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 EOn.
1. Introduction
Let p be a fixed prime number. Let En be Lubin and Tate's [8]
ring representing deformations to complete local W(Fpn)-algebras of a
standard formal group of the Morava K(n) theory, see, for example,
[10 ]. Then there is a 2-periodic ring spectrum En such that ssoddEn = 0
and ss0En ~= En. Lubin and Tate show in ([8], 3.4) that the Morava
stabilizer group Sn acts on En. We call this representation the Lubin-
Tate representation. Recently Hopkins and Miller show that this action
is induced by an action of Sn on the spectrum En. Let Gn denote a
maximal finite subgroup of the Morava stabilizer group Sn. Hopkins
and Miller define EOn to be the homotopy fixed point spectrum of
the action of Gn on En. There is a spectral sequence which abuts to
ss*EOn with E2 term H*(Gn; En)Gal . EO2 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 EOn for n = p - 1, and describe explicitly the Lubin-Tate
representation of Gn. 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 Fp 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 Gp-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 Gp-1 on the ring
1
2 V. GORBOUNOV AND M. MAHOWALD
Ep-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]]0n= K[[x1; :::; xm ]]0n
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 vp 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 oe of
K and a power q of p such that aoe= aq mod p for any a 2 v. Introduce
a variable T such that T a = aoeT for any a 2 v, and define the twisted
power series ring Mn(vp)[[T ]]oeas the ring of series Mn(vp)[[T ]] in which
T and Mn(vp) 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[[x1; : :;:xm ]] by
j qj
(u * f)(x) = 1j=0Cjfoe(x );
where u = jCjT j, f 2 K[[x]]0n.
We say that an element u 2 Mn(vp)[[T ]]oeis special for f 2 K[[x]]0n
if u ~= ssIn mod degree 1 and (u * f)(x) 0 mod p, where ss is a
uniformizer for p and In is the n x n identity matrix. Let F be an
n-dimensional abelian formal group law over v and f = (f1; : :;:fn)
denote its logarithm. An element u 2 Mn(vp)[[T ]]oeis 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 u1; u2 2 Mn(vp)[[T ]]oe, equivalent if u1 = vu2,
where v 2 Mn(vp)[[T ]]oeis a unit.
The following theorem is proved in [4]
Theorem 2.1. Suppose that K is a discrete totally unramified valua-
tion 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 2 Mn(vp)[[T ]]oeof the form
u ~=ssIn 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
FORMAL COMPLETION OF THE JACOBIANS OF PLANE CURVES 3
the formal completion of the Jacobian of a curve and the space of
holomorphic differentials on it. Let be a curve of genus n > 0 over
K. Denote by J the Jacobian of , and by the canonical map !
J. Let !1; : :;:!n be a basis of holomorphic differentials on , each
defined over K. Choose a local parameter z 2 K() at some point
P , and denote by !i(z) the expansion of !i at P . There are i(z) 2
K[[z]], 1 i n, such that i(0) = 0 and !i(z) = d i(z). Let
y = (y1; : :;:yn) K(J) be a system of local parameters at the origin
of J and let 1; : :;:n be the invariant differentials on J such that
iO = !i (1 i n):
Denote by i(y) the expansion of i at the origin. There are OEi(y) 2
K[[y]], 1 i n, such that OEi(0) = 0 and i(y) = dOEi(y). The
vector OE(y) = (OEi(y)) is the logarithm of the formal group law F of J,
associated to the system of local parameters y. Let S1 be the set of
primes at which , , J, z, or y has a bad reduction, and S0 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:
o S contains S0 [ S1.
o If p 62 S and u 2 Mn(vp)[[T ]]oeis a special element for (z) =
( i(z)), then u is also a special element for the vector (OEi(y)), so
that F is of type u as a formal group law over vp.
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 oe : K ! K, such
that oe(a) = ap mod pA for all a 2 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 2 K[[x]]0n, 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 2
Mn(A)[[T ]]oeof the form u ssIn 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 .
4 V. GORBOUNOV AND M. MAHOWALD
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 G2,
a maximal finite subgroup of S2. 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 G2 on the moduli space of defor-
mations, we need to choose a specific deformation of a formal group of
height two. Consider the following elliptic curve C
y2 = x3 - 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 F3 is isomorphic to S2. We will present a convenient model for the
moduli space of lifts of C to Artinian rings.
