ON THE IMAGE OF GALOIS L-ADIC REPRESENTATIONS FOR ABELIAN VARIETIES OF TYPE III ...

24 Pages·2012·0.23 MB·English

ON THE IMAGE OF GALOIS L-ADIC REPRESENTATIONS
FOR ABELIAN VARIETIES OF TYPE III
G. Banaszak, W. Gajda, P. Krason(cid:19)
Abstract. In this paper we investigate the image of the l-adic representation at-
tached to the Tate module of an abelian variety de(cid:12)ned over a number (cid:12)eld. We
considersimpleabelian varietiesoftype III intheAlbertclassi(cid:12)cation cf.[M,Th.2,
p. 201]. We compute the image ofthe l-adic and modl Galoisrepresentations and
weverifytheMumford-Tateconjectureforawideclassofsimpleabelianvarietiesof
typeIII.
1. Introduction. In this paperwecompute the imageof l-adicandmod l Galois
representationsattachedtocertainabelianvarietiesof typeIIIaccordingtoAlbert
classi(cid:12)cation list. We also prove the Mumford-Tate conjecture for these varieties.
To be more precise, the main results of this paper concern the following class of
abelian varieties:
De(cid:12)nition of class B.
Abelian variety A=F; de(cid:12)ned over a number (cid:12)eld F is of class B; if the following
conditions hold:
(i) A is a simple, abelian variety of dimension g
(ii) R = EndF(cid:22)(A) = EndF(A) and the endomorphism algebra D = R(cid:10)Z Q;
is of type III in the Albert list of division algebras with involution (cf. [M],
Th.2, p. 201).
(iii) the (cid:12)eld F is such that for every l the Zariski closure Galg of (cid:26) (G ) in
l l F
GL =Q is a connected algebraic group
Vl(A) l
(iv) g = hed; where h is an odd integer, e = [E : Q] is the degree of the center
E of D and d2 =[D : E]:
In sections 2 and 3 we give an explicit description of the endomorphism alge-
bra and its involution for an abelian variety of type III as well as the relation to
various bilinear forms coming from Weil pairing. This detailed treatment of en-
domorphism algebras and bilinear forms di(cid:11)ers signi(cid:12)cantly from the approach of
[C2] and [BGK2]. Due to our approach the proof of Theorem 3.29, in section 3,
is achieved in a very simple and explicit way. Theorem 3.29 is an important tool
which shows how to extract a symmetric form out of the Weil pairing, which is a
symplecticform. Insection4wecomputeLiealgebrasthatleadtotheproofofThe-
orem 4.19. In section 5 we apply Theorem 4.19 in the proof of the Mumford-Tate
conjecture for the abelian varieties of class B:
TypesetbyAMS-TEX
1
2 G. BANASZAK, W. GAJDA, P. KRASON(cid:19)
Theorem 5.11. If A is an abelian variety of class B, then
Galg =MT(A)(cid:10)Q ;
l l
for every prime number l, where MT(A) denotes the Mumford-Tate group of A,
i.e., the Mumford Tate conjecture holds true for A:
This generalizes the result of Tankeev [Ta] who proved the Mumford-Tate conjec-
ture for abelian varieties of type III, with similar dimension restrictions, such that
End(A)(cid:10)Q has center equal to Q:
On the wayof the proof of Mumford-Tate conjecturewealso compute explicitly
the Hodge group and prove that it is equal to the Lefschetz group. However it is
not enough to get directly the Hodge conjecture for abelian varieties of type III of
classB cf. [Mu]. TheproofofMumford-TateconjectureandequalityofHodgeand
Lefschetz groupsfor abelianvarieties of type I and II of classA in [BGK2]gaveus
the Hodge and Tate conjectures for these abelian varieties. In section 6 (Theorem
6.22,Corollary6.26)wecomputetheimagesofl-adicandmod lrepresentationsfor
abelianvarietiesofclassB:Finallyinsection7(Theorem7.2)weproveananalogue
ofthe openimageTheoremofSerrecf. [Se1-Se4],[Se6]forabelianvarietiesofclass
B:
Theorem7.2. IfAisanabelianvarietyof classB;thenforeveryprime numberl;
(cid:26) (G )isopeninthegroup C (GSp )(Z ): Inaddition, forl(cid:29)0thesubgroup
l F R ((cid:3);(cid:20)) l
(cid:26) (G0 ) of C (Sp )(Z ) is of index dividing 2r(l); where r(l) is the number of
l F R ((cid:3);(cid:20)) l
primes over l in O : More precisely (cf. 6.23)
E
(cid:25) (Spin (O )) (cid:26) (cid:26) (G0 ) (cid:26) SO (O );
(cid:21) (T(cid:21); (cid:21)) (cid:21) l F (T(cid:21); (cid:21)) (cid:21)
