Large 2-adic Galois image and non-existence of certain abelian surfaces over Q

0.3 MB·

by  Armand Brumer

LARGE 2-ADIC GALOIS IMAGE AND NON-EXISTENCE OF
CERTAIN ABELIAN SURFACES OVER Q
7
1 ARMANDBRUMERANDKENNETHKRAMER
0
2
Abstract. Motivatedbyourarithmeticapplications,werequiredsometools

n thatmightbeofindependent interest.
a Let E be an absolutely irreducible group scheme of rank p4 over Zp. We
J provide a complete description of the Honda systems of p-divisible groups G
7 such that G[pn+1]/G[pn]≃E foralln. Thenwe findabound forthe abelian
conductor of the second layer Qp(G[p2])/Qp(G[p]), stronger in our case than

] canbededucedfromFontaine’s bound.
T Let π: Sp2g(Zp)→Sp2g(Fp)be the reduction map andlet G bea closed
N subgroup of Sp2g(Zp) with G=π(G) irreducible and generated by transvec-
tions. Wefillagapinthe literaturebyshowingthatifp=2andGcontains
.
h atransvection,thenGisaslargeaspossibleinSp2g(Zp)withgivenreduction
at G,i.e. G=π−1(G).
m One simpleapplication arises when A=J(C) is the Jacobian of ahyper-
elliptic curve C: y2+Q(x)y = P(x), where Q(x)2+4P(x) is irreducible in
[ Z[x]ofdegreem=2g+1or2g+2,withGaloisgroupSm⊂Sp2g(F2). Ifthe
1 IgusadiscriminantI10 ofC isoddandsomeprimeq exactlydividesI10,then
v G=Gal(Q(A[2∞])/Q)isπ˜−1(Sm),whereπ˜: GSp2g(Zp)→Sp2g(Fp).
0 When m=5, Q(x)=1and I10 =N isaprime, A=J(C) isan example
of a favorable abelian surface. We use the machinery above to obtain non-
9
existence resultsforcertainfavorableabeliansurfaces,evenforlargeN.
8
1
0
.
1
0
7
Contents
1
: 1. Introduction 2
v
i 2. A large image result 2
X
2.1. Review 2
r 2.2. The case G= 3
a 2.3. The case G SOm±(F ) 4
≃ 2g 2
3. Preliminaries on Honda systems 6
4. Our p-divisible groups 8
5. Exponent p2 11
5.1. Baer sums for exponent p2 11
5.2. Field of points and conductor for exponent p2 12
5.3. Finding the Honda parameters for A[4] 14
6. Global Applications 17
References 19
2010 Mathematics Subject Classification. Primary11G10, 14K15;Secondary17B45,20G25.
Keywordsandphrases. abeliansurface,semistablereduction,groupscheme,p-divisiblegroup,
Hondasystem,conductor, symplecticgroup.
1
2 A.BRUMERANDK.KRAMER
1. Introduction
LetA beag-dimensionalabelianvarietyandG=Gal(Q(A[p∞])/Q)theGalois
/Q
group of its p-division tower. Serre’s work on the open image problem for abelian
varietieshasstimulateda largeliterature. Forinstance,[ALS,H,KM,V,ZS]show
that, under suitable hypotheses, G is an open subgroup of GSp (Z ), at least for
2g p
large p. We concentrate on p = 2, for which more residual images exist. Theorem
2.1.1 describes our group theoretical conclusions. As usual, suitable abelian vari-
eties thereby give rise to large Galois extensions with controlledramification,as in
Proposition 2.3.4.
Given an integer N and a group scheme over Z[1] of exponent p, one may
E N
ask for the existence (or even the uniqueness up to isogeny) of an abelian variety
Awith A[p] . In [BK1],we found non-existencecriteriawhen is reducible. In
≃E E
this paper, we treat non-existence criteria when dimA = 2 and is absolutely
E|Zp
irreducible. Then A has a polarizationof degree prime to p and Gal(Q(A[p∞])/Q)
is contained in GSp (Z ). This requires a delicate study of the extensions of
4 p W E
by of exponent p2. Non-existence of Q( )/Q implies that of A with A[p2] .
E W ≃W
Let be an absolutely irreducible group scheme of rank p4 over Z . In 4, we
p
E §
give a complete description of the Honda systems of p-divisible groups such that
G
[pn+1]/ [pn] foralln. In 5,westudythefieldofpointsof = [p2],thereby
G G ≃E § W G
obtaining a bound for the abelian conductor of Q ( )/Q ( ), stronger in our
p p
W E
special case than can be deduced from Fontaine’s bound. When A is the Jacobian
of a genus 2 curve over Q , the parameters associated to A[4] are determined in
2
Proposition 5.3.1.
For the global applications in 6, let p=2 and recall the following definition.
§
Definition 1.1 ([BK3]). A quintic field is favorable if its discriminant is 16N
±
with N prime and the ramificationindex overthe prime 2 is 5. An abelian surface
A of conductor N is favorable if its 2-division field is the Galois closure of a
/Q
favorable quintic field.
