Abelian varieties with l-adic Galois representation of Mumford's type

Abstract

This paper is devoted to the study of 4-dimensional abelian varieties over number fields with the property that the Lie algebra of the image of some associated -adic Galois representation is Q -isomorphic to c ⊕ (sl 2) 3 . Such varieties will be referred to as 'abelian varieties with -adic Galois representation of Mumford's type'. We show that such abelian varieties have potentially good reduction at all prime ideals and we determine the possible Newton polygons and the possible isogeny types of these reductions.

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Abelian varieties with ?-adic Galois representation of
Mumford’s type
Rutger Noot
Abstract
This paper is devoted to the study of 4-dimensional abelian varieties over number
fields with the property that the Lie algebra of the image of some associated ?-adic
Galois representation is Q?-isomorphic to c ⊕ (sl2)3. Such varieties will be referred to
as ‘abelian varieties with ?-adic Galois representation of Mumford’s type’. We show
that such abelian varieties have potentially good reduction at all prime ideals and
we determine the possible Newton polygons and the possible isogeny types of these
reductions.
Introduction
A Shimura variety ‘of Hodge type’ parametrizes a family (or families) of abelian varieties with
certain properties. The fact of being of Hodge type offers some powerful tools for the study of
a Shimura variety. If a Shimura variety of Hodge type is a moduli space for abelian varieties
of PEL type (that is, abelian varieties characterized by a polarization, endomorphisms and
a level structure), this is even more useful, since it then is the solution to a moduli problem
that can be formulated in any characteristic. This can be particularly useful for the study
of models in mixed characteristic.
An example due to Mumford shows that not all families of Hodge type are of PEL type.
He constructs 1-dimensional families associated to representations V of algebraic groups
G of the following type. One has Lie(G)Q∼= cQ⊕ sl3
1-dimensional centre, and the induced representation of Lie(G)Qon V ⊗ Q is the tensor
product of the standard representations. In this paper, we will refer to such a pair (G,V )
as being of Mumford’s type, cf. 1.2. If X/C is a fibre of one of Mumford’s families, then the
representation of its Mumford–Tate group on H1
of Mumford’s type. If the fibre in question is not of CM type, then this representation is
itself of Mumford’s type. Conversely, it can be shown that if X/C is an abelian variety
such that the representation of its Mumford–Tate group on H1
representation of Mumford’s type, then it arises as a fibre in one of Mumford’s families.
2,Q, where c ⊂ Lie(G) denotes the
B(X(C),Q) factors through a representation
B(X(C),Q) factors through a
1991 Mathematics Subject Classification 14G25, 11G10, 14K15
1

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2 Noot, Galois representations of Mumford’s type
If X is a fibre over a number field of a family of Mumford’s type associated to a pair
(G,V ), then, for any prime number ?, the associated ?-adic Galois representation factors
through G(Q?). Unless X is of CM type, the image is Zariski dense. Conversely, let F be
a number field and let X/F be an abelian variety such that Zariski closure of the image
of the Galois representation on V? = H1
(G?,V?) is of Mumford’s type. In what follows we will simply say that the ?-adic Galois
representation associated to X is of Mumford’s type. The Mumford–Tate conjecture predicts
that the Mumford–Tate group of X then is of the above kind and therefore that X is a fibre
in a family of the type we consider. This means that, conjecturally, the abelian varieties
arising from Mumford’s construction are exactly those for which the associated ?-adic Galois
representations are of Mumford’s type. Abelian varieties with a Galois representation of
this kind are the examples of the smallest dimension where the Mumford–Tate conjecture
is unsettled. Some evidence supporting the Mumford–Tate conjecture is provided by the
fact that if X is an abelian variety over a number field such that one associated ?-adic
representation is of Mumford’s type, then all associated Galois representations are of this
kind, cf. lemma 1.3.
Shimura curves of Mumford’s type have been extensively studied. For the study of the
reduction of such a curve and of its and zeta- or L-functions, a comparison to a Shimura
curve of PEL type can be used. This has been done in work of Shimura, Morita and, more
recently, Reimann. The results have some implications for the reduction of a fibre over a
number field of a family of Mumford’s kind.
In the present paper, we study the reduction properties of abelian varieties over number
fields such that the associated Galois representations are of Mumford’s type. It is therefore
not unrelated to the Mumford–Tate conjecture, showing that many properties of the abelian
varieties in question can be derived from their Galois representations.
We briefly sketch the contents of the article.
In section 1, we introduce the notions of a representation of Mumford’s type and of
an abelian variety with associated ?-adic Galois representation of Mumford’s type. Some
fundamental properties are established, notably the lemma on ?-independence referred to
above. This lemma in fact states some well known results due to Serre and to Pink.
In section 2, we study the properties of good reduction of an abelian variety for which
the associated Galois representations are of Mumford’s type. It is shown in corollary 2.2
that any such abelian variety has potentially good reduction everywhere.
In section 3 we investigate the Newton polygons of certain crystalline p-adic representa-
tions of Gal(Qp/Qnr
shown in proposition 3.2 that there are only four possibilities. Imposing a finer condition
on the image of the Galois representation permits a refinement of the result. For example,
if Gpis split, then the only possible Newton polygons are 4 × 0,4 × 1 and 8 × 1/2. The
precise results can be found in propositions 3.5 and 3.6. These results have rather obvious
implications for an abelian variety for which the associated Galois representations are of
´ et(X ¯F,Q?) is an algebraic group G?/Q? such that
p) factoring through a representation (Gp,Vp) of Mumford’s type. It is