Y Y
(cid:21)jl (cid:21)jl
2. Abelian varieties of type III and their endomorphisms algebras.
LetA=F beasimpleabelianvarietyofdimensiongsuchthatD=EndF(cid:22)(A)(cid:10)ZQ=
End (A)(cid:10) Q and the polarization of A is de(cid:12)ned over F: We assume that A=F
F Z
is an abelian variety over F of type III according to the Albert’s classi(cid:12)cation list
(see [M], p. 201). Hence D is an inde(cid:12)nite quaternion algebra over E with center
E; a totally real extension of Q of degree e such that for every imbedding E (cid:26)R
D(cid:10) R = H:
E
Observe that in this case [D : E] = 4 so g = 2eh where e = [E : Q] and h is
an integer. We take l (cid:29) 0 such that A has good reduction at all primes over l
(cf. [ST]) and the algebra D splits over all primes over l and l does not divide the
degree of the polarization. Let RD be a maximal order in D: Since R=EndF(cid:22)(A)
is an order in D; we observe that R(cid:10) Z = R (cid:10) Z for l that does not divide
Z l D Z l
the index [R : R]. Since R is (cid:12)nitely generated, free Z module we check that
D
R\E =O0 is an order in O :
E E
To get explicit information about the algebra D we start with a more general
framework. Let D be a division algebra with two involutions (cid:3) and (cid:3) and center
1 2
E: For each x 2 D we will denote x(cid:3)i to be the image of the involution (cid:3)i acting
THE IMAGE OF GALOIS L-ADIC REPRESENTATIONS, 3
on x: By Skolem-Noether Theorem [R] p. 103 there is an element a2D such that
for each x2D we have:
(2.1) x(cid:3)2 =ax(cid:3)1a(cid:0)1:
Because (cid:3) (cid:14)(cid:3) = id ; applying (cid:3) to (2.1) we get
i i D 2
(2.2) a(cid:3)1 =(cid:15)a
for (cid:15) 2 E and applying (cid:3) we check that (cid:15)2 = 1 hence (cid:15) = 1 or (cid:15) = (cid:0)1 c.f. [M] p.
1
195. Let E =fc2E; c(cid:3) =cg: Then E=E is an extension of degree at most 2:
0 0
For a simple abelian variety of type III, E = E and E is totally real cf. [M] p.
0
194. Also in this case (cid:15) = 1 in (2.2) (cf. [M] pp. 193-196 ). Hence a 2 E and
(cid:3) =(cid:3) : Therefore the division algebra D coming from a simple abelian variety of
2 1
type III has a unique positive involution (cid:3), the Rosati involution. Moreover the
map D ! D given by (cid:11) ! (cid:11)(cid:3) is an isomorphism of E algebras so by [R] Cor.
7.14 p. 96 the algebra D gives an element of order 1 or 2 in Br(E): Since D is a
noncommutative division algebra, it gives an element of order 2 in Br(E):
By theorem of Suslin and Merkurjev [SM] for any (cid:12)eld L such that (cid:22) (cid:26) L and
m
(charL; m)=1 there is a natural isomorphism
(cid:24)= //
(2.3) K (L)=mK (L) Br(L)[m](cid:10)(cid:22)
2 2 m
If L is a number (cid:12)eld, then by a result of Lenstra [Le] every element of K (L) is a
2
single Steinberg symbol fc; dg for some c;d2L:
Therefore for m=2 and L=E the isomorphism (2.3) shows that D is isomorphic
as an E-algebra to a division algebra
D(c;d) := fa +a (cid:11)+a (cid:12)+a (cid:11)(cid:12); (cid:11)2 =c; (cid:12)2 =d; (cid:11)(cid:12) =(cid:0)(cid:12)(cid:11)g
0 1 2 3
This isomorphisminduces the unique positiveinvolution on D(c;d) which will also
be denoted by (cid:3): Therefore (cid:3) must be the following natural positive involution
(a +a (cid:11)+a (cid:12)+a (cid:11)(cid:12))(cid:3) =a (cid:0)a (cid:11)(cid:0)a (cid:12)(cid:0)a (cid:11)(cid:12)
0 1 2 3 0 1 2 3
onD(c;d):FromnowonweidentifyDwithD(c;d):PutL=E((cid:11)):Let(cid:17) =a +a (cid:11)
0 1
and (cid:13) =a +a (cid:11): Hence
2 3
(cid:17)+(cid:13)(cid:12) =a +a (cid:11)+a (cid:12)+a (cid:11)(cid:12):
0 1 2 3
For an element (cid:14) = e+f(cid:11) 2 L; with e;f 2 E; put (cid:14)(cid:22)= e(cid:0)f(cid:11): The (cid:12)eld L splits
the algebra D(c;d): Namely we have an isomorphism of L algebras:
(2.4) D(c;d)(cid:10) L!M (L)
E 2;2
(cid:17) (cid:13)
((cid:17)+(cid:13)(cid:12))(cid:10)1 !
