REMARKS ON A THEOREM OF PINK IN PRESENCE OF BAD
REDUCTIONWOJCIECH GAJDA AND MARC HINDRY

Abstract. In this note we prove new cases of the Mumford-Tate conjecture by extendinga theorem of Richard Pink for abelian varieties without nontrivial endomorphisms andwith bad semistable reduction. We use quadratic pairs introduced by J.G.Thompson in
the seventies, an important tool in the program of classifying all simple finite groups.
Proof of our main result applies the classification of the quadratic pairs as described byPremet and Suprunenko. Along the way we reprove and generalize a theorem of ChrisHall on the image of Tate module representation of abelian variety as above, to all possible
values of its toric dimension.

1. Introduction
Let A be a simple abelian variety of dimension g defined over a number field K. Supposethat A/K has bad (split) semistable reduction at some place v of the field K, denote Fv
the finite residual field. Let A → spec(OK ) be the Néron model; the connected componentof its special fibre lies in an exact sequence of algebraic Fv -groups:(1)0 −→ T = Gsm −→ A0v −→ Bv −→ 0,

where Bv is an abelian variety over Fv . We call s = dim T the toric dimension of A at v.
The Mumford-Tate group MT(A) is an algebraic Q-group (cf. for example [Pi98], forthe definition) defined in terms of the complex structure of A(C), that comes together
with a representation on V :=H 1 (A(C), Q). When the abelian variety is of type I, II or III
in Albert classification, the Hodge group Hdg(A) is the derived group of MT(A) and wehave MT(A)=Gm Hdg(A). Denote T` (A) the Tate module and V` (A):=T` (A) ⊗Z` Q` ; thecentral object of our study is the Galois representation ρ` : Gal(K̄/K) → GL(V` (A)). We
denote G` the connected component of the Zariski closure of ρ` (Gal(K̄/K)) in GL(V` (A)) ∼=GL2g,Q` , it is an algebraic Q` -group and we still denote ρ` its representation on V` . Via thecomparison isomorphism V` ∼=V ⊗Q Q` we can compare MT(A)⊗Q Q` and G` inside GL2g,Q` .
A deep result due to Borovoï [Bor74], Deligne [Del82] and Piatetski-Shapiro [Pia] showsthat G` is an algebraic subgroup of MT(A) ⊗Q Q` , the Mumford-Tate conjecture asserts
the equality of these two algebraic groups. We refer the reader to a variety of surveys on
the current state of arts of the Mumford-Tate conjecture, which is available online.
It is known that the Mumford-Tate conjecture holds for abelian varieties over number
fields, if and only if, it holds for simple ones cf. [Com19]. Further, it is easy to see that2010 Mathematics Subject Classification. 11G10, 14K15.
Key words and phrases. Abelian variety, Galois representation, Mumford-Tate conjecture.
1

2

WOJCIECH GAJDA AND MARC HINDRY

replacing K by a finite extension or A by an isogenous abelian variety does not alter thevalidity of Mumford-Tate conjecture; it is a deeper result (cf. [LaPi95], Theorem 4.3) that
Mumford-Tate conjecture does not depend on the chosen prime `. Hence, without loosing
any generality, we can concentrate on simple abelian varieties over K, assume semistability
and pick any convenient `.
Main Theorem. Let A be an abelian variety defined over a number field K. Suppose that
End(A) = Z and there is a place of bad semistable reduction with toric dimension s. Then
MT(A)Q` = GSp2g,Q` = G` , and the Mumford-Tate conjecture holds for A, except may bein following exceptional cases.
(1) There is an odd integer r ≥ 3 such that g = 12 2rr and s = 2r−2.
r−1t(2) There is an integer t ≥ 4 with t≡0 or 1 mod 4, such that g = 2 and s = g or g/2.
We remark that the hypotheses in the theorem are generic in the sense that a genericabelian variety of dimension g is simple with endomorphism ring Z, whereas abelian varieties with some bad reduction are far more numerous that abelian varieties with (potential)
good reduction everywhere, the typical case being for g = 1 where integers are rarer thanrational numbers. The first instances of the first exceptional case in Main Theorem occurfor (g, s) = (10, 6), (84, 70), (1716, 924) etc.The first instances of the second exceptionalcase occur for (g, s) = (16, 8), (16, 16), (32, 16), (32, 32), (256, 128), (256, 256) etc.
Comparison with former results
Compared to former attemps at the conjecture (which were often based on computation of
Lie algebras of algebraic groups, e.g., [BGK03, BGK06, BGK10]) our method is different.
The novelty of this note approach depends on applying theory of representations of finitegroups of Lie type directly to comparison of special fibres of two Chevalley group schemesin question, and then lifting equality to their Q` −points (cf. Lemma 2.4) in order toconfirm the conjecture. Our main theorem shall be compared to a result of Pink (under
the assumption End(A)=Z) and to the following results of Noot, and Chris Hall.
Theorem 1.1. (Pink [Pi98], Theorem 5.14) Let A be an abelian variety of dimension gwith endomorphism ring Z. Assume that g is neither• half of a k th power for any odd k > 1, nor
1 2m• of the shape 2 m for any odd m ≥ 3.
Then G` = GSp2g,Q` and the Mumford-Tate conjecture holds for A.
Theorem 1.2. (Noot [No00]) Let A be an abelian variety of dimension 4 with endomorphism ring Z. Assume that A is defined over a number field and has a place with badsemistable reduction. Then G` = GSp8,Q` and the Mumford-Tate conjecture holds for A.
Theorem 1.3. (C. Hall [Ha11]) Let A be an abelian variety of dimension g with endomorphism ring Z. Assume that A has a place of bad semistable reduction with toric dimensions = 1. Then G` = GSp2g,Q` and the Mumford-Tate conjecture holds for A.

