TOWARD THE HOMOTOPY GROUPS OF THE
HIGHER REAL K-THEORY EO2
VASSILY GORBOUNOV AND PETER SYMONDS
Abstract. In this note we compute the initial term of the spectral
sequence which converges the homotopy groups of the higher real
K-theory EO2 at the prime 3. We also describe precisely the way
in which the integral modular forms are embedded in this initial
term.
1. Introduction
Our main result is the calculation of the initial term of the spectral
sequence H*(G; E)Gal ) ss*EO2, which converges to the homotopy
groups of the higher real K-theory EO2 at the prime 3. Here G is the
group Z=3 o Z=4 and E is an infinitely generated module for G over
Z3, which arises from the theory of formal groups. We also show how
the integral modular forms of Deligne appear naturally in this initial
term. This calculation was originally sketched by Hopkins and Miller
[5], but the details were never published, so we have proceeded by our
own methods.
For each prime p and natural number n there is a p-adic Lie group
Sn, which is the group of absolute endomorphisms of the Honda formal
group law F over Fp. There is also a ring E0n, which can be considered
as the moduli space of lifts of F to a formal group law of height n
over a p-local ring. Thus Sn acts on E0n. This action is extended to an
action on En = E0n[u; u-1]. According to the Morava Change of Rings
Theorem, there is a spectral sequence H*c(Sn; En)Gal ) ss*LK(n), where
LK(n) is the localization of the sphere spectrum at the nth Morava K-
theory K(n) [7].
Morava [10 ] introduced a spectrum En such that ss*En = En. Later,
Hopkins and Miller [5] sketched how to construct a nice action of Sn
on En that induces the original action on En.
It is known that if n = p - 1, then Sn has a unique maximal finite
subgroup Gn, up to conjugation. When p = 3 and n = 2, this is the
group G of order 12 mentioned above. Hopkins and Miller define EOn
to be the homotopy fixed point spectrum of the action of Gn on En,
so there is a spectral sequence H*(Gn; En)Gal ) ss*EOn. When p = 2
1
2 VASSILY GORBOUNOV AND PETER SYMONDS
and n = 1, EO1 turns out to be KO, ordinary real K-theory, whilst
K(1) is ordinary complex K-theory. This justifies the name "higher
real K-theories". When n = 2 this construction can be generalized to
give an integral version of elliptic cohomology, which agrees with the
usual elliptic cohomology when 6 is inverted [6]. The spectrum which
represents this cohomology theory is called the spectrum of topological
modular forms, TMF .
The calculation presented here is for the case p = 3 and n = 2,
and can be viewed as a computation of the initial term of the spectral
sequence which converges to ss*TMF(3). It has the interesting property
that (EG )Gal , which is part of the initial term, is related to the ring
of integral modular forms of Deligne [1]. A problem here is that the
Morava Change of Rings Theorem concerns a specific formal group of
height n over the field Fp, the Honda formal group. When n = 2,
a supersingular elliptic curve yields another formal group, which is
naturally connected to the integral modular forms. These two formal
groups only become isomorphic after a field extension.
The authors wish to thank Mark Mahowald for helpful discussions.
2. The moduli space
As a general reference for this section we recommend [3]. Let Fu1 be
the formal group law over Z3 which is obtained from the usual universal
commutative 3-typical formal group law over Z3 [v1; v2; : :]:by setting
v1 = u1, v2 = 1, vi = 0, i 3.
Fu1(x; y) = x + y - u1(x2y + xy2) - (x6y3 + x3y6) + : :::
Its reduction modulo 3 and u1 is
F (x; y) = x + y - (x6y3 + x3y6) + : :::
Now S2 F9 [[x]] is the ring of absolute endomorphisms of F : it
also can be characterized as the group of units in the maximal order
O of a certain division algebra. This O, in turn, can be described as
follows. Let W be the ring of Witt vectors over F9, ie the integers
of the unramified extension of degree 2 of Q3. Then S2 contains an
element S such that, additively, O ~= W SW , and the multiplication
is described by S2 = 3, wS = Sw , w 2 W . Thus S2 contains a finite
iS - 1 2
group G ~=Z=3 o Z=4 generated by a = _______, of order 3, and b = i ,
2
of order 4, where i 2 W is a primitive 8th root of unity.