IfAisfavorable,thentheimageinSp (F )oftherepresentationofG onA[2]is
4 2 Q
O−(F ) ,withtransvectionscorrespondingtotranspositions. Inaddition,A[2]
4 2 ≃S5
is biconnected, absolutely simple and Cartier self-dual over Z . Let S = π−1( ),
2 5
S
where π : GSp (Z/4Z) Sp (F ) is the natural projection. By Remark 6.1,
4 → 4 2
Q(A[4]) is a favorable S-field with F =Q(A[2]), as in the following definition.
Definition1.2. FixtheGaloisclosureF ofafavorablequinticfieldofdiscriminant
16N and let L be a field containing F. Then L is a favorable S-field if L/Q is
±
Galois, with Gal(L/Q) S and
≃
i) L/Q is unramified outside 2N ;
∞
ii) the abelian conductor exponent at primes over 2 in L/F is 6 and
iii) the inertia groupat eachprime over N in L/Q is generatedby a transvection.
Proposition6.2givesatestablerayclassfieldcriterionnecessaryfortheexistence
of a favorable S-field L. This explains the non-existence results in [BK3, Table 3].
2. A large image result
2.1. Review. ForclosedsubgroupsΓofGL (Z ),setΓ(n) = g Γ g I (pn)
m p m
{ ∈ | ≡ }
and Γ=Γ/Γ(1). A closed subgroup G of Γ is saturated in Γ if G(1) =Γ(1), so that
LARGE GALOIS IMAGE AND NON-EXISTENCE 3
G is as large as possible in Γ, subject to its reduction being G. When there is a
symplectic pairing [ , ]: M M Z on M = Z2g, transvections in Sp(M) have
× → p p
the form σ(x)=x λ[y,x]y, with y in M and λ in Z×.
− p
Theorem 2.1.1. Let G be a closed subgroup of Sp (Z ) containing transvections.
2g p
If G is irreducible and generated by transvections, then G is saturated in Sp (Z ).
2g p
Thisassertioniswell-knownwhenG=Sp (F ),soweneedonlyconsiderp=2
2g p
and proper subgroups, thanks to a classical result of McLaughlin.
Proposition 2.1.2 ([McL]). For g 2, let H be an irreducible proper subgroup of
≥
Sp (F ) generated by transvections. Then p=2 and H is one of the following:
2g p
i) the symmetric group with m=2g+1 or 2g+2,
m
S
ii) the orthogonal group O+(F ) with g 3, or O−(F ).
2g 2 ≥ 2g 2
Orthogonal groups and theta characteristics are reviewed in [D, GH, BK2]. Set
sp (F ) = A Mat (F ) AtJ +JA = 0 , where J is the Gram matrix of a
2g p { ∈ 2g p | }
basis e ,...,e for M. Then dim sp (F )=2g2+g, since
1 2g Fp 2g p
(2.1.3) sp (F )= A= a b b=bt,c=ct,d= at when J = 0g Ig .
2g p c d | − −Ig 0g
(cid:8) (cid:2) (cid:3) (cid:9) h i
To prove the Theorem, one verifies that sending an element 1 + pA of G(1)
to Amodp induces an isomorphism : G(1)/G(2) sp (F ). It follows that
L → 2g p
1+pnA Amodp gives an isomorphism G(n)/G(n+1) sp (F ) for all n 1.
7→ → 2g p ≥
Then one shows by induction that G(1) Sp(1)(Z/pnZ) is surjective and passes
→ 2g
to the limit. For the groups in the Theorem, the transvections s in G form one
conjugacyclass. SinceGcontainsatransvection,sliftstoatransvectionσinG,say
σ(x) = x λ[y,x]y. Furthermore, (σp) sp (F ) is the matrix representation
− L ∈ 2g p
of the endomorphism of M =M/pM given by
(2.1.4) f : x (s 1)(x)= λ[y,x]y,
y
7→ − −
wherex xdenotestheprojectionmapM M andthepairingisinducedonM.
7→ →
Hence it suffices to show that the f ’s generate sp (F ). This is done in Lemmas
y 2g p
2.2.1 and 2.3.3.
2.2. The case G= . Let act by permuting the coordinates of
m m
S S
W = (a ,..,a ) Fm a + +a =0
{ 1 m ∈ 2 | 1 ··· m }
and let V = W/ (a,...,a) a F or V = W if m is even or odd, respectively.
2
Then G is sympl{ectic for th|e p∈airin}g on V induced by [(a ),(b )]= ma b .
i i 1 i i
Let v in V be represented by the vector in W with non-zero entries only in
ij
P
coordinatesiandj. ForxinV,thetransvections (x)=x+[v ,x]v corresponds
ij ij ij
to π = (ij) in . Let f (x) = (s 1)(x) = [v ,x]v be the endomorphism of
m ij ij ij ij
S −
V as in (2.1.4).
