Quasiisogenies and Morava stabilizer groups
Niko Naumann
Abstract
For every prime p and integer n > 3 we explicitly construct an abelian vari*
*ety A=Fpn
of dimension n such that for a suitable prime l the group of quasiisogenie*
*s of A=Fpn of
lpower degree is canonically a dense subgroup of the nth Morava stabilize*
*r group at p.
We also give a variant of this result taking into account a polarization. T*
*his is motivated
by a perceivable generalization of topological modular forms to more genera*
*l topological
automorphic forms.
For this, we prove some results about approximation of local units in maxim*
*al orders
which is of independent interest. For example, it gives a precise solution *
*to the problem
of extending automorphisms of the pdivisible group of a simple abelian var*
*iety over a
finite field to quasiisogenies of the abelian variety of degree divisible *
*by as few primes as
possible.
1. Introduction
One of the most fruitful ways of studying the stable homotopy category is the c*
*hromatic approach:
After localizing, in the sense of Bousfield, at a prime p, one is left with an *
*infinite hierarchy of primes
corresponding to the Morava Ktheories K(n), n > 0, see [R2 ]. The successive l*
*ayers in the resulting
filtration are the K(n)local categories [HS ] the structure of which is govern*
*ed to a large extend by
(the continuous cohomology of) the nth Morava stabilizer group Gn,_i.e. the au*
*tomorphism group
of the onedimensional commutative formal group of height n over Fp. A fundamen*
*tal problem in
this context is to generalize the fibration
LK(1)S0 ! EhF1! EhF1,
c.f. the introduction of [GHMR ], to a resolution of the K(n)local sphere for*
* n > 2. This has been
solved completely for n = 2 and in many other cases as well by clever use of ho*
*mological algebra for
Gnmodules [GHMR ],[H ]. Recently, pursuing a question of M. Mahowald and C. R*
*ezk, M. Behrens
was able to give a modular interpretation of one such resolution in the case n *
*= 2 [B1]:
A basic observation is that G2 is the automorphism group of the pdivisible gro*
*up of a super
singular elliptic curve E over a finite field k. Hence it seemed plausible, an *
*was established in loc.
cit., that the morphisms in the resolution of LK(2)S0 should have a description*
* in terms of suitable
endomorphisms of E. A key result for seeing this was to observe that for suitab*
*le primes l
_______________________________________________________________________________*
*__
2000 Mathematics Subject Classification primary: 55P42, secondary: 14L05
Keywords: Morava group, maximal orders, pdivisible groups
Niko Naumann
` ~ ~'*
1
(1) Endk(E) _ G2
l
is a dense subgroup [BL , Theorem 0.1].
One of our main results, Theorem 12, is the direct generalization of (1) to arb*
*itrary chromatic level
n > 3 in which E is replaced by an abelian variety of dimension n which is know*
*n to be the minimal
dimension possible.
In Theorem 13 we give a sharpening of Theorem 12 in which on the lefthandside*
* of (1) we only
allow endomorphisms which are unitary with respect to a given Rosatiinvolution*
*. The motivation
for this stems from current attempts [B2] to bring suitable Shimura varieties t*
*o bear on problems
of arbitrary chromatic level, generalizing the role of topological modular form*
*s for problems of
chromatic level at most two.
In deriving these results we take some intermediate steps as we now explain rev*
*iewing the individual
sections in more detail:
In subsection 2.1 (resp. 2.2) we reduce the problem of approximating a local un*
*it of a maximal
order in a skewfield D, finitedimensional over Q, (and carrying an involution*
* of the second kind)
to a similar approximation problem for numberfields, Theorem 1 (resp. Theorem 6*
*). The basic tool
here is strong approximation for an inner (resp. outer) form of Sld.
The resulting problem in the numberfield case is solved in subsection 2.3 using*
* class field theory,
Theorem 7.
In subsection 3.1 we explain the application of the results obtained so far to *
*the following problem:
Given a simple abelian variety A over a finite field k one would like to extend*
* an automorphism of
the pdivisible group A[p1 ] of A to a quasiisogeny of A the degree of which s*
*hould be divisible by
as few primes as possible. The additional ingredient needed here is the classic*
*al result of J. Tate
that End k(A) Z Zp ' Endk(A[p1 ]).
Finally, subsections 3.2 and 3.3 are devoted to the proofs of Theorems 12 and 1*
*3 reviewed above.
Acknowledgements
I would like to thank U. Jannsen and A. Schmidt for useful discussion concernin*
*g subsection 2.3
and M. Behrens and T. Lawson for pointing out to me the abelian varieties used *
*in subsection 3.2.
2.Padic approximation
2.1 Forms of type Ad1
In this subsection we consider the problem of padically approximating local un*
*its of a maximal
order O D where D is a finite dimensional skewfield over Q. The center of D,*
* denoted k, is a
number field and we denote by d the reduced dimension of D, i.e. dimkD = d2. Th*
*e title of this
subsection refers to the fact that the relevant algebraic group will turn out t*
*o be Sl1(D), i.e. an
inner form of type Ad1. We denote by Ok k the ring of integers and note that*
* k \ O = Ok as an
immediate consequence of [D , VI,x11, Satz 7].
Recall that D is determined by its local invariants as follows [PR , 1.5.1]. Fo*
*r any place v (finite or
infinite) of k there is a local invariant invv(D) 2 1_dZ=Z Q=Z and invv(D) = *
*0 for almost all v.