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Noot, Galois representations of Mumford’s type3
Mumford’s type, cf. 4.4.
In section 4, we investigate the possible isogeny types of a reduction of such an abelian
variety. The final result is stated in corollary 4.4, which is derived from proposition 4.1.
In 4.1, we use arguments going back to Serre to deduce the possible isogeny types of a
reduction at a finite place from properties of an associated ?-adic Galois representation
satisfying appropriate conditions.
Acknowledgements. First of all, I thank Frans Oort for encouraging me to look into
the matters studied in this paper and Johan de Jong for the discussions we had during the
‘Semestre p-adique’ in Paris and in particular for the suggestion to use the Galois represen-
tation for the proof of potential good reduction. I thank the referee of this paper for his
detailed remarks, for urging me to use the Newton cocharacter in section 3 and especially for
the idea to use the Galois representation on the Lie algebra in the proof of proposition 3.2.
This last idea permits to exclude one more Newton polygon from the list of possibilities.
Several other people have commented on earlier versions of this paper and I thank all of
them for their contribution.
Contents
1 Representations of Mumford’s type.3
2 Good reduction5
3 Newton polygons6
4 Isogeny types11
References15
1 Representations of Mumford’s type.
1.1
GF= Gal(¯F/F). If ¯ v is a valuation of¯F, then IF,¯ v⊂ GFis the inertia group of ¯ v. In case F
is a local field, we write just IF⊂ GF for the inertia subgroup. For a number field or local
field F, we write OF for the ring of integers or the valuation ring, respectively.
Let p be a prime number. Denote by Qpthe p-adic completion of Q, by Qnr
unramified extension of Qpcontained in Qp, by Qph the unique extension of Qpof degree h
contained in Qnr
the fields Qnr
If F is a field of characteristic 0 and p is a prime number, then GF acts on the set of
p-power roots of unity in¯F. This action defines the cyclotomic character χp: GF → Z∗
When there can be no confusion as to the value of p, we write χ instead of χp. For i ∈ Z, we
Notations. For any field F, we denote by¯F an algebraic closure of F and we write
pthe maximal
pand by Cpthe completion of Qp. The Frobenius automorphism of any of
pand Qph will be denoted by σ.
p.

