FORMAL COMPLETION OF THE JACOBIANS OF PLANE CURVES AND HIGHER REAL K-THEORIES

V. GORBOUNOV AND M. MAHOWALD

Abstract. In this note we will study the formal completion of the Jacobian of a certain class of curves over p-adic rings. These curves generalize the Legendre family of elliptic curves. As an immediate application, we will describe the representation which is crucial for calculating the initial term of the spectral sequence, converging to the homotopy groups of the higher real K-theories EO_{n}.

1. Introduction

Let G_{n} be a maximal finite subgroup of the Morava stabilizer group 𝕊_{n}. Hopkins and Miller define EO_{n} to be the homotopy fixed point spectrum of the action of G_{n} on 𝔼_{n}, which extends the action of 𝕊_{n} on the ring E_{n}. The later is defined using the deformation theory of formal groups [7]. We call this representation the Lubin-Tate representation. These spectra come equipped with a spectral sequences for calculating its homotopy groups:

H^{∗}(G_{n}, E_{n})Gal ⇒ π_{∗}EO_{n}.

EO_{2} agrees with the usual elliptic cohomology when 6 is inverted [6]. In this paper we will study a formal group which can be used to define EO_{n} for n = p − 1, and describe the Lubin-Tate representation of G_{n}. It was suggested to the authors by Mike Hopkins that a universal lift of a formal group of height n over 𝔽_{p} should come as a summand in the formal completion of the Jacobian of a certain curve with p marked points. There is an action of G_{n} on such a curve, which can be expressed in terms of permutations of the marked points. This leads to a precise description of the Lubin-Tate action of G_{n} on the spectrum 𝔼_{n}. The authors wish to thank Mike Hopkins for useful comments during the preparation of this paper.

2. Overview of the Honda theory

We recall briefly the Honda classification of formal groups from [4]. Let K be a discrete valuation field. Denote by 𝔳, 𝔭 and 𝔳_{𝔭} the ring of integers of K, the maximal ideal of 𝔳, and the completion of 𝔳 with respect to 𝔭 respectively. We assume that the residue class field is of characteristic p > 0. Suppose also that there is an endomorphism σ of K and a power q of p such that a^{σ} = a^{q} modulo 𝔭 for any a ∈ 𝔳.

Let F be an n-dimensional abelian formal group over 𝔳 and f = (f_{1}, …, f_{n}) denote its logarithm. Define an action of the twisted power series ring M_{n}(𝔳_{𝔭})[[T]]^{σ} on the vector space of n-dimensional vectors over the ring K[[x_{1}, …, x_{n}]] by

(u ∗ f)(x) = Σ_{η=0}^{∞}C_{η}f^{σ^{η}}(x^{q^{η}}),

where u = Σ_{η}C_{η}T^{η}. An element u in the twisted power series ring M_{n}(𝔳_{𝔭})[[T]]^{σ} is said to be special for F if u ≅ πI_{n} modulo degree 1 and (u ∗ f)(x) = 0 modulo 𝔭, where π is a uniformizer for 𝔭 and I_{n} is the n × n identity matrix. The following theorem is proved in [4]

Theorem 2.1. Suppose that K is a discrete totally unramified valuation field (in addition to the conditions stated above), then the strict isomorphism classes of formal group laws over 𝔳 correspond bijectively to left associate classes of elements u ∈ M_{n}(𝔳_{𝔭})[[T]]^{σ} of the form u ≅ πI_{n} modulo degree 1.

We intend to study the formal completion of the Jacobian of a certain class of curves. Let us state the connection, established in [3], between the formal completion of the Jacobian of a curve and the space of holomorphic differentials on it. Let Γ be a curve of genus n > 0 over K. Denote by J the Jacobian of Γ, and by Λ the canonical map Γ → Λ. Let ω_{1}, …, ω_{n} be a base of holomorphic differentials on Γ, each defined over K. Choose a local parameter z ∈ K(Γ) at some point P, and denote by ω_{i}(z) the expansion of ω_{i} at P. There are ψ_{i}(z) ∈ K[[z]], 1 ≤ i ≤ n, such that ψ_{i}(0) = 0 and ω_{i}(z) = dψ_{i}(z). Let y = (y_{1}, …, y_{n}) ⊂ K(J) be a system of local parameters at the origin of J and let ν_{1}, …, ν_{n} be the invariant differentials on J such that

ν_{i} ∘ Λ = ω_{i} (1 ≤ i ≤ n).