REMARKS ON A THEOREM OF PINK IN PRESENCE OF BAD REDUCTION3

Our proof of Main Theorem uses quadratic pairs introduced by J.G.Thompson in [Th71].
We apply the classification of the quadratic pairs described by Premet and Suprunenko in[PS83]. Note that along the way we reprove and generalize Theorem 1.3 to all possible
values of s.
2. Quadratic pairs and minuscule representations
Definition 2.1. A quadratic pair (G, ρ) is formed by a group G, a faithful irreducible
representation ρ : G → GL(V ) such that the subset Q := {g ∈ G \ {1} | (ρ(g) − 1V )2 = 0}
is non empty and generates G. (This is Thompson’s definition [Th71] when we assume
G finite and V a Fp -vector space; otherwise we understand “generates" as “topologicallygenerates"). A quadratic pair is polarised (orthogonal or symplectic) if G leaves invariant
a symmetric or antisymmetric non degenerate bilinear form. A quadratic pair is simple if
G is equal to its commutator group and the quotient of G by its centre is simple.
The main properties of quadratic pairs are a decomposition theorem into (almost) simple
blocs and a classification of these simple blocs which are due to Thompson [Th71], see also
[Ho76, Sa76] and complemented by Premet-Suprunenko in [PS83].
Proposition 2.2. Let (G, ρ) be a quadratic pair with V an Fp -vector space.
(1) (Thompson’s central product theorem) There exist simple quadratic pairs (Gi , ρi )such that G=G1 · · · Gt , and (G, ρ) is isomorphic to ρ1 ⊗ · · · ⊗ ρt : G1 · · · Gt →GL(V1 ⊗ . . . ⊗ Vt ). Further, if the pair (G, ρ) is polarised, then each (Gi , ρi ) ispolarised.
(2) (Thompson, Premet-Suprunenko) A simple quadratic pair belongs to the following
list which is given indicating the root system associated to a Chevalley group and ashort (generally accepted) name for the representation.
• (An , ∧j Std)
• (Bn , Std) or (Bn , Spin)• (Cn , Wj ) with a subrepresentation Wj ⊆ ∧j Std• (Dn , Std) or (Dn , Spin± )• Explicitly given finite set of representations of exceptional groups of type E6 ,E7 , F4 and G2 .
We will require the following elementary lemma. An element g ∈ End(V ) is k-unipotent
if (g − 1)k = 0 but (g − 1)k−1 6= 0; for example a quadratic element is, by definition,
2-unipotent.
Lemma 2.3. Let g1 (resp. g2 ) be a k1 -unipotent element in End(V1 ) (resp. a k2 -unipotent
element in End(V2 )), then g1 ⊗ g2 is a (k1 + k2 − 1)-unipotent element in End(V1 ⊗ V2 ). In
particular, quadratic elements of a quadratic pair (G1 · · · Gt , ρ1 ⊗ · · · ⊗ ρt ) lie in some Gi .
Proof. See [Pa04], Lemma 2.2.1.