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4Noot, Galois representations of Mumford’s type
write Qp(i) for the one dimensional Qp-linear representation of GF where GF acts through
χi
p.
1.2
be an algebraic group over K and let V be a faithful K-linear representation of G. We will
say that the pair (G,V ) is of Mumford’s type if
Representations of Mumford’s type. Let K be a field of characteristic 0, let G
– Lie(G) has one dimensional centre c,
– Lie(G) ¯ K∼= c ¯ K⊕ sl3
2,¯ Kand
– Lie(G) ¯ Kacts on V ¯ Kby the tensor product of the standard representations.
We do not require G to be connected.
If (G,V ) is of Mumford’s type, then there exists an alternating bilinear form on V
(unique up to a scalar) which is fixed by G up to a character. If moreover G is connected,
then there exists a central isogeny N :?G → G where?G is an algebraic group over K such
isomorphic to the tensor product of the standard representations. The Shimura varieties
and families of abelian varieties constructed by Mumford in [Mum69, §4] are associated to
Q-linear representations of Mumford’s type.
Let F be a field, ? a prime number and V?a Q?-vector space. We say that a continuous
Galois representation ρ: GF→ GL(V?)(Q?) is of Mumford’s type if the Zariski closure G?⊂
GL(V?) of ρ(GF) has the property that (G?,V?) is of Mumford’s type. If X/F is a fibre of
one of the families of abelian fourfolds from [Mum69] which is not of CM type, then the
associated ?-adic representations are of Mumford’s type.
that?G∼= Gm,K×?Gssand?Gss
¯ K∼= SL3
2,¯ K. The induced representation of?G ¯ Kon V ¯ Kis
1.3
prime number ?, let G?be the Zariski closure of the image of the ?-adic Galois representation
of G?on V?= H1
Mumford’s type. Then (G?,V?) is of Mumford’s type for all ? and there are infinitely many
prime numbers ? such that Lie(G?)ssis Q?-simple.
Lemma. Let X be a 4-dimensional abelian variety over a number field F. For each
´ et(X ¯F,Q?). Suppose that there exists prime number p such that (Gp,Vp) is of
Proof. It follows from [Ser81] that, after replacing F by a finite extension, we can assume
that all G?are connected. Since
End(XF?) ⊗ZQp∼= EndGF?(Vp) = EndGp(Vp) = Qp
for any finite extension F?of F, one has End(X ¯F) = Z, so EndG?(V?) = EndGF(V?) = Q?
for all ?. For any prime number ?, one deduces from Faltings’ theorem that G?is reductive
and from the facts that X is polarizable and End(X) = Z together with the classification
of semi-simple Lie algebras that Lie(G?)∼= c ⊕ sp8or Lie(G?)∼= c ⊕ sl3
(G?,V?) is of Mumford’s type and in the first case G?= GSp8,Q?. As [Ser81] and [Chi92, 3.10]
2. In the second case,

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Noot, Galois representations of Mumford’s type5
show that the rank of G?is independent of ?, one concludes that (G?,V?) is of Mumford’s
type for every ?.
It follows from [Pin98, 5.13] that there exist a connected reductive group G over Q and
a representation V of G such that (G,V ) ⊗ Q?∼= (G?,V?) for all primes ? belonging to a
set of primes of Dirichlet density 1. Of course, Lie(G) has one dimensional centre c and
Lie(G)Q∼= cQ⊕ sl3
simple. The conditions imply that for any maximal torus of Gss, the group GQacts on the
associated root datum through {(±1,±1,±1)}?S3and that the projection of the image to
S3contains the 3-cycles. The Lie algebra Lie(G)ss
to S3of the image of GQ?⊂ GQin {(±1,±1,±1)} ? S3still contains these cycles. It thus
follows from the Chebotarev density theorem that the set of primes ? such that GQ?has the
desired properties and where (G,V ) ⊗ Q?∼= (G?,V?) has density 1/3 (if the image of GF in
{(±1,±1,±1)} ? S3projects onto S3) or 2/3 (in the other case).
2,Q. The theorem of Pink cited above also implies that Lie(G)ssis Q-
Q?is Q?-simple if and only if the projection
2Good reduction
2.1
such that the residue characteristic at v is different from ?. Suppose that X/F is an abelian
variety such that the representation of GF on V = H1
ρ: GF→ G(Q?) for an algebraic group G ⊂ GL(V ) such that
Proposition. Let F be a field, v a discrete valuation of F and let ? be a prime number
´ et(X ¯F,Q?) factors through a morphism
– the pair (G,V ) is of Mumford’s type and
– Lie(G)ssis Q?-simple.
Then X has potentially good reduction at v.
Proof. Write Ovfor the local ring of F at v and IF,¯ vfor the inertia group of a valuation ¯ v of
¯F extending v. After replacing F by a finite extension, we can assume that G is connected.
Let Y be any abelian variety over F and let ρY: GF→ GL(H1
?-adic Galois representation. It follows from [GRR72, Expos´ e I, 3.6] that there exists a
finite extension F?of F such that the inertia group IF?,¯ v acts unipotently and such that
(ρY(α) − id)2= 0 for each α ∈ IF?,¯ v.
We compare this with the unipotent elements of G(Q?). If x ∈ G(Q?) is unipotent but
x ?= id, then, viewed as an element of G(Q?), it is the image by ‘tensor product’ of an element
(y1,y2,y3) ∈ SL2(Q?)3, with each yiunipotent. Since the image of (y1,y2,y3) is GQ?-invariant
and since there is at least one yi?= id, it follows that all yi?= id. Up to conjugation, we can
assume that for i = 1,2,3, one has
?
0
´ et(Y ¯F,Q?)) be the associated
yi=
1 ai
1
?