Lemma 2.2.1. The set f 1 i<j 2g+1 spans sp (F ).
{ ij| ≤ ≤ } 2g 2
Proof. The vectors b = v for 1 i 2g form a basis for V. By definition,
i i,2g+1
≤ ≤
v =b +b for 1 i<j 2g. To prove that the 2g2+g elements in the Lemma
ij i j
≤ ≤
4 A.BRUMERANDK.KRAMER
span, we show that they are linearly independent. If not, there are constants
α ,β F satisfying
ij i 2
∈
2g
α [x,v ]v + β [x,b ]b =0 for all x V.
ij ij ij i i i
∈
1≤i<j≤2g i=1
X X
Fix k in 1,...,2g and evaluate at x=b , using
k
{ }
1 if i=k or j =k, 1 if i=k,
[b ,v ]= and [b ,b ]= 6
k ij k i
(0 otherwise (0 if i=k,
to obtain
(2.2.2) α (b +b )+ α (b +b )+ β b =0.
kj k j ik i k i i
j>k i<k i6=k
X X X
Matchcoefficientsofb withi<k tofindthatα =β foralli<k andthoseofb
i ik i j
withj >k tofindthatα =β forallj >k. Henceα =β =β =γ isconstant.
kj j ij i j
From the coefficients of b in (2.2.2) we have (2g 1)γb =0 and so γ =0. (cid:3)
k k
−
2.3. The case G O±(F ). LetV be asymplectic spaceofdimension2g overF
≃ 2g 2 2
with basis b and Gram matrix J in (2.1.3). Consider the theta characteristic
i
{ }
g−1
(2.3.1) θ (x)=Q (x ,x )+ x x for x=(x ,...,x ) in V,
ǫ ǫ g 2g j g+j 1 2g
i=1
X
with Q (x ,x ) = x x in the even case and Q (x ,x ) = x2 +x x +x2
+ g 2g g 2g − g 2g g g 2g 2g
in the odd case. We have θ (x+y) = θ (x)+θ (y)+[x,y], i.e. θ belongs to the
ǫ ǫ ǫ ǫ
pairing [ , ] on V. The transvection s: x x+[v,x]v in Sp (F ) acts on theta
7→ 2g 2
characteristics by s(θ)(x) = θ(x) +(1+θ(v))[v,x]2. Hence s is in the stabilizer
Oǫ (F ) of θ in Sp (F ) exactly when θ (v)=1.
2g 2 ǫ 2g 2 ǫ
Let f :x [v,x]v be the endomorphism of V as in (2.1.4). For ǫ= , let
v
7→ ±
ǫ(V)= f sp (F ) v V and θ (v)=1
Fg { v ∈ 2g 2 | ∈ ǫ }
and write ǫ(V) for its span in sp (F ).
hFg i 2g 2
Remark 2.3.2. −(V) =sp (F )= A Mat (F ) trace(A)=0 , with
hF1 i 2 2 { ∈ 2 2 | }
f =[01], f =[10] and f =[11].
b1 00 b2 00 b1+b2 11
Also,dim +(V) =1,dim +(V) =6and +(V) =sp (F )is21-dimensional.
hF1 i hF2 i hF3 i 6 2
There are symplectic isomorphisms O+(F ) and O−(F ) .
6 2 ≃S8 4 2 ≃S5
g 1 if ǫ= ,
Lemma 2.3.3. We have ǫ(V) =sp (F ) for ≥ −
hFg i 2g 2 (g 3 if ǫ=+.
≥
Proof. For g 2, the subspace V = span b j = 1,g+1 of V is isomorphic to
1 j
≥ { | 6 }
the symplectic space of dimension 2g 2 whose theta characteristicθ′ and pairing
− ǫ
are obtained by restriction from θ and [ , ]. Given a linear map h : V V ,
ǫ 1 1 1
→
let h = δ (h ) denote its extension to V satisfying h(b ) = h(b ) = 0. Then
1 1 1 g+1
δ : sp (F ) ֒ sp (F ). In particular, δ (x′ [v′,x′]v′) is the matrix of the
1 2g−2 2 → 2g 2 1 7→
linearmapfv′ onV givenbyfv′(x)=[v′,x]v′. HenceY1 =δ1(Fgǫ(V1))iscontained
in ǫ(V).
Fg
LARGE GALOIS IMAGE AND NON-EXISTENCE 5
Remark 2.3.2 treats the small values of g, so we assume g 2 for −(V) and
≥ Fg
g 4 for +(V). Let V = span e j = 2,g + 2 and extend h : V V
≥ Fg 2 { j| 6 } 2 2 → 2
to a linear map h = δ (h ) on V by setting h(b ) = h(b ) = 0. Then δ :
2 2 2 g+2 2
sp (F ) ֒ sp (F ) and Y = δ ( ǫ(V )) also is contained in ǫ(V). By the
2g−2 2 → 2g 2 2 2 Fg 2 Fg
induction hypothesis,
dimY =dimsp (F )=2(g 1)2+(g 1)=2g2 3g+1 for i=1,2.
i 2g−2 2 − − −
We have dimY Y dimsp (F ) = 2(g 2)2+g 2 = 2g2 7g+6. But
1 ∩ 2 ≤ 2g−4 2 − − −
dimsp (F )=2g2+g, so the codimension of Y +Y in sp (F ) is at most 4.