Our typical example of quadratic pair is obtained by considering the Galois representation on A[`] ∼= (Z/`Z)2g and restricting it to the subgroup R generated by inertia and4

WOJCIECH GAJDA AND MARC HINDRY

their conjugates. One sees that this representation is still irreducible for ` large enough.
Minuscule representation appear when considering the `-adic representation.
The link between quadratic pairs and minuscule representations is suggested in [Se79](see the remark after Corollaire, page 180), where it is observed that root elements of a(semi-simple) group act quadratically. To pass from representations mod ` to representations over Z` or Q` we’ll use the following elementary lemma.
Lemma 2.4. Let G be a smooth algebraic subgroup of GLN over Z` and H a closed subgroupof G(Z` ) such that π(H) = G(F` ), where π is the reduction mod ` map. Assume ` ≥ 5 andthe Lie algebra of GF` is spanned by quadratic elements. Then H = G(Z` ).
Proof. This is a special case of Lemme 2.6 in [HiRa16].



The lemma applies to SLm , to Sp2m loc.cit. and more generally to any quadratic representation ! Thus the property of being spanned by null square matrices is one of thecharacterisations of minuscule representation, see for example [Pa04] (definition 2.1.3),[Se79] (Lemme 5 and Corollaire after Proposition 5).
Below we recall the table of minuscule weights of height one (reference: [Se79]) giving
the type of root system (A,B,C,D), the label of the highest weight describing the representation (labeled as in loc. cit.), a short name for the representation, the dimension of
the representation and finally the “sign" of the representation: +1 for an orthogonal representation, −1 for a symplectic representation, and 0 for a non auto-dual representation.
Table of minuscule representations
Lie type
weightrepresentation dimension
sign
j

(−1)
if
n = 2j − 1
n+1
An (n ≥ 1) $j (1 ≤ j ≤ n)
Λj (Std)
j
0
else
+1
if
n
≡
0,
3 mod 4Bn (n ≥ 2)
$n
Spin
2n
−1 if ≡ 1, 2 mod 4
Cn (n ≥ 2)
$1
Std
2n
−1
+1
Std
$1
2n

Dn (n ≥ 3)
+1 if n ≡ 0 mod 4
−
n−1
$n−1
Spin
2
−1 if n ≡ 2 mod 4
$n
2n−1
Spin+
0
if n odd

3. Proof of Main Theorem
Existence of a place of bad reduction with toric dimension s implies that a topological
generator of (the image of) inertia provides a quadratic element g with drop equal to s.
Recall that a quadratic element of (G, ρ) is an element such that (ρ(g) − 1)2 = 0 andthe drop is the dimension of the image of ρ(g) − 1. Let us denote by G0` /Q` the derived

REMARKS ON A THEOREM OF PINK IN PRESENCE OF BAD REDUCTION

5

Zargroup of the algebraic monodromy group G` := ρ` (Gal(K̄/K)) which, without loss ofgenerality, can be assumed connected. For simplicity we treat the case End(A)=Z although
our methods apply to more general situations (cf. Section 4). The algebraic group G0` andthe natural representation ρ` : G0` → GL(T` (A)) ⊂ GL2g,Q` have very special features.
• The representation is faithful symplectic.
• (Faltings, [Fa83]) The `-adic representation is absolutely irreducible; further, for `large enough, the representation mod ` is still irreducible.
• (Pink, [Pi98]) The representation is a tensor product of representations ρi : Gi →GL(Vi ), where Gi are absolutely irreducible groups, G0` = G1 · · · Gs (an almost
direct product), V` ∼= V1 ⊗ · · · ⊗ Vs and each of these factors is a classical minuscule
symplectic representation. Further, the decomposition is isotypical with a transitive
action of the Galois group.
We distinguish two cases, using quadratic pairs in the first case and minuscule representations for the second:• (A) s > 1 (hence odd and ≥ 3).
• (B) s = 1 (i.e. the group G0` is absolutely irreducible).
Case (A) Assumption on s implies (denoting m= dim Vi ) that 2g=ms as in [Pi98], but wewill show that, when there is a place of bad reduction, this case does not occur. We choose` large enough so that the representation modulo `, which we denote ρ̄` : Gal(K̄/K) →GL(A[`]) ∼= GL2g (F` ), is still irreducible. Let R̄ be the subgroup generated by the imageof the inertia group at the place of bad reduction and its conjugates. We first claim thatthe restriction of ρ̄` to R̄ is still irreducible. Indeed by Clifford’s classical theorem, therepresentation is induced from an irreducible subrepresentation W , but (R̄, W ) forms aquadratic pair and then R̄ is a finite group of Lie type and has no normal subgroups ofsmall index, at least if ` is chosen sufficiently large, therefore V =W . Next, R̄ is generated
by the conjugacy class of a quadratic element, therefore it cannot be an almost direct
product G1 · · · Gs with s>1. The argument for this is that a quadratic element g has tooccur in only one factor, i.e., g=(1, . . . , gi , . . . , 1) ∈ G1 · · · Gs . We now lift R̄ to R ⊂ G0` ,where R denotes the subgroup generated by inertia in the `-adic representation. Since the
Galois group permutes the factors, the inertia cannot sit in only one factor (cf. Lemma
2.3) and we conclude that s=1. Note that the case of dim A=4 was initially treated by
Noot in Prop. 2.1 and Cor. 2.2, [No00].
Case (B). The list of symplectic minuscule representations of dimension 2g of a classicalgroup is short; see the table in previous section; we denote (G1 , ρ1 ) ∼ (G2 , ρ2 ) when thetwo simple algebraic groups Gi have the same root system and ρi correspond to the same
weight.
(1) (G0` , ρ) ∼ (Sp2g , Std).
(2) (G0` , ρ) ∼ (SL2r , ∧r Std), with odd r ≥ 3 and 2g = 2rr .
(3) (G0` , ρ) ∼ (SO2r+1 , Spin), with 2g = 2r and r ≡ 1, 2 mod 4.
(4) (G0` , ρ) ∼ (SO2r , Spin± ), with 2g = 2r−1 and r ≡ 2 mod 4.

