The splitting of reductions of an abelian variety - Cornell University

27 Pages·2013·0.44 MB·English

by  David Zywina

THE SPLITTING OF REDUCTIONS OF AN ABELIAN VARIETY
DAVID ZYWINA
Abstract. Consider an absolutely simple abelian variety A defined over a number field K. For
most places v of K, we study how the reduction A of A modulo v splits up to isogeny. Assuming
v
the Mumford–Tate conjecture for A and possibly increasing the field K, we will show that A is
v
isogenoustothem-thpowerofanabsolutelysimpleabelianvarietyforallplacesv ofK awayfrom
asetofdensity0,wheremisanintegerdependingonlyontheendomorphismringEnd(A ). This
K
proves many cases, and supplies justification, of a conjecture of Murty and Patankar. Under the
same assumptions, we will also describe the Galois extension of Q generated by the Weil numbers
of A for most v.
v
1. Introduction
Let A be a non-zero abelian variety defined over a number field K. Let Σ be the set of finite
K
places of K, and for each place v ∈ Σ let F be the corresponding residue field. The abelian
K v
variety A has good reduction at all but finitely many places v ∈ Σ . For a place v ∈ Σ for which
K K
A has good reduction, the reduction A modulo v is an abelian variety A defined over F . We know
v v
that A is isogenous to a product of simple abelian varieties (all defined over F ). The goal of this
v v
paper is study how A factors for “almost all” places v, that is, for those v away from a subset of
v
Σ with (natural) density 0. In particular, we will supply evidence for the following reformulation
K
of a conjecture of Murty and Patankar [MP08].
Conjecture 1.1 (Murty–Patankar). Let A be an absolutely simple abelian variety over a number
field K. Let V be the set of finite places v of K for which A has good reduction and A /F is simple.
v v
Then, after possibly replacing K by a finite extension, the density of V exists and V has density 1
if and only if End(A ) is commutative.
K
The conjecture in [MP08] is stated without the condition that K possibly needs to be replaced
by a finite extension. An extra condition is required since one can find counterexamples to the
original conjecture. (For example, let A/Q be the Jacobian of the smooth projective curve defined
by the equation y2 = x5 −1. We have End(AQ)⊗Z Q = Q(ζ5), and in particular A is absolutely
simple. For each prime p ≡ −1 (mod 5), the abelian variety A has good reduction at p and A is
p
isogenous to E2, where E is an elliptic curve over F that satisfies |E (F )| = p+1. Conjecture 1.1
p p p p p
will hold for A with K = Q(ζ ), equivalently, A is simple for all primes p ≡ 1 (mod 5) away from
5 p
a set of density 0.)
We shall relate Conjecture 1.1 to the arithmetic of the Mumford–Tate group of A, see §2.5 for
a definition of this group and §2.6 for a statement of the Mumford–Tate conjecture for A. In our
work, we will first replace K by an explicit finite Galois extension Kconn. The field Kconn is the
A A
smallest extension of K for which all the (cid:96)-adic monodromy groups associated to A over Kconn are
A
2000 Mathematics Subject Classification. Primary 14K15; Secondary 11F80.
Key words and phrases. Reductions of abelian varieties, Galois representations.
This material is based upon work supported by the National Science Foundation under agreement No. DMS-
1128155.
1
connected, cf. §2.3. We now give an alternative description of this field due to Larsen and Pink
[LP95]. For each prime number (cid:96), let A[(cid:96)∞] be the subgroup of A(K) consisting of those points
whose order is some power of (cid:96). Let K(A[(cid:96)∞]) be the smallest extension of K in K over which all
the points of A[(cid:96)∞] are defined. We then have
(cid:92)
Kconn = K(A[(cid:96)∞]).
A
(cid:96)
Theorem 1.2. Let A be an absolutely simple abelian variety defined over a number field K such
that Kconn = K. Define the integer m = [End(A)⊗ZQ : E]1/2 where E is the center of the division
A
algebra End(A)⊗ZQ.
((i)) For all v ∈ Σ away from a set of density 0, A is isogenous to Bm for some abelian variety
K v
B/F .
v
((ii)) Suppose that the Mumford–Tate conjecture for A holds. Then for all v ∈ Σ away from a
K
set of density 0, A is isogenous to Bm for some absolutely simple abelian variety B/F .
v v
Corollary 1.3. Let A be an absolutely simple abelian variety defined over a number field K such
that Kconn = K. Let V be the set of finite places v of K for which A has good reduction and A /F
A v v
is simple. If End(A) is non-commutative, then V has density 0. If End(A) is commutative and the
Mumford–Tate conjecture for A holds, then V has density 1.
WewillobservelaterthatEnd(A )⊗ZQ = End(A)⊗ZQwhenKconn = K; sotheabovecorollary
K A
shows that Conjecture 1.1 is a consequence of the Mumford–Tate conjecture. Using Theorem 1.2,
we will prove the following general version.
Theorem 1.4. Let A be a non-zero abelian variety defined over a number field K such that Kconn =
A
K. The abelian variety A is isogenous to An11 ×···×Anss, where the abelian varieties Ai/K are
simple and pairwise non-isogenous. For 1 ≤ i ≤ s, define the integer mi = [End(Ai)⊗ZQ : Ei]1/2,
where Ei is the center of End(Ai)⊗ZQ.