Now we must identify these elements in F9[[x]]. Any w 2 W corre-
sponds to f-1 (wf(x)), where f(x) is the logarithm of F ,
HOMOTOPY GROUPS OF THE HIGHER REAL K-THEORY 3
f(x) = x + 1_3x9 + 1_9x81+ : :::
Therefore f(ix) = if(x), and so multiplication by i corresponds to ix
and b corresponds to i2x. Now S corresponds to x3 and iS - 1 has
series
ix3 +F (-x) = -x + ix3 + i3x15+ : :::
Also [2]F = x +F x = -x + x9 + : :,:so [1_2]F = -x - x9 - x81 - : : :
and thus a has series x - ix3 + x9 - i3x15+ : :.:
A theorem of Lubin and Tate [9, 3] shows that the group S2 acts
continuously on E0 = W [[u1]] according to the formula
Fu1(^gffx; ^gffy) = ^gffFg(u1)(x; y);
where g 2 S2 F9 [[x]], ^gis a lift of g to E0[[x]], and ff 2 E0[[x]] is
such that its reduction to F9[[x]] is the identity. Here g(u1) 2 E0. Both
g(u1) and ^gff are uniquely determined by this equation, which we shall
simplify by writing "g= ^gff so that
Fu1("gx; "gy) = "gFg(u1)(x; y):
The action can be extended to E = E0[u; u-1] by the formula g(u) =
("g0(0))u.
Proposition 2.1. Suppose that g 2 S2, "g= fl1x + fl2x2 + fl3x3 + : :,:
fli 2 E0. Then g(u1) = fl21u1 + 3fl3fl-11and fl2 = 0 in E0.
Proof. Notice that
Fu1("gx; "gy) = "gx + "gy - u1fl31(x2y + xy2)
modulo degree 4 in x and y and
"gFg(u1)(x; y) = fl1(x + y - g(u1)(x2y + xy2))+
fl2(x2 + y2 + 2xy) + fl3(x3 + y3 + 3(x2y + xy2))
modulo degree 4 in x and y. Examing the coefficients of x2y and xy __
yields the result. |__|
Corollary 2.2. All of S2 fixes u-2u1 2 E .
3. The Action of G on E
Let E = F9 W E = F9[[u1]][u; u-1].
4 VASSILY GORBOUNOV AND PETER SYMONDS
Proposition 3.1. In E we have:
a(u1) = u1 + i3u21 mod u31
and
a(u) = u(1 - i3u1) mod u21:
Proof. Let "a(x) = ff1x + ff2x2 + : :,:ffi 2 E0 . Then ff1 = 1, ff3 = -i,
ff9 = 1 modulo u1, and the other ffi, i 9 satisfy ffi = 0 modulo u1.
We work in E throughout.
Fu1("a(x); "a(y)) = "ax+"ay-u1[(x-ix3)2(y-iy3)+(x-ix3)(y-iy3)2]-
ff91(x6y3 + x3y6)
modulo u21and degree 4 in x and y.
The coefficient of x6y3 is -1 + i3u1 modulo u21. Now
"aFa(u1)(x; y) = "a[x + y - a(u1)(x2y + xy2)] - ff1(x6y3 + x3y6)
modulo degree 10 in x and y.
The only terms in "a which could possibly yield an x6y3 term in
"a[x+y-a(u1)(x2y+xy2)] without a u21(remember that a(u1) is divisible
by u1) are the powers 1, 3 and 9. It is easy to check that these do not
infact do so, hence the coefficient of x6y3 is -ff1. We have -ff1 =
-1 + i3u1 mod u21and so ff21= 1 + i3u1 mod u21. The result follows __
from Proposition 2.1 and the definition of the action on u. |__|
Proposition 3.2. We have b(u1) = -u1, b(u) = i2u in E.