For a given place v, we denote by kv the completion of k with respect to v. The*
*n Dv := D kkv is a
central simple kvalgebra which determines a class [Dv] 2 Br(kv) in the Brauer *
*group of kv. There
2
Quasiisogenies and Morava stabilizer groups
are specific isomorphisms
8
< Q=Z , v finite
ov : Br(kv) ~! 1_Z=Z , v real
: 2 0 , v complex
such that invv(D) = ov([Dv]). In particular, Dv is a skewfield if and only if *
*the order of invv(D) 2
1_
dZ=Z is exactly d.
We now fix a prime 0 6= p Ok at which we wish to approximate. There is a uniq*
*ue prime P O
above p [D , VI, x12, Satz 1] and we denote by OP the Padic completion of O, c*
*.f. [D , VI, x11].
To describe the denominators we allow the approximating global units to have, w*
*e fix a finite set
of places S of k such that
p 62 S and there exists a placev0 2 S such thatDv0is not a skewfie*
*ld.
We write Sfinfor the set of finite places contained in S and consider the ring *
*OSfinof Sfinintegers
Ok OSfin:= {x 2 k  v(x) > 0 for all finitev 62 S} k.
Since p 62 S we have OSfin Ok,p, the completion of Ok with respect to p. Thus
1 *
(2) X := {x 2 O*Sfin v infinite andinvv(D) = _ implyv(x) > 0} Ok,pand
2
(O Ok OSfin)* O*P.
Denoting by N the reduced norm of k D [PR , 1.4.1] we can state our first res*
*ult as follows.
Theorem 1. The closure of (O Ok OSfin)* inside O*Pequals
{x 2 O*P N(x) 2 O*k,plies in the closureXof}.
Example 2. 1) If d = 1, i.e. D is commutative, then X = O*Sfin= (O Ok OSfin)*,*
* O*P= O*k,pand
the assertion of Theorem 1 is tautological.
2) For k = Q and D a definite quaternion algebra, i.e. d = 2 and invv(D) = 1_2f*
*or the unique infinite
place v of Q, we can choose S = {l} for any prime l 6= p at which D splits, i.e*
*. invl(D) = 0. Then
O*Sfin= { 1} x lZ and X = lZ O*k,p= Z*p. For p 6= 2 we can choose l as above *
*such that in
addition X Z*pis dense and conclude that in this case O[1_l]* O*Pis dense. *
*For p = 2 we can
choose l such that the closure of X equals 1 + 4Z2 and conclude that the closur*
*e of O[1_l]* inside O*P
equals
ker(O*PN! Z*2! Z*2=(1 + 4Z2) ' { 1}),
c.f. Remark 11. In the special case in which D is the endomorphism algebra of a*
* supersingular
elliptic curve in characteristic p, i.e. invv(D) = 0 for all v 6= p, 1, this re*
*sult has been established
by different means in [BL , Theorem 0.1].
3) See Theorem 7 in subsection 2.3 for a further discussion of the closure of X*
* O*k,p.
3
Niko Naumann
The rest of this subsection is devoted to the proof of Theorem 1 which is an ap*
*plication of strong
approximation for algebraic groups.
The groupvalued functor G on Okalgebras R
G(R) := (O Ok R)*
is represented by an affine algebraic group scheme G=Spec(Ok). The reduced norm*
* N gives an exact
sequence of representable fppfsheaves on Spec(Ok)
(3) 1 ! G0! G N!Gm ! 1.
Proposition 3. The subgroup G0(OSfin) G0(Ok,p) is dense.
Proof.First note that G0=Spec(Ok) is represented by an affine algebraic group s*
*cheme, hence the
injectivity of the homomorphism G0(OSfin) ! G0(Ok,p) follows from the injectiv*
*ity of OSfin,! Ok,p.
Secondly, G0(Ok,p) is canonically a topological group [We , Chapter I] and we c*
*laim density with
respect to this topology. We have that G0k:= G0 Ok k = Sl1(D) [PR , 2.3] is an *
*inner form of Sld
and thus is semisimple and simply connected. Furthermore, G0k k kv0= Sln(D") f*
*or some central
skewfield "Dover kv0and some n > 1. Since Dv0is not a skewfield by assumption*
*, we have n > 2
and rkkv0G0k k kv0 = n  1 > 1 [PR , Proposition 2.12], i.e. G0kis isotropic at*
* v0. From strong
approximation [S, Theorem 5.1.8] we conclude that
(4) G0(k) . G0(kv0) G0(Ak) is dense,
where Ak denotes the ad`elering of k. Fix x 2 G0(Ok,p) and an open subgroup Up*
* G0(Ok,p). Denote
by "x2 G0(Ak) the ad`ele having pcomponent x and all other components equal to*
* 1. Then
Y Y
U := Upx G0(Ok,v) x G0(kv) G0(Ak)
v6=p finite v infinite
is an open subgroup and by (4) there exist fl 2 G0(k) and ffi 2 G0(kv0) such th*
*at flffi 2 "xU. Since
p 6= v0 this implies that flp 2 "xpUp = xUp, where flp is the pcomponent of th*
*e principal ad`ele fl,
equivalently, the image of fl under the inclusion G0(k) G0(kp). Since x and U*
*p are arbitrary, we
will be done if we can show that fl 2 G0(OSfin) G0(k), i.e. that for every fi*
*nite v 62 S we have
flv 2 G0(Ok,v). For v = p this is clear since xUp G0(Ok,p) whereas for v 6= p*
* we have, using that
ffiv = 1 since v 6= v0 2 S,
(flffi)v = flv 2 ("xU)v = "xv. G0(Ok,v) = G0(Ok,v).