The Legendre curve is a plane curve defined by the equation
y2 = x(x - 1)(x - )
over the ring E = Z[1_2; ]. Denote this curve by C and rewrite its
equation as
y2 = x(x - 1)(x - u1 + 1): (3.2)
Complete E with respect to the ideal (3; +1) and denote the resulting
ring E.
Lemma 3.3. The pair (E; FC ), where FC denotes the formal comple-
tion of C, is a universal deformation of the formal completion of C .
Proof. Choose z = x_yas a local parameter at infinity. As a simple com-
putation shows the formal completion of C is a series in two variables
F (z1; z2) = z1 + z2 + u1(z21z2 + z1z22) mod (z1; z2)4
__
According to [8] this is a universal deformation. |__|
The pair (E; FC ) is the model of the Lubin-Tate lift we will use.
Now we are ready to describe the Lubin-Tate representation of G2.
FORMAL COMPLETION OF THE JACOBIANS OF PLANE CURVES 5
Lemma 3.4. G2 is isomorphic to the group of automorphisms of the
elliptic curve C over the field F9 via the functor of the formal comple-
tion.
Proof. G2 is known to be isomorphic to Z=3 o Z=4. The group of
automorphisms of C over the field F9 is given by the substitutions
s1: x = x0+ 1 y = y0
s2: x = -x0 y = -iy0
where i4 = 1. This groups is isomorphic to Z=3 o Z=4. The functor of
formal completion provides a monomorphism Aut F9(C ) ! S2. __
|__|
Proposition 3.5. There are six automorphisms of the complete local
W (F9)-algebra W (F9) ^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; }. Any automorphism g of the moduli space takes
the triple {0; 1; } to an unordered triple {0; 1; g()}. Let
x0= w2x + r
y0 = w3y
be an algebraic transformation from C to g*C. It, in its turn, takes
the triple {0; 1; } to an unordered triple {r; w2 + r; w2 + r}. There
are six way to match these two unordered triples, which produce the
following values for :
1 - 1 1
; 1 - ; __; ______; ______; ______
1 - - 1
__
|__|
All we need now is to replace with u1 - 1 and take into account the
fact that g fixes elements of the base ring W (F9). Here are possible
values for g(u1):
n u 2u - 3 u - 3 2u - 3 o
u1; 3 - u1; ___1___; __1_____; _1_____; __1_____
u1 - 1 u1 - 1 u1 - 2 u1 - 2
6 V. GORBOUNOV AND M. MAHOWALD
Remark 3.6. By definition, g(u) = u_w. The appropriate values of w
are easy to calculate. For the substitutions 3.5 they are
r ____ r __
1 1
-__ ; __
.
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 o 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 G2 for
p = 2 in a similar fashion.
4. The generalized Legendre family
Fix a prime number p from now on. Then any maximal finite sub-
group of Sp-1, Gp-1, is isomorphic to ZpoZ=(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 ( + 1) has G2 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
a-1 p
yp = x - x;
defined over Fp. 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 Zp[[u1; : :;:up-2]].
Definition 4.1. Let the generalized Legendre curve C be the plane
curve over E defined by the following equation:
yp-1 = xp + u1xp-1 + : :+:up-2x2 + (-1 - p-2i=1ui)x
(4.2)
Denote -1 - p-2i=1ui by b.
FORMAL COMPLETION OF THE JACOBIANS OF PLANE CURVES 7
If we factor out the ideal I = (p; u1; : :u:p-2), then the reduction of
the curve C will be a curve C over Fp defined by the equation
yp-1 = xp - 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:
yp-1 = x(x - 1)(x - e1) . : :.:(x - ep-2)
where ei = 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 Fpp-1 the group of
automorphisms of C contains Gp-1. Indeed, Z=p and Z=(p - 1)2 act on
the curve C via a substitutions:
x = x0+ 1; y = y0; x = ip-1x0; y = ipy0; (4.4)
2
where i(p-1) = 1.