__
Proof. iFi2u1(x; y) = Fu1(ix; iy) by construction. So "b(x) = i2x. |__|
4. The Module Structure of E
Let k = F9 and consider the group algebra of . Let A = a - 1 2
k. Then k = k[A]=(A3). We know that a(u1) = u1 + i3u21+ Xu31,
X 2 E0, so Au1 = i3u21+ Xu31, and A2u1 = -i2u31mod deg 4.
Let y = A2u1 6= 0. Since y is invariant under a, so is k[[y]] E0 .
Let F be the k-submodule generated by u1, which has k-basis u1,
Au1, A2u1. Then as F has dimension 3 it must be free and E0 =
k (F k[[y]]). Note that b(y) = -y, and that F is easily checked to
be a kG-module.
It will be convenient for us to discuss the ring LE = E[_1_u1], and
similarly LE . Note that LE0 = F k[[y]][_1y].
By Proposition 2.1, u2LE0 contains an invariant element u2u-11, and
multiplication by u2u-11yields an isomorphism of kG-modules LE0 !
HOMOTOPY GROUPS OF THE HIGHER REAL K-THEORY 5
u2LE0 . The invariants of u2LE0 under are therefore u2u-11k[[y]][_1y]
so occur in u1-filtration -1 modulo 3. Now if w 2 uLE0 is invariant
under a and is an eigenvector for b, then so is w2, and so w must have
u1 filtration 1 modulo 3. After multiplying by a suitable power of y
we may assume that w is in filtration 1, so its leading term is uu1 and
the eigenvalue under the action of b must be -i2, ie b(w) = -i2w.
Multiplication by wr yields an isomorphism of k-modules LE0 !
urLE0 .
Finally set x = y-1 w3, which has u1-filtration 0. To sum up we have:
A = a-1, y = A2u1, Aw = Ax = Ay = 0, and bu1 = -u1, bw = -i2w,
bx = -i2x, by = -y.
This is represented pictorially in figure 1. The blocks in unbroken
lines represent indecomposable summands of E as a kG-module, and
each block has a k-basis with u1-filtration and u-grading correspond-
ing to the points in the block with integral rectangular coordinates.
The dotted lines indicate the blocks of LE . Each complete block of
dimension 3 is projective.
The picture has vertical periodicity 3 as an -module (given by
multiplication by x) and periodicity 12 as a G-module (multiplication
by x4).
This decomposition into blocks also lifts to E and to LE. This is
because there is a one-to-one correspondence between the finite dimen-
sional indecomposable modules for G over W and over k. It can be
seen explicitly as follows: Let m be the left-most basis element in of
any block of LE . Lift m to an element "m of LE that is an eigenvector
under the action of b. Then the W G-lattice generated by "m is a direct
summand of LE that is equal to the original block modulo 3.
Let B denote the submodule of E with basis {1; wy-1 Au1; w}. In
other words B consists of the two incomplete blocks in u-grading 0 and
1. Then E = B k[x; 1_x] (projective).
Now x 2 E lifts to "x2 E such that a"x= "x, b"x= -i2"x, and B lifts
to B". Thus E = (B" Z3["x; 1_"x]) (projective).
Remark 4.1. We have a product of projectives because E involves
power series in u1. Over k a product of projectives is also projective,
but over W this is not true, since since it is not even projective over
W (According to [8], Zp is not a free Zp-module when is infinite).
However (projective) is at least cohomologically trivial.
5. Cohomology
In Tate cohomology, H"*(; E) = ^H*(; B) Z3["x; "x-1], where we
identify "xwith its image in H^0(; E).