2g 2 1 2 2g 2
Next wefill ina 4-dimensionalsubspaceofsp (F ) independent ofY +Y . For
2g 2 1 2
matrices in Y , all entries in the row or column numbered k or g+k are 0. If A is
k
in Y +Y , then A =0 for the eight pairs (i,j) with
1 2 ij
i,j = 1,2 , 1,g+2 , g+1,2 or g+1,g+2 .
{ } { } { } { } { }
By the symplectic condition (2.1.3) on A, we have
A =A , A =A , A =A , A =A .
1,2 g+2,g+1 1,g+2 2,g+1 g+1,2 g+2,1 g+1,g+2 2,1
Define j∗ =j+g if j <g and j∗ =j g otherwise. Fix (i,j) in
−
S = (1,2), (1,g+2), (g+1,2), (g+1,g+2)
{ }
andletv =vij =bi+bj∗+bg+b2g. Since θǫ(v)=1,thelinearmapfv isinFgǫ(V).
Also,Ai,j =1andAi′,j′ =0forallotherpairs(i′,j′)inS. Hencethe fv’sgenerate
the desired 4-dimensional space as (i,j) ranges over the 4 pairs in S. (cid:3)
In the following Proposition, a group is said to be McL if it is isomorphic to
Sp (F ) or one of the groups in Proposition 2.1.2.
2g 2
Proposition 2.3.4. Let A be a g-dimensional abelian variety of odd conductor
/Q
N, with q N for a prime q ramifying in F = Q(A[2]) and let F = Q(A[2∞]). If
∞
k
G= Gal(F/Q) is McL, then G = Gal(F /Q) is saturated in GSp (Z ). Also, G
∞ 2g 2
is McL if A[2] is irreducible and one of the following holds:
i) the conductor of A[2] is square-free and √ 1 is not in F, or
−
ii) F contains no proper extension of Q unramified at q.
Proof. Since A[2] is irreducible in all cases, a minimal polarization of A has odd
degree. HencetheWeilpairinginducesaperfectpairingontheTatemoduleT (A)
2
and G is a closed subgroup of GSp (Z ). The symplectic similitude ν: G Z×
2g 2 → 2
giving the action of G on µ2∞ is surjective and so F∞ contains Q(µ2∞). By
Grothendieck’s monodromy theorem, inertia at v q is generated topologically by a
|
transvection σ , since the toroidal dimension of A at q is 1 and q ramifies in F.
v
Assume (i). For each prime w dividing cond(A[2]), inertia at w is generated by
a transvection s in G. Thus the fixed field k of the normal subgroup generated
w
by all s is unramified outside 2 . As in [BK2, Prop. 6.2], Fontaine’s bound on
w
∞
ramification at 2 implies that k is contained in Q(i) and thus k = Q. Hence G is
generatedby transvectionsandso G is McL. In case (ii), the subfield of F fixed by
the normal closure of σ in G is unramified over q, so equals Q and G is McL.
v
IfGisMcL,transvectionsformoneconjugacyclassgeneratingG. Since σ fixes
v
F Q(µ ), the latter equals Q and the restriction of ν to G(1) = Gal(F /F)
∩ 2∞ ∞
surjects ontoZ×. For H =G Sp (Z ), we find thatH =G soH(1) =Sp(1)(Z )
2 ∩ 2g 2 2g 2
by Theorem 2.1.1. Hence G(1) =GSp(1)(Z ). (cid:3)
2g 2
6 A.BRUMERANDK.KRAMER
Example2.3.5. LetAbetheJacobianofahyperellipticcurvey2+Q(x)y =P(x),
where Q(x)2 +4P(x) is irreducible in Z[x] of degree m = 2g+1 or 2g+2, with
Galois group Sp (F ). If the Igusa discriminant I of C is odd and some
Sm ⊂ 2g 2 10
primeq exactlydividesI ,thenG=Gal(Q(A[2∞])/Q)issaturatedinGSp (Z ).
10 2g 2
3. Preliminaries on Honda systems
The basic material on Honda systems may be found in [BC, C2, F1] and is
summarized in [BK3]. We review the required notation and recall some finite
Honda systems constructed in [BK3].
Let k a perfect field of prime characteristic p and W = W(k) the ring of Witt
vectorsoverk. FixanalgebraicclosureK ofthefieldoffractionsK ofW,letWbe
its ring of integers and write G = Gal(K/K). Let σ: W W be the Frobenius
K
→
automorphism characterized by σ(x) xp (mod p) for x in W. The Dieudonn´e
≡
ring D = W[F,V] is generated by the Frobenius operator F and Verschiebung
k
operator V, with FV=VF=p, Fa=σ(a)F and Va=σ−1(a)V for all a in W.
