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 p be a fixed prime number. Let E_{n} be Lubin and Tate’s [8] ring representing deformations to complete local 𝕎(F_{p^{n}})-algebras of a standard formal group of the Morava K(n) theory, see, for example, [10]. Then there is a 2-periodic ring spectrum 𝔼_{n} such that π_{odd}𝔼_{n} = 0 and π_{0}𝔼_{n} ≅ E_{n}. Lubin and Tate show in ([8], 3.4) that the Morava stabilizer group 𝕊_{n} acts on E_{n}. We call this representation the Lubin-Tate representation. Recently Hopkins and Miller show that this action is induced by an action of 𝕊_{n} on the spectrum 𝔼_{n}. Let 𝔾_{n} denote 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}. There is a spectral sequence which abuts to π_{∗}𝔼O_{n} with E_{2} term H^{∗}(𝔾_{n}, E_{n})Gal. 𝔼O_{2} agrees with the usual elliptic cohomology when 2 is inverted [7].

In this paper we will study a formal group which can be used to construct 𝔼O_{n} for n = p − 1, and describe explicitly the Lubin-Tate representation of 𝔾_{n}. In [9] Manin proved that every formal group of finite height defined over a field of finite characteristic is a summand in the formal completion of the Jacobian of a certain curve. 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_{p−1} 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_{p−1} on the ring E_{p−1}. This representation was also studied using different techniques in [5], [6], and [7].

The authors wish to thank Mike Hopkins for useful comments, Avinash Sathaye for helpful discussions during the preparation of this paper and the referee for carefully reading the paper and suggesting a number of improvements.

2. Overview of the Honda theory

We recall briefly the Honda classification of formal group laws from [4]. If K is a ring or a field we denote by K[[x]]_{n}^{0} = K[[x_{1}, ..., x_{m}]]_{n}^{0} the set of n-dimensional vectors over the ring of series over K with constant term equal to zero.

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} mod 𝔭 for any a ∈ 𝔳. Introduce a variable T such that T a = a^{σ}T for any a ∈ 𝔳, and define the twisted power series ring M_{n}(𝔳_{𝔭})[[T]]^{σ} as the ring of series M_{n}(𝔳_{𝔭})[[T]] in which T and M_{n}(𝔳_{𝔭}) commute according to this rule. Define an action of this twisted power series ring on the vector space of n-dimensional vectors over the ring K[[x_{1}, …, x_{m}]] by

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

where u = Σ_{η}C_{η}T^{η}, f ∈ K[[x]]_{n}^{0}.

We say that an element u ∈ M_{n}(𝔳_{𝔭})[[T]]^{σ} is special for f ∈ K[[x]]_{n}^{0} if u ≅ πI_{n} mod degree 1 and (u ∗ f)(x) ≡ 0 mod 𝔭, where π is a uniformizer for 𝔭 and I_{n} is the n × n identity matrix. Let F be an n-dimensional abelian formal group law over 𝔳 and f = (f_{1}, …, f_{n}) denote its logarithm. An element u ∈ M_{n}(𝔳_{𝔭})[[T]]^{σ} is said to be special for F if it is a special element for f. We say in this case that F is of type u. Call elements u_{1}, u_{2} ∈ M_{n}(𝔳_{𝔭})[[T]]^{σ}, equivalent if u_{1} = vu_{2}, where v ∈ M_{n}(𝔳_{𝔭})[[T]]^{σ} is a unit.

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 the equivalence classes of elements u ∈ M_{n}(𝔳_{𝔭})[[T]]^{σ} of the form u ≅ πI_{n} mod 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 → J. Let ω_{1}, …, ω_{n} be a basis 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).