*
* __
*
*__
To proceed, we apply (3) to OSfin,! O*k,pto obtain
4
Quasiisogenies and Morava stabilizer groups
(5) 1____//_G0(OSfin)_//_"G`(OSfin)"N`//_O*Sfin" `
  
 ' 
fflffl fflfflNp fflffl
1_____//G0(Ok,p)__//G(Ok,p)___//_O*k,p.
By definition, G(OSfin) = (O Ok OSfin)* and G(Ok,p) = (O Ok Ok,p)* = O*P[D , *
*VI, x11, Satz 6],
so Theorem 1 is concerned with the closure of the image of '. Recall the subgro*
*up X O*Sfinfrom
(2).
Proposition 4. In (5) we have im(N) = X O*Sfin.
Proof.Firstly, the commutative diagram of groups
(6) D*O_____N____//_k*OOO
 
 
?O N ?O*
(O Ok OSfin)*___//_OSfin
is cartesian, i.e. x 2 D* and N(x) 2 O*Sfinimply x 2 (O Ok OSfin)*. This follo*
*ws from the similar
statement, applied to both x and x1, that x 2 D* and N(x) 2 OSfinimply x 2 (O *
* Ok OSfin): By
[PR , Theorem 1.15] we have
"
O Ok OSfin= OQ
Q O prime andQ\Ok62S
and are reduced to seeing that x 2 D*Qand N(x) 2 Ok,Q\Okimply x 2 OQ , which is*
* true [PR ,
1.4.2].
Since (6) is cartesian we see that
1*
* *
N((O OkOSfin)*) = N(D*)\O*Sfin= {x 2 k* v(x) > 0 forv infinite withinvv(D) = _*
*}\OSfin= X.
2
Here, the second equality is Eichler's norm Theorem [PR , Theorem 1.13].
*
* __
*
*__
Since in (5) Np is surjective [PR , 1.4.3] we can, using Proposition 4, rewrite*
* (5) as
(7) 1 ____//_G0(OSfin)//_"(`O" Ok`OSfin)*N//_X"_`//_1
ff ' 
fflffl fflfflNp fflffl
1 ____//_G0(Ok,p)______//_O*P_______//O*k,p_//_1.
5
Niko Naumann
Since the image of ff is dense by Proposition 3 and O*Pis compact, all that rem*
*ains to be done to
conclude the proof of Theorem 1 is to apply Proposition 5 below to (7).
__X
For a subset Y of a topological space X we denote by Y the closure of Y in X.
Proposition 5. Let
ae
1____//_H0___//_"H`"/`/_H00"/`/_1
  
  
fflfflfflfflfflfflss
1____//_G0___//G____//G00__//_1
be a commutative diagram of first countable topological groups with exact rows,*
* G compact, and
such that H0 G0is dense. Then
__G 1 ___G00
H = ss (H00 ).
__G *
* ___G00
Proof.Assume that g 2 H . Then g = limnhn for suitable hn 2 H and ss(g) = limn*
*ss(hn) 2 H00 .
Conversely, given g 2 G with ss(g) = limnhn for suitable hn 2 H00, choose yn 2 *
*H with ae(yn) = hn.
The sequence (yng1)n in G has a convergent subsequence, z := limynig1 2 G. Th*
*en ss(z) = 1,
i
i.e. z 2 G0 and we have z = limwi for suitable wi 2 H0. The sequence (w1iyni)i*
* in H satisfies
i __
limw1iyni= z1zg = g, hence g 2 HG .
i
*
* __
*
*__
2.2 Forms of type 2Ad1
Here we consider a variant of the problem addressed in subsection 2.1 which tak*
*es into account an
involution. Let D be a finite dimensional skewfield of reduced dimension d > *
*1 over Q carrying a
positive involution * of the second kind, i.e. for all x 2 D* we have trDQ(*xx)*
* > 0 (positivity) and *
restricted to the center L of D is nontrivial. Then L is a CMfield with k := *
*{x 2 L  x =* x} L
as its maximal real subfield [Mu , page 194]. Note that * is klinear. We assum*
*e that O D is a
maximal order which is invariant under *. Then O \ L = OL and O \ k = Ok are th*
*e rings of
integers of L and k. We consider the affine algebraic groupschemes G and T ove*
*r Spec(Ok) defined
for any Okalgebra R by
G(R) = {g 2 (O Ok R)* *gg = 1} and
T (R) = {g 2 (OL Ok R)* NLk(g) = 1}.
There is a homomorphism N : G ! T over Spec(Ok) given on points by N(g) = NDL(*
*g) and we
put SG := ker(N) to obtain an exact sequence over Spec(Ok)
(8) 1_____//SG___//_GN__//_T__//_1.
Over Spec(k), this is the sequence
1 ____//_SU1(D,_1)_//_U1(D,_1)N//_ResLk(Gm_)(1)//_1,
6
Quasiisogenies and Morava stabilizer groups
where "1" denotes the standard rank one Hermitian form on D and
NLk
ResLk(Gm )(1):= ker(ResLk(Gm ) ! Gm,k) ;
c.f. [PR , 2.3] for notation and general background on the unitary groups SU an*
*d U.