AHonda systemconsistsofafinitely generatedfreeW-module ,asubmodule
M
of and a Frobenius-semilinear injective endomorphism F of such that
L M M
p F and inclusion induces an isomorphism /p /F . Then
M ⊆ M L L → M M M
becomes a D -module with Verschiebung defined by Vx=F−1(px) for all x in .
k
M
Let be a D -module, finitely generatedandfree as a W-module and let be
k
M L
a W-submodule of . Then ( , ) is a Honda system if and only if the following
M M L
sequence is exact:
V F
(3.1) 0 / 0.
→L−→M−→M L→
Lemma 3.2. Given a Honda system ( , ), let ∗ =Hom ( ,W) and let ∗
W
M L M M L
be the annihilator of in ∗. Define F and V on elements ψ of ∗ by
L M M
(3.3) F(ψ)(x)=σ(ψ(Vx)) and V(ψ)(x)=σ−1(ψ(Fx)) for all x .
∈M
Then ( ∗, ∗) forms a Honda system.
M L
Proof. Since F = p , the quotient / is torsion-free, so is a direct
L ∩ M L M L L
summand of . The pairing , : ∗ W induces perfect pairings:
M h− −i M ×M→
∗/ ∗ W and ∗ / W.
M L ×L→ L ×M L→
By dualizing (3.1), the sequence 0 ∗ V ∗ F ∗/ ∗ 0 is exact. (cid:3)
→L −→M −→M L →
If F is topologically nilpotent, then ( , ) is connected. If both F and V are
M L
topologically nilpotent, then ( , ) is biconnected.
M L
A finite Honda system is a pair (M,L) consisting of a D -module M of finite W-
k
lengthandaW-submoduleLwithV: L MinjectiveandthemapL/pL M/FM
→ →
induced by the identity is an isomorphism. If ( , ) is a Honda sytem then
M L
( /pn , /pn ) is a finite Honda system.
M M L L
Let CW denote the formal k-group scheme associated to the Witt covector
k
group functor CW , cf. [C2, F2]. In particular, if k′ is a finite extension of k
k
and K′dis the field of fractions of W(k′), we have CW (k′) K′/W(k′). If R
k
≃
is a k-algebra, then D = W[F,V] acts on elements a = (...,a ,...,a ,a )
k −n −1 0
of CW (R) by Fa = (...,ap ,...,ap ,ap), Va = (...,a ,...,a ,a ) and
k −n −1 0 −(n+1) −2 −1
c˙a = (...,cp−na ,...,cp−1a ,ca ), where c˙ is the Teichmu¨ller lift of c. It is
−n −1 0
LARGE GALOIS IMAGE AND NON-EXISTENCE 7
convenient to write (~0,a ,...,a ) for an element of CW (W/pW) with a =0
−n 0 k −m
for all m>n.
The Hasse-Witt exponential map is a homomorphisdm of additive groups:
ξ : CW (W/pW) K/pW given by (...,a ,...,a ,a ) p−na˜pn
k → −n −1 0 7→ −n
X
indepedndent of the choice of lifts a˜ in W.
−n
We generallyusecalligraphicletters,e.g. forfinite flatgroupschemesandthe
V
corresponding roman letter, e.g. V for the associated Galois module. If (M,L) is
the finite Honda system of , the points of V correspond to D -homomorphisms
k
V
ϕ: M CW (W/pW)suchthatξ(ϕ(L))=0,withtheactionofG onV induced
k K
→
from its action on CW (W/pW).
k
For thedstudy of p-divisible groups in the next section, recall the finite Honda
system E introducded in [BK3, 4] and our classification of extensions of exponent
λ
§
p of E by E .
λ λ
Notation 3.4. Fix λin k× andletE =(M,L) be the finite Hondasystemwitha
λ
standard k-basis x ,x ,x ,x for M such that L=span x ,x and Verschiebung
1 2 3 4 1 2
{ }
and Frobenius are represented by the matrices:
0 0 00 0000
V= 1 0 00 , F= 0000 .
0λ00 0001
(cid:20)0 0 00(cid:21) (cid:20)1000(cid:21)
Denote the corresponding group scheme by and its Galois module by E .
λ λ
E
Proposition 3.5 ([BK3, Prop. 5.1.1]). Let λ˙ be the Teichmu¨ller lift of λ and let
R = a W/pW λp2ap4 ( p)p+1a (mod pp+2W) . For ain R , defineb=b
λ λ a
and c={ c∈ in W/pW| by b ≡ −1λpap3 (mod pW) and}c λap2 (mod pW).
a ≡−p ≡
i) Let x ,...,x be a standard basis for the finite Honda system E of . A
1 4 λ λ
E
D -map ψ represents a point of if and only if ψ(x ) = (~0,c,b,a) for some
k λ 1
E
a in R . If so, ψ(x )=(~0,c,b), ψ(x )=(~0,λ−1c) and ψ(x )=(~0,ap).
