On the reex norm
Krzysztof Klosin
March 11, 2005
1 The reex eld
Let F denote a CM eld, [F : Q] = 2g. Denote by F + the maximal totally real subeld of F . Let C be
a CM eld which is Galois over Q and splits F . We denote by R the maximal totally real subeld of C .
To ease notation, every time we write A B , for two Q-algebras A and B , we mean A Q B . There is a
canonical map of F C -algebras
F C !
Y
2HomQ (F;C )
C
(1)
given by c f 7! (c(f )) , which is an isomorphism; the notation C means C made into an F -algebra via .
We
decomposition for F R. Indeed, F R is canonically isomorphic to the (nite) product
Q have a similar
(
F
R)=p, where each of the quotients is non-canonically R-isomorphic to C . We naturally
p2Spec(F R)
have (F R)=p = (F R)p . We let Fp denote (F R)p and P denote Spec(F R). We also write Fp+ for
the corresponding factor eld of F + R. We have a natural identication:
(ResF=Q Gm )R !
Y
p
which for any R-algebra A is given by
2P
ResF =R Gm
(F A) ! (F R R A) !
p
Y
p
2P
(Fp R A) :
Let (F; ) be a CM type (over C , cf. Brian's talk), i.e., HomQ (F; C ) is a set of representatives for
HomQ (F; C ) mod Gal(F=F + ). Every 2 HomQ (F; C ) uniquely extends to an R-algebra map ~ : F R ! C
(f r 7! (f )r) which must factor through exactly one of the factor elds Fp , yielding an R-algebra
isomorphism p : Fp ! C . Hence, we can think of as the set fp : Fp ! C j p 2 P g. If p 2 we will
denote the other R-algebra isomorphism Fp ! C by p since it is related to p via Gal(C=R)..
Q
Let S := ResC=R Gm . We denea map h :S ! (ResF=Q Gm )R = p2P ResF =R Gm , which for every
Q
Q
R-algebra A is given by h (A) = 2 p 1 1 : (C R A) ! (F A) = p2P Fp R A.
p
Example 1.1. On R-points, the map h is the map C ! Qp2P Fp given by z 7! (p 1 (z))p . In fact, since
S is a torus and R is innite, S(R) is Zariski-dense in S and hence knowing h on R-points determines it
p
completely.
Let T be the kernel of the composite ResF=Q Gm ! ResF + =Q Gm ! ResF + =Q Gm =Gm , where the rst
arrow is the norm map NF=F + and the second is the natural projection. In Nick's talk it was shown that the
kernel of the rst arrow is a torus, so connectedness of Gm implies connectedness of T , hence T ResF=Q Gm
is a Q-torus.
Lemma 1.2. The map h factors through TR.
1
Proof. We want to show that h 1 (TR ) = S. As rational points are Zariski-dense in any torus over an innite
eld, it is enough to check the equality holds on R-points, i.e., we want to show that h (S(R)) T (R). By
Example 1.1, this says that the map
C
p 2 p
Q
lands in R diagonally embedded in
1
!
Q
p
Y
p 2
F
Q
Y
NFp =Fp+ ()
!
p
p
2P
(Fp+ )
+ + 2P (Fp ) . However this is clear as the
canonical maps R ! (Fp )
are isomorphisms and for every p 2 P we have that p NF =(F + ) (p 1 (z )) = NC=R (z ) 2 R is independent
of p.
p
p
Choose a totally negative element 2 R for which there exists 2 C such that 2 = . (The existence
of such and was shown in Tong's talk via weak approximation.) We dene a map
C
: Gm=C ,! SC (h )!
TC ;
where the rst arrow j : Gm=C ,! SC is dened on A-points (for any C -algebra A) by
j (a) = 21 (a + 1) + 2(a 1) :
It is easy to see a priori (using that R is the maximal totally real subeld of the CM eld C ) that j is
independent of and and that j (aa0 ) = j (a)j (a0 ), so j really is unit-valued and is a group morphism.
More concretely, if we identify SC with Gm c(Gm ) (with c denoting the non-trivial element of Gal(C=R))
then j is given (on Yoneda points) by z 7! (z; 1).
Denition 1.3. The subeld E(F;) of C which is the eld of denition of is called the reex eld of
(F; ).
We will denote the descent of to E := E(F;) by ;E , or simply by when it cannot cause confusion.
We have Gal(C=E(F;) ) = f 2 Gal(C=Q) j ( ) = g.
Remark
talk the reex eld F was dened to be the subeld of C generated by elements
P (a),1.4.a 2InF .Eiji's
He also proved that F is the xed eld of the group f 2 Gal(C=Q) j = g. The
2
next two lemmas imply that the notion of the reex eld in the sense of Denition 1.3 coincides with that
dened in Eiji's talk.