We fix a prime 0 6= p Ok and a finite set of finite places S of k with p 62 S*
* and denote by OS k
the ring of Sintegers.
Theorem 6. The closure of G(OS) G(Ok,p) equals
{g 2 G(Ok,p)  N(g) lies in the closureTof(OS) T (Ok,p)}.
See subsection 3.3 for the computation of the closure of T (OS) T (Ok,p) in a*
* special case.
Note that
G(OS) = {g 2 (O Ok OS)* *gg = 1}
but the structure of G(Ok,p) depends on the splitting behavior of p in L:
If there is a unique prime q OL over p, then invq(D) = 0 [Mu , page 199, (B)]*
* and
G(Ok,p) ' {(xi,j) 2 Gld(OL,q)  (___xji)(xij) = 1},
where  denotes the nontrivial automorphism of Lq over kp. If pOL = qq0 with q*
* 6= q0 then
invq(D)+invq0(D) = 0 [Mu , page 197, (A)], Dq ' Doppq0as kp = Lq = Lq0algebras*
* and the involution
on D kkp = DqxDq0exchanges the factors. Then U1(D, 1) kkp ' Gl1(Dq) ' Gl1(Dq0)*
*, the latter
isomorphism since for any group X, x 7! x1 : X ! Xoppis an isomorphism. From *
*this we get
G(Ok,p) ' O*Dq' O*Dq0
in this case.
In the rest of this subsection we give the proof of Theorem 6 which is similar *
*to the proof of Theorem
1 and we will limit ourselves to indicating the relevant modifications.
Firstly, in analogy with Proposition 3, we have that SG(OS) SG(Ok,p) is a den*
*se subgroup: Since
SU1(D, 1) is an (outer) form of Sld, it is semisimple and simply connected. Fo*
*r any infinite place
(necessarily real) v of k we have SU1(D, 1) k kv ' Ud, the standard compact fo*
*rm of Gld over
kv ' R. Since rkRUd = d  1 > 1, SU1(D, 1) is anisotropic at v and one proceeds*
* as in the proof of
Proposition 3 using v0 = v there.
Next, we explain why N : G(Ok,p) ! T (Ok,p) is surjective:
One reduces to seeing that N : G(kp) ! T (kp) is surjective as at the beginnin*
*g of the proof of
Proposition 4 and, for later reference, we will prove the surjectivity of N : G*
*(kv) ! T (kv) for any
(not necessarily finite) place v of k. If v splits into w and w0in L, then v is*
* finite (since k is totally
real and L is totally imaginary, no infinite place of k is split in L) and we h*
*ave a commutative
diagram
G(kv)__N_//_T (kv)
' '
fflfflNDvfflfflkv
D*v_______//k*v
and the lower horizontal arrow is surjective by [PR , 1.4.3]. If there is a uni*
*que place w of L over v
then we get
7
Niko Naumann
G(kv)__________N_________//T (kv)
' '
fflffl___ fflffl __
{(xij) 2 Gld(Lw)  (xji)(xij)d=e1}t//_{x 2 L*w xx = 1}
and the lower horizontal arrow is surjective since it is split by x 7! diag(x, *
*1, . .,.1).
Finally, we show that N : G(OS) ! T (OS) is surjective:
Again, it is enough to see that N : G(k) ! T (k) is surjective and to this end*
* we contemplate the
following diagram:
G(k)____N____//_T (k)_______//H1(k, SG)
  
  '
Q fflfflQ Q fflffl Q fflffl
G(kv) __Nv//_ T (kv)___//_ H1(kv, SG).
v2 1k v2 1k v2 1k
Here, 1k denotes the set of infinite places of k, the horizontal lines are par*
*t of the cohomology
sequence associated with (8) and the rightmost vertical arrow is an isomorphis*
*m by the Hasse
principle for SG Ok k = SU1(D, 1) [PR , Theorem 6.6]. Hence the surjectivity o*
*f N follows from the
surjectivity of Nv for all v 2 1kwhich has already been established.
To sum up, we have shown the existence of a diagram
1_____//SG(OS)_____//_"G`(OS)_//_"T`(OS)_//"1`
  
  
fflffl fflffl fflffl
1____//_SG(Ok,p)__//_G(Ok,p)__//_T (Ok,p)//_1
fulfilling the assumptions of Proposition 5 an application of which concludes t*
*he proof of Theorem 6.
2.3 The commutative case
In subsection 2.1 the problem of approximating a local unit in a maximal order *
*was reduced to a
similar problem involving solely numberfields:
Let k be a numberfield, 0 6= p Ok a prime dividing the rational prime p and S*
*1 a possibly empty
set of real places of k. For a finite set of finite places S of k not containin*
*g p we consider
XS := {x 2 O*S v(x) > 0 for allv 2 S1 } O*S
and wish to understand when XS O*k,p=: Up is a dense subgroup. The oneunits
U(1)p:= 1 + pOk,p Up
8
Quasiisogenies and Morava stabilizer groups
(1)p
are canonically a finitely generated Zpmodule and Up=Up is a finite abelian *
*group. It follows from
Nakayama's lemma that a subgroup Y Up is dense if and only if the composition*
* Y ,! Up !