6 VASSILY GORBOUNOV AND PETER SYMONDS
The cohomology of "Bis easy to calculate: One part is ^H*(; W ) =
k[z; z-1 ], deg z = 2, and b(z) = -z, as is known from the calculation
of the cohomology of the dihedral group of order 6. The cohomology of
the other part is calculated using dimension shifting in the block of LE
that contains it to reduce to the case of the 1-dimensional W G-lattice
on wy-1 u1. The result is a free k[z; z-1 ]-module on a generator t of
degree 1, and bt = -i2t.
Hence H^*(; E) = k["x; "x-1; z; z-1 ] E(t).
Now H^*(G; E) = ^H*(G; E)**, and calculating the invariants yields:
Theorem 5.1. H^*(G; E) = F9["x4; "x-4; "x2z; ("x2z)-1] E("x-1t)
There is an action of Gal(W=Z3) = ~=Z=2 on G by conjugation
with S. There is also an action on E via the action on the coefficients
of the series. But conjugation by S has the same effect on a as conju-
gation by b, so does not affect H^*(G; Z3), and thus Gal(W=Z3) acts on
^H*(G; E) via its action on E only. Now H^*(G; W ) = ^H*(G; Z3) W ,
so z could have been chosen invariant under the action. Consider the
action of oe on "x. We seek a replacement ^xfor x that is invariant under
oe + 1 oe - 1
oe. Now "x= (______)x + (______)x, so at least one of the two terms on
2 2
the right hand side, taken modulo 3, has u1-filtration 0. If it is the
oe + 1 2 oe - 1
first, take ^x= (______)x. If it is the second, take ^x= i (______)x. It
2 2
is easy to check that ^x4is still invariant under G. A similar argument
applies to ^x-1t and we obtain
Theorem 5.2. We have:
^H*(G; E)Gal = F3[^x4; ^x-4; ^x2z; (^x2z)-1] E(^x-1t);
H*(G; E)Gal = F3[^x4; ^x-4; ^x2z] E(^x-1t); * > 0:
6. Invariants
In E , y is not invariant under G, but y2 is. Also r = u-2u1 is
invariant. Together with x, these generate the invariants (see figure 2).
Theorem 6.1. EG = F9[[y2]][x4; x-4; r]=(r6 = x4y2)
There are no invariant elements with odd u-grading.
When we lift this to W , each invariant element lifts to an invariant
element in the corresponding block of LE, but its u1-filtration may
increase by 1 or 2. In particular y lifts to "y2 E, invariant under a,
b"y = -"y, and we have seen that x4 lifts to "x4 2 E. But r cannot
HOMOTOPY GROUPS OF THE HIGHER REAL K-THEORY 7
lift to an invariant element of E because the 2-dimensional integral
representation above it contains no non-zero invariants. On the other
hand p = r2 lifts to "pand q = r3 lifts to "q= "x-2"y. Thus the image
of EG in EG misses precisely the part with basis the monomials rxi
(this could also be seen from the long exact sequence in cohomology
for 3E ! E ! E ). By construction, "p3= "q2(mod 3).
Theorem 6.2. EG = W [["y2]]["x4; "x-4; "p; "q], where "q2= "x-4"y2, and "p3-
"q2= 3ffi for some ffi 2 EG .
(see figure 3). As before,
Theorem 6.3. (EG )Gal = Z3[[^y2]][^x4; ^x-4; ^p; ^q], where ^q2= ^x4^y2, and
^p3- ^q2= 3ffi for some ffi 2 (EG )Gal.
7. The Ring Structure of the Invariants
Theorem 7.1. The invariant elements p and q can be chosen in such
a way that p3 - q2 is divisible by 33. This is the best possible: they
can not be chosen so as to make p3 - q2 divisible by 34. In fact, if
ffi = (p3 - q2)=33 2 E, then the leading term of ffi is a unit times u-12.
Proof. The first part will follow from section 8 using modular forms,
but we give an elementary proof here. Chose p 2 E0u-4 to be any
element invariant under both G and Gal that reduces to r2 modulo 3.
Normalize it so that the coefficient of u21u-4 is 1.