Proposition 4.5. The curves C and C are non-singular curves of
genus m = (p - 1)(p - 2)=2. The differentials
xidx
!i;j= _____; 0 i j - 1 p - 3
yj
form a basis of the space of holomorphic differentials.
Proof. Direct computation shows that the curves C and C are non-
singular. To calculate the genus note that
div(x) = (p - 1)(0) - (p - 1)(1)
div(y) = (0) + (1) + (e1) + : :+:(ep-2) - p(1)
div(dx) = (p-2)(0)+(p-2)(1)+(p-2)(e1)+: :+:(p-2)(ep-2)-p(1)
So the above differentials are indeed holomorphic. A standard argu-
ment shows that these form a basis for the space of holomorphic dif-
ferentials. 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 !i;jis equal to
p(j - 1) - (p - 1)i. If we choose z = x_yas a local parameter at infinity
than the expansion of !i;jis a power series k0 gkznkdz where gk is in
E, and nk j - 1 mod p - 1.
8 V. GORBOUNOV AND M. MAHOWALD
x 1
Proof. Introduce the following coordinates z = __ and w = __. In these
y y
coordinates the equation of the curve is
w = zp + u1zp-1w + : :+:up-2z2wp-2 + bzwp-1 (4.7)
Out of this equation we immediately get that
w = zpk0 akzmk ;
y = z-p k0 bkzlk ;
x = z1-pk0 bkzlk ;
dx = z-p k0 ckzlk ;
where mk; lk 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 oe of E Z(p)Q by the formulas:
oe(ui) = upi; oe(a) = a; a 2 Qp
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 i;j(z) the formal anti-derivative of the formal expansion
of !i;jaround infinity, and by the vector with the coordinates i;j(z),
ordered in such a way that (i; j) > (i1; j1) if j > j1 . Recall that
m = (p - 1)(p - 2)=2 is the number of entries in the vector .
Theorem 5.1. i Define the formal group law F of J(C) in terms of
the coordinates on J(C) determined by . Then this is a formal group
law over E, therefore, there is a special element for F , which is also
a special element for the vector .
ii If u 2 Mm (E)[[T ]]oeis a special element for the the vector , then
it also is a special element for F , so that F is of type u as a formal
group over E.
FORMAL COMPLETION OF THE JACOBIANS OF PLANE CURVES 9
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 Cp
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 . Pulling the
logarithm of F back using the canonical embedding C ! J(C), we
see that it is also a special element for .
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 be as above. If u; v 2 Mm (E)[[T ]]oeare special
elements for , then there is a unit in Mm (E)[[T ]]oesuch that v = tu.
Proof. Suppose
u = pIm + 1k=1CkT k; v = pIm + 1k=1DkT k
Ck, Dk 2 Mm (E). We will show that if Ck = Dk, 0 k l - 1 with
some l 1, then Cl= Dl mod p. We have
k pk
1k=l(Ck - Dk) oe(z ) 0 mod p
It follows from 4.6 by direct computation that all the entries of Cl- Dl
are divisible by p. Put now
1 l
tl= Im - __(Cl- Dl)T
p
tlis a unit in Mm (E)[[T ]]oeand tlu = v mod degree l +1. In this way we
can find units t1; : :;:tlsuccessively for each l > 0, so that tl: :t:1u = v
mod degree l + 1. The limit of tl: :t:1clearly satisfies the requirement_
of our lemma. |__|
__
|__|
Theorem 5.3. The isomorphism class of F 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 . To prove the theorem we will show that there is a spe-
cial element for which is made out of diagonal blocks of dimensions
1; 2; : :;:p - 2. Let u = pIm + 1k=1CkT kbe a special element for ,
which exists according to 5.1. Recall that the order of zero at infinity
xidx
of !i;j, reduced mod p - 1, is equal to j - 1. Expanding !i;j= _____
yj
10 V. GORBOUNOV AND M. MAHOWALD
around infinity into a power series and integrating the result, we see
from 4.6 that i;jis 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 * is
a sum of p - 2 power series rj, such that rj is a power series in z where
the only powers of z occurring are congruent to j mod p - 1. Since
u * = 0 mod p, we must have rj = 0 mod p for each j. This implies
that the element u0= pIm +1k=1C0kT k, where each C0kis a block matrix
with the blocks of dimensions 1; : :;:p - 2 along the diagonal, obtained
from Ck by changing the off-diagonal-block entries to zero, is a special
element for the vector .