Up=U(1)ppis surjective. We denote by
diagM * M * *,+
E+ := ker(O*k,! kv ! kv=kv )
v2S1 v2S1
the group of global units which are positive at all places in S1 . Here k*,+vde*
*notes the connected
component of 1 inside k*v, i.e. k*,+v' R+ (resp. k*,+v' C*) if v is real (resp.*
* complex). We write
_ : E+ O*k,! Up
for the inclusion. Then Up=_(E+ )U(1)ppis a finite abelian group the minimal nu*
*mber of generators
of which we denote by g(p, S1 ).
Theorem 7. In the above situation:
i) If XS Up is dense then S > g(p, S1 ).
ii) Given a set T of places of k of density 1, there exists S as above such tha*
*t XS Up is dense,
S = g(p, S1 ) and S T.
iii)
ae
1(kp) = {1}
g(p, S1 ) 6 1 [kp+:[Qp]k, if~p
p,:iQp]f~p1(kp) 6= {1}.
p _
Remark 8. In general, the inequalities in iii) are strict: For k = Q( 2), p di*
*viding 7 and S1 = ;
one can check that g(p, S1 ) = 0, i.e. O*k Up is dense.
Proof.We consider the following subgroups of Ik, the id`eles of k:
Y (1)p Y Y
U1 := Uv x Up x k*,+vx k*v,
v61,v6=p v2S1 v1,v62S1
Y Y Y
U := Uv x k*,+vx k*vand
v61 v2S1 v1,v62S1
Y Y
U+ := Uv x k*,+v.
v61 v1
Then U1 U and k*U1 Ik is of finite index. Class field theory, e.g. [N , VI]*
*, yields finite abelian
extensions k H L and the upper part of diagram (9) below. The field corresp*
*onding to k*U+ is
the big Hilbert class field of k which we denote by H+ . Since k*U1.k*U+ = k*U *
*we have H+ \L = H
and we put K := H+ L. We have the following diagram of fields
K @
 @@@
 @@@
 @
H+ C "L
CCC """
CCC """
C "
H



k
9
Niko Naumann
and some of the occurring Galois groups are identified as follows:
(9) 1_______//Gal(L=H)__'__//_Gal(L=k)ss//_Gal(H=k)__//_1OOOOOO
fi' ' '
  
1_______//k*U=k*U1______//Ik=k*U1____//Ik=k*U____//_1OO
ff'

Up=_(E+ )U(1)pp
The isomorphism ff is induced by the inclusion Up ,! k*U. Since the proof that *
*ff is an isomorphism
is virtually identical to the argument of [W , Theorem 13.4] we only give a ske*
*tch: One has k*U =
k*UpU1 hence
k*U=k*U1 = k*UpU1=k*U1o'o_Up=(Up\_k*U1)
and one checks that Up\ k*U1 = _(E+ )U(1)ppas in [W , Lemma 13.5].
To prove i), assume that XS Up is dense; then XS Up ! Up=U(1)ppis surjecti*
*ve, hence
so is XS=E+ ! Up=_(E+ )U(1)pp. Dirichlet's unit Theorem implies that XS=E+ '*
* ZS, hence
S > g(p, S1 ).
To prove ii), fix generators xi 2 Up=_(E+ )U(1)pp(1 6 i 6 g(p, S1 )). Let oei 2*
* Gal(K=k) be the
unique element such that oeiH+ = idand oeiL = ('fiff)(xi). Note that ('fiff)(*
*xi)H = (ss'fiff)(xi) = id
by (9). By Chebotarev's density Theorem [N , VII, Theorem 13.4], there is a fin*
*ite place vi 2 T,
unramified in K=k and such that oei= Frob1vi, where Frobvidenotes the Frobeniu*
*s at the place vi,
in Gal(K=k). Then ('fiff)(xi) = Frob1viin Gal(L=k). Since FrobviH+ = oe1iH+*
* = id, the prime
ideal pi Ok corresponding to viis principal: pi= ssiOk [N , VI, Theorem 7.3] w*
*here we can, and
do, choose ssi2 Ok to be totally positive. We claim that the image of ssiin Up=*
*_(E+ )U(1)ppequals
xi:
To see this, we apply the Artinmap (, L=k) : Ik ! Gal(L=k) to the identity s*
*si = ssi,p. (_ssi_ssi,p)
in Ik, where ssi,pdenotes the id`ele having ssias its pcomponent and all other*
* components equal to
1. By Artinreciprocity we obtain 1 = (ssi,p, L=k)(_ssi_ssi,p, L=k). Denoting y*
* := _ssi_ssi,pwe have (y, L=k) =
Q
(yv, Lv=kv) [N , VI, Satz 5.6] and evaluate the local terms (yv, Lv=kv) as fo*
*llows:
v
For v = p we obtain 1 since yp = 1; for v 6= p, vi finite we obtain 1 since yv *
*2 O*vand v is
unramified in L=k; for v = vi we obtain Frobvisince L=k is unramified at vi and*
* yvi2 Ok,viis a
local uniformizer; finally, for v1 we obtain 1 since yv > 0 because ssiis tota*
*lly positive.