Suppose that the Mumford–Tate conjecture for A holds. Then for all places v ∈ Σ away from
K
a set of density 0, A is isogenous to a product (cid:81)s Bmini, where the B are absolutely simple
v i=1 i i
abelian varieties over F that are pairwise non-isogenous and satisfy dim(B ) = dim(A )/m .
v i i i
Observe that the integer s and the pairs (n ,m ) from Theorem 1.4 can be determined from the
i i
endomorphism ring End(A)⊗ZQ.
1.1. The Galois group of characteristic polynomials. Let A be a non-zero abelian variety
over a number field K. Fix a finite place v of K for which A has good reduction. Let π be
Av
the Frobenius endomorphism of A and let P (x) be the characteristic polynomial of π . The
v Av Av
polynomial P (x) is monic of degree 2dimA with integral coefficients and can be characterized
Av
by the property that P (n) is the degree of the isogeny n−π of A for each integer n.
Av Av v
Let W be the set of roots of P (x) in Q. Honda–Tate theory shows that A is isogenous to
Av Av v
a power of a simple abelian variety if and only if P (x) is a power of an irreducible polynomial;
Av
equivalently, if and only if the action of GalQ on WAv is transitive.
The following theorem will be important in the proof of Theorem 1.2. Let G be the Mumford–
A
Tate group of A; it is a reductive group over Q which we will recall in §2.5. Let W(G ) be the
A
Weyl group of G . We define the splitting field k of G to be the intersection of all the subfields
A GA A
L ⊆ Q for which the group G is split.
A,L
Theorem 1.5. Let A be an absolutely simple abelian variety over a number field K that satisfies
Kconn = K. Assume that the Mumford–Tate conjecture for A holds and let L be a finite extension
A
of k . Then
GA ∼
Gal(L(W )/L) = W(G ).
Av A
2
for all places v ∈ Σ away from a set of density 0.
K
Moreover, we expect the following conjecture to hold.
Conjecture 1.6. Let A be a non-zero abelian variety over a number field K that satisfies Kconn =
A
K. There is a group Π(G ) such that Gal(Q(W )/Q) ∼= Π(G ) for all v ∈ Σ away from a set
A Av A K
with natural density 0.
We shall later on give an explicit candidate for the group Π(G ); it has W(G ) as a normal
A A
subgroup. We will also prove the conjecture in several cases, cf. §8.
1.2. Some previous results. We briefly recall a few earlier known cases of Theorems 1.2 and 1.5.
Let A be an abelian variety over a number field K such that End(A ) = Z and such that
K
2dim(A) is not a k-th power and not of the form (cid:0)2k(cid:1) for every odd k > 1. Under these as-
k
sumptions, Pink has shown that G is isomorphic to GSp and that the Mumford–Tate
A 2dim(A),Q
conjecture for A holds [Pin98, Theorem 5.14]. We will have Kconn = K, so Theorem 1.2 says that
A
A /F is absolutely simple for all places v ∈ Σ away from a set of density 0. We have k = Q
v v K GA
since G is split, so Theorem 1.5 implies that Gal(Q(W )/Q) is isomorphic to the Weyl group
A Av
∼
W(GSp2dim(A),Q) = W(Sp2dim(A),Q) = W(Cdim(A)) for all v ∈ ΣK away from a set of density 0.
TheseresultsareduetoChavdarov[Cha97, Cor.6.9]inthespecialcasewheredim(A)is2, 6orodd
(thesedimensionsareusedtociteatheoremofSerrewhichgivesamod(cid:96)versionofMumford–Tate).
Now consider the case whereA is an absolutely simple CM abelian variety defined over a number
fieldK;soF := End(A )⊗ZQisanumberfieldthatsatisfies[F : Q] = 2dim(A). AfterreplacingK
K
by a finite extension, we may assume that F = End(A)⊗ZQ. We have GA ∼= ResF/Q(Gm,F), where
Res denotes restriction of scalars from F to Q. The theory of complex multiplication shows
F/Q
that A satisfies the Mumford–Tate conjecture and hence Theorem 1.2 says that A /F is absolutely
v v
simple for almost all places v ∈ Σ ; this is also [MP08, Theorem 3.1] where it is proved using L-
K
functions and Hecke characters. Theorem 1.5 is not so interesting in this case since W(G ) = 1.
A
SeveralcasesofTheorem1.2wereprovedbyJ.Achter[Ach09,Ach11]; forexample, thoseabelian
varieties A/K such that F := End(A )⊗Z Q is a totally real number field and dim(A)/[F : Q]
K
is odd. A key ingredient is some of the known cases of the Mumford–Tate conjecture from the
papers [Vas08], [BGK06] and [BGK10]. Achter’s approach is very similar to this paper and boils
down to showing that P (x) is an appropriate power of an irreducible polynomial for almost all
Av
places v ∈ Σ . In the case where End(A ) is commutative, Achter uses the basic property that if
K K
P (x) mod (cid:96)isirreducibleinF [x], thenP (x)isirreducibleinZ[x]; unfortunately, thisapproach
Av (cid:96) Av
will not work for all absolutely simple abelian varieties A/K for which Kconn = K and End(A )
A K
is commutative (see §8.1 for an example). Corollary 1.6 in the non-commutative case also follows
from [Ach09, Theorem B].