Denote by ν_{i}(y) the expansion of ν_{i} at the origin. There are ϕ_{i}(y) ∈ K[[y]], 1 ≤ i ≤ n, such that ϕ_{i}(0) = 0 and ν_{i}(y) = dϕ_{i}(y). The vector ϕ(y) = (ϕ_{i}(y)) is the logarithm of the formal group law F of J, associated to the system of local parameters y. Let S_{1} be the set of primes at which Γ, Λ, J, z, or y has a bad reduction, and 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 ∈ M_{n}(𝔳_{𝔭})[[T]]^{σ} is a special element for ψ(z) = (ψ_{i}(z)), then u is also a special element for the vector (ϕ_{i}(y)), so that F is of type u as a formal group law over 𝔳_{𝔭}.

Let A be a ring of characteristic zero that is also a ℤ_{(p)}-algebra, and K = A ⊗ ℚ. Suppose there is a ring endomorphism σ : K → K, such that σ(a) = a^{p} mod pA for all a ∈ A.

Lemma 2.3. Let A be a ring as above.

i: There is a special element for every formal group law over A. ii: If there is a special element u for a series f ∈ K[[x]]_{n}^{0}, f ≡ x mod

degree 2, then the formal group law f^{−1}(f(x) + f(y)) is a formal group law over A.

iii: The strict isomorphism classes of formal group laws over A correspond bijectively to the equivalence classes of elements u ∈ M_{n}(A)[[T]]^{σ} of the form u ≡ πI_{n} mod degree one.

Proof. i: It is proved in [1] that every formal group F over A has a logarithm defined by a functional equation, and [1] (20.3.11) shows that it is equivalent to the existence of a special element for F. ii: The direct check shows that the proofs of the Lemmas 2.3 and

2.4, and Theorem 2 from [3] go without a change when K and A are as above.

iii: The proof of the Proposition 2.6 from [3] goes through without change as well.

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

In this section we assume that p = 3. We will use the properties of the Legendre curve to describe the Lubin-Tate representation of 𝔾_{2}, a maximal finite subgroup of 𝕊_{2}. 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.

In order to describe the action of 𝔾_{2} on the moduli space of deformations, we need to choose a specific deformation of a formal group of height two. Consider the following elliptic curve C̄

y^{2} = x^{3} − x  (3.1)

The formal completion of this elliptic curve 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 C̄ to Artinian rings.

The Legendre curve is a plane curve defined by the equation

y^{2} = x(x − 1)(x − λ)

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

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

Complete Ē with respect to the ideal (3, λ+1) and denote the resulting ring E.

Lemma 3.3. The pair (E, F_{C}), where F_{C} denotes the formal completion of C, is a universal deformation of the formal completion of C̄.

Proof. Choose z = \frac{x}{y} as a local parameter at infinity. As a simple computation shows the formal completion of C 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}

According to [8] this is a universal deformation.

The pair (E, F_{C}) is the model of the Lubin-Tate lift we will use. Now we are ready to describe the Lubin-Tate representation of 𝔾_{2}.

Lemma 3.4. 𝔾_{2} is isomorphic to the group of automorphisms of the elliptic curve C̄ over the field 𝔽_{9} via the functor of the formal completion.

Proof. 𝔾_{2} is known to be isomorphic to ℤ/3 ⋊ ℤ/4. The group of automorphisms of C̄ over the field 𝔽_{9} is given by the substitutions

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

where i^{4} = 1. This groups is isomorphic to ℤ/3 ⋊ ℤ/4. The functor of formal completion provides a monomorphism Aut_{𝔽_{9}}(C̄) → 𝕊_{2}.

Proposition 3.5. There are six automorphisms of the complete local W (𝔽_{9})-algebra W (𝔽_{9}) ˆ⊗E which preserve the isomorphism class of the curve C.

Proof. This is a standard fact in elliptic curve theory, (cf [11], p 54). 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. 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 algebraic transformation from C to g^{∗}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 fact that 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 3.6. By definition, g(u) = \frac{u}{w}. The appropriate values of w are easy to calculate. For the substitutions 3.5 they are

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

.

Remark 3.7. The fact that we found only six automorphisms of the moduli space E means that the center of the group ℤ/3 ⋊ ℤ/4 acts trivially on it. It always happens this way as is proved in [6].