6

WOJCIECH GAJDA AND MARC HINDRY

In the first case we conclude G0` = Sp2g,Q` = Hdg(A)Q` . In the three last cases we use
the computation by Premet-Suprunenko of the drop of a quadratic element(cf. [PS83],
1 2r
Section 3). More precisely, in the second case we see that dim A = 2 r is very specific.
Further, the drop of a non-trivial quadratic element in this representation is s := 2r−2
r−1
(see [PS83], Lemma 18 and the calculation at p.78, beginning of §2), therefore this casecan only happen if the toric dimension at a bad place is equal to s. In the third case we
see that dim A = 2r−1 is very specific. Further, the drop of a non-trivial quadratic elementin this representation is s := 2r−1 or 2r−2 (see [PS83] Lemma 20), therefore this case canonly happen if the toric dimension at a bad place is equal to s = g or g/2. For the spin
representation to be symplectic, we need r ≡ 1, 2 mod 4. In the fourth case we see that
dim A = 2r−2 is very specific. Further, the drop of a non-trivial quadratic element in thisrepresentation is s := 2r−2 or 2r−3 (see [PS83] Lemma 21 and Note 2, at p.86), therefore
this case can only happen, if the toric dimension at a bad place is equal to s = g or g/2.
For the half-spin representation to be symplectic, we need r ≡ 2 mod 4. This completes
the proof of Main Theorem.
Remark 3.1. Our proof shows that inertia subgroups at bad places generate the derived
group of the image of Galois. Indeed the above arguments show that the algebraic subgroup
R generated by the inertia subgroup and their conjugates is a non trivial normal subgroupand the only such subgroup of Sp2g is Sp2g itself.
4. Conclusion
As a final remark we add the observation that the same method can handle the casewhen the endomorphism algebra End0 (A) := End(A) ⊗ Q is a quaternion algebra withcenter Q. However, note that the combinatorics, when the center is larger, is much more
involved. We plan to address this issue in a future work.
Theorem 4.1. Let A be an abelian variety defined over a number field K. Suppose that
End0 (A) := End(A) ⊗ Q is a quaternion algebra with centre Q and there is a place of bad
reduction with toric dimension s (note s is always even). Then,• when the algebra D is indefinite, i.e., A has type II, we have MT(A)Q` = GSpg,Q` =G`
and the Mumford-Tate conjecture holds for A, except may be in following exceptional
cases:(1) there is an odd integer r ≥ 3 such that g = 2rr and s = 2 2r−2,
r−1t(2) there is an integer t≥ 5 with t≡1 or 2 mod 4 such that g=2 and s=g or g/2,
• when the algebra D is definite, i.e., A has type III, we have MT(A)Q` = GSOg,Q` =G`
and the Mumford-Tate conjecture holds for A, except may be in following exceptional
cases:(1) there is an even integer r ≥ 2 such that g = 2rr and s = 2 2r−2,
r−1t(2) there is an integer t≥4 with t≡0 or 3 mod 4 such that g=2 and s=g or g/2.
The extension of Theorem 1.1 (and Theorem 4.1) when s = 2 to abelian varieties of type
II is worked out in [HiRa16], Théorèmes 10.6 and 10.7.