1.3. Overview. We set some notation. The symbol (cid:96) will always denote a rational prime. If X is
a scheme over a ring R and we have a ring homomorphism R → R(cid:48), then we denote by X the
R(cid:48)
scheme X× SpecR(cid:48) over R(cid:48). The homomorphism is implicit in the notation; it will usually be
SpecR
a natural inclusion or quotient homomorphism; for example, Q → Q , Z → Q , Z → F , K (cid:44)→ K.
(cid:96) (cid:96) (cid:96) (cid:96) (cid:96)
For a field K, we will denote by K a fixed algebraic closure and define the absolute Galois group
Gal := Gal(K/K).
K
For a number field K, a topological ring R and a finitely generated R-module M, consider a Ga-
lois representation ρ: Gal → Aut (M); our representations will always be continuous (for finite
K R
R, we always use the discrete topology). If v is a finite place of K for which ρ is unramified, then
we denote by ρ(Frob ) the conjugacy class of ρ(Gal ) arising from the Frobenius automorphism
v K
3
at v.
Let A be a non-zero abelian variety over a number field K that satisfies Kconn = K. Fix an
A
embedding K ⊆ C. In §2, we review the basics about the (cid:96)-adic representations arising from the
action of Gal on the (cid:96)-power torsion points of A. To each prime (cid:96), we will associate an algebraic
K
group G over Q . Conjecturally, the connected components of the groups G are isomorphic
A,(cid:96) (cid:96) A,(cid:96)
to the base extension of a certain reductive group G defined over Q; this is the Mumford–Tate
A
group of A. The group G comes with a faithful action on H (A(C),Q). In §3, we review some
A 1
facts about reductive groups and in particular define the group Π(G ) of Conjecture 1.6.
A
Let us hint at how Theorems 1.2 and 1.5 are connected; further details will be supplied later.
For the sake of simplicity, suppose that End(A ) = Z. Fix a maximal torus T of G and a number
K A
field L for which T is split. Let X(T) be the group of characters of T and let Ω ⊆ X(T) be
L Q
the set of weights arising from the representation of T ⊆ G on H (A(C),Q). The Weyl group
A 1
W(G ,T) has a natural faithful action on the set Ω.
A
Using the geometry of G and our additional assumption End(A ) = Z, one can show that
A K
action of W(G ,T) on Ω is transitive. Assuming the Mumford–Tate conjecture, we will show that
A
for all v ∈ Σ away from a set of density 0 there is a bijection W ↔ Ω such that the action
K Av
of Gal on W corresponds with the action of some subgroup of W(G ,T) on Ω (this will be
L Av A
described in §6.2 and it makes vital use of a theorem of Noot described in §4). So for almost all
v ∈ Σ , wefindthatGal(L(W )/L)isisomorphictoasubgroupofW(G ,T). IfGal(L(W )/L)
K Av A Av
is isomorphic to W(G ,T), then we deduce that Gal acts transitively on W and hence P (x)
A L Av Av
is a power of an irreducible polynomial. The assumption End(A ) = Z also ensures that P (x)
K Av
is separable for almost all v, and thus we deduce that P (x) is almost always irreducible (and
Av
hence A is almost always simple). To show that Gal(L(W )/L) is maximal for all v ∈ Σ away
v Av K
fromasetofdensity0,wewilluseaversionofJordan’slemmawithsomelocalinformationfrom§5.
TheproofofTheorem1.4canbefoundin§7; itiseasilyreducedtotheabsolutelysimplecase. In
§8 we discuss Conjecture 1.6 further and give an extended example. Finally, we will prove effective
versions of Theorems 1.2 and 1.5 in §9.
2. Abelian varieties and Galois representations: background
Startingin§2.2,wefixanon-zeroabelianvarietyAdefinedoveranumberfieldK. Inthissection,
we review some theory concerning the (cid:96)-adic representations associated to A. In particular, we will
definetheMumford–TategroupofAandstatetheMumford–Tateconjecture. Forbasicsonabelian
varieties,see[Mil86]. Thepapers[Ser77]and[Ser94]supplyoverviewsofseveralmotivicconjectures
for A and how they conjecturally relate with its (cid:96)-adic representations.
2.1. Characteristic polynomials. Fix a finite field F with cardinality q. Let B be a non-zero
q
abelian variety defined over F and let π be the Frobenius endomorphism of B. The characteristic
q B
polynomialofB istheuniquepolynomialP (x) ∈ Z[x]forwhichtheisogenyn−π ofB hasdegree
B B
P (n) for all integers n. The polynomial P (x) is monic of degree 2dimB. We define W to be
B B B
the set of roots of P (x) in Q. The elements of W are algebraic integers with absolute value q1/2
B B
under any embedding Q (cid:44)→ C. We define Φ to be the subgroup of Q× generated by W .
B B
Thefollowing lemmasays that, undersome additional conditions, thefactorizationofP (x)into
B
irreducible polynomials corresponds to the factorization of B into simple abelian varieties.