Hence (ssi,p, L=k) = Frob1vi= ('fiff)(xi) in Gal(L=k). Denoting by o : Up ! U*
*p=_(E+ )U(1)ppthe
projection we have (ssi,p, L=k) = ('fiffo)(ssi,p) by construction, hence xi= o(*
*ssi,p) by the injectivity
of 'fiff. This establishes the above claim saying that the global elements ssi2*
* Ok have the prescribed
image xiin Up=_(E+ )U(1)pp. To conclude the proof of ii), put S := {vi 1 6 i 6*
* g(p, S1 )} and note
that ssi2 XS with this choice of S, hence XS ! Up=_(E+ )U(1)ppis surjective an*
*d since E+ XS,
so is XS ! Up=U(1)pp, i.e. XS Up is dense.
To see iii) we use
Up = ~q1x U(1)p' ~q1x ~p1(kp) x Z[kp:Qp]p,
10
Quasiisogenies and Morava stabilizer groups
where q = Ok,p=pOk,p [N , II, Satz 5.7, i)] which implies that the upper boun*
*d claimed in iii) is in
fact the minimal number of generators of Up=U(1)ppwhich obviously is greater th*
*an or equal to the
*
* __
minimal number of generators of Up=_(E+ )U(1)pp, i.e. g(p, S1 ). *
* __
3. Applications
3.1 Extending automorphisms of pdivisible groups
Here we explain the application of the results from subsections 2.1 and 2.3 to *
*the following problem:
Let k be a finite field of characteristic p and A=k a simple abelian variety su*
*ch that End k(A) is
a maximal order in the skewfield D := End k(A) Z Q. The center K of D is a nu*
*mberfield and
K \ Endk(A) = OK is its ring of integers.
The pdivisible group of A=k [T ] splits as
Y
(10) A[p1 ] = A[p1 ],
pp
the product extending over all primes p of OK dividing p. By [MW , Theorem 6] *
*the canonical
homomorphism
(11) End k(A) Z Zp '!End k(A[p1 ])
is an isomorphism. We have
Y Y
End k(A) Z Zp ' End k(A) OK OK,p' End k(A)P
pp pp
with P the unique prime of End k(A) lying over p. Similarly, (10) implies that
Y
End k(A[p1 ]) ' End k(A[p1 ]).
pp
These decompositions are compatible with (11) in that the canonical homomorphism
Endk(A) OK OK,p'! Endk(A[p1 ])
is an isomorphism for every pp. We fix some pp and ask for a finite set S of *
*finite primes of K such
that p =2S and
(12) (Endk(A) OK OK,S)*,! Autk(A[p1 ])
is a dense subgroup. Note that this density is equivalent to the following asse*
*rtion:
For every ff 2 Aut k(A[p1 ]) and n > 1 there is an isogeny OE 2 End k(A) of deg*
*ree divisible by
11
Niko Naumann
primes in S only and some x 2 O*K,Ssuch that
OExA[pn]= ffA[pn],
i.e. the quasiisogeny OEx of A extends the truncation at arbitrary finite leve*
*l n of ff.
By Theorem 1, the inclusion (12) is dense if and only if X Up is dense where *
*X O*Kis the
subgroup of units which are positive at all real places of K at which D does no*
*t split. The density
of X Up in turn is firmly controlled by Theorem 7. We would like to illustrat*
*e all of this with
some examples:
According to the Albertclassification [Mu , Theorem 2, p. 201], note that type*
*s I and II do not
occur over finite fields, there are two possibilities:
Type III: Here, K is a totally real numberfield and D=K is a totally definite q*
*uaternion algebra.
The simplest such case occurs if A=k is a supersingular elliptic curve with En*
*d k(A) = End _k(A).
In this case, it follows from Example 2,2) that for a suitable prime l
` ~ ~'*
1 1
End k(A) _ ,! Autk(A[p ])
l
is dense.
To see another example of this type, let A=Fp correspond to a pWeil number ss *
*with ss2 = p. Then
dim(A)p=_2 and A FpFp2ispisogeneous_to the square of a supersingular elliptic*
* curve. We have
K = Q( p) and p = ( p)OK , hence A[p1 ] = A[p1 ]. Furthermore, O*K= { 1} x ff*
*lZ for a funda
mental unit ffl and X O*Kis of index 4. To find a small set S such that (12) *
*is dense one first
needs to compute the minimal number of generators of Up=XU(1)pp, denoted g(p, S*
*1 ) in Theorem
7 where, in the presentpsituation,_S1 consists of both the infinite places of K*
*. For p = 2 one can
choose ffl = 1 + 2, then X = ffl2Z. Since Up=U(1)2p' F32and ffl2 =2U(1)2pone g*
*ets g(p, S1 ) = 2.
p_ (1)3
For p = 3 we may take ffl = 2 + 3, then X = ffl2Z again. Since now Up=Up = ~*
*2x F23' Z=6 x Z=3
the fact that ffl2 =2U(1)3pis not enough to conclude that g(p, S1 ) = 1. Howeve*
*r, one checks in addition
that ffl2 2 U(1)p, and concludes that Up=XU(1)3p' Z=6 and hence indeed g(p, S1 *
*) = 1.
For p > 5 one has Up=U(1)pp= ~p1x F2pand since ~p1 6 K the image of a genera*
*tor of X in
Up=U(1)ppwill have nontrivial projection to F2pand one concludes that g(p, S1 *
*) = 1.