Then p = (u21+ 3s)u-41for some s 2 E0 which is an even function of
u1, and so
p3 = (u61+ 3u41s)u-12 mod 33.
2 -1 3 2 3
Let "q= (u31+ 3_2u1s)u , so that p - "q= 0 (mod 3 ).
Now since p is invariant, so is "q2(mod 33). In other words a("q2)-"q2=
0 (mod 33) or 33|(a("q) - "q)(a("q) + "q). But a("q) + "q= (2u31+ : :):u-6
(mod3), so we must have 33|a("q) - "q.
Now q" 2 E0u-6, and from the decomposition of E as a sum of
indecomposables we can see that H1(G; E0u-6) = 0. Let H = G o ,
so that H1(H; E0u-6) = 0.
The long exact sequence for cohomology becomes
0 ! (E0u-6)H ! (E0u-6)H ! (E0u-6=33)H ! 0;
and so "qlifts to an invariant element q 2 E0u-6 such that q = "q(mod
33) and p3 - q2 = 0 (mod 33).
To show that this is the best possible power of 3, note that p must
be an even function of u1 and that q must be odd, so it suffices to prove
that the constant term (in u1) of p is not divisible by 32.
8 VASSILY GORBOUNOV AND PETER SYMONDS
Let m denote the maximal ideal of E0: it is generated over W by 3
and u1. Then p = (c + u21)u-4 (mod m3) for some constant c, and an
argument similar to the proof of 3.1, but working modulo m2 instead of
3, shows us that a(u1) = -3i + u1 (mod m2), so a(u1) = -3i + u1 + e,
say, where the constant term of e is divisible by 32.
Also a(u) = ff1u (mod m), where ff1 = 1 (mod m). Thus
a(p) = (c + (-3i + u1 + e)2)ff-41u-4 mod m3:
But since p is invariant this must be equal to (c + u21)u-4 (mod m3),
and in particular the constant terms should be equal (mod 33). Using
the subscript 0 to denote the constant term, this yields
(c + (-3i + e0)2)(ff1)-40= c (mod 33)
and hence
c((ff1)40- 1) = (-3i + e0)2 (mod 33):
Now 32|e0 and 3|((ff1)40- 1) so we see that if 32|c then 3|i, a contra-_
diction. |__|
According to 7.1, if p3 - q2 = 33ffi then the leading term of ffi is u-12,
up to a unit. This enables us to express ^x-4in terms of ffi and y2, hence:
Theorem 7.2.
(EG )Gal = Z3[[q2ffi-1 ]][ffi; ffi-1 ; p; q]=(p3 - q2 = 33ffi)
8. Modular Forms
In this section we describe a connection between the ring of integral
modular forms computed by Deligne [1] and the ring of invariants EG .
We recall some standard material about elliptic curves and modular
forms. An elliptic curve over a scheme S is a smooth morphism p :C !
S, whose geometric fibers are connected curves of genus one, together
with a section e :S ! C. We denote by !_C=S the invertible sheaf
p*(1C=S) on S. The following definition is taken from [1]:
Definition 8.1. An integral modular form of weight k 2 Z and level
one is a rule which assigns to any elliptic curve over any scheme S a
section f(C=S) of (!_C=S)k over S such that the following two condi-
tions are satisfied.
1. f(C=S) depends only on the S-isomorphism class of the elliptic
curve C=S.
2. The formation of f(C=S) commutes with arbitrary change of base
g :S0 ! S (meaning that f(C0=S0) = g*f(C=S).
HOMOTOPY GROUPS OF THE HIGHER REAL K-THEORY 9
We are interested in the situation in which the base S is an affine
scheme, S = Spec(R), R is a ring, and !_C=R is a free R-module of
rank one with basis !. Then any section of !_kC=Rcan be written as
f(C=R; !) . !k , where f is a rule which assigns to every pair (C=R; !)
consisting of an elliptic curve C over a ring R and a nowhere vanishing
section of 1C=Ron C, an element f(C=R; !) 2 R, such that the follow-
ing three conditions are satisfied:
1. f(C=R; !) depends only on the R-isomorphism class of the pair
(C=R; !).