Lemma 2.1. Let B be a non-zero abelian variety defined over F , where p is a prime. Assume
p
that PB(x) is not divisible by x2−p. If PB(x) = (cid:81)si=1Qi(x)mi, where the Qi(x) are distinct monic
4
irreducible polynomials in Z[x], then B is isogenous to (cid:81)s Bmi, where B is a simple abelian
i=1 i i
variety over F satisfying P (x) = Q (x).
p Bi i
Proof. This is a basic application of Honda–Tate theory; see [WM71]. We know that B is isoge-
nous to (cid:81)t Bni, where the B /F are simple and pairwise non-isogenous. We have P (x) =
i=1 i i p B
(cid:81)ti=1PBi(x)ni. Honda–Tate theory says that PBi(x) = Qi(x)ei where the Qi(x) are distinct irre-
ducible monic polynomials in Z[x] and the e are positive integers. After possibly reordering the
i
B , we have the factorization of P (x) in the statement of the lemma with m = n e and s = t. It
i B i i i
thus suffices to show that e = 1 for 1 ≤ i ≤ t.
i
Fix 1 ≤ i ≤ t. Since Bi is simple, the ring E := End(Bi)⊗Z Q is a division algebra with center
Φ := Q(π ). By Waterhouse and Milne [WM71, I. Theorem 8], we have e = [E : Φ]1/2. If the
Bi i
number field Φ has a real place, then π = ±p1/2 since |π | = p1/2. However, if π = ±p1/2, then
Bi Bi Bi
x2−p divides P (x) ∈ Q[x]. So by our assumption that x2−p does not divide P (x), we conclude
Bi B
that Φ has no real places. Using that Q(π ) has no real places and F has prime cardinality,
Bi p
[Wat69, Theorem 6.1] implies that E is commutative and hence e = [E : Φ]1/2 = 1. (cid:3)
i
2.2. Galois representations. For each positive integer m, let A[m] be the m-torsion subgroup
of A(K); it is a free Z/mZ-module of rank 2dimA. For a fixed rational prime (cid:96), let T (A) be the
(cid:96)
inverse limit of the groups A[(cid:96)e] where the transition maps are multiplication by (cid:96). We call T (A)
(cid:96)
the Tate module of A at (cid:96); it is a free Z(cid:96)-module of rank 2dimA. Define V(cid:96)(A) = T(cid:96)(A)⊗Z(cid:96) Q(cid:96).
There is a natural action of Gal on the groups A[m], T (A), and V (A). Let
K (cid:96) (cid:96)
ρA,(cid:96): GalK → AutQ(cid:96)(V(cid:96)(A)).
be the Galois representation which describes the Galois action on the Q -vector space V (A).
(cid:96) (cid:96)
Fix a finite place v of K for which A has good reduction. Denote by A the abelian variety over
v
F obtained by reducing A modulo v. If v (cid:45) (cid:96), then ρ is unramified at v and satisfies
v A,(cid:96)
P (x) = det(xI −ρ (Frob ))
Av A,(cid:96) v
where P (x) ∈ Z[x] is the degree 2dimA monic polynomial of §2.1. Furthermore, ρ (Frob ) is
Av A,(cid:96) v
the conjugacy class of a semisimple element of AutQ(cid:96)(V(cid:96)(A)) ∼= GL2dimA(Q(cid:96)).
2.3. (cid:96)-adic monodromy groups. Let GL be the algebraic group defined over Q for which
V (A) (cid:96)
(cid:96)
GLV(cid:96)(A)(L) = AutL(L⊗Q(cid:96)V(cid:96)(A))forallfieldextensionsL/Q(cid:96). TheimageofρA,(cid:96)liesinGLV(cid:96)(A)(Q(cid:96)).
Let G be the Zariski closure in GL of ρ (Gal ); it is an algebraic subgroup of GL
A,(cid:96) V (A) A,(cid:96) K V (A)
(cid:96) (cid:96)
called the (cid:96)-adic algebraic monodromy group of A. Denote by G◦ the identity component of G .
A,(cid:96) A,(cid:96)
Let Kconn be the fixed field in K of ρ−1(G◦ (Q )); it is a finite Galois extension of K that does
A A,(cid:96) A,(cid:96) (cid:96)
not depend on (cid:96), cf. [Ser00, 133 p.17]. Thus, for any finite extension L of Kconn in K, the group
A
ρ (Gal(K/L)) is Zariski dense in G◦ (equivalently, G = G◦ ). We have Kconn = K if and
A,(cid:96) A,(cid:96) AL,(cid:96) A,(cid:96) A
only if all the (cid:96)-adic monodromy groups G are connected.
A,(cid:96)
Proposition 2.2. Assume that Kconn = K.
A
((i)) The commutant of GA,(cid:96) in EndQ(cid:96)(V(cid:96)(A)) is naturally isomorphic to End(A)⊗ZQ(cid:96).
((ii)) The group G is reductive.
A,(cid:96)
((iii)) We have End(A )⊗ZQ = End(A)⊗ZQ.
K
Proof. Faltings proved that the representation ρ is semisimple and that the natural homomor-
A,(cid:96)
phism
End(A)⊗ZQ(cid:96) → EndQ(cid:96)[GalK](V(cid:96)(A))
is an isomorphism, cf. [Fal86, Theorems 3–4]. So the commutant of GA,(cid:96) in EndQ(cid:96)(V(cid:96)(A)) equals
End(A)⊗ZQ(cid:96). It easily follows that GA,(cid:96) is reductive.
5
Let L be a finite extension of K for which End(A ) = End(A ). Since G = G , we obtain
K L AL,(cid:96) A,(cid:96)
an isomorphism between their commutants; End(AL) ⊗Z Q(cid:96) = End(A) ⊗Z Q(cid:96). By comparing
dimensions, we find that the injective map End(A)⊗ZQ → End(AL)⊗ZQ = End(AK)⊗ZQ is an
isomorphism. (cid:3)
The following result of Bogomolov [Bog80,Bog81] says that the image of ρ in G is large.