Type IV: In this case K is a CMfield and X = O*K. The easiest such example occ*
*urs for an
ordinary elliptic curve and we give two examples:
A solutionpof_ss2+ 5 = 0 ispa_5Weil number to which there corresponds an ellip*
*tic curve E=F5 with
K = D = Q( 5). For p = ( 5)OK one has Up=U(1)pp= ~4 x F25an since O*K= { 1} o*
*ne gets
Up=XU(1)pp' Z=10 x Z=5, hence g(p, S1 ) = 2.
Similarly, a solution of ss2 4ss + 5 = 0 gives an elliptic curve over F5 with *
*D = K = Q(i) and since
5 splits in K one has Up=XU(1)pp' Z=10, hence g(p, S1 ) = 1 in this case.
Finally, we leave it as an easy exercise to an interested reader to check that *
*for any prime p and
integer N > 1 there exists a simple abelian variety A=Fp such that any S for wh*
*ich (12) is dense
necessarily satisfies S > N.
12
Quasiisogenies and Morava stabilizer groups
3.2 A dense subgroup of quasiisogenies in the Morava stabilizer group
Let p be a prime and n > 1 an integer. The nth Moravastabilizer group Gn is t*
*he group of units
of the maximal order of the central skewfield over Qp of Hasseinvariant 1_n.
In this section we will construct an abelian variety A=k over a finite field k *
*of characteristic p such
that for a suitable prime l the group (End k(A)[1_l])* is canonically a dense s*
*ubgroup of Gn. We will
completely ignore the case n = 1 as it is very well understood. In case n = 2 o*
*ne can take for A
a supersingular elliptic curve [BL ] and the resulting dense subgroup of G2 ha*
*s been used to great
advantage in the construction of a "modular" resolution of the K(2)local spher*
*e [B1].
For general n we remark that, since_Endk(A)_ Z Zp ' Endk(A[p1 ]), in order that*
* Endk(A) have a
relation with Gn one needs A[p1 ] kk to have an isogeny factor of type G1,n1[M*
*a , IV,x2,2.]. By the
symmetry of pdivisible groups of abelian varieties [Ma , IV, x3, Theorem 4.1],*
* there must then also
be a factor of type Gn1,1which shows that n = 2 is somewhat special since (1, *
*n  1) = (n  1, 1)
in this case. For n > 3 the above considerations imply that the sought for abel*
*ian variety must
be of dimension at least n, as already remarked by D. Ravenel [R1 , Corollary 2*
*.4 (ii)]. Following
suggestions of M. Behrens and T. Lawson we will be able to construct A having t*
*his minimal possible
dimension. We start by constructing a suitable isogenyclass as follows.
Proposition 9. Let p be a prime and n > 3 an integer. Then there is a simple ab*
*elian variety
A=Fpnsuch that the center of End Fpn(A) ZQ is an imaginary quadratic field in w*
*hich p splits into,
say, p and p0such that invp(End Fpn(A) Z Q) = 1_n, invp0(End Fpn(A) Z Q) = 1*
*_nand dim(A) = n.
Furthermore, A is geometrically simple with End ___Fpn(A) Z Q = EndFpn(A) Z Q.
__
Proof.We use HondaTate theory, see [MW ] for an exposition. Let ss 2 Q be a *
*root of f :=
x2 px + pn 2 Z[x]. Since the discriminant of f is negative, ss is a pnWeil nu*
*mber and we choose
A=Fpn simple associated with the conjugacy class of ss. Then Q(ss) is an imagin*
*ary quadratic field
and is the center of EndFpn(A) Z Q. Since n > 3 the Newton polygon of f at p h*
*as different slopes
1 and n  1 which shows that f is reducible over Qp [N , II, Satz 6.4], hence p*
* splits in Q(ss) into
p and p0and, exchanging ss and __ssif necessary, we can assume that vp(ss) = 1 *
*and vp(__ss) = n  1.
Then [MW , Theorem 8, 4.]
vp(ss) 1
invp(End Fpn(A) Z Q) = ______[Q(ss)p : Qp] = __and similarly
vp(pn) n
n  1 1
invp0(End Fpn(A) Z Q) = _____= ___.
n n
Furthermore [MW , Theorem 8, 3.], 2 . dim(A) = [End Fpn(A) Z Q : Q(ss)]1=2. [*
*Q(ss) : Q] = 2n. The
final statement follows easily from the fact that ssk 62 Q for all k > 1, c.f. *
*[HZ , Proposition_3(2)],
which in turn is evident since vp(ss) 6= vp(__ss). *
* __
Remark 10. The statement of Proposition 9 should also hold for n = 2 but we cou*
*ld only verify
this in case p 1 (4) (using f = x2+ p2) or p 1 (12) (using f = x2 px + p2*
*).
Since the properties of A=Fpn in Proposition 9 are invariant under Fpnisogenie*
*s, we can, and do,
choose A=Fpn having these properties such that in addition End Fpn(A) End Fpn*
*(A) Z Q is a
maximal order [Wa , proof of Theorem 3.13]. Denoting by P EndFpn(A) the uniqu*
*e prime above
the prime p constructed above, Proposition 9 implies that (End Fpn(A))*P= Gn. W*
*e choose a prime
l as follows: If p 6= 2 we take l to be a topological generator of Z*p. For p =*
* 2 we take l = 5.