2. f is homogeneous of degree -k in the second variable; for any
2 Rx , f(C=R; !) = -k f(C=R; !).
3. The formation of f(C=R; !) commutes with arbitrary extension of
scalars g : R ! R0 (meaning that f(CR0=R0; !0) = g(f(C=R; !))).
Let (C=R; !) be a pair of the type considered above. Choose coordi-
nate functions x and y such that the image of y in !_-1C=R !_-1C=R !_-1C=R
is !-1 !-1 !-1 and the image of x in !_-1C=R !_-1C=Ris !-1 !-1 .
Consider the corresponding Weierstrass equation
y2 + a1xy + a3y = x3 + a2x2 + a4x + a6
of the curve C. The symbols c4, c6 and denote the usual polynomials
in the ai's [1]. An elementary computation shows that the rules which
assign to the pair (C=R; !) the differential forms c4!4, c6!6 and !12
are indeed modular forms. Denote the ring of the integral modular
forms of level one by M.
Theorem 8.2. (Deligne) M = Z[c4; c6; ]=(c34- c26= 123)
Let us also recall the Serre-Tate Theorem. A good reference is [2].
Let R be a ring and I an ideal such that the ideal (p; I) is nilpotent.
Theorem 8.3. The following categories are equivalent:
a. The category A of elliptic curves over R and R-homomorphisms.
b. The category B of triples (F; C0; i) where F is a p-divisible group
over R, C0 is an elliptic curve over R=(p; I), i is an isomorphism
between the reduction of F modulo (p; I) and FC0. Morphisms are pairs
(r1; r2), where r1 is an R-homomorphism of p-divisible groups and r2 is
an R=(p; I)-homomorphism of elliptic curves which make the obvious
diagram commutative.
Denote by F G and Ell the functors which provide the equivalence
of the above categories.
The argument we will present below works for all primes, but for the
purposes of this paper we restrict ourselves to the case when the prime
p is equal to 3. Denote by Fsep the separable closure of F3 and by S
10 VASSILY GORBOUNOV AND PETER SYMONDS
the ring of integers of the maximal unramified extension of Z3, (which
has residue class field Fsep). From now on take R to be the ring S[[u1]].
Note that R is the inverse limit of rings Rn to which the Serre-Tate
Theorem applies. Take for example Rn to be R=(3; u1)n. Denote by
rn and pn the canonical homomorphisms Rn ! Rn-1 and R ! Rn
respectively. The ideal (3; u1) is invariant under G, so G also acts on
Rn.
Take C0 to be a supersingular elliptic curve over Fsep. There is only
one such up to isomorphism. Recall [1] that Aut(C0) is isomorphic to
G ~=Z=3 o Z=4, and denote by F0 the formal group of C0.
First we construct a map from M to Rn[u-1; u]G for each n. Ac-
cording to the classification of formal groups over Fsep, see for example
[3], F0 is isomorphic to F . Let i be such an isomorphism, and con-
sider the triple (Fu1; F0; i). Denote by Tn the triple (p*nFu1; F0; i). Let
r = (r1; r2) : Tn ! F G(Ell(Tn) be part of the natural isomorphism
idRn ! F G O Ell. Let Cn = Ell(Tn).
Lemma 8.4. There is a choice of coordinates on the elliptic curve Cn
over Rn such that the corresponding formal group law Fn is strictly
isomorphic to p*nFu1. Furthermore, this can be done in such a way that
Fn has the form
Fn(x; y) = x + y - u1(x2y + xy2) (mod deg 4 )
modulo 3.
Proof. Choose coordinates x and y on Cn. Then r1 becomes a series
over Rn such that r01(0) is an invertible element of Rn. Then
x0= (r01(0))2x;
y0 = (r01(0))3y
provide the required coordinates.