A,(cid:96) A,(cid:96)
Proposition 2.3. The group ρ (Gal ) is an open subgroup of G (Q ) with respect to the (cid:96)-adic
A,(cid:96) K A,(cid:96) (cid:96)
topology.
Using that ρ (Gal ) is an open and compact subgroup of G (Q ), we find that the algebraic
A,(cid:96) K A,(cid:96) (cid:96)
group G describes the image of ρ up to commensurability. For each place v ∈ Σ for which
A,(cid:96) A,(cid:96) K
A has good reduction, we have a group Φ as defined in §2.1.
Av
Proposition 2.4. Assume that Kconn = K.
A
((i)) The rank of the reductive group G does not depend on (cid:96).
A,(cid:96)
((ii)) Let r be the common rank of the groups G . Then the set of v ∈ Σ for which Φ is a
A,(cid:96) K Av
free abelian group of rank r has density 1.
Proof. Fix a prime (cid:96). For a place v where A has good reduction, let T be the Zariski closure in
v
G of the subgroup generated by a fixed (semisimple) element in the conjugacy class ρ (Frob ).
A,(cid:96) A,(cid:96) v
The set of places v for which T is a maximal torus of G has density 1; this follows from [LP97a,
v A,(cid:96)
Theorem 1.2]. Observe that T is a maximal torus of G if and only if Φ (the multiplicative
v A,(cid:96) Av
group generated by the eigenvalues of ρ (Frob )) is a free abelian group whose rank equals the
A,(cid:96) v
reductive rank of G . Parts (i) and (ii) follow since the rank of Φ does not depend on (cid:96). (cid:3)
A,(cid:96) Av
2.4. The set S . WedefineS tobethesetofplacesv ∈ Σ thatsatisfythefollowingconditions:
A A K
• A has good reduction at v;
• F has prime cardinality;
v
• Φ is a free abelian group whose rank equals the common rank of the groups G .
Av A,(cid:96)
Using Proposition 2.4(ii), S has density 1 if Kconn = K. Since we are willing to exclude a set of
A A
places with density 0 from our main theorems, it will suffice to restrict our attention to the places
v ∈ S .
A
2.5. The Mumford–Tate group. Fix a field embedding Kconn ⊆ C. The homology group V =
A
H (A(C),Q)isavectorspaceofdimension2dimAoverQ. ItisnaturallyendowedwithaQ-Hodge
1
structure of type {(−1,0),(0,−1)}, and hence a decomposition
V ⊗QC = H1(A(C),C) = V−1,0⊕V0,−1
such that V0,−1 = V−1,0. Let
µ: Gm,C → GLV⊗QC
be the cocharacter such that µ(z) is the automorphism of V ⊗QC which is multiplication by z on
V−1,0 and the identity on V0,−1 for each z ∈ C× = G (C).
m
Definition 2.5. The Mumford–Tate group of A is the smallest algebraic subgroup of GL , defined
V
over Q, which contains µ(Gm,C). We shall denote it by GA.
The endomorphism ring End(AC) acts on V; this action preserves the Hodge decomposition, and
hencecommuteswithµandthusalsoGA. Moreover,theringEnd(AC)⊗ZQisnaturallyisomorphic
to the commutant of GA in EndQ(V). The group GA/Q is reductive since the Q-Hodge structure
for V is pure and polarizable. Using our fixed embedding Kconn ⊆ C and Proposition 2.2(iii), we
A
have a natural isomorphism End(AC)⊗ZQ = End(AKconn)⊗ZQ.
A
6
2.6. The Mumford–Tate conjecture. The comparison isomorphism V(cid:96)(A) ∼= V ⊗QQ(cid:96) induces an
isomorphism GLV(cid:96)(A) ∼= GLV,Q(cid:96). The following conjecture says that G◦A,(cid:96) and GA,Q(cid:96) are the same
algebraic group when we use the comparison isomorphism as an identification, cf. [Ser77, §3].
Conjecture 2.6 (Mumford–Tate conjecture). For each prime (cid:96), we have G◦A,(cid:96) = GA,Q(cid:96).
The Mumford–Tate conjecture is still open, however significant progress has been made in show-
ing that several general classes of abelian varieties satisfy the conjecture; we simply refer the reader
to[Vas08,§1.4]forapartiallistofreferences. TheMumford–TateconjectureforAholdsifandonly
if the common rank of the groups G◦ equals the rank of G [LP95, Theorem 4.3]; in particular,
A,(cid:96) A
the conjecture holds for one prime (cid:96) if and only if it holds for all (cid:96). The following proposition says
that one inclusion of the Mumford–Tate conjecture is known unconditionally, see Deligne’s proof
in [DMOS82, I, Prop. 6.2].
Proposition 2.7. For each prime (cid:96), we have G◦A,(cid:96) ⊆ GA,Q(cid:96).
Using this proposition, we obtain a well-defined Galois representation ρA,(cid:96): GalKconn → GA(Q(cid:96))
A
for each prime (cid:96).