Remark 3.8. We can handle the Lubin-Tate representation of 𝔾_{2} for p = 2 in a similar fashion.

4. The generalized Legendre family

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

i) C is a 2-cover of the projective line with three marked points on it. The reduction of this curve mod the ideal (λ + 1) has 𝔾_{2} as a subgroup of its group of automorphisms via 3.5. ii) The formal completion of the Jacobian of C 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 [9] suggests considering curves of the form

y^{p^{a}−1} = x^{p} − x,

defined over 𝔽_{p}. He showed in [9] that, at least up to isogeny, the formal completion of the Jacobians of such a curve contains a one dimensional summand, whose height divides a(p − 1). Here we will study the case when a = 1.

Denote by E the ring ℤ_{p}[[u_{1}, …, u_{p−2}]].

Definition 4.1. Let the generalized Legendre curve C be the plane curve over E 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)

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 will be a curve C̄ 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.

Hensel’s lemma implies the following lemma

Lemma 4.3. All the roots of the polynomial in the right hand side of 4.2 are in E, so the curve C 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 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 contains 𝔾_{p−1}. Indeed, ℤ/p and ℤ/(p − 1)^{2} act on the curve C̄ 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. The curves C and C̄ are non-singular curves of genus m = (p − 1)(p − 2)/2. The differentials

ω_{i,j} = \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 and C̄ are nonsingular. To calculate the genus note that

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

div(dx) = (p−2)(0)+(p−2)(1)+(p−2)(e_{1})+…+(p−2)(e_{p−2})−p(∞) 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 that the genus is as indicated in the proposition.

Corollary 4.6. The order of the zero at infinity of ω_{i,j} 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_{i,j} is a power series Σ_{k≥0}g_{k}z^{n_{k}}dz where g_{k} is in E, and n_{k} ≡ j − 1 mod p − 1.

Proof. Introduce the following coordinates z = \frac{x}{y} and w = \frac{1}{y}. In these

coordinates the equation of the curve is

w = z^{p} + u_{1}z^{p−1}w + … + u_{p−2}z^{2}w^{p−2} + bzw^{p−1}  (4.7)

Out of this equation we immediately get that

w = z^{p}Σ_{k≥0}a_{k}z^{m_{k}},

y = z^{−p}Σ_{k≥0}b_{k}z^{l_{k}},

x = z^{1−p}Σ_{k≥0}b_{k}z^{l_{k}},

dx = z^{−p}Σ_{k≥0}c_{k}z^{l_{k}},

where m_{k}, l_{k} ≡ 0 mod p−1. This implies the statement of the corollary.

5. The splitting of the formal completion of the Jacobian

The ring E is of the type described in 2.3. Indeed, we can define the required endomorphism σ of E ⊗_{ℤ_{(p)}} ℚ by the formulas:

σ(u_{i}) = u_{i}^{p}, σ(a) = a, a ∈ ℚ_{p}

Therefore, the formal group law associated to a choice of coordinates on J(C) is determined by the equivalence class of special elements for its logarithm. We will study the properties of this class of special elements.

Denote by ψ_{i,j}(z) the formal anti-derivative of the formal expansion of ω_{i,j} around infinity, and by ψ the vector with the coordinates ψ_{i,j}(z), ordered in such a way that (i, j) > (i_{1}, j_{1}) if j > j_{1} . Recall that m = (p − 1)(p − 2)/2 is the number of entries in the vector ψ.

Theorem 5.1. i Define the formal group law F_{ψ} of J(C) in terms of the coordinates on J(C) determined by ψ. Then this is a formal group law over E, therefore, there is a special element for F_{ψ}, which is also a special element for the vector ψ. ii If u ∈ M_{m}(E)[[T]]^{σ} is a special element for the the vector ψ, then it also is a special element for F_{ψ}, so that F_{ψ} is of type u as a formal group over E.

Proof. i. We need to prove that the Jacobian is smooth at the origin. This follows from the general theory of reduction of algebraic varieties [12]. 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 [12] it means that the objects we deal with are I-simple.