(cid:20)d(cid:13)(cid:22) (cid:17)(cid:22)(cid:21)
4 G. BANASZAK, W. GAJDA, P. KRASON(cid:19)
from this isomorphism it is clear that
((cid:17)+(cid:13)(cid:12))(cid:3) =Tr0((cid:17)+(cid:13)(cid:12))(cid:0)((cid:17)+(cid:13)(cid:12))
because by de(cid:12)nition
(cid:17) (cid:13)
Tr0((cid:17)+(cid:13)(cid:12))=Tr = 2a ;
(cid:20)d(cid:13)(cid:22) (cid:17)(cid:22)(cid:21) 0
where Tr0 denotes the reduced trace (see [R] pp.112-116) from D(c;d) to E: The
involution on M (L) induced by (cid:3) is of the following form:
2;2
(2.5) B(cid:3) =JBtJ(cid:0)1
where B 2M (L) and
2;2
0 1
J =
(cid:20)(cid:0)1 0(cid:21)
Remark 2.6. It is clear that if we take in the above computations instead of L=
E((cid:11))the(cid:12)eldE((cid:12))orthe(cid:12)eldE((cid:11)(cid:12))thentheyalsosplitthealgebraDbyaformula
similar to (2.4) and the involution (cid:3) will induce on M (E((cid:12))) and M (E((cid:11)(cid:12)))
2;2 2;2
the involution given by formula (2.5).
Note that any maximal commutative sub(cid:12)eld of D(c;d) has form E(a (cid:11)+a (cid:12) +
1 2
a (cid:11)(cid:12)) for some a ;a ;a 2E not all equal to zero. If Nr0 : D(c;d) !E denotes
3 1 2 3
the reduced norm, then for every (cid:17)+(cid:13)(cid:12) 2D(c;d):
(cid:17) (cid:13)
Nr0((cid:17)+(cid:13)(cid:12))=det = ((cid:17)+(cid:13)(cid:12))(cid:3)((cid:17)+(cid:13)(cid:12)) =
(cid:20)d(cid:13)(cid:22) (cid:17)(cid:22)(cid:21)
(2.7) = a2(cid:0)a2c(cid:0)a2d+a2cd = a2(cid:0)(a (cid:11)+a (cid:12)+a (cid:11)(cid:12))2:
0 1 2 3 0 1 2 3
For some a ;a ;a 2E not all equal to zero put (cid:11)0 :=a (cid:11)+a (cid:12)+a (cid:11)(cid:12): If (cid:12)0 :=
1 2 3 1 2 3
b (cid:11)+b (cid:12)+b (cid:11)(cid:12);isanelementofD(c;d);putc :=a b (cid:0)a b ; c :=a b (cid:0)a b ;
1 2 3 1 3 2 2 3 2 1 3 3 1
c :=a b (cid:0)a b : Then
3 1 2 2 1
(2.8) (cid:11)0(cid:12)0 =a b c+a b d(cid:0)a b cd+c d(cid:11)+c c(cid:12)+c (cid:11)(cid:12)
1 1 2 2 3 3 1 2 3
and
a a a
1 2 3
(2.9) det 2 b1 b2 b33=(cid:0)dc21(cid:0)cc22+c23 (cid:21)0
dc cc c
1 2 3
4 5
Since c < 0 and d < 0; the determinat in (2.9) is zero if and only if elements (cid:11)0
and (cid:12)0 are linearly dependent over E: Hence it is possible to (cid:12)nd (cid:12)0 in such a way
that a b c+a b d(cid:0)a b cd=0and the determinantin (2.9)isnonzero. With this
1 1 2 2 3 3
choice of (cid:12)0 we see that c0 :=(cid:11)02 <0; d0 :=(cid:12)02 <0 and (cid:11)0(cid:12)0 =(cid:0)(cid:12)0(cid:11)0: We observe
that for any a0;a0;a0;a0 2E
0 1 2 3
(2.10) (a0 +a0(cid:11)0+a0(cid:12)0+a0(cid:11)0(cid:12)0)(cid:3) = a0 (cid:0)a0(cid:11)0(cid:0)a0(cid:12)0(cid:0)a0(cid:11)0(cid:12)0:
0 1 2 3 0 1 2 3
Hence D(c;d)=D(c0;d0) and we can use the (cid:12)eld L=E((cid:11)0) and the isomorphism
(2.4) for this (cid:12)eld to split our algebra D(c0;d0):
Foraprime numberl throughoutthe paper(cid:21) will denote anideal inO such that
E
(cid:21)jl and w will denote an ideal of O such that wj(cid:21):
L
THE IMAGE OF GALOIS L-ADIC REPRESENTATIONS, 5
3. Bilinear forms associated with abelian varieties of type III.
Put R = R(cid:10)Z and D = D (cid:10)Q : The polarization of A gives a Z-bilinear,
l l l l
non-degenerate, alternating pairing
(3.1) (cid:20) : (cid:3)(cid:2)(cid:3)!Z:
which upon tensoring with Z ([Mi], diagram on page 133) becomes Z -bilinear,
l l
non-degenerate, alternating pairing
(3.2) (cid:20) : T (A)(cid:2)T (A)!Z ;
l l l l
easily derived from the Weil pairing. By assumption on l; for any (cid:11) 2 R we get
l
(cid:11)(cid:3) 2 R ; (see [Mi] chapter 13 and 17) where (cid:11)(cid:3) is the image of (cid:11) via the Rosati
l
involution. Hence for any v;w 2 T (A) we have (cid:20) ((cid:11)v;w) = (cid:20) (v;(cid:11)(cid:3)w) (see loc.