2.7. A conjecture on Frobenius conjugacy classes. Let R be the affine coordinate ring of
G . The group G acts on R by composition with inner automorphisms (more precisely, G (k)
A A A
acts on R⊗Q k for each extension k/Q). We define RGA to be the Q-subalgebra of R consisting
of those elements fixed by this G -action; it is the algebra of central functions on G . Define
A A
Conj(GA) := Spec(RGA); it is a variety over Q which we call the variety of (semi-simple) conjugacy
classes of G . We define cl : G → Conj(G ) to be the morphism arising from the inclusion
A GA A A
RGA (cid:44)→ R of Q-algebras.
Let L be an algebraically closed extension of Q. Each g ∈ G (L) can be expressed uniquely in
A
the form g g with commuting g ,g ∈ G (L) such that g is semisimple and g is unipotent. For
s u s u A s u
g,h ∈ G (L), we have g = h if and only if cl (g) = cl (h).
A s s GA GA
Assume that Kconn = K. We can then view ρ as having image in G (Q ). The following
A A,(cid:96) A (cid:96)
conjecture says that the conjugacy class of G containing ρ (Frob ) does not depend on (cid:96); see
A A,(cid:96) v
[Ser94, C.3.3] for a more refined version.
Conjecture 2.8. Suppose that Kconn = K. Let v be a finite place of K for which A has good
A
reduction. Then there exists an F ∈ Conj(G )(Q) such that cl (ρ (Frob )) = F for all
v A GA A,(cid:96) v v
primes (cid:96) satisfying v (cid:45) (cid:96).
Remark 2.9. The algebra of class functions of GL is Q[a ,...,a ] where the a are the morphisms
V 1 n i
of GL that satisfy det(xI −g) = xn+a (g)xn−1+...+a (g)x+a (g) for g ∈ GL (Q). The
V 1 n−1 n V
inclusion G ⊆ GL induces a morphism f: Conj(G ) → Conj(GL ) := SpecQ[a ,...,a ] ∼=
A V A V 1 n
An. The morphism f ◦cl can thus be viewed as mapping an element of G to its characteristic
Q GA A
polynomial.
Let v be a finite place of K for which A has good reduction. Conjecture 2.8 implies that for any
prime(cid:96)satisfyingv (cid:45) (cid:96),f(cl (ρ (Frob ))) = f(F )belongstoConj(GL )(Q)andisindependent
GA A,(cid:96) v v V
of (cid:96); this consequence is true, and is just another way of saying that det(xI − ρ (Frob )) has
A,(cid:96) v
coefficients in Q and is independent of (cid:96).
In §4, we will state a theorem of Noot that gives a weakened version of Conjecture 2.8.
2.8. Image modulo (cid:96). Let GL be the group scheme over Z for which GL (R) =
T (A) (cid:96) T (A)
(cid:96) (cid:96)
AutR(R⊗Z(cid:96) T(cid:96)(A)) for all (commutative) Z(cid:96)-algebras R. Note that the generic fiber of GLT(cid:96)(A) is
GL and the image of ρ lies in GL (Z ). Let G be the Zariski closure of ρ (Gal )
V (A) A,(cid:96) T (A) (cid:96) A,(cid:96) A,(cid:96) K
(cid:96) (cid:96)
in GL ; it is a group scheme over Z with generic fiber G .
T (A) (cid:96) A,(cid:96)
(cid:96)
7
Let ρ : Gal → Aut (A[(cid:96)]) be the representation describing the Galois action on the (cid:96)-
A,(cid:96) K Z/(cid:96)Z
torsion points of A. Observe that ρ (Gal ) is naturally a subgroup of G (F ); the following
A,(cid:96) K A,(cid:96) (cid:96)
results show that these groups are almost equal.
Proposition 2.10. Suppose that Kconn = K and that A is absolutely simple.
A
((i)) For (cid:96) sufficiently large, G is a reductive group over Z .
A,(cid:96) (cid:96)
((ii)) There is a constant C such that the inequality [G (F ) : ρ (Gal )] ≤ C holds for all
A,(cid:96) (cid:96) A,(cid:96) K
primes (cid:96).
((iii)) For (cid:96) sufficiently large, the group ρ (Gal ) contains the commutator subgroup of G (F ).
A,(cid:96) K A,(cid:96) (cid:96)
Proof. In Serre’s 1985-1986 course at the Coll`ege de France [Ser00, #136], he showed that the
groups ρ (Gal ) are essentially the F -points of certain reductive groups. For each prime (cid:96), he
A,(cid:96) K (cid:96)
∼
constructs a certain connected algebraic subgroup H(cid:96) of GLT(cid:96)(A),F(cid:96) = GL2dimA,F(cid:96). There exists a
finite extension L/K for which the following properties hold for all sufficiently large primes (cid:96):
• H is reductive.
(cid:96)
• ρ (Gal ) is a subgroup of H (F ) and the index [H (F ) : ρ (Gal )] can be bounded
A,(cid:96) L (cid:96) (cid:96) (cid:96) (cid:96) A,(cid:96) L
independently of (cid:96).
• ρ (Gal ) contains the commutator subgroup of H (F ).
A,(cid:96) L (cid:96) (cid:96)
Detailed sketches of Serre’s results were supplied in letters that have since been published in his
collected papers; see the beginning of [Ser00], in particular the letter to M.-F. Vign´eras [Ser00,
#137]. The paper of Wintenberger [Win02] also contains everything we need.