It follows from 2.3 that there is a special element for F_{ψ}. Pulling the logarithm of F_{ψ} back using the canonical embedding C → J(C), we see that it is also a special element for ψ. ii. We will show that the following version of Lemma 1 from [3] holds in our setting. Then the proof of part ii will follow exactly as the proof of Proposition 2.2 of [3].

Lemma 5.2. Let ψ be as above. If u, v ∈ M_{m}(E)[[T]]^{σ} are special elements for ψ, then there is a unit in M_{m}(E)[[T]]^{σ} such that v = tu.

Proof. Suppose

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_{m}(E). We will show that if C_{k} = D_{k}, 0 ≤ k ≤ l − 1 with some l ≥ 1, then C_{l} = D_{l} mod p. We have

Σ_{k=l}^{∞}(C_{k} − D_{k})ψ^{σ^{k}}(z^{p^{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}(E)[[T]]^{σ} and t_{l}u = v mod 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 mod degree l + 1. The limit of t_{l} … t_{1} clearly satisfies the requirement of our lemma.

Theorem 5.3. The isomorphism class of F_{ψ} over E contains a formal group law, which splits into p−2 summands of dimensions 1, 2, … p−2 respectively. The reduction of the one-dimensional summand mod I has the height p − 1.

Proof. According to (5.1, ii), we need to study special elements for the vector ψ. To prove the theorem we will show that there is a special element for ψ which is made out of diagonal blocks of dimensions 1, 2, …, p − 2. Let u = pI_{m} + Σ_{k=1}^{∞}C_{k}T^{k} be a special element for ψ, which exists according to 5.1. Recall that the order of zero at infinity

of ω_{i,j}, reduced mod p − 1, is equal to j − 1. Expanding ω_{i,j} = \frac{x^{i}dx}{y^{j}} around infinity into a power series and integrating the result, we see from 4.6 that ψ_{i,j} is a formal series in powers of z congruent to j mod p − 1, 1 ≤ j ≤ p − 2. Using the definition of the ∗-product and the fact that p ≡ 1 mod p−1, we see that each component of the vector u∗ψ is a sum of p − 2 power series r_{j}, such that r_{j} is a power series in z where the only powers of z occurring are congruent to j mod p − 1. Since u ∗ ψ = 0 mod p, we must have r_{j} = 0 mod p for each j. This implies that the element u′ = pI_{m}+Σ_{k=1}^{∞}C′_{k}T^{k}, where each C′_{k} is a block matrix with the blocks of dimensions 1, …, p − 2 along the diagonal, obtained from C_{k} by changing the off-diagonal-block entries to zero, is a special element for the vector ψ.

Consider now the vector f(x) = (p/u′) ∗ i(x), where x is a vector of variables (x_{1}, … x_{m}), and i(x) = x. The remark 2.3 implies that the formal group law H(x, y) = f^{−1}(f(x)+f(y)) is defined over E, and, by construction, is a sum of p − 2 summands of dimensions 1, …, p − 2. The theorem 5.1 guarantees that it is isomorphic to F.

According to [9] (page 74, Theorem 4.2), the reduction of F_{ψ} mod I contains a one-dimensional summand of the height p − 1. Since the maximal dimension of an indecomposable summand of the reduction of F_{ψ} is less or equal to p − 2, using [9] ((4.7), page 71) we see that the height of the reduction of the one-dimensional summand must be p − 1.

Now we present a certain deformation of a formal group of height p − 1 over 𝔽_{p}. Later we will prove that this is a universal deformation in the sense of Lubin-Tate [8].

Corollary 5.4. Denote by F_{C} the one-dimensional formal group law with the logarithm ψ_{0,1}(z). This is a formal group law over E, and the height of the reduction of F_{C} mod the ideal I is p − 1.