l l l
cit.). Incaseof anabelianvarietywhichisnotprincipallypolarized,wetakel that
doesnotdividethedegreeofthepolarisationofAtoget(cid:11)(cid:3)(cid:10)12R(cid:10)Z for(cid:11)2R:
l
Let V (A) = T (A)(cid:10) Q ; and let (cid:20)0 : V (A)(cid:2)V (A) ! Q be the bilinear form
l l Zl l l l l l
(cid:20) (cid:10) Q : By [BGK2] lemma 3.1 there is a unique O -bilinear form
l Zl l El
(3.3) (cid:30) : T (A)(cid:2)T (A)!O
l l l El
suchthat Tr ((cid:30) (v ;v ))=(cid:20) (v ;v )forallv ;v 2T (A): Put (cid:30)0 =(cid:30) (cid:10) Q ;
El=Ql l 1 2 l 1 2 1 2 l l l Zl l
(3.4) (cid:30)0 : V (A)(cid:2)V (A)!E :
l l l l
By uniqueness of the form (cid:30) for each (cid:11)2R and for all v ;v 2T (A)
l l 1 2 l
(3.5) (cid:30) ((cid:11)v ; v )=(cid:30) (v ; (cid:11)(cid:3)v )
l 1 2 l 1 2
hence (cid:30)0((cid:11)v ; v )=(cid:30)0(v ; (cid:11)(cid:3)v ) for each (cid:11)2D and for all v ;v 2V (A): De(cid:12)ne
l 1 2 l 1 2 l 1 2 l
T~l(A)=Tl(A)(cid:10)O0 OL; V~l(A)=Vl(A)(cid:10)E L and (cid:30)~l =(cid:30)l(cid:10)O0 OL
E E
(3.6) (cid:30)~ : T~(A)(cid:2)T~(A) ! O :
l l l Ll
Hence (cid:30)~0 =(cid:30)~ (cid:10) L is the L bilinear form:
l l OL l
(3.7) (cid:30)~0 : V~(A)(cid:2)V~(A) ! L :
l l l l
The form (cid:30)~ is non-degenerate i(cid:11) (cid:30) is non-degenerate.
l l
By (2.4) we get the following isomorphism
(3.8) D(cid:10) L (cid:24)=M (L ):
E l 2;2 l
We treat R(cid:10)OE0 OLl as the subring of M2;2(Ll) via isomorphism (3.8). Since R is
a (cid:12)nitely generated O0 module and O is a PID we note that for all l (cid:29) 0 and
E w
all wjl the isomorphism (3.8) gives an imbedding R(cid:10)O0 Ow (cid:26) M2;2(Ow) of free
E
O modules of rank 4. Since R is an order of D the matrices e 2 M (L ) are
w ij 2;2 w
actually in R(cid:10)O0 Ow for l(cid:29)0 by (2.4), where eij has ij entry equal to 1 and all
other entries equaEl to 0: So (3.8) induces an isomorphism R(cid:10)O0 Ow (cid:24)= M2;2(Ow)
E
hence also an isomorphism
(3.9) R(cid:10)OE0 OLl (cid:24)=M2;2(OLl)
From now on l (cid:29) 0 be such that (3.9) holds. In particular for such l the matrix
J 2M2;2(Ll) is an element of R(cid:10)OE0 OLl:
6 G. BANASZAK, W. GAJDA, P. KRASON(cid:19)
Remark 3.10. We should note that an isomorphism between both sides of (3.9)
can be obtained by Corollary 11.6 p. 134 and Theorem 17.3 p. 171 of [R] for all
l(cid:29)0: However these results give an isomorphism which comes from a conjugation
by an elementof D(cid:10) L (cid:24)=M (L )(see [R] loc. cit.). To keep track of the action
E l 2;2 l
of the involution (cid:3) we prefer to use the isomorphism (3.9) induced by (3.8).