Next we want to show that we can make a choice of coordinates
such that the strict isomorphism between p*nFu1 and Fn has the form
r1(z) = z mod deg 3 and modulo 3. Then Fn has to be of the form given
in the statement of the lemma. Suppose that the strict isomorphism
had the form r1(z) = z + ffz2 mod deg 3 and modulo 3. An elementary
computation shows that the change of coordinates x0 = x and y0 =
y - ffx on Cn yields the change of the coordinate on Fn given by the
series Z(z) = z - ffz2 mod deg 3 and modulo 3. The resulting formal
group law is strictly isomorphic to p*nFu1 and the strict isomorphism_is
Z(r(z)), which is equal to z mod deg 3 and modulo 3. |__|
Let g 2 G. We denote by the same letter the element of AutR associ-
ated to it in Section 2. Let g(x) be the representative of g in Fsep[[x]]
HOMOTOPY GROUPS OF THE HIGHER REAL K-THEORY 11
and denote by "g(x) the Lubin-Tate lift of g(x). Recall that it has the
following property
"g-1Fu1("g(x); "g(y)) = g*F(u1)(x; y):
Denote by "gn(x) the series "g(x) with the coefficients reduced modulo the
ideal (3; u1)n. The following proposition is an immediate consequence
of the Serre-Tate Theorem. We are using standard notation from the
theory of elliptic curves (see [1]).
Proposition 8.5. For every g 2 G the elliptic curves Cn and g*Cn =
C0nare isomorphic. In terms of the Weierstrass equations of Cn and
C0nsuch an isomorphism is given by a substitution ln(g):
x = a2x0+ b
y = a3y0+ a2cx0+ d; a; b; c; d 2 Rn:
With coordinates on Cn chosen as in 8.4 and corresponding coordinates
on C0n, the substitution ln(g) which provides an isomorphism between
C0nand Cn has the property "g0n(0) = a-1.
Proof. If F G(Cn) = (F1; E1; i1), then by definition of F G we have
F G(g*Cn) = (g*F1; E1; i1). There is the following diagram, all the
maps of which are isomorphisms in the category B which corresponds
to the ring Rn:
Tn - - - g*Tn
? ("gn;g) ?
(r1;r2)?y g*(r1;r2)?y (8.6)
(F1; E1; i1) (g*F1; E1; i1):
This shows that F G(Cn) and F G(g*Cn) are isomorphic in B, therefore
we obtain the required Rn-isomorphism ln(g) of Cn and g*Cn from
Serre-Tate Theorem and we have a commutative diagram
Tn - - - g*Tn
? ("g;g) ?
(r1;r2)?y g*(r1;r2)?y (8.7)
(F1; E1; i1)- - - - - - -(g*F1; E1; i1):
(FG(ln(g));g)
By 8.4 we can choose coordinates on Cn such that the series r1 provides
a strict isomorphism of the corresponding formal group laws. For this
choice of coordinates F G(ln(g)) becomes a series such that its derivative
at zero is equal to a-1, as follows from the definition of the functor_
F G. |__|
12 VASSILY GORBOUNOV AND PETER SYMONDS
We now construct a ring homomorphism jn :M ! Rn[u; u-1]. Let
Cn be the elliptic curve defined above. Choose coordinates on Cn as
in 8.4 and denote by ! the corresponding section of !_. Let f be a
modular form of weight k. Define jn(f) to be equal to f(Cn; !)uk.
Corollary 8.8. jn(M) Rn[u; u-1]G .
Proof. Indeed, taking into account the properties of modular forms and
the formula for the action of ln(g) on the invariant differential from [1],
we obtain:
g(f(Cn; !)uk) = f(g*Cn; !)g(uk) = f(Cn; ln(g)-1!)a-k uk =
f(Cn; a-1!)a-k uk = f(Cn; !)uk:
__
|__|
It is hard to identify the images of the generators of M in Rn[u; u-1]
directly, because we know very little about the curve Cn. We can
however say something in this direction if we change the base to Rn =
Rn F3. First we need to recall the structure of the ring of modular
forms in characteristic 3 as computed in [1].