In[Win02,§3.4],itisshownthatSerre’sgroupH equalsthespecialfiberofG forallsufficiently
(cid:96) A,(cid:96)
large (cid:96). Parts (ii) and (iii) then follow from the properties of H . For part (i), see [Win02, §2.1]
(cid:96)
and [LP95]. (cid:3)
2.9. Independence. Combining all our (cid:96)-adic representations together, we obtain a single Galois
representation
(cid:89)
ρA: GalK → AutQ(cid:96)(V(cid:96)(A))
(cid:96)
which describes the Galois action on all the torsion points of A. The following theorem shows that,
after possibly replacing K by a finite extension, the Galois representations ρ will be independent.
A,(cid:96)
Proposition 2.11. (Serre [Ser00, #138]) There is a finite Galois extension K(cid:48) of K in K such
(cid:81)
that ρ (Gal ) equals ρ (Gal ).
A K(cid:48) (cid:96) A,(cid:96) K(cid:48)
We will need the following straightforward consequence:
Proposition 2.12. Fix an extension K(cid:48)/K as in Proposition 2.11. Let Λ be a finite set of ra-
tional primes. For each prime (cid:96) ∈ Λ, fix a subset U of the group ρ (Gal ) that is stable under
(cid:96) A,(cid:96) K
conjugation. Let S be the set of v ∈ Σ such that ρ (Frob ) ⊆ U for all (cid:96) ∈ Λ. Then S has
K A,(cid:96) v (cid:96)
density
(cid:88) |C| (cid:89) |ρA,(cid:96)(ΓC)∩U(cid:96)|
·
|Gal(K(cid:48)/K)| |ρ (Γ )|
A,(cid:96) C
C (cid:96)∈Λ
where C varies over the conjugacy classes of Gal(K(cid:48)/K) and Γ is the set of σ ∈ Gal for which
C K
σ| ∈ C.
K(cid:48)
(cid:81) (cid:81)
Proof. Set m := (cid:96), and define U := U , which we view as a subset of Aut (A[m]).
(cid:96)∈Λ m (cid:96)|m (cid:96) Z/mZ
Let ρ : Gal → Aut (A[m]) be the homomorphism describing the Galois action on A[m].
A,m K Z/mZ
Let µ and µ(cid:48) be the Haar measures on Gal normalized so that µ(Gal ) = 1 and µ(cid:48)(Gal ) = 1.
K K K(cid:48)
8
TheChebotarevdensitytheoremsaysthatthedensityδ ofS isdefinedandequalsµ({σ ∈ Gal :
K
ρ (σ) ∈ U }). Let{σ } beasubsetofGal consistingofrepresentativesofthecosetsofGal
A,m m i i∈I K K(cid:48)
in Gal . We then have
K
[K(cid:48) : K]δ = (cid:88)µ(cid:48)({σ ∈ σ Gal : ρ (σ) ∈ U }) = (cid:88) |ρA,m(σiGalK(cid:48))∩Um|.
i K(cid:48) A,m m |ρ (σ Gal )|
A,m i K(cid:48)
i∈I i∈I
We have ρ (Gal ) = (cid:81) ρ (Gal ) by our choice of K(cid:48) and hence ρ (σ Gal ) equals
A,m K(cid:48) (cid:96)|m A,(cid:96) K(cid:48) A,m i K(cid:48)
(cid:81)
ρ (σ Gal ) for all i ∈ I. Therefore,
(cid:96)|m A,(cid:96) i K(cid:48)
[K(cid:48) : K]δ = (cid:88)(cid:89) |ρA,(cid:96)(σiGalK(cid:48))∩U(cid:96)|.
|ρ (σ Gal )|
A,(cid:96) i K(cid:48)
i∈I (cid:96)|m
Since U is stable under conjugation, we find that |ρ (σ Gal ) ∩ U |/|ρ (σ Gal )| depends
(cid:96) A,(cid:96) i K(cid:48) (cid:96) A,(cid:96) i K(cid:48)
onlyontheconjugacyclassC ofGal(K(cid:48)/K)containingσ | andequals|ρ (Γ )∩U |/|ρ (Γ )|.
i K(cid:48) A,(cid:96) C (cid:96) A,(cid:96) C
Using this and grouping the σ by their conjugacy class when restricted to K(cid:48), we deduce that
i
[K(cid:48) : K]δ = (cid:88)|C|(cid:89) |ρA,(cid:96)(ΓC)∩U(cid:96)|
|ρ (Γ )|
A,(cid:96) C
C (cid:96)|m
where C varies over the conjugacy classes of Gal(K(cid:48)/K). (cid:3)
3. Reductive groups: background
Fix a perfect field k and an algebraic closure k.
3.1. Tori. An (algebraic) torus over k is an algebraic group T defined over k for which T is
k
isomorphic to Gr for some integer r. Fix a torus T over k. Let X(T) be the group of characters
m,k
T → G ; it is a free abelian group whose rank equals the dimension of T. Let Aut(T ) be the
k m,k k
groupofautomorphismsofthealgebraicgroupT . Foreachf ∈ Aut(T ), wehaveanisomorphism
k k
f : X(T) → X(T), α (cid:55)→ α◦f−1; this gives a group isomorphism
∗
∼
Aut(T ) −→ Aut(X(T)), f (cid:55)→ f
k ∗
which we will often use as an identification. There is a natural action of the absolute Galois group
Gal on X(T); it satisfies σ(α(t)) = σ(α)(σ(t)) for all σ ∈ Gal , α ∈ X(T) and t ∈ T(k). Let
k k
ϕ : Gal → Aut(X(T)) be the homomorphism describing this action, that is, ϕ (σ)α = σ(α)
T k T
for σ ∈ Gal and α ∈ X(T). Note that the torus T, up to isomorphism, can be recovered from
k
the representation ϕ . We say that the torus T is split if it is isomorphic to Gr ; equivalently, if
T m,k
ϕ (Gal ) = 1.