Hence by (2.5) for each B 2R(cid:10)OE0 OLl and for all v1;v2 2T~l(A)
(cid:30)~(Bv ; v )=(cid:30)~(v ;B(cid:3)v )=(cid:30)~(v ; JBtJ(cid:0)1v ):
l 1 2 l 1 2 l 1 2
Therefore for each B 2M (L ) and for all v ;v 2V~(A)
2;2 l 1 2 l
(cid:30)~0(Bv ; v )=(cid:30)~0(v ;B(cid:3)v )=(cid:30)~0(v ; JBtJ(cid:0)1v ):
l 1 2 l 1 2 l 1 2
De(cid:12)nition 3.11. De(cid:12)ne a new bilinear form ~ as follows.
l
~ : T~(A)(cid:2)T~(A) ! O
l l l Ll
~(v ;v )=(cid:30)~(Jv ;v ):
l 1 2 l 1 2
Let ~0 = ~ (cid:10) L:
l l OL
Proposition 3.12. The form ~ ( ~0 resp.) is non-degenerate i(cid:11) (cid:30)~ ((cid:30)~0 resp.) is
l l l l
non-degenerate. For each v;w 2 T~l(A) and each B 2R(cid:10)OE0 OLl
(3.13) ~(Bv ; v )= ~(v ; Btv ):
l 1 2 l 1 2
Similarly, for each v ;v 2V~ and each B 2M (L )
1 2 l 2;2 l
(3.14) ~0(Bv ; v )= ~0(v ; Btv ):
l 1 2 l 1 2
Moreover ~ ( ~0 resp.) is symmetric (resp. antisymmetric) if and only if (cid:30)~ ((cid:30)~0
l l l l
resp.) is antisymmetric (resp. symmetric).
Proof.
~(Bv ;v )=(cid:30)~(JBv ;v )=(cid:30)~(v ; JBtJtJ(cid:0)1v )=(cid:30)~(v ; (cid:0)JBtv )=
l 1 2 l 1 2 l 1 2 l 1 2
=(cid:30)~(v ; JJtJ(cid:0)1Btv )=(cid:30)~(Jv ; Btv )= ~(v ; Btv ):
l 1 2 l 1 2 l 1 2
The remaining claim follows by De(cid:12)nition 3.11 and by the observation that Jt =
J(cid:0)1 =(cid:0)J and JJtJ(cid:0)1 =(cid:0)J: (cid:3)
Lemma 3.15. For l (cid:29) 0 the ideal (cid:21)jl splits completely in one of the maximal
commutative sub(cid:12)elds of D=D(c;d):
Proof. Invertinga(cid:12)nitesetS ofprimesofZifnecessarywecanassumethatRisa
maximalorderofDwithR\E =O andR=O +O (cid:11)+O (cid:12)+O (cid:11)(cid:12):
E;S E;S E;S E;S E;S
We assume that 2 2 S: Take l (cid:29) 0 such that l 2= S and such that all primes of
O are unrami(cid:12)ed primes for the algebra D cf. [R], Th. 32.1. Let t > 0 be an
E;S
element of (cid:21)(cid:0)(cid:21)2: By [R], Th. 22.4, Th. 22.15 and Th. 24.13 there is a maximal
ideal M (cid:26) R such that Nr0(M) = (cid:21): Let m = a +a (cid:11)+a (cid:12) +a (cid:11)(cid:12) 2 M be
0 1 2 3
such that Nr0(m) =t: By formula (2.7) t=a2(cid:0)(a (cid:11)+a (cid:12)+a (cid:11)(cid:12))2: Put (cid:11)0 :=
0 1 2 3
(a (cid:11)+a (cid:12)+a (cid:11)(cid:12)):LetL:=E((cid:11)0):TheringofS-integersofLisO =O [(cid:11)0]:
1 2 3 L;S E;S
It follows by [R], Th. 32.1 that a 2= (cid:21): Indeed, if a 2 (cid:21); then ((cid:11)0)2 2 (cid:21)(cid:0)(cid:21)2:
0 0
Hence (cid:21) rami(cid:12)es in O : But this contradicts [R], Th. 32.1 (ii). Hence ((cid:11)0)2 is a
L;S
square in O(cid:2): Hence (cid:21) splits in L: (cid:3)
(cid:21)
THE IMAGE OF GALOIS L-ADIC REPRESENTATIONS, 7
Remark 3.16. WeneedLemma3.15 tohave veryexplicit formula for thesplitting
isomorphism for our algebra D (cf. 2.4) and for the extension of the involution (cid:3)
to this split algebra cf. (2.5).
Proposition 3.17. Let w be a prime of L over a prime (cid:21) which splits in L: Then
the involution (cid:3) induced on R(cid:10)O0 O(cid:21) (on M2;2(E(cid:21)) resp.) from D has the form
E
B(cid:3) =JBtJ(cid:0)1 for any B 2R(cid:10)O0 O(cid:21) (for any B 2M2;2(E(cid:21)) resp.)