Proof. We have proved above that the vector ψ(z) = (ψ_{i,j}(z)) has a special element u = pI_{m} + Σ_{k=1}^{∞}C_{k}T^{k}, such that for all k, C_{k} is a block diagonal matrix with the blocks of dimensions 1, …, p − 2. By definition of the ∗-product it means that ψ_{0,1} has a special element. It is made out of the blocks C_{k}^{1} of C_{k} of the size one by one. In other words, it is given by the formula u = p+Σ_{k=1}^{∞}C_{k}^{1}T^{k}. The Honda theory [4] and the remark 2.3 imply that the formal group law F_{C}(x, y) = ψ_{0,1}^{−1}(ψ_{0,1}(x)+ψ_{0,1}(y)) is defined over E. Moreover, the very same series u is a special element for the one-dimensional summand of the formal group law H(x, y), defined above. Therefore, according to the Honda theory, these two one-dimensional formal group laws are isomorphic over E. Thus, the height of the reduction mod I of F_{C} is p − 1.

6. A useful representation of 𝔾_{p−1}

From now on log_{F} denotes the logarithm of a formal group F. Denote by E_{1} the ring W (𝔽_{p^{p−1}})[[u_{1}, …, u_{p−2}]]. In this section we will describe a certain action of 𝔾_{p−1} on the ring E_{1}. The result of the next section will imply that this action is the Lubin-Tate action. In order to obtain the formulas for this representation of 𝔾_{p−1} we will use the generalized Legendre curve C in the same way we used the Legendre curve in section three.

Let F be a formal group law of dimension one over the ring E_{1}. Denote by F̄ the reduction of F mod I = (p, u_{1}, …, u_{p−2}), and suppose that it has the height greater than or equal to one. Let G be a subgroup of the group of automorphisms of F̄ over 𝔽_{p^{p−1}}. Recall, that we can view G as a subgroup of the group of power series over 𝔽_{p^{p−1}} which are invertible under composition.

Denote g^{−1}F (g(x), g(y)) by g^{−1}(F).

Definition 6.1. We say that F has the Lubin-Tate property with respect to G if for any ḡ ∈ G, there is a unique lift g of ḡ to the ring E_{1}[[x]], and an unique automorphism α_{g} of E_{1} as a complete local W (𝔽_{p^{p−1}})-algebra, such that

g^{−1}(F) = α_{g}^{∗}(F (x, y))

This lets us define an action of G on the ring E_{1} as follows. For ḡ ∈ G, define a W (𝔽_{p^{p−1}})-linear ring automorphism of E_{1} by the formula:

ḡ(u_{i}) = α_{g}(u_{i})

We call this representation of G, the representation defined by F.

Remark 6.2. For purposes in homotopy theory we need to consider formal groups over the ring E_{1}[u^{−1}, u]. If a formal group F defines a representation of G on E_{1} as in the definition 6.1, then we can extend it to an action of G on E_{1}[u^{−1}, u] by the formula

α_{g}(u) = g′(0)u

Remark 6.3. A formal group law F has the Lubin-Tate property if and only if for all ḡ ∈ G there is a unique lift g of ḡ to the ring E_{1}[[x]], and an unique automorphism α_{g} of E_{1} as a complete local W (𝔽_{p^{p−1}})- algebra, such that

α_{g}^{∗}(log_{F}) = \frac{log_{F} ∘g}{g′(0)}

Let C be a plane curve over the ring A, given by the equation C(x, y) = 0, α is a ring homomorphism of A, and f is a transformation of A^{2} given by a pair of polynomial functions f(x, y) = (f_{1}(x, y), f_{2}(x, y)).

We denote by α^{∗}(C) the inverse image of C under α, and by f(C) the curve, defined by the equation C(f_{1}(x, y), f_{2}(x, y)) = 0.

Set e_{0} = 1 and e_{p−1} = 0 and denote by [m] the reduction of the integer m mod p. With this notation we have e_{i} ≡ [i + 1] mod I, 0 ≤ i ≤ p − 1.

