ i − 1 we have If n ≥ i + 1 we have The case n = i is similar.which settles the case n = 1.For the induction step, assuming the statement true for some n ≥ 1, we have Proof.To check assertion 1 it is enough to check it for m = 1.Now assertion 1 follows from the following computations in which i + j = ν: Σ * (p i (x (s) ) p j ) = p i (δ s (x p )) p j ∈ p i I [p j+1 ] ⊂ I [p i+j+1 ] = I [p ν+1 ] ; in the above we used the fact that since x p ∈ I [p] we have δ s (x p ) ∈ I [p] hence (δ s (x p )) p j ∈ I [p j+1 ] ; cf.Lemma 3.1.To prove assertion 2 it is enough, by the compatibility with composition, to prove these two statements for m = 1 and n ≥ 1 which we now assume.Now by Theorem 1.1 we have δ n (x p ) ∈ A {n−1} for n ≥ 1.Consequently, for n ≥ 1 and F ∈ A n+i we have Σ * (p i F (x, ..., x (n+i) )) = p i F (x p , ..., δ n+i (x p )) ∈ p i A {n+i−1} ⊂ A {n−1} which proves assertion 2.

Lemma 5 . 4 .
If A is a Noetherian ring, I is an ideal, A is I-adically complete, and (L n ) is a descending sequence of ideals such that L n ⊂ I n then A = lim ← A/L n .Documenta Mathematica 18 (2013) 943-969
Then I [p ν ] coincides