E
Proof. By (2.4) and (2.5) for any B 2M (L) we get B(cid:3) =JBtJ(cid:0)1: But also by
2;2
(2.4) and (3.8) we get:
(3.18) D(cid:10) L (cid:24)=M (L)(cid:10) L (cid:24)=M (L );
E w 2;2 L w 2;2 w
hence for any B 2 M (L ) we get B(cid:3) =JBtJ(cid:0)1: Since (cid:21) splits completely in L
2;2 w
the isomorphism (3.18) is just the isomorphism D(cid:10) E (cid:24)= M (E ) induced by
E (cid:21) 2;2 (cid:21)
(2.4). Hence for any B 2M2;2(E(cid:21)) we get B(cid:3) =JBtJ(cid:0)1 so identifying R(cid:10)O0 O(cid:21)
with a subring of M2;2(E(cid:21)) we get B(cid:3) =JBtJ(cid:0)1 for any B 2R(cid:10)O0 O(cid:21): (cid:3)E
E
Let e be the idempotent corresponding to the decomposition O (cid:24)= O :
(cid:21) El (cid:21)jl (cid:21)
PutT (A)=e T (A)(cid:24)=T (A)(cid:10) O ; V (A)=T (A)(cid:10) E :De(cid:12)neO -Qbilinear
(cid:21) (cid:21) l l OEl (cid:21) (cid:21) (cid:21) O(cid:21) (cid:21) (cid:21)
form (cid:30)(cid:21) = (cid:30)l (cid:10)OE0 O(cid:21): Put (cid:30)~w := (cid:30)~l (cid:10)OLl Ow and ~w := ~l (cid:10)OLl Ow: For l (cid:29) 0
such that (cid:21) splits in L; we have O = O : Hence for such an l we get (cid:30) = (cid:30)~ :
(cid:21) w (cid:21) w
This allows us to de(cid:12)ne the O - bilinear form
(cid:21)
(3.19) : T (A)(cid:2)T (A)!O ;
(cid:21) (cid:21) (cid:21) (cid:21)
(v ;v ) = (cid:30) (Jv ;v )
(cid:21) 1 2 (cid:21) 1 2
for all v ;v 2 T (A): Observe that = ~ : This gives us corresponding k -
1 2 (cid:21) (cid:21) w (cid:21)
bilinearform = (cid:10) k andE - bilinearform 0 = (cid:10) E respectively:
(cid:21) (cid:21) O(cid:21) (cid:21) (cid:21) (cid:21) (cid:21) O(cid:21) (cid:21)
(3.20) : A[(cid:21)](cid:2)A[(cid:21)]!k ;
(cid:21) (cid:21)
(3.21) 0 : V (A)(cid:2)V (A)!E :
(cid:21) (cid:21) (cid:21) (cid:21)
Corollary 3.22. Let (cid:21) be a prime of E such that (cid:21) splits in L: Then for any
v1;v2 2T(cid:21)(A) and any B 2R(cid:10)O0 O(cid:21)
E
(Bv ;v ) = (v ;Btv )
(cid:21) 1 2 (cid:21) 1 2
Hence for any v1;v2 2 A[(cid:21)] and any B 2 R(cid:10)O0 k(cid:21) (cid:24)= M2;2(k(cid:21)) (resp. for any
E
v ;v 2V (A) and any B 2M (E ))
1 2 (cid:21) 2;2 (cid:21)
(Bv ;v ) = (v ;Btv )
(cid:21) 1 2 (cid:21) 1 2
( resp: 0(Bv ;v ) = (v ;Btv ) )
(cid:21) 1 2 (cid:21) 1 2
Proof. The Corollaryfollows from Propositions 3.12 and 3.17. (cid:3)
8 G. BANASZAK, W. GAJDA, P. KRASON(cid:19)
Remark 3.23. All bilinear forms ; and 0 are symmetric
(cid:21) (cid:21) (cid:21)
and non-degenerate. This follows by results of this section, Lemmas 3.1 and 3.2 of
[BGK2] and by the non-degeneracy of the independent of l pairing (3.1).