Let S_{1} be the set of primes at which at least one the above algebrogeometric objects has a bad reduction. Let S_{0} be the set of ramified primes of K.

Proposition 2.2. ([3], Theorem 1) There is a finite set S of primes of K satisfying the following conditions:

• S contains S_{0} ∪ S_{1}.

• If 𝔭 ∉ S and u is a special element of M_{n}(𝔳_{𝔭})[[T]]^{σ} which kills ψ(z) modulo 𝔭, then u also kills the logarithm of F mod 𝔭, so that F is of type u as a formal group over 𝔳_{𝔭}.

Remark 2.3. As it was pointed out in [1] (20.3.11) the Honda classification result applies in the following situation: A is a ring of characteristic zero that is also a ℤ_{(p)}-algebra, K = A⊗ℚ, σ : K → K, such that σ(a) = a^{p} modpA for all a ∈ A. Note that the ring A = ℤ_{p}[[u_{1}, …, u_{n}]] is of that type.

3. Motivating example: points on elliptic curves and Lubin Tate representation

In this section we will use the properties of the Legendre curve to describe the Lubin-Tate representation of G_{2} for p = 3. We can handle the case p = 2 in a similar fashion. The general strategy here will be to work with lifts of elliptic curves to the appropriate moduli space rather than with lifts of formal groups.

We choose to describe the action of G_{2} on the moduli space, E_{2}, of the formal completion of the elliptic curve

y^{2} = x^{3} − x  (3.1)

The fromal completion is a formal group of height two, therefore the group of its automorphisms over a suitable extension of 𝔽_{3} is isomorphic to 𝕊_{2}. We will present a convenient model for the moduli space of lifts of (3.1) to Artinian rings.

The Legendre curve is a plane curve defined by the equation

y^{2} = x(x − 1)(x − λ)

over the ring e = ℤ[\frac{1}{2}, λ]. Denote this curve by C_{2} and rewrite its equation as

y^{2} = x(x − 1)(x − u_{1} + 1).  (3.2)

Complete e with respect to the ideal (3, λ + 1) and denote the completion E_{2}^{0}. The formal completion F_{C_{2}} of (3.2) is a Lubin-Tate lift of the formal completion of (3.1). Indeed, as a simple computation shows, the formal completion of C_{2} is a series in two variables

F (z_{1}, z_{2}) = z_{1} + z_{2} + u_{1}(z_{1}^{2}z_{2} + z_{1}z_{2}^{2}) mod (z_{1}, z_{2})^{4}

The pair (E_{2}^{0}, F_{C_{2}}) is the model of Lubin-Tate lift we will use.

Now we are ready to describe the Lubin-Tate representation of G_{2}. For that we consider G_{2} as the group of automorphisms of the elliptic curve (3.1) over the field 𝔽_{9}. This representation of G_{2} is given by the substitutions

\begin{matrix}s_{1} : & x = x′ & y = y′ \\ s_{2} : & x = −x′ & y = −iy′ \end{matrix}

where i^{4} = 1.

Proposition 3.3. There are six automorphisms of the moduli space E_{2}^{0} which preserve the isomorphism class of the curve C_{2}.