13
Niko Naumann
Remark 11. Note that for p 6= 2 a prime l 6= p topologically generates Z*pif an*
*d only if (l mod p2)
generates (Z=p2)*. Hence, by Dirichlet's Theorem on primes in arithmetic progre*
*ssions, the set of
all such l has a density equal to ((p  1)OE(p  1))1 > 0 and is thus infinite*
*. Such an l can be found
rather effectively: Given l 6= p, compute ffk := (lp(p1)=kmodp2) for all prime*
*s k dividing p(p  1).
If for all k, ffk 6 1 (p2), then l is suitable.
Theorem 12. In the above situation
1 * *
(End Fpn(A)[_ ]) ,! (End Fpn(A))P = Gn
l
is a dense subgroup.
Proof.We apply Theorem 1 with O := End Fpn(A), k := Q(ss), p the prime of Ok co*
*nstructed in
Proposition 9 and S := {1, l} the set consisting of the unique infinite place 1*
* of k and all places
dividing l. Clearly, p 62 S and D := O Z Q is not a skewfield at 1 since k1 *
*' C and n > 1.
Using the notation of Theorem 1 we have OSfin= Ok[1=l] and X = (Ok[1=l])* since*
* k has no real
place. Theorem 1 applies and shows that the claim of Theorem 12 is equivalent t*
*o the density of
(Ok[1=l])* O*k,p' Z*p. Since l 2 (Ok[1=l])*, this density is clear for p 6= 2*
* by our choice of l whereas
*
* __
for p = 2 we have that { 1} x 5Z Z*2is dense and 1, 5 2 (Ok[1=5])*. *
* __
3.3 Density of unitary quasiisogenies
In this subsection we prove a sharpening of Theorem 12. Fix a prime p, an integ*
*er n > 3 and let
A=Fpn be an abelian variety as in Theorem 12, i.e. A=Fpn satisfies the conclusi*
*on of Proposition 9
and End Fpn(A) is a maximal order. Assume in addition that there is a polarizat*
*ion of A=Fpn such
that the corresponding Rosatiinvolution * [Mu , p. 189] on End Fpn(A) Z Q sta*
*bilizes End Fpn(A).
Let P EndFpn(A) denote the unique prime lying over the prime p specified in P*
*roposition 9.
Theorem 13. In the above situation, there is a prime l 6= p such that
* *
OE 2 EndFpn(A)[1=l]  OEOE,=!1EndFpn(A)P = Gn
is a dense subgroup.
Proof.Let k be the center of End Fpn(A) Z Q, it is the imaginary quadratic fie*
*ld considered in
Proposition 9, and put T := ResOkZ(Gm ). According to Theorem 6 , for a given p*
*rime l 6= p, the
conclusion of Theorem 13 is equivalent to the density of
(13) T (Z[1=l]) = {ff 2 Ok[1=l]* ff__ff= 1} Up ' T (Zp),
where  denotes complex conjugation. The proof of the existence of such a prime*
* l is similar to the
argument of Theorem 7,ii) but some extra_care is needed to deal with the norm c*
*ondition ff__ff= 1.
Remember that p splits in k, say pOk = pp, and consider the following subgroups*
* of the id`eles of k:
Y (1)p (1)p Y
U1 := Uv x Up x U_p x k*vand
v6=p,_pfinite v1
14
Quasiisogenies and Morava stabilizer groups
Y Y
U := Uv x k*v.
v finite v1
We have a corresponding tower of abelian extensions k H K and since U1 is s*
*table under
Gal(k=Q), the extension K=Q is Galois, though rarely abelian. We have an isomor*
*phism
Up=U(1)ppx U_p=U(1)p_p'
OE : UpU_p=U(1)ppU(1)p_pO*k' _________________*! Gal(K=H)
Ok
induced by the Artinmap, where O*kis embedded diagonally. Since p splits in k,*
* Up=U(1)ppO*kis
cyclic and we fix a generator x. By Chebotarev's Theorem applied to K=Q there e*
*xists a rational
prime l 6= p, unramified in K=Q and such that for a suitable prime of K over *
*l we have
Frob1=lOE([(x, 1)]) inGal(K=H) Gal(K=Q).
We claim that l satisfies the conclusion of Theorem 13:
Put ~ := k; since (Frob l)H = id, ~ is a principal ideal of Ok a generator *
*of which we denote by
ss. Then
ss * __
fi := ___2 {ff 2 Ok[1=l]  ffff=,1}
ss
which we claim goes to x under the map induced by (13). This will show that (13*
*) is a dense
subgroup and conclude the proof. As in the proof of Theorem 7,ii) one sees that
(ssp, ss_p) = [(x, 1)] and similarly
Up=U(1)ppx U_p=U(1)p_p
((__ss)p, (__ss)_p) = [(1, x)] in_________________*,
Ok
hence indeed
(fip, fi_p) = [(x, x1)]
and a fortiori fip = x in Up=U(1)ppO*k.
*
* __
*
*__
Remark 14. Given p and n as in Theorem 13, a prime l satisfying its conclusion *
*can be found
effectively using complex multiplication. The computations however can get rath*
*er time consuming
if the imaginary quadratic field involved has elevated class number. This is in*
* contrast with the
simpler variant Theorem 12 which ignores the polarization; c.f. Remark 11.
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Niko Naumann niko.naumann@mathematik.uniregensburg.de
NWF I Mathematik, Universit"at Regensburg, 93040 Regensburg
16