Recall that the group 𝔾_{p−1} is isomorphic to ℤ_{p} ⋊ ℤ/(p − 1)^{2}. The last can be viewed as a group of automorphisms over 𝔽_{p^{p−1}} of the curve C̄, defined in section 4. Choose i from the set {1, …, p − 1}. One can see directly that the substitution ā of order p

x_{1} = x + i
y_{1} = y

and b̄ of order (p − 1)^{2}

x_{1} = ζ^{p−1}x
y_{1} = ζ^{p}y

where ζ^{(p−1)^{2}} = 1, generate this group. A general form of a transformation a of C, which reduces mod I to the automorphism ā of C̄ is:

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

where r_{i} ≡ i and v_{i} ≡ 1 mod I, and a general form of a transformation b of C, which reduces mod I to an automorphism of C̄ of order (p−1)^{2} is

x_{1} = v^{p−1}x
y_{1} = v^{p}y

where v_{i} ≡ 1, r_{i} ≡ i, v ≡ ζ mod I. We can always assume that ζ^{p−1} = k^{−1} were k is a generator of 𝔽_{p}^{×}.

Proposition 6.4. For every ḡ ∈ 𝔾_{p−1} there is a W (𝔽_{p^{p−1}})-linear automorphisms α_{g} of the ring E, and a transformation g of the curve C, such that

α_{g}^{∗}(C_{p}) = g^{−1}(C_{p})

Proof. Since 𝔾_{p−1} is generated by ā and b̄, we will look for automorphisms α_{a} and α_{b} of the ring E_{1}, and transformations a and b of the form described above, such that

α_{a}^{∗}(C) = a^{−1}(C), α_{b}^{∗}(C) = b^{−1}(C)

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 under α_{a}, α_{b} and a, b. If we match the images of these points 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}  α_{b}(e_{j}) = v^{p−1}e_{[(j+1)k−1]}
α_{a}(e_{j}) = v_{i}^{p−1}e_{[j−i]} + r_{i}

1 ≤ j ≤ p − 2, we can solve the resulting equations for α_{a} and α_{b}. From the way we solved these equations it is clear that the solution is unique. In terms of the roots e_{1}, …, e_{p−2} these automorphisms are given by the formulas:

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

Remark 6.5. The series (1 + x)^{\frac{1}{p−1}} exists in ℤ_{p}[[x]].

Corollary 6.6. The formal group law F_{C} has the Lubin-Tate property with respect to 𝔾_{p−1}.

Proof. Let ḡ ∈ 𝔾_{p−1} and its lift, obtained in 6.4, be represented by the following substitution

x_{1} = v^{p−1}x + r
y_{1} = v^{p}y

Expanding this transformation we obtain a series g(z_{1}), such that z = g(z_{1}).

Recall that the series ω_{0,1} is the expansion with respect to the pa-

rameter z = \frac{x}{y} of the differential \frac{dx}{y} on the curve C_{p}. Denote by ω^{g}

the expansion with respect to the parameter z_{1} = \frac{x_{1}}{y_{1}} of the differential

\frac{dx_{1}}{y_{1}} on the curve g(C_{p}). Then

\frac{dx}{y} = \frac{d(v^{1−p}(x_{1} + r))}{v^{−p}y_{1}} = v \frac{dx_{1}}{y_{1}}

Expanding both sides with respect to the parameter z_{1}, we obtain that

v^{−1}(g)′(z_{1})ω_{0,1}(g(z_{1})) = ω^{g}(z_{1})

.

On the one hand, the proposition 6.4 implies that α_{g}^{∗} \frac{dx}{y} = \frac{dx_{1}}{y_{1}} ,

therefore, after expanding both sides into series we obtain

α_{g}^{∗}ω_{0,1}(z_{1}) = ω^{g}(z_{1})

To finish the proof we need to formally integrate both the identities for ω^{g}(z_{1}), and note that v = (g)′(0).

Corollary 6.7. Let F be a formal group law and γ : F_{C} → F be a strict isomorphism of formal group laws, both defined over E_{1}, then F has the Lubin-Tate property with respect to 𝔾_{p−1}. The representation of 𝔾_{p−1} defined by F coincides with the one defined by F_{C}.