λ 2 3 4
ii) F = K(E ) is the splitting field of λ˙p2xp4−1 ( p)p+1 over K. The maximal
λ
− −
subfield of F unramified over K is F = K(µ ,ξ), where ξ is any root of
0 p4−1
xp+1 λ˙. Moreover F/F is tamely ramified of degree t=(p2+1)(p 1). For
0
a=0−we have ord (a)= 1, ord (b)= p2−p+1, ord (c)= p2. −
6 p t p t p t
iii) Rλ is an Fp4-vector space under the usual operations in W/pW and a Pa
defines an F [G ]-isomorphism R ∼ E . 7→
p K λ λ
−→
Let Ext1(E ,E ) be the group of classes of extensions of finite Honda systems:
λ λ
ι π
(3.6) 0 E (M,L) E 0
λ λ
→ −→ −→ →
under Baer sum. The subgroup Ext1 (E ,E ) of those classes such that pM = 0
[p] λ λ
was determined in [BK3, Prop. 4.5], as follows.
Proposition 3.7. If (M,L) represents a class in Ext1 (E ,E ), then there is a
[p] λ λ
k-basis e ,...,e for M such that ι(x )=e , π(e )=x , L=span e ,e ,e ,e ,
1 8 1 1 5 1 1 2 5 6
{ }
8 A.BRUMERANDK.KRAMER
0 0 0 0 0 λs 0 0 0 0 0 0 0 0 0 0
2
 1 0 0 0 0 λs 0 0   0 0 0 0 0 0 0 0 
3
0 λ 0 0 0 λs 0 0 0 0 0 1 0 −sp −sp 0
V= 0 0 0 0 s1 λs45 0 0 and F= 1 0 0 0 0 01 −s5p2 0 
 0 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0 
   
 0 0 0 0 1 0 0 0   0 0 0 0 0 0 0 0 
   
 0 0 0 0 0 λ 0 0   0 0 0 0 0 0 0 1 
   
 0 0 0 0 0 0 0 0   0 0 0 0 1 0 0 0 
with s ,s ,s ,s ,s in k. For k˜ = k/(σ4 1)(k), the map (M,L) (s ,...,s )
1 2 3 4 5 1 5
induces an isomorphism of additive groups−s: Ext1 (E ,E ) ∼ k k k k˜ k.
[p] λ λ −→ ⊕ ⊕ ⊕ ⊕
Proposition3.8([BK3,Prop.5.2.16]). Let beanextensionof by killedby
λ λ
W E E
p. ThefieldofpointsL=K( )isanelementaryabelianp-extensionofF =K( )
λ
W E
whose conductor exponent satisfies f(L/F) p2.
≤
4. Our p-divisible groups
We classify Honda systems ( , ) associated to p-divisible groups whose first
M L
layeris ,asinNotation3.4,withαink×. We alsodeterminethe Hondasystems
α
E
of the Cartier duals of such p-divisible groups.
Proposition 4.1. Let ( , ) be a Honda system as above. Then there is a basis
M L
e ,e ,e ,e for over W and parameters λ in W× and s ,s ,s ,s in W such
1 2 3 4 1 2 3 5
M
that λ α (mod pW), =span e ,e ,
1 2
≡ L { }
0 pλs 0 p 0 p p2s 0
2 3
−
1 pλs 0 0 0 0 p/λ 0
V= 3  and F=σ .
0 λ 0 0 0 ps p2s s ps 1
1 1 3 5
− −
 ps1 pλs5 p 0   1 0 ps2 0 
   − 
   
Proof. Choose a lift u in W× of α−1 and lifts e in of a standardbasis for E ,
i,1 α
M
suchthate =Fe ande =Fe . Weprovebyinductionthatthereisabasis
4,1 1,1 3,1 4,1
e ,...,e for satisfying e =Fe , e =Fe ,
1,n 4,n 4,n 1,n 3,n 4,n
M
(4.2) e Ve pnspan e +pspan e and e uVe +p .
2,n 1,n 3,n 4,n 3,n 2,n
− ∈ { } { } ∈ M
Substituting the last relation into the first relation in (4.2), we get
e Ve pn(auVe +p )+pspan e
2,n 1,n 2,n 4,n
− ∈ M { }
pnauVe +pn+1 +pspan e .
2,n 4,n
∈ M { }
Let e =e σ(au)pne . Then
1,n+1 1,n 2,n
−
(4.3) e Ve pn+1 +pspan e .
2,n 1,n+1 4,n
− ∈ M { }
Set e =Fe . We have Fe p by Notation 3.4 and Fe =e , so
4,n+1 1,n+1 2,n 1,n 4,n
∈ M
e =Fe is in Fe +pnspan Fe e +pn+1 .
4,n+1 1,n+1 1,n 2,n 4,n
{ }⊆ M
Hence (4.3) is equivalent to
(4.4) e Ve pn+1 +pspan e .