We proceed to investigate some natural Galois actions. From now on, we assume
that R=EndF(cid:22)(A)=EndF(A): Consider the representations:
(cid:26) : G !GL(T (A))
l F l
(cid:26)0 : G !GL(V (A))
l F l
(cid:26) : G !GL(A[l])
l F
Let Galg be the Zariski closure of (cid:26) (G ) in GL and let Galg be the Zariski
l l F Tl(A) l
closureof (cid:26)0(G ) in GL :Let G(l)alg be the special(cid:12)berof Galg=Z :Note that
l F Vl(A) l l
Galg is the general (cid:12)ber of Galg=Z : This gives natural representations:
l l l
(cid:26) : G !GL(T (A))
(cid:21) F (cid:21)
(cid:26)0 : G !GL(V (A))
(cid:21) F (cid:21)
(cid:26) : G !GL(A[(cid:21)])
(cid:21) F
Wecanalsode(cid:12)neGalg tobetheZariskiclosureof(cid:26) (G )inGL andletGalg
(cid:21) (cid:21) F T(cid:21)(A) (cid:21)
be the Zariski closure of (cid:26)0(G ) in GL : Let G((cid:21))alg be the special (cid:12)ber of
(cid:21) F V(cid:21)(A)
Galg=O : Then, Galg is the general (cid:12)ber of Galg=O :
(cid:21) (cid:21) (cid:21) (cid:21) (cid:21)
Lemma 3.24. Let (cid:31) : G ! Z (cid:26) O be the composition of the cyclotomic
(cid:21) F l (cid:21)
character with the natural imbedding Z (cid:26) O : Let l (cid:29) 0 be such that (cid:21)jl be a
l (cid:21)
prime of E which splits in L:
(i) For any (cid:27) 2G and all v ;v 2T (A)
F 1 2 (cid:21)
((cid:27)v ;(cid:27)v )=(cid:31) ((cid:27)) (v ;v ):
(cid:21) 1 2 (cid:21) (cid:21) 1 2
(ii) For any B 2R(cid:10)O0 O(cid:21) and all v1;v2 2T(cid:21)(A)
E
(Bv ;v )= (v ;Btv ):
(cid:21) 1 2 (cid:21) 1 2
Proof. The part (i) follows by [C2 Lemma 2.3] or [BGK2, Lemma 4.7] and also by
(3.19) and by de(cid:12)nition 3.11 because the G action commutes with the action of
F
R on T (A): Indeed [C2 Lemma 2.3] and [BGK2, Lemma 4.7] concern the pairing
l
(cid:30) but (v;w) = (cid:30) (Jv;w) and J commutes with the G -action by assumption
(cid:21) (cid:21) (cid:21) F
so we get immediately statement (i) for : The part (ii) of the Lemma follows by
(cid:21)
Corollary3.22. (cid:3)
Let D(cid:21) := D(cid:10)E E(cid:21) R(cid:21) := R(cid:10)O0 O(cid:21): By [Fa], Th. 3 and [BGK2] lemma 4.17
E
G acts on both V (A) and A[(cid:21)] semi-simply and Galg and G((cid:21))alg are reductive
F (cid:21) (cid:21)
algebraicgroups. HenceGalg isareductivegroupschemeoverO forl(cid:29)0by[LP]
(cid:21) (cid:21)
Prop. 1.3 cf. [Wi] Th. 1.
THE IMAGE OF GALOIS L-ADIC REPRESENTATIONS, 9
Take (cid:21)jl a prime of E which splits in L: Then by (2.7) D (cid:24)= M (E ): By (2.8)
(cid:21) 2;2 (cid:21)
R (cid:24)=M (O ), so R (cid:10) k (cid:24)=M (k ): Let
(cid:21) 2;2 (cid:21) (cid:21) O(cid:21) (cid:21) 2;2 (cid:21)
1 0 0 1
t= ; u= :
(cid:18)0 (cid:0)1(cid:19) (cid:18)1 0(cid:19)
Let f = 1(1 + u); X = fT (A); Y = (1 (cid:0) f)T (A): Put X = X(cid:10) E ;
2 (cid:21) (cid:21) O(cid:21) (cid:21)
Y =Y(cid:10) E ;X =X(cid:10) k ;Y =Y(cid:10) k :Becausetft=1(cid:0)f;thematrixtgives
O(cid:21) (cid:21) O(cid:21) (cid:21) O(cid:21) (cid:21)
an O [G ]-isomorphism between X and Y; hence it gives an E [G ]-isomorphism
(cid:21) F (cid:21) F
between X andY and ak [G ]-isomorphismbetween X and Y: Usingthe compu-
(cid:21) F
tations of endomorphism algebrasby [Fa], Satz 4 and [Za], Cor. 5.4.5 we get:
(3.25) End (X)=O
O(cid:21)[GF] (cid:21)
(3.26) End (X)=E
E(cid:21)[GF] (cid:21)
(3.27) End (X)=k :
k(cid:21)[GF] (cid:21)
SotherepresentationsofG onthespacesX andY (resp. X andY )areabsolutely
F
irreducible over E (resp. over k ). Hence, the bilinear form 0 (resp. ) when
(cid:21) (cid:21) (cid:21) (cid:21)
restrictedtoanyofthespacesX;Y (resp. spacesX andY)iseithernon-degenerate
or isotropic.
Lemma 3.28. The modules X and Y are orthogonal with respect to : Conse-
(cid:21)
quently the modules X and Y (resp. X and Y) are orthogonal with respect to 0
(cid:21)
(resp. ).
(cid:21)
Proof. Note that uf =f and u(1(cid:0)f)=(cid:0)(1(cid:0)f): Hence for every v 2X and for
every w 2Y we get uv =v and uw=(cid:0)w: Hence
(v;w)= (uv;w)= (v;utw)= (v;uw)= (v;(cid:0)w)=(cid:0) (v;w):
(cid:21) (cid:21) (cid:21) (cid:21) (cid:21) (cid:21)
Hence (v;w)=0 for every v 2X and for every w2Y: (cid:3)
(cid:21)
Theorem 3.29. Let A be of type III and l (cid:29) 0: Then there is a free O -module
(cid:21)
W (A) of rank 2h such that
(cid:21)
(i) T (A)(cid:24)=W (A)(cid:8)W (A) as O [G ]- modules
(cid:21) (cid:21) (cid:21) (cid:21) F
(ii) there exists a symmetric, non-degenerate pairing
: W (A)(cid:2)W (A)!O
(cid:21) (cid:21) (cid:21) (cid:21)
(ii’) For W (A)=W (A)(cid:10) E the induced symmetric pairing
(cid:21) (cid:21) O(cid:21) (cid:21)
0 : W (A)(cid:2)W (A) !E is non-degenerate. The G module W (A) is
(cid:21) (cid:21) (cid:21) (cid:21) F (cid:21)
absolutely irreducible.