Consider now the vector f(x) = (p=u0) * i(x), where x is a vector of
variables (x1; : :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 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 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 Fp. Later we will prove that this is a universal deformation in the
sense of Lubin-Tate [8].
Corollary 5.4. Denote by FC the one-dimensional formal group law
with the logarithm 0;1(z). This is a formal group law over E, and the
height of the reduction of FC mod the ideal I is p - 1.
Proof. We have proved above that the vector (z) = ( i;j(z)) has a
special element u = pIm + 1k=1CkT k, such that for all k, Ck is a
block diagonal matrix with the blocks of dimensions 1; : :;:p - 2. By
definition of the *-product it means that 0;1has a special element. It
is made out of the blocks C1kof Ck of the size one by one. In other
words, it is given by the formula u = p+1k=1C1kT k. The Honda theory
[4] and the remark 2.3 imply that the formal group law FC (x; y) =
-10;1( 0;1(x)+ 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 FC is p - 1. |__|
FORMAL COMPLETION OF THE JACOBIANS OF PLANE CURVES 11
6. A useful representation of Gp-1
From now on logF denotes the logarithm of a formal group F . Denote
by E1 the ring W (Fpp-1)[[u1; : :;:up-2]]. In this section we will describe
a certain action of Gp-1 on the ring E1. 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 Gp-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 E1.
Denote by F the reduction of F mod I = (p; u1; : :;:up-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 F over Fpp-1. Recall, that we can
view G as a subgroup of the group of power series over Fpp-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 re-
spect to G if for any g 2 G, there is a unique lift g of g to the ring
E1[[x]], and an unique automorphism ffg of E1 as a complete local
W (Fpp-1)-algebra, such that
g-1 (F ) = ffg*(F (x; y))
This lets us define an action of G on the ring E1 as follows. For g 2 G,
define a W (Fpp-1)-linear ring automorphism of E1 by the formula:
g(ui) = ffg(ui)
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 E1[u-1; u]. If a formal group F defines a
representation of G on E1 as in the definition 6.1, then we can extend
it to an action of G on E1[u-1; u] by the formula
ffg(u) = g0(0)u
Remark 6.3. A formal group law F has the Lubin-Tate property if
and only if for all g 2 G there is a unique lift g of g to the ring E1[[x]],
and an unique automorphism ffg of E1 as a complete local W (Fpp-1)-
algebra, such that
logF Og
ff*g(logF ) = _______
g0(0)
Let C be a plane curve over the ring A, given by the equation
C(x; y) = 0, ff is a ring homomorphism of A, and f is a transformation
of A2 given by a pair of polynomial functions f(x; y) = (f1(x; y); f2(x; y)).
12 V. GORBOUNOV AND M. MAHOWALD
We denote by ff*(C) the inverse image of C under ff, and by f(C) the
curve, defined by the equation C(f1(x; y); f2(x; y)) = 0.
Set e0 = 1 and ep-1 = 0 and denote by [m] the reduction of the
integer m mod p. With this notation we have ei [i + 1] mod I,
0 i p - 1.
Recall that the group Gp-1 is isomorphic to Zp o Z=(p - 1)2. The
last can be viewed as a group of automorphisms over Fpp-1of the curve
C , defined in section 4. Choose i from the set {1; : :;:p - 1}. One can
see directly that the substitution a of order p
x1 = x + i
y1 = y
and b of order (p - 1)2
x1 = ip-1x
y1 = ipy
2
where i(p-1) = 1, generate this group. A general form of a trans-
formation a of C, which reduces mod I to the automorphism a of C
is:
x1 = vp-1ix + ri
y1 = vpiy
where ri i and vi 1 mod I, and a general form of a transformation
b of C, which reduces mod I to an automorphism of C of order (p - 1)2
is
x1 = vp-1x
y1 = vpy
where vi 1, ri i, v i mod I. We can always assume that
ip-1 = k-1 were k is a generator of Fxp.