Theorem 8.9. The ring M3 of modular forms in characteristic 3 is
isomorphic to
F3[b2; ]
and c4 = b22, c6 = -b32.
An argument like the one presented above provides us with a map
j(3)n:M3 ! Rn [u; u-1]G :
Proposition 8.10. j(3)3(b2) = u1u2
Proof. From Proposition 8.4 we know that there is a choice of coordi-
nates over Rn for Cn such that its formal group law has the form
Fn0(x; y) = x + y - u1(x2y + xy2) (mod deg 4 )
modulo 3. On the other hand according to a computation presented in
[11 ], p115, this means that the element b2 of the Weierstrass equation__
which corresponds to this choice of coordinates is equal to u1. |__|
Finally we show that the maps jn-1 and rnjn are equal for all n
and, therefore, that we have constructed a map j :M ! R[u-1; u]G .
Indeed, since the triples Tn-1 and r*nTn coincide, the elliptic curves
Cn-1 and r*nCn are isomorphic over Rn-1 by the Serre-Tate theorem.
Furthermore, since this isomorphism induces a strict isomorphism of
the formal groups laws of Cn-1 and Cn (with respect to the coordinates
chosen), the maps jn-1 and rnjn coincide by construction. The same
HOMOTOPY GROUPS OF THE HIGHER REAL K-THEORY 13
argument shows that the maps j(3)nand rnj(3)nare equal, where rn is
the reduction of rn modulo 3.
This implies that j(c4) and j(c6) are suitable candidates for the ele-
ments p and q of Theorem 7.2, and since (c34- c26)=33 = 43, Theorem
7.1 shows that j() is a suitable replacement for ffi. Identifying M with
its image under j, we can rewrite 7.2 as:
Corollary 8.11. (SE)G = S[[c26-1]][; -1; c4; c6]=(c34-c26= 123):
This can be expressed more succinctly as follows.
Corollary 8.12. (S E)G is isomorphic as a ring to M[-1] S
completed with respect to c26-1.
References
[1]P. Deligne, Courbes Elliptiques: Formulaire (d'apres J. Tate), in Modular
Functions of One Variable IV, Springer LNM 476, 53-74.
[2]V. Drinfeld, Coverings of p-adic symmetric domains(Russian), Func. Anal. i
Prilozen, 10, no. 2, (1976), 29-40.
[3]M. Hazewinkel, Formal Groups and Applications, Academic Press, 1978.
[4]M. J. Hopkins, B. H. Gross, The rigid analytic period mapping, Lubin-Tate
space, and stable homotopy theory, Bull. Amer. Math. Soc. 30 (1994), 76-86.
[5]M. Hopkins, H. Miller, Midwest Topology Seminar, Fall 1994.
[6]M. Hopkins, H. Miller, AMS Summer Research Institute, Seattle, 1996.
[7]M. Hopkins, M. Mahowald, H. Sadofsky, Constructions of elements in Picard
Groups, in Topology and Representation Theory, Contemporary Mathematics,
158, (1994).
[8]Kaplansky, Infinite Abelian Groups, Univ. Michigan Press 1954.
[9]J. Lubin, J. Tate, Formal moduli for one parameter formal Lie groups, Bull.
Soc. Math. France 94 (1966), 49-60.
[10]J. Morava, Noetherian localisation of categories of cobordism comodules, An*
*n.
Math., 121 (1985), 1-39.
[11]J. H. Silverman, The Arithmetic of Elliptic Curves, Springer Graduate Texts
in Mathematics, 106, 1986.
Department of Mathematics, University of Kentucky, Lexington,
KY 40506-0027
E-mail address: vgorb@ms.uky.edu, symonds@ms.uky.edu
14 VASSILY GORBOUNOV AND PETER SYMONDS
HOMOTOPY GROUPS OF THE HIGHER REAL K-THEORY 15
16 VASSILY GORBOUNOV AND PETER SYMONDS
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