(ii") For W (A)=W (A)(cid:10) k the induced symmetric pairing
(cid:21) (cid:21) O(cid:21) (cid:21)
: W (A)(cid:2)W (A)!k is non-degenerate. The G module W (A) is
(cid:21) (cid:21) (cid:21) (cid:21) F (cid:21)
absolutely irreducible.
10 G. BANASZAK, W. GAJDA, P. KRASON(cid:19)
Pairings (ii), (ii’) and (ii") are compatible with the G -action in the same way as
F
the pairing in Lemma 3.24 (i).
Proof. Part(i) followsbytakingW (A)=X: Weget part(ii) restricting toX:
(cid:21) (cid:21)
To (cid:12)nish the proof observe that the form (3.2) is non-degenerate so = (cid:10)F
l l l
is non-degenerate for any abelian variety with polarization degree prime to l: By
Lemma3.2[BGK2]the form isnon-degenerateforall(cid:21)hencethe forms 0 and
(cid:21) (cid:21)
are simultaneously non-degenerate. Hence (ii’) and (ii") follow by and (3.26)
(cid:21)
and (3.27) and also by Remark 3.23, Lemma 3.28. (cid:3)
4. Representations associated with Abelian varieties of type III.
Let A=F be an abelian variety of type III. The (cid:12)eld of de(cid:12)nition F is such that
Galg isaconnected algebraicgroup. Let usput T =W (A); V =T (cid:10) E and
l (cid:21) (cid:21) (cid:21) (cid:21) O(cid:21) (cid:21)
A = V =T :WiththisnotationbyTheorem3.29wehave V (A)= V (cid:8)V :
(cid:21) (cid:21) (cid:21) l (cid:21)jl (cid:21) (cid:21)
We put L (cid:0) (cid:1)
(4.1) V = V
l (cid:21)
M
(cid:21)jl
LetV be the spaceV consideredoverQ :Thenthereisthe followingequalityof
(cid:8)(cid:21) (cid:21) l
Q -vector spaces:
l
(4.2) V = V
l (cid:8)(cid:21)
M
(cid:21)jl
The l-adic representation
(4.3) (cid:26)0 : G (cid:0)!GL(V (A))
l F l
induces the following representations (note that we use the notation (cid:26)0 for both
l
representations (4.4) and (4.5) cf. Remark 5.13 [BGK2]):
(4.4) (cid:26)0: G (cid:0)!GL(V )
l F l
(4.5) (cid:26)0: G (cid:0)!GL(V )
(cid:21) F (cid:21)
The representation (cid:26) was de(cid:12)ned in [BGK2]:
(cid:8)(cid:21)
(4.6) (cid:26) : G (cid:0)!GL(V ):
(cid:8)(cid:21) F (cid:8)(cid:21)
ByTheorem3.29(cf. [BGK2],Remark5.13)thegroupschemeGalg (resp. Galg) is
l (cid:21)
naturallyisomorphictotheZariskiclosureinGL (resp. GL )oftheimageofthe
Vl V(cid:21)
representation (cid:26) of (4.4) (resp. (cid:26) of (4.5)). Let Galgdenote the Zariski closure
l (cid:21) (cid:8)(cid:21)
in GL of the image of the representation (cid:26) of (4.6). Let g = Lie(Galg);
V(cid:8)(cid:21) (cid:8)(cid:21) l l
g = Lie(Galg) and let g = Lie(Galg): By de(cid:12)nition Galg (cid:26) Galg so g (cid:26)
(cid:21) (cid:21) (cid:8)(cid:21) (cid:8)(cid:21) l (cid:21)jl (cid:8)(cid:21) l
g : This implies: Q
(cid:21)jl (cid:8)(cid:21)
L
(4.7) (Galg)0 (cid:26) (Galg)0
l (cid:21)jl (cid:8)(cid:21)
Y
(4.8) gss (cid:26) gss :
l (cid:21)jl (cid:8)(cid:21)
M
SimilarlytopreviossectionweworkwithabelianvarietiesoftypeIII.Inthissection
we compute the Lie algebras corresponding to representations we consider. Some
resultsthat weprovedin [BGK2]for abelian varietiesof type I and II workaswell
forabelianvarietiesoftypeIII.Sincethedetailedproofsoftheseresultsweregiven
in [BGK2] we will merely reformulate them for abelian varieties of type III.