Proof. This is a standard fact in elliptic curve theory, (cf [9] ). In particular, if one adds to the moduli space e a forth root of unity i, then it admits six automorphisms which preserve the isomorphism class of C_{2}. Because this example illustrates our latter work, we will outline the proof. We have three marked points on our curve with x coordinates {0, 1, λ}. Any automorphism g of the moduli space takes the triple {0, 1, λ} to an unordered triple {0, 1, g(λ)}. Let

x′ = w^{2}x + r
y′ = w^{3}y

be an isomorphism between g^{∗}C and C. It, in its turn, takes the triple {0, 1, λ} to an unordered triple {r, w^{2}+r, w^{2}λ+r}. There are six way to match these two unordered triples, which produce the following values for λ:

\{λ, 1 − λ, \frac{1}{λ}, \frac{λ − 1}{λ}, \frac{1}{1 − λ}, \frac{λ}{λ − 1}\}

All we need now is to replace λ with u_{1} − 1 and take into account the g fixes elements of the base ring W (𝔽_{9}). Here are possible values for g(u_{1}):

\{u_{1}, 3 − u_{1}, \frac{u_{1}}{u_{1} − 1}, \frac{2u_{1} − 3}{u_{1} − 1}, \frac{u_{1} − 3}{u_{1} − 2}, \frac{2u_{1} − 3}{u_{1} − 2}\}

Remark 3.4. By definition, g(u) = \frac{u}{w}. The appropriate values of w are easy to calculate. For the substitutions 3.3 they are

\sqrt{−\frac{1}{λ}}, \sqrt{\frac{1}{λ}}

.

Remark 3.5. The fact that we found only six automorphisms of the moduli space E_{2}^{0} means that the center of the group ℤ/3 ⋊ ℤ/4 acts trivially on it. It always happens this way as is proved in [5].

4. The generalized Legendre family

Fix a prime number p. Then a maximal finite subgroup of 𝕊_{n}, G_{p−1}, is isomorphic to ℤ_{p} ⋊ ℤ/(p − 1)^{2} [2]. The important points in the example with the Legendre curve C_{2} are:

i) C_{2} is a 2-cover of the projective line with three marked points on it. The reduction of this curve modulo the ideal (λ + 1) has G_{2} as a subgroup of its group of automorphisms via 3.3. ii) The formal completion of the Jacobian of C_{2} is a universal Lubin-Tate lift of a formal group of height two.

We would like to realize universal Lubin-Tate lifts of formal groups of other heights in a similar “algebraic” fashion. In particular, we would like to obtain them as summands in the formal completion of the Jacobian of curves. The work of Manin [8] suggests considering curves of the form

y^{p^{a}−1} = x^{p} − x,

defined over 𝔽_{p}. He showed in [8] that, at least up to isogeny, the formal completion of the Jacobians of such a curve contains one dimensional summands, whose heights divide a(p − 1). Here we will study the case when a = 1.

Definition 4.1. We will call a plane curve C_{p} over E_{p−1}^{0} ≅ ℤ_{p}[[u_{1}, …, u_{p−2}]] defined by the following equation:

y^{p−1} = x^{p} + u_{1}x^{p−1} + … + u_{p−2}x^{2} + (−1 − Σ_{i=1}^{p−2}u_{i})x (4.2)

the generalized Legendre curve. Denote −1 − Σ_{i=1}^{p−2}u_{i} by b.

If we factor out the ideal I = (p, u_{1}, … u_{p−2}), then the reduction of the curve C_{p} will be a curve C̄_{p} over 𝔽_{p} defined by the equation

y^{p−1} = x^{p} − x

The roots of the right hand side are 0, 1, 2, … p − 1 mod p.

Remark 4.3. Hensel’s lemma implies that all the roots of the polynomial in the right hand side of 4.2 are in E_{p−1}^{0}, so the curve C_{p} may be defined by the following equation:

y^{p−1} = x(x − 1)(x − e_{1}) · . . . · (x − e_{p−2})

where e_{i} = i + 1 mod I.

As we see from the above remark C_{p} is a p − 1 covering of the projective line with p marked points on it. Over the ring 𝔽_{p^{p−1}} the group of automorphisms of C̄_{p} contains G_{p−1}. Indeed, ℤ/p and ℤ/(p−1)^{2} act on the curve C̄_{p} via a substitutions:

x = x′ + 1, y = y′, x = ζ^{p−1}x′, y = ζ^{p}y′,  (4.4)

where ζ^{(p−1)^{2}} = 1.