Proof. Let ḡ ∈ G, then there are g and α_{g}, described above, such that g^{−1}(F_{C}) = α_{g}^{∗}F_{C}. Then we have (γ ∘ g ∘ α_{g}^{∗}(γ^{−1}))^{−1}F = α_{g}^{∗}F Since γ was a strict isomorphism, we have (α_{g}^{∗}(γ)g(γ^{−1}))′(0) = (g^{−1})′(0).

7. On the properties of the one dimensional summand

Since our base ring E is a Z_{(p)}-algebra, F_{C} is strictly isomorphic over E to a p-typical formal group law, which we denote by G_{C}. We also denote by Ḡ_{C} (F̄_{C}) the reductions of G_{C} (F_{C}) modulo the ideal I, and by G_{C}^{0} (F_{C}^{0}) the reductions of G_{C} (F_{C}) modulo the ideal I_{0} = (u_{1}, …, u_{p−2}). So Ḡ_{C} and F̄_{C} are formal group laws defined over 𝔽_{p}, and G_{C}^{0} and F_{C}^{0} are formal group laws defined over ℤ_{p}.

Remark 7.1. According to [1], (16.4.14), log_{G_{C}} is obtained from log_{F_{C}} by crossing out the terms m_{i}z^{i} with i ≠ p^{k}. Moreover, if ϕ(z) ∈ E[[z]] is the strict isomorphism between G_{C} and F_{C}, then log_{G_{C}}(z) = log_{F_{C}}(ϕ(z)).

The purpose of this section is to prove the theorem

Theorem 7.2. The formal group law G_{C} is a universal Lubin-Tate lift of a formal group of height p − 1 over 𝔽_{p}.

The proof will follow directly from the next two propositions.

Proposition 7.3. There is an automorphism ḡ of Ḡ_{C}(x, y) of order p, defined over 𝔽_{p}, such that ḡ(z) = z + z^{p} mod degree p + 1.

Proof. The formal group laws G_{C} and F_{C} are strictly isomorphic over E by construction, therefore Ḡ_{C} and F̄_{C} are isomorphic over 𝔽_{p}. We claim that there is an isomorphism between Ḡ_{C} and F̄_{C}, such that as a series in z it is equal to z mod degree p + 1.

Note that log_{G_{C}^{0}}(z) = log_{F_{C}^{0}}(ϕ^{0}(z)), where ϕ^{0}(z) is the reduction of ϕ(z) mod I_{0}. The logarithm of F_{C}^{0} is the formal integral of the following differential form:

ω = \frac{(1 − p)dz}{1 + (p − 1)zw^{p−2}}  (7.4)

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

w = z^{p} − zw^{p−1}

This functional equation implies that w is a series in powers of z congruent to p modulo (p − 1)^{2} and, therefore, ω = ω(z) dz is the differential form such that ω(z) is a series in powers of z congruent to zero modulo (p − 1)^{2}. From this we conclude that the series log_{F_{C}^{0}} and log_{G_{C}^{0}} are congruent to z mod degree p + 1. Therefore, ϕ^{0}(z) is congruent to z mod degree p + 1. Its reduction mod p gives an isomorphism between Ḡ_{C} and F̄_{C} with the stated above properties.

F̄_{C} has an automorphism of order p, such that as a series, it is congruent to z mod degree p + 1. Take, for example, a_{1} from 6.4, expand it into a series with respect to a local parameter \frac{x}{y} and reduce the resulting series mod I. This proves the proposition.

Proposition 7.5.

log_{G_{C}} = z + Σ_{i=1}^{p−2} \frac{t_{i}}{p} u_{i}z^{p^{i}}  (7.6)

t_{i} ∈ ℤ_{p}^{×}, mod I_{0}^{2} and mod degree p^{p−1}. This implies that G_{C} is a universal deformation of the formal group law Ḡ_{C}.

Proof. Recall that the logarithm of G_{C} is 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 ω = \frac{dx}{y} on the curve C_{p}.