T k
Let G be a reductive group over k. A maximal torus of G is a closed algebraic subgroup that
is a torus (also defined over k) and is not contained in any larger such subgroup. If T and T(cid:48) are
maximal tori of G, then T and T(cid:48) are maximal tori of G and are conjugate by some element of
k k k
G(k). The group G has a maximal torus whose dimension is called the rank of G. We say that G
is split if it has a maximal torus that is split.
3.2. Weyl group. Let G be a connected reductive group over k. Fix a maximal torus T of G.
The (absolute) Weyl group of G with respect to T is the finite group
W(G,T) := N (T)(k)/T(k)
G
where N (T) is the normalizer of T in G. For an element g ∈ N (T)(k), the morphism T →
G G k
T , t (cid:55)→ gtg−1 is an isomorphism that depends only on the image of g in W(G,T); this induces a
k
9
faithful action of W(G,T) on T . So, we can identify W(G,T) with a subgroup of Aut(T ) and
k k
hence also of Aut(X(T)).
There is a natural action of Gal on W(G,T). For σ ∈ Gal and w ∈ W(G,T), we have
k k
ϕ (σ)◦w◦ϕ (σ)−1 = σ(w). In particular, note that the action of Gal of W(G,T) is trivial if T
T T k
is split.
We define Π(G,T) to be subgroup of Aut(X(T)) generated by W(G,T) and ϕ (Gal ). The
T k
Weyl group W(G,T) is a normal subgroup of Π(G,T). Up to isomorphism, the groups W(G,T)
and Π(G,T) are independent of T (for the group Π(G,T), see Proposition 2.1 of [JKZ11]); we
shall denote the abstract groups by W(G) and Π(G), respectively.
3.3. Maximal tori over finite fields. We now assume that k is a finite field F with q elements.
q
Let G be a connected reductive group defined over F . Assume further that G is split and fix a
q
splitmaximaltorusT. LetT(cid:48) beanymaximaltorusofG. Thereisanelementg ∈ G(F )suchthat
q
gT(cid:48) g−1 = T . Since T and T(cid:48) are defined over F , we find that Frob (g)T(cid:48) Frob (g)−1 = T
Fq Fq q q Fq q Fq
and hence gFrob (g)−1 belongs to N (T)(F ). Let θ (T(cid:48)) be the conjugacy class of W(G,T)
q G q GA
containing the coset represented by gFrob (g)−1. These conjugacy classes have the following inter-
q
pretation:
Proposition 3.1. The map T(cid:48) (cid:55)→ θ (T(cid:48)) defines a bijection between the maximal tori of G up to
G
conjugation in G(F ) and the conjugacy classes of W(G,T).
q
Proof. This is [Car85, Prop. 3.3.3]. Note that the action of Frob on W(G,T) is trivial since T is
q
split, so the Frob -conjugacy classes of W(G,T) in [Car85] are just the usual conjugacy classes of
q
W(G,T). (cid:3)
Let G(F ) be the set of g ∈ G(F ) that are semisimple and regular in G. Each g ∈ G(F ) is
q sr q q sr
contained in a unique maximal torus T of G. We define the map
g
θ : G(F ) → W(G,T)(cid:93), g (cid:55)→ θ (T )
G q sr G g
where W(G,T)(cid:93) is the set of conjugacy classes of W(G,T). We will need the following equidistri-
bution result later.
Lemma 3.2. Let G be a connected and split reductive group over a finite field F , and fix a split
q
maximal torus T. Let C be a subset of W(G,T) that is stable under conjugation and let κ be a
subset of G(F ) that is a union of cosets of the commutator subgroup of G(F ). Then
q q
|{g ∈ κ∩G(F ) : θ (g) ⊆ C}| |C|
q sr G
= +O(1/q)
|κ| |W(G,T)|
where the implicit constant depends only on the type of G (and in particular, not on q, T, C and
κ).
Proof. We shall reduce to a special case treated in [JKZ11] which deals with semisimple groups.
Let Gad be the quotient of G by its center and let ϕ: G → Gad be the quotient homomorphism.
Let Tad be the image of T under ϕ; it is a split maximal torus of Gad. The homomorphism ϕ
induces a group isomorphism ϕ : W(G,T) −→∼ W(Gad,Tad). An element g ∈ G(F ) is regular and
∗ q
semisimple in G if and only if ϕ(g) is regular and semisimple in Gad. For g ∈ G(F ) , one can
q sr
check that θ (g) ⊆ C if and only if θ (ϕ(g)) ⊆ ϕ (C). Therefore,
G Gad ∗
|{g ∈ κ∩G(F ) : θ (g) ⊆ C}| |{g ∈ ϕ(κ)∩Gad(F ) : θ (g) ⊆ ϕ (C)}|
(3.1) q sr G = q sr Gad ∗ .
|κ| |ϕ(κ)|
10