Proposition 4.5. For all primes p the curves C_{p} and C̄_{p} are nonsingular curves of genus m = (p − 1)(p − 2)/2. The following differentials

ω_{ij} = \frac{x^{i}dx}{y^{j}}, 0 ≤ i ≤ j − 1 ≤ p − 3

form a basis of the space of holomorphic differentials.

Proof. Direct computation shows that the curves C_{p} and C̄_{p} are nonsingular. To calculate the genus note that

div(x) = (p − 1)P_{0} − (p − 1)P_{∞}
div(y) = P_{0} + P_{1} + P_{e_{1}} + … + P_{e_{p−2}} − pP_{∞}

div(dx) = (p − 2)P_{0} + (p − 2)P_{1} + (p − 2)P_{e_{1}} + … + (p − 2)P_{e_{p−2}} − pP_{∞} So the above differentials are indeed holomorphic. A standard argument shows that these form a basis for the space of holomorphic differentials. The Riemann-Roch theorem implies the proposition.

Corollary 4.6. The order of the zero at infinity of ω_{ij} is equal to p(j− 1) − (p − 1)i. If we choose z = \frac{x}{y} as a local parameter at infinity than the expansion of ω_{ij} is a power series in z congruent to j − 1 modulo p − 1.

5. The splitting of the formal completion of the Jacobian

Denote by ψ_{ij}(z) the formal anti-derivative of the formal expansion of ω_{ij}. We will prove that in our case the prime number p belongs to the set S from 2.2. In other words we will show that the formal completion of J(C_{p}) is defined by a special element for a vector ψ = (ψ_{ij}(z)), and we will study the properties of this special element. Recall that m = (p − 1)(p − 2)/2.

Theorem 5.1. For all prime numbers p: i The formal completion F of J(C_{p}) is a formal group law over ℤ_{p}[[u_{1}, …, u_{p−2}]]. ii If a special element u of M_{m}(ℤ_{p}[[u_{1}, …, u_{p−2}])[[T]]]^{σ} kills the vector (ψ_{ij}) mod p, then it also kills the logarithm of F modulo p, so that F is of type u as a formal group over ℤ_{p}[[u_{1}, …, u_{p−2}]].

Proof. i. We can use the general theory of reduction of algebraic varieties [10]. In our case the statement follows from the fact that the curve C_{p} is non-singular and its reduction is non-singular. In the terminology of [10] it means that the objects we deal with are I-simple. Note also that in our case F is killed by a special element according to 2.3. ii. The proof repeats word for word the proof of Theorem 1 from [3], if we show that the following version of Lemma 1 from [3] holds in our case.

Lemma 5.2. Let ψ = (ψ_{ij}) be as above. Then, if u and v are special elements of M_{m}(ℤ_{p}[[u_{1}, …, u_{p−2}]])[[T]]^{σ} such that u ∗ ψ = v ∗ ψ = 0 mod p, there is a unit of M_{m}(ℤ_{p}[[u_{1}, …, u_{p−2}]])[[T]]^{σ} such that v = tu.

Proof.

u = pI_{m} + Σ_{k=1}^{∞}C_{k}T^{k}, v = pI_{m} + Σ_{k=1}^{∞}D_{k}T^{k}

C_{k}, D_{k} ∈ M_{k}(ℤ_{p}[[u_{1}, …, u_{p−2}]]). We will show that if C_{k} = D_{k}, 0 ≤ k ≤ l − 1 with some l ≥ 1, then C_{l} = D_{l} modulo p. We have

Σ_{k=l}^{∞}(C_{k} − D_{k})T^{k} ≡ 0 mod p

It follows from 4.6 by direct computation that all the entries of C_{l} − D_{l} are divisible by p. Put now

t_{l} = I_{m} − \frac{1}{p}(C_{l} − D_{l})T^{l}

t_{l} is a unit in M_{m}(ℤ_{p}[[u_{1}, …, u_{p−2}]])[[T]]^{σ} and t_{l}u = v modulo degree l + 1. In this way we can find units t_{1}, …, t_{l} successively for each l > 0 so that t_{l} … t_{1}u = v modulo degree l + 1 and t_{l} … t_{1} converges as l approaches infinity. The limit of t_{l} … t_{1} clearly satisfies the requirement of our lemma.