If ω = ω(z) dz, then:

ω(z) = \frac{(1 − p)}{1 − (Σ_{i=1}^{p−2}iu_{i}z^{p−i}w^{i−1} + (p − 1)bzw^{p−2})}  (7.7)

where w satisfies the following functional equation

w = z^{p} + Σ_{i=1}^{p−2}u_{i}z^{p−i}w^{i} + bzw^{p−1}  (7.8)

Integrate ω(z) and cross out the terms m_{i}z^{i} with i not equal to a power of p.

We claim that 7.6 holds for some t_{i} ∈ ℚ_{p}, mod I_{0}^{2}. This follows from the following facts: a. In the series w(z) and ω(z), reduced mod I_{0}, the powers of z are congruent to p and zero mod (p − 1)^{2} respectively. b. In the series w(z) the coefficient of z^{k} is equal to u_{i} mod degree two if and only if k = p − i + pi or zero mod (p − 1)^{2}. c. In the series ω(z) the coefficient of z^{k} is equal to u_{i} mod degree two if and only if k is congruent to pi − i or zero mod (p − 1)^{2}. d. p^{i} − 1 = pi − p mod (p − 1)^{2}, 1 ≤ i ≤ p − 1. The proof is direct and uses 7.7 and 7.8.

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 t_{1} ∈ ℤ_{p}^{×}. Suppose

we have proved the proposition for j < i. Since F_{C} is isomorphic to G_{C}, the corollary 6.7 implies that G_{C} has the Lubin-Tate property with respect to 𝔾_{p−1}. In particular, there is a lift g of ḡ from 7.3, such that

α_{g}^{∗}(G_{C}) = g(G_{C}).

For log_{G_{C}} we get the identity

α_{g}^{∗}(log_{G_{C}})(z) = \frac{log_{G_{C}}(g(z))}{g′(0)}

and taking into account that g(z) = z + z^{p} mod degree p + 1, we see that

α_{g}^{∗}(t_{i}u_{i}) = t_{i}u_{i} + t_{i−1}u_{i−1},

mod I_{0}^{2}, and mod p, u_{1}, …, u_{i−2}, and t_{i−1} ∈ ℤ_{p}^{×}. On the other hand, the formulas 6.4 tell that

α_{g}^{∗}(u_{i}) = u_{i} + c_{i−1}u_{i−1}

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

The Proposition 1.1 from [8] implies now that G_{C} is a universal deformation of Ḡ_{C}.

The following corollary shows that the formulas from 6.4 can be used for calculations with the cohomology theories represented by the spectra 𝔼O_{p−1}.

Corollary 7.9. The representation of 𝔾_{p−1} defined by F_{C} is the Lubin-Tate representation.

References

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

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

[3] T. Honda, On the formal structure of the Jacobian variety of the Fermat curve over a p-adic integer ring, Symp. Math. INDAM 11 (1973), 271-284.

[4] T. Honda, On the theory of commutative formal groups, J. Math. Soc. Japan, 22 (1970), 213-246.

[5] E. Devinatz, M. J. Hopkins, The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts, American J Math, v 117, No 3, (1995), 669-710.

[6] M. J. Hopkins, B. H. Gross, The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory, Bull. Amer. Math. Soc., v 30, No 1, (1994), 76-86.

[7] H. Miller, AMS Summer Research Institute, Seattle, 1996.

[8] J. Lubin, J. Tate, Formal moduli for one parameter formal Lie groups, Bull. Soc. Math. France 94 (1966), 49-60.

[9] Yu. I. Manin, The theory of commutative formal groups over fields of finite characteristic, Russian Math. Surveys, 18, (1963), 1-81.

[10] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, New York, (1986).

[11] J. H. Silverman, The Arithmetic of Elliptic Curves, Springer Graduate Texts in Mathematics, 106, 1986.

[12] G. Shimura and Y. Taniyama, Complex multiplication of abelian varieties and its application to number theory, Publ. Math. Soc. Japan, 6, 1960.

University of Kentucky, Lexington KY 40506

Northwestern University, Evanston IL 60208