REMARKS ON A THEOREM OF PINK IN PRESENCE OF BAD REDUCTION

7

Acknowledgements. Both authors thank professor Ronald Solomon for his guidance
through the literature on quadratic pairs. The authors were partially supported by aresearch grant UMO-2018/31/B/ST1/01474 of the National Centre of Sciences of Poland.

References[BGK03][BGK06][BGK10][Bor74][Com19][Del82][Fa83][Ha11][HiRa16][Ho76][LaPi95][No00][Pa04][Pia][Pi98][PS83][Sa76][Se79][Th71]

Banaszak, G., Gajda, W., Krasoń, P., On Galois representations for abelian varieties
with complex and real multiplications J. Number Theory 100 (2003), 117-132.
Banaszak, G.; Gajda, W.; Krasoń, P., On the image of l-adic Galois representations
for abelian varieties of type I and II. Doc. Math. 2006, Extra Vol., 35–75.
Banaszak, G.; Gajda, W.; Krasoń, P., On the image of Galois l-adic representations
for abelian varieties of type III. Tohoku Math. J. 62 (2010), 163–189.
Borovoi, M., The action of the Galois group on the rational cohomology classes of type(p, p) of Abelian varieties. Mat. Sb. (N.S.) 94(136) (1974), 649-652, 656.
Commelin, J., The Mumford-Tate conjecture for products of abelian varieties. Algebr.
Geom. 6 (2019), 650–677.
Deligne, P., et al., Hodge cycles, motives, and Shimura varieties. Lecture Notes inMathematics, 900. Springer-Verlag, Berlin–New York, 1982.
Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Inventiones
Mathematicae 73, (1983). 349–366.
Hall, C., An open image theorem for a general class of abelian varieties. With an appendix
by Emmanuel Kowalski. Bull. Lond. Math. Soc. 43 (2011), 703–711.
Hindry, M.; Ratazzi, N., Torsion pour les variétés abéliennes de type I et II. Algebra &Number Theory 10 (2016), 1845–1891.
Ho, C.Y., On the quadratic pairs, J. Algebra 43 (1976), 338–358.
Larsen, M, Pink, R.. Abelian varieties, `-adic representations, and `-independence. Math.
Ann. 302, (1995), 561–579.
Noot, R., Abelian varieties with `-adic Galois representation of Mumford’s type. J. für die
reine und ang. Math. 519 (2000), 155–169.
Paugam, F. , Galois representations, Mumford-Tate groups and good reduction of abelianvarieties. Math. Ann. 329 (2004), 119–160.
Piatetski-Shapiro, I. , Interrelations between the Tate and Hodge hypotheses for abelianvarieties. Mat. Sb. (N.S.) 85(127) (1971), 610–620.
Pink, R., `-adic algebraic monodromy groups cocharacters, and the Mumford-Tate conjecture. J. für die reine und ang. Math. 495 (1998), 187–237.
Premet, A.A.; Suprunenko, I.D., Quadratic modules for Chevalley groups over fields of
odd characteristics. Math. Nachr. 110 (1983) 65–96.
Salzberg Stark, B., Another Look at Thompson’s Quadratic Pairs, J. of Algebra 45(1977), 334–342.
Serre, J.-P., Groupes algébriques associés aux modules de Hodge-Tate. Astérisque 65(1979),155–188.
Thompson, J.G., Quadratic pairs. In Actes du Congrès International des Mathématiciens(Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, 375-376.

8

WOJCIECH GAJDA AND MARC HINDRY

Wojciech Gajda
Faculty of Mathematics and Computer Science
Adam Mickiewicz University
Uniwersytetu Poznańskiego 461-614 Poznań, Poland
E-mail adress: gajda@amu.edu.pl

Marc Hindry
Institut de Mathematiques de Jussieu - Paris rive gauche (Imj-Prg),
Université Paris Cité (Paris Diderot),
Bâtiment Sophie Germain,
Boite Courrier 7012,
8 place Aurélie Nemours,
F-75205 Paris CEDEX 13, France.
E-mail address: marc.hindry@imj-prg.fr