Theorem 5.3. For all prime numbers p, F splits into p−2 summands of dimensions 1, 2, … p − 2 respectively. The differential y^{−1}dx corresponds to a summand of F of dimension one. Denote this summand by F_{C_{p}}. The height of the reduction of F_{C_{p}} modulo the ideal I is p − 1.

Proof. Let u be a special element for the vector ψ = (ψ_{ij}(z)). To prove the splitting we will show that there is a special element for (ψ_{i,j}(z)), which is made out of blocks of dimensions 1, 2, …, p − 2. Let u = pI_{m} + Σ_{k=1}^{∞}C_{k}T^{k} be a special element which kills ψ modulo p. The coordinates of the vector ψ are naturally split into p − 2 groups, according to the order of the zero at infinity of ω_{ij} modulo p−1. Recall that the order of zero at infinity of ω_{ij} is equal to j − 1. Denote these subsets Ω_{k}, k = 1, …, p − 2, then ψ_{ij} ∈ Ω_{j}, and is a formal series in powers of z congruent to j modulo p − 1. We can assume that the coordinates of ψ = (ψ_{ij}) are arranged in such a way that j is increasing. Using the definition of the operation ∗ and the fact that p = 1 modulo p−1, we see that each component r of the vector u∗ψ is a sum of p−2 power series r_{j} such that r_{j} is a power series of powers of z congruent to j mod p − 1. Since u ∗ ψ = 0 mod p we get that each r_{j} = 0 mod p. The above shows that the element u′ = pI_{m} +Σ_{k=1}^{∞}C′_{k}T^{k} where each C′_{k} is a block submatrix of C_{k} with the blocks of dimensions 1, …, p − 2 along the the diagonal, kills vector (ψ_{ij}(z)) modulo p. This together with 2.2 proves the splitting.

¿From the Honda theory [4] of special elements it follows that the formal expansion of the differential y^{−1}dx is the invariant differential of the one dimensional summand in the above splitting. Denote by F_{C_{p}}^{0} the reduction of F_{C_{p}} modulo the ideal I_{0} = (u_{1}, …, u_{p−2}). So now F_{C_{p}}^{0} is a formal group over Z_{p}. Its differential is given by the formula

ω = \frac{(1 − p)dz}{1 + (p − 1)zw^{p−2}}  (5.4)

where the series w(z) obeys the following functional equation:

w = z^{p} − zw^{p−1}

This shows that w is a series in powers of z congruent to p modulo (p − 1)^{2} and, therefore, ω is a series in powers of z congruent to zero modulo (p − 1)^{2}. This implies that the extension of degree p − 1 of Z_{p}, obtained by adding the (p − 1)^{2} root of unity acts as endomorphisms of F_{C_{p}}^{0}. Hence, according to [11], p − 1 divides the height of F_{C_{p}}^{0}. Then, according to [8], page 71 (4.7), the reduction of F_{C_{p}}^{0} modulo p must contain an indecomposable summand of dimension k(p − 1) − 1, for some natural number k. But as we saw above the maximal dimension of an indecomposable summand of F can be p − 2, therefore k = 1, and the height of F_{C_{p}}^{0} is equal to p − 1.

6. Lubin-Tate representation of G_{p−1}

Denote by E_{p−1} the ring W (𝔽_{p^{p−1}}) ⊗_{ℤ_{p}} E_{p−1}^{0}[u^{−1}, u]. In order to obtain formulas for the Lubin-Tate representation of G_{p−1} we will use the curve C_{p} in the same way we used C_{2} in section three.

Set e_{0} = 1 and e_{p−1} = 0 and denote by [m] the reduction of the integer m mod p. With this notations we have e_{i} ≡ [i + 1] mod I, 0 ≤ i ≤ p−1. G_{p−1} is generated by two elements of order p and (p−1)^{2} respectively. Let 1 < k < p be a generator of 𝔽_{p}^{×}.