2,n 1,n+1 4,n+1
− ∈ M { }
Define e =Fe and use Fe =e to obtain
3,n+1 4,n+1 4,n 3,n
e =Fe Fe +pn+1F e +pn+1 .
3,n+1 4,n+1 4,n 3,n
∈ M⊆ M
LARGE GALOIS IMAGE AND NON-EXISTENCE 9
Clearly =span e , e , e , e . By(4.4),therearescalarsb ,b ,b ,b
1,n 2,n 3,n+1 4,n+1 1 2 3 4
M { }
such that
e Ve =pn+1σ(b )e +pn+1σ(b )e +pn+1b e +pb e .
2,n 1,n+1 1 1,n 2 2,n 3 3,n+1 4 4,n+1
−
Setting e = e pn+1σ(b )e pn+1σ(b )e gives the induction step for
2,n+1 2,n 1 1,n 2 2,n
− −
the first relation in (4.2). For the second part,
e is in e +pn+1 uVe +p uVe +p .
3,n+1 3,n 2,n 2,n+1
M⊆ M⊆ M
Then e , e , e , e converge to a basis e , e , e , e for . By the first
1,n 2,n 3,n 4,n 1 2 3 4
M
part of (4.2), Ve = e +ps e for some s in W. By the second part, Ve is in
1 2 1 4 1 2
λe +pspan e ,e ,e for some λ in W× lifting α. Also, Ve = VFe = pe and
3 1 2 4 3 4 4
{ }
Ve = VFe = pe . This verifies the matrix for V and that for F = pV−1 follows
4 1 1
by semi-linearity. (cid:3)
Definition4.5. Abasis = e ,e ,e ,e asinthePropositionisastandardbasis
1 2 3 4
B { }
for( , ). AstandardbasisforafiniteHondasystem(M,L)=( /pn , /pn )
M L M M L L
is the reduction of a standard basis for ( , ) viewed over W/pnW. Denote the
M L
associated parameters by s =[λ;s ,s ,s ,s ].
B 1 2 3 5
Corollary 4.6. Another basis ′ = e′ for ( , ) is standard if and only if there
B { i} M L
is an a in W× such that e′ =σ2(a)e , e′ =σ(a)e , e′ =σ4(a)e and e′ =σ3(a)e .
1 1 2 2 3 3 4 4
Then sB′ is given by
a σ(a) σ4(a) σ4(a) σ4(a)
λ′ = λ, s′ = s , s′ = s , s′ = s and s′ = s .
σ4(a) 1 σ3(a) 1 2 σ2(a) 2 3 σ(a) 3 5 σ3(a) 5
Proof. Since =span e ,e =span e′,e′ , we can find a,b,c,d in W such that
L { 1 2} { 1 2}
e′ =σ2(a)e +σ(b)e and e′ =ce +σ(d)e .
1 1 2 2 1 2
Then e′ =Fe′ =σ3(a)e +σ2(b)(pe pσ(s )e ) and
4 1 4 1− 1 3
Ve′ = σ(a)V(e )+bV(e )
1 1 2
= σ(a)(e +ps e )+b(pλs e +pλs e +λe +pλs e ).
2 1 4 2 1 3 2 3 5 4
From the matrix for V on the new basis, we have:
Ve′ =e′ +ps′e′ =ce +σ(d)e +ps′(σ3(a)e +σ2(b)(pe pσ(s )e )).
1 2 1 4 1 2 1 4 1− 1 3
Comparing coefficients of e in Ve′ gives λb = p2s′σ(s )σ2(b), so b = 0 or else
3 1 − 1 1
ord (b) 2+ord (b). By comparing coefficients of e , we find that c = 0. Hence
p p 1
≥
the coefficients of e and e give d=a and s′σ3(a)=σ(a)s . We now have:
2 4 1 1
e′ =σ2(a)e , e′ =σ(a)e , e′ =σ3(a)e and e′ =Fe′ =σ4(a)e .
1 1 2 2 4 4 3 4 3
Compare coefficients in Ve′ =aV(e )=aλ(ps e +ps e +e +ps e ) and
2 2 2 1 3 2 3 5 4
Ve′ = λ′(ps′e′ +ps′e′ +e′ +ps e′)
2 2 1 3 2 3 5 4
= λ′(ps′σ2(a)e +ps′σ(a)e +σ4(a)e +ps′σ3(a)e ). (cid:3)
2 1 3 2 3 5 4
Corollary 4.7. Let ( ∗, ∗) be the dual of ( , ) as in Lemma 3.2. There is a
M L M L
standard basis ˜= ξ for ∗ with s =[λ′;s′,s′,s′,s′] related to s by:
B { i} M B˜ 1 2 3 5 B
λ′ = 1 , s′ = 1 s , s′ = σ2(λ)s , s′ =σ2(λ)σ−1(ps s s ),
σ2(λ) 1 −σ(λ) 1 2 − 2 3 1 3− 5
ps σ2(λ)
s′ = σ2(λ)σ(s ) 1 σ−1(ps s s ).