Proposition 6.4. For every g 2 Gp-1 there is a W (Fpp-1)-linear au-
tomorphisms ffg of the ring E, and a transformation g of the curve C,
such that
ff*g(Cp) = g-1 (Cp)
Proof. Since Gp-1 is generated by a and b, we will look for automor-
phisms ffa and ffb of the ring E1, and transformations a and b of the
form described above, such that
ff*a(C) = a-1(C); ff*b(C) = b-1(C)
It is enough to look at the images of the marked points
(0; 0); (1; 0); (e1; 0); : :;:(ep-2; 0)
FORMAL COMPLETION OF THE JACOBIANS OF PLANE CURVES 13
of C under ffa, ffb and a, b. If we match the images of these points in
the following way:
0 = vp-1ie[-1-i]+ ri 1 = vp-1e[k-1]
1 = vp-1ie[-i]+ ri ffb(ej) = vp-1e[(j+1)k-1]
ffa(ej) = vp-1ie[j-i]+ ri
1 j p - 2, we can solve the resulting equations for ffa and ffb.
From the way we solved these equations it is clear that the solution
is unique. In terms of the roots e1; : :;:ep-2 these automorphisms are
given by the formulas:
e[j-i]- e[-i-1] e[(j+1)i-1]
ffa(ej) = ______________ ffb(ej) = _________
e[-i]- e[-1-i] e[k-1]
_1_ _1_
ffa(u) = (e[-i]- e[-1-i])p-1uffb(u) = (e[k-1])p-1u
__
|__|
_1_
Remark 6.5. The series (1 + x)p-1 exists in Zp[[x]].
Corollary 6.6. The formal group law FC has the Lubin-Tate property
with respect to Gp-1.
Proof. Let g 2 Gp-1 and its lift, obtained in 6.4, be represented by the
following substitution
x1 = vp-1x + r
y1 = vpy
Expanding this transformation we obtain a series g(z1), such that z =
g(z1).
Recall that the series !0;1is the expansion with respect to the pa-
x dx g
rameter z = __ of the differential ___ on the curve Cp. Denote by !
y y
x1
the expansion with respect to the parameter z1 = ___of the differential
y1
dx1_
on the curve g(Cp). Then
y1
dx_ d(v1-p(x1 + r)) dx1
= _______________ = v ____
y v-p y1 y1
Expanding both sides with respect to the parameter z1, we obtain that
v-1 (g)0(z1)!0;1(g(z1)) = !g(z1)
.
14 V. GORBOUNOV AND M. MAHOWALD
dx dx1
On the one hand, the proposition 6.4 implies that ff*g__ = ____,
y y1
therefore, after expanding both sides into series we obtain
ff*g!0;1(z1) = !g(z1)
To finish the proof we need to formally integrate both the identities for_
!g(z1), and note that v = (g)0(0). |__|
Corollary 6.7. Let F be a formal group law and fl : FC ! F be a
strict isomorphism of formal group laws, both defined over E1, then F
has the Lubin-Tate property with respect to Gp-1. The representation
of Gp-1 defined by F coincides with the one defined by FC .
Proof. Let g 2 G, then there are g and ffg, described above, such that
g-1 (FC ) = ff*gFC . Then we have (fl O g O ff*g(fl-1 ))-1F = ff*gF Since fl
__
was a strict isomorphism, we have (ff*g(fl)g(fl-1 ))0(0) = (g-1 )0(0). |__|
7. On the properties of the one dimensional summand
Since our base ring E is a Z(p)-algebra, FC is strictly isomorphic
over E to a p-typical formal group law, which we denote by GC . We
also denote by GC (FC ) the reductions of GC (FC ) modulo the ideal
I, and by G0C(FC0) the reductions of GC (FC ) modulo the ideal I0 =
(u1; : :;:up-2). So GC and FC are formal group laws defined over Fp,
and G0Cand FC0are formal group laws defined over Zp.
Remark 7.1. According to [1], (16.4.14), logGC is obtained from logFC
by crossing out the terms mizi with i 6= pk. Moreover, if OE(z) 2
E[[z]] is the strict isomorphism between GC and FC , then logGC (z) =
logFC(OE(z)).
The purpose of this section is to prove the theorem
Theorem 7.2. The formal group law GC is a universal Lubin-Tate lift
of a formal group of height p - 1 over Fp.
The proof will follow directly from the next two propositions.