A general form of a transformation C_{p}, which reduces to an automorphism of order p of C̄_{p} is:

x′ = v_{i}^{p−1}x + r_{i}
y′ = v_{i}^{p}y

where r_{i} = i, 1 ≤ i ≤ p − 1, and v_{i} ≡ 1 mod I, and a general form of a lift an automorphism of C̄_{p} of order (p − 1)^{2} is

x′ = v^{p−1}x
y′ = v^{p}y

where v ≡ ζ mod I, ζ^{(p−1)^{2}} = 1. We can always assume that ζ^{p−1} = k^{−1}.

Proposition 6.1. There are automorphisms ã_{i}, i = 1, …, p − 1, and b̃ of the space E_{p−1}, and transformations a_{i} and b of the form described above, such that

ã_{i}^{∗}(C_{p}) = a_{i}(C_{p}), b̃^{∗}(C_{p}) = b(C_{p})

Here ã_{i}^{∗}(C_{p}), and b̃^{∗}(C_{p}) denote the inverse images of C_{p} under ã and b̃ respectively.

Proof. It is enough to look at the images of the marked points (0, 0), (1, 0), (e_{1}, 0), …, (e_{p−2}, 0) of C_{p}. If we match the images of these points under a_{i} and ã_{i}, and b and b̃ in the following way:

0 = v_{i}^{p−1}e_{[−1−i]} + r_{i}  1 = v^{p−1}e_{[k−1]}
1 = v_{i}^{p−1}e_{[−i]} + r_{i} b̃(e_{j}) = v^{p−1}e_{[(j+1)k−1]}
ã_{i}(e_{j}) = v_{i}^{p−1}e_{[j−i]} + r_{i}

1 ≤ j ≤ p − 2, we can solve the resulting equations for ã_{i} and b̃. In terms of the roots e_{1}, …, e_{p−2} these automorphisms are given by the formulas:

\begin{matrix}ã_{i}(e_{j}) = \frac{e_{[j−i]} − e_{[−i−1]}}{e_{[−i]} − e_{[−1−i]}} & b̃(e_{j}) = \frac{e_{[(j+1)i−1]}}{e_{[k−1]}} \\ ã_{i}(u) = (e_{[−i]} − e_{[−1−i]})^{\frac{1}{p−1}} u & b̃(u) = (e_{[k−1]})^{\frac{1}{p−1}} u \end{matrix}

Remark 6.2. The series (1 + x)^{\frac{1}{p−1}} exists in ℤ_{p}[[x]].

Denote by a_{i}^{F} and b^{F} the expansions of a_{i} and b in terms of the local parameter z. Then the one dimensional summand F_{C_{p}} has the following property.

Corollary 6.3.

ã_{i}^{∗}(F_{C_{p}}) = a_{i}^{F} (F_{C_{p}}), b̃^{∗}(F_{C_{p}}) = b^{F} (F_{C_{p}})

Remark 6.4. Let γ : F_{C_{p}} → G be an isomorphism of formal group laws, then

ã_{i}^{∗}(G) = ã_{i}^{∗}(γ) ∘ a_{F} ∘ γ^{−1}(G), b̃^{∗}(G) = b̃^{∗}(γ) ∘ b_{F} ∘ γ^{−1}(G)

.

For the purposes of computations with the above formulas let us state the following proposition.

Proposition 6.5.

e_{j} = j + 1 + a_{1}u_{1} + … + a_{p−2}u_{p−2}

modulo degree 2, where a_{i} = (j + 1)^{p−i} − (j + 1) modulo p.

7. On the properties of the one dimensional summand

The purpose of this section is to prove the theorem

Theorem 7.1. Formal group law F_{C_{p}} is a universal Lubin-Tate lift of the formal group of height p − 1 over 𝔽_{p}.

In this section log(F) always denotes the logarithm of a formal group law F (x, y).

Since our base ring is a Z_{(p)}-algebra, F_{C_{p}} is strictly isomorphic to a p-typical formal group law, which we denote by G_{C_{p}} again. We also denote by GC̄_{p} and by G_{C_{p}}^{0} the reductions of F_{C_{p}} modulo the ideals I and I_{0} respectively.