5 − 3 − σ(λ) 1 3− 5
10 A.BRUMERANDK.KRAMER
Proof. By (3.3) and the Proposition, the matrices for Verschiebung and Frobenius
on ∗ in terms of its dual basis e∗,...,e∗ are given by
M 1 4
0 0 0 1 0 1 0 ps
1
p 0 ps 0 pλs pλs λ pλs
V= − 1 , F=σ 2 3 5 .
p2s p/λ p2s s ps ps 0 0 0 p
3 1 3 5 2
− − −
 0 0 1 0   p 0 0 0 
   
Since∗ istheannihilatorof ,wehave ∗=span e∗,e∗ . Forthestandardbasis,
L L L { 3 4}
ξ is in ∗ while Fξ is in p ∗, so ξ = xe∗ +pwe∗ with x in W× and w in W.
2 L 2 M 2 4 3
But ξ ,ξ is a basis for ∗, so ξ =ye∗+ze∗, with y in W×. By scaling, assume
{ 1 2} L 1 3 4
that y =1 and ξ =e∗+ze∗. Apply V and let t =ps s s , to find that
1 3 4 5 1 3− 5
V(ξ )=σ−1(z)e∗ ps e∗+p[t σ−1(z)s ]e∗+e∗.
1 1− 1 2 5− 2 3 4
Proposition 4.1 gives F and V on the standard basis ˜for ∗. Thus:
B M
ξ = F(ξ )=pσ(s z)e∗+σ(λ+pλs z)e∗+pσ(z)e∗,
4 1 1 1 5 2 3
V(ξ ) = ξ +ps′ξ =xe∗+pwe∗+ps′(pσ(s z)e∗+σ(λ+pλs z)e∗+pσ(z)e∗).
1 2 1 4 4 3 1 1 1 5 2 3
Equating the coefficient of e∗ gives σ−1(z) = ps′σ(s z), whose valuation implies
1 1 1
thatz =0. Comparingtheothercoefficientsgivesx=1,w=t andσ(λ)s′ = s ,
5 1 − 1
so ξ = e∗, ξ = e∗+pt e∗, ξ = σ(λ)e∗, ξ = Fξ = σ2(λ)(e∗+pσ(λs )e∗). The
1 3 2 4 5 3 4 2 3 4 1 3 2
remaining formulas relating s and s result from a comparison of
B˜ B
V(ξ ) = V(e∗)+pσ−1(t )V(e∗)
2 4 5 3
= e∗ ps e∗+pσ−1(t )( ps e∗+pt e∗+e∗)
1− 2 3 5 − 1 2 5 3 4
= e∗ p2s σ−1(t )e∗ p(s pσ−1(t )t )e∗+pσ−1(t )e∗
1− 1 5 2− 2− 5 5 3 5 4
with
V(ξ )=λ′(ps′ξ +ps′ξ +ξ +ps′ξ )
2 2 1 3 2 3 5 4
=λ′ ps′e∗+ps′(e∗+pt e∗)+σ2(λ)(e∗+pσ(λs )e∗)+ps′σ(λ)e∗ .
2 3 3 4 5 3 1 3 2 5 2
=λ′ σ2(λ)e∗+p(σ2(λ)σ(λs )+σ(λ)s′)e∗+p(s′ +pt s′)e∗+ps′e∗ .
(cid:0) 1 3 5 2 2 5 3 3 3(cid:1)4
Corollary 4.(cid:0)8. Let be a standard basis for and sB =[λ;s1,s2,s3,s5]. T(cid:1)hen
B M
( ∗, ∗) and ( , ) are isomorphic if and only if there are a,b in W× such that
M L M L
σ2(a) σ3(a)
λ= b, s = σ(bs )+ps s
− a 5 σ2(a) 3 1 3
and one of the following holds: i) s =0 (or s =0) and b=1; ii) s =s =0
1 2 1 2
6 6
and b= 1; iii) all s =0 and bσ2(b)=1.
j
±
Proof. Assume that ( , ) and ( ∗, ∗) are isomorphic and use the previous
Corollaries for the relaMtionLship betwMeenLs and s . In particular, λσ2(λ) = σ4(a)
B˜ B a
for some a in W×. Define b in W× by λ = σ2(a)b. Then bσ2(b) = 1 and the
− a
requirement on s′ implies our claimed formula for s . In case (i), the condition on
3 5
s′ (or s′) forces b=1. In case (ii), use s′ to find that b= 1. Conversely,in each
o1f these2cases, there is an isomorphism. 3 ± (cid:3)
Example 4.9. Since the finite Honda system with parameters s = [λ;0,0,0,0]
B
plays an important role in later conductor estimates, note that it occurs naturally
for p=2. Let A=J(C) be the Jacobian of the curve C: y2+ay =x5+b with a
and b units in W. Then A[2] is isomorphic to as in Notation 3.4. Without loss
λ
E
of generality, we may assume that a primitive fifth root of unity ζ is in W. The
5