Proposition 7.3. There is an automorphism g of GC (x; y) of order
p, defined over Fp, such that g(z) = z + zp mod degree p + 1.
Proof. The formal group laws GC and FC are strictly isomorphic over
E by construction, therefore GC and FC are isomorphic over Fp. We
claim that there is an isomorphism between GC and FC , such that as
a series in z it is equal to z mod degree p + 1.
FORMAL COMPLETION OF THE JACOBIANS OF PLANE CURVES 15
Note that logG0C(z) = logF0C(OE0(z)), where OE0(z) is the reduction of
OE(z) mod I0. The logarithm of FC0is the formal integral of the following
differential form:
(1 - p)dz
! = _________________ (7.4)
1 + (p - 1)zwp-2
where the series w(z) obeys the following functional equation:
w = zp - zwp-1
This functional equation implies that w is a series in powers of z congru-
ent to p modulo (p - 1)2 and, therefore, ! = !(z) dz is the differential
form such that !(z) is a series in powers of z congruent to zero modulo
(p - 1)2. From this we conclude that the series logF0Cand logG0C are
congruent to z mod degree p + 1. Therefore, OE0(z) is congruent to z
mod degree p + 1. Its reduction mod p gives an isomorphism between
GC and FC with the stated above properties.
FC has an automorphism of order p, such that as a series, it is con-
gruent to z mod degree p + 1. Take, for example, a1 from 6.4, expand
it into a series with respect to a local parameter x_yand reduce the
__
resulting series mod I. This proves the proposition. |__|
Proposition 7.5.
ti pi
logGC = z + p-2i=1_uiz (7.6)
p
ti 2 Zxp, mod I20and mod degree pp-1. This implies that GC is a
universal deformation of the formal group law GC .
Proof. Recall that the logarithm of GC is obtained in the following
way: choosing a local parameter z as in 4.6 we obtain the following
dx
formula for the expansion of the differential ! = ___ on the curve Cp.
y
If ! = !(z) dz, then:
(1 - p)
!(z) = ______________________________________p-2 (7.7)
1 - (i=1iuizp-iwi-1 + (p - 1)bzwp-2)
where w satisfies the following functional equation
w = zp + p-2i=1uizp-iwi+ bzwp-1 (7.8)
Integrate !(z) and cross out the terms mizi with i not equal to a power
of p.
We claim that 7.6 holds for some ti 2 Qp, mod I20. This follows from
the following facts:
a. In the series w(z) and !(z) , reduced mod I0, the powers of z are
16 V. GORBOUNOV AND M. MAHOWALD
congruent to p and zero mod (p - 1)2 respectively.
b. In the series w(z) the coefficient of zk is equal to ui mod degree two
if and only if k = p - i + pi or zero mod (p - 1)2.
c. In the series !(z) the coefficient of zk is equal to ui mod degree two
if and only if k is congruent to pi - i or zero mod (p - 1)2.
d. pi- 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 ti 2 Zxp. Examining the above formula for the
dx x
expansion of the differential ___, we easily see that t1 2 Zp . Suppose
y
we have proved the proposition for j < i. Since FC is isomorphic to
GC , the corollary 6.7 implies that GC has the Lubin-Tate property with
respect to Gp-1. In particular, there is a lift g of g from 7.3, such that
ff*g(GC ) = g(GC ):
For logGC we get the identity
logGC (g(z))
ff*g(logGC )(z) = ____________
g0(0)
and taking into account that g(z) = z + zp mod degree p + 1, we see
that
ff*g(tiui) = tiui+ ti-1ui-1;
mod I20, and mod p; u1; : :;:ui-2, and ti-1 2 Zxp. On the other hand,
the formulas 6.4 tell that
ff*g(ui) = ui+ ci-1ui-1
ci-1 2 Zxp, mod p; u1 : :;:ui-2, and I20. This implies that ti must be a
unit in Zp.
The Proposition 1:1 from [8] implies now that GC is a universal __
deformation of GC . |__|
The following corollary shows that the formulas from 6.4 can be used for
calculations with the cohomology theories represented by the spectra
EOp-1.
Corollary 7.9. The representation of Gp-1 defined by FC is the Lubin-
Tate representation.
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