Remark 7.2. According to [1], (16.4.14), log(G_{C_{p}}) is obtained from log(F_{C_{p}}) by crossing out the terms m_{i}z^{i} with i ≠ p^{k}. This, together with 5.3, implies that

log(G_{C_{p}}^{0}) = z + \frac{a}{p}z^{p^{p−1}}

modulo degree p^{p−1} + 1, a ∈ ℤ_{p}, a ≠ 0 mod p.

Proposition 7.3. There is an automorphism α of GC̄_{p}(x, y) of order p such that α(z) = z + z^{p} modulo degree p + 1.

Proof. GC̄_{p} and FC̄_{p} are equal modulo degree p + 1, as it follows from 5.4. FC̄_{p} has an automorphism of order p with the desired property. Take, for example, a_{1}^{F} from 6.1 and reduce it modulo I. This implies the statement of the proposition.

The theorem 7.1 follows from the following proposition.

Proposition 7.4.

log(G_{C_{p}}) = z + Σ_{i=1}^{p−2} \frac{t_{i}}{p} u_{i}z^{p^{i}} + tz^{p^{p−1}}  (7.5)

t_{i} ∈ ℤ_{p}^{×}, modulo I_{0}^{2}.

Proof. According to the above remark the logarithm of G_{C_{p}} can be obtained in the following way: choosing a local parameter z as in 4.6 we obtain the following formula for the expansion of the differential \frac{dx}{y} on the curve C_{p}:

ω = \frac{(1 − p)dz}{1 − (Σ_{i=1}^{p−2}iu_{i}z^{p−i}w^{i−1} + (p − 1)bzw^{p−2})}

where w satisfies the equation

w = z^{p} + Σ_{i=1}^{p−2}u_{i}z^{p−i}w^{i} + bzw^{p−1}

Integrate it and cross out the terms m_{i}z^{i} with i not equal to a power of p. This is the logarithm of G_{C_{p}}. We claim that 7.5 holds for some t_{i} ∈ ℚ_{p}, modulo I_{0}^{2}. This follows from the following facts: a. In the series w(z) and \frac{1}{ω(z)}, reduced modulo I_{0}, the powers of z are congruent to p and zero modulo (p − 1)^{2} respectively. b. In the series w(z) the coefficient of z^{k} is equal to u_{i} modulo degree two iff k = p − i + pi or zero modulo (p − 1)^{2}. c. In the series ω(z) the coefficient of z^{k} is equal to u_{i} modulo degree two iff k is congruent to pi − i or zero modulo (p − 1)^{2}. d. p^{p−1} − 1 = 0 mod (p − 1)^{2}, p^{i} − 1 = pi − p mod (p − 1)^{2} Now we will prove that t_{i} ∈ ℤ_{p}^{×}. Examining the above formula for the expansion of the differential \frac{dx}{y}, we easily see that it is true for i = 1. Suppose we have proved the proposition for j < i. F_{C_{p}} is isomorphic to G_{C_{p}} by construction. 6.4 implies that there is a lift α′ of α such that

ã_{1}^{∗}(G_{C_{p}}) = α′(G_{C_{p}})

That implies that

ã_{1}(\frac{t_{i}}{p} u_{i}) = \frac{t_{i}}{p} u_{i} + \frac{t_{i−1}}{p} u_{i−1} + n_{i}u_{i} + n_{i−1}u_{i−1} + n,

modulo I_{0}^{2}, and modulo u_{1}, …, u_{i−2}, and n_{i}, n_{i−1}, n ∈ ℤ_{p}. On the other hand, the formulas 6.1 tell that

ã_{1}(u_{i}) = u_{i} + c_{i−1}u_{i−1}

c_{i−1} ∈ ℤ_{p}^{×}, modulo p, u_{1} …, u_{i−2}, and I_{0}^{2}. This implies that t_{i} must be a unit.

References

[1] M. Hazewinkel, Formal groups and applications, Academic Press, (1978).

[2] T. Hewett, Finite subgroups of the Morava stabilizer algebra, Preprint.

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