Lemma 1.5. ( ) = Proof. Since is a map between connected algebraic groups in characteristic zero, it is uniquely determined
by the induced map on the Lie algebras. For any ane group scheme G=k , where k is a eld, the Lie algebra
Lie(G) is in canonical bijection with the points in G(k[]) mapping to 1 2 G(k) (here 2 = 0). The map
= (h )C j induces a map on C []-points
C [] ! (C R C []) ! (F C []) =
Y
p
2P
(Fp R C [])
(2)
where the rst arrow satises 1 + c 7! 21 (2 + c) + 2c and the second arrow satises
1 + (c c0 )(1 ) 7! (1 + (p 1 (c) c0 )(1 ))p :
We now investigate the eect of the -operation on the C -linear map Lie . We consider the following
situation. Let G, H be two ane group schemes of nite type over Q and let V and W be the tangent
spaces at identity of G and H respectively. A C -group map GC ! HC induces a C -linear transformation
C V T! C W that is Lie( ). On the other hand, for 2 Gal(C=Q) the map induces the Lie
2
algebra map C V T! C W that is the scalar extension of T via , so we get a commutative diagram
of C -vector spaces
id T
C C O C V
c
v7!c
1
v o
/
C C OC W
o c
w7!c
1
w
T
C V
(3)
C W
P
P
Hence, if v 2 V and T (v) = ci wi 2 C W , then ( T )(v) = (ci ) wi = ( 1) T (v). So in
/
our case, ( )(1 + ) = (1 )( (1 + )) for v 2 Q[] C [] , where 1 : (F C ) ! (F C ) is
induced from 1 : F C ! F C . We have the following commutative diagram of rings:
F C
o f z7!(f z)v
Q
p
p
2P C
›
p
/
(fp zp )p 7!(fp (zp ))p
2P Fp R C
o (f z ) 7!( (f )z ; (f )z )
›
p
Q
1
p
Q
p p
C
p
p
p
p
p
F C
(4)
o
Q
/
2P Fp R C
›
p
p
o (f z ) 7!( (f )z )
p
p
=
(z ) 7!((z 1 ))
:F,!C C
›
/
Q
p p
:F,!C C
›
where f := fp and z := zp if and are the two ways to identify Fp with C as R-algebras.
Q
Let v = 1 + 2 Q[] . We rst calculate the image
( :F,!C C []) .
of1 (1 + 1())2
(F C []) inside
Q
(1 ) 2 p2P (Fp C )p which
By the denition of maps (2), we have (1+ ) = 1 + 2 1 + p
Q
gives the element (1 + ; 1) 2 p2P C C , i.e., (1 + ) = ( ) , where = 1 + if 2 and = 1
otherwise.
Q
Now, ( )(1 + ) = (1 )( (1 + )) viewed as an element of ( F,!C C []) is identied with
(( ) ) by commutativity of diagram (4), where is the right vertical arrow in diagram (4). We have
(( ) ) = (( 1 )) , so by the denition of we have ( 1 ) = 1 + if 1 2 and ( 1 ) = 1
otherwise. Hence ( ) = .
p
p
p
Lemma 1.6. = 0 if and only if = 0.
Proof. One implication is trvial. The other implication follows immediately Q
from the description of in the
proof of Lemma 1.5. Indeed, we calculated there that (1 + ) = ( ) 2 ( :F,!C C ) , where = 1 + if 2 and = 1 otherwise. Hence if = 0 , we must have = 0 .
2 The reex norm
We will now dene the notion of the reex norm. Let L0 =L be a nite separable eld extension (not
necessarily Galois), and X a commutative ane L-group of nite type. Fix a separable closure Lsep=L.
There is a canonical \trace" map
which for an L-algebra A is given by
ResL0 =L XL0 ! X;
XL0 (A L L0 ) = X (A L L0 ) ! X (A L Lsep)Gal(Lsep =L) = X (A)
3
(5)
a0 7!
This is independent of the choice of Lsep
we obtain a canonical map
Y
2HomL (L0 ;Lsep )
(a0 ):
and is natural in X . Setting L = Q, L0 = E = E(F;) , and X = T ,
: ResE=Q TE ! T:
Remark 2.1. Such a map can also be obtained with ResF=Q Gm replacing T , and the two are compatible via
the inclusion : T ,! ResF=Q Gm since (5) is natural in X . We will use this fact in the proof of Proposition
2.5. On C -points with X = ResF=Q Gm , the \trace" ResL0 =Q XL0 ! X is the map (F L0 C ) ! (F C )
induced by the product of F C -algebra maps x y z 7! x (y)z over all 2 HomQ (L0 ; C ), i.e., this is
the usual ring-theoretic norm on units.
Denition 2.2. The map r = ResE=Q : ResE=Q Gm ! T is the reex norm.
We will now give an alternative denition of the reex norm and show that it is equivalent to the one given
above.
Let A be an abelian variety over C , with i : F ,! End (A) of type (F; ). Let E be the reex eld of
(F; ). The abelian
Q the tangent space at identity tA=C
Q variety (A; i) typically does not descend to E . However
is isomorphic 2 C as an F C -module and this concrete model 2 C has a natural C -semilinear
action of Gal(C=E ) via ((a ) ) = ((a 1 )) , so it descends to E in the following sense.
Lemma 2.3. Up to isomorphism, there exists a unique F E -module t such that t E C = Q2 C as
F C -modules compatible with the natural Gal(C=E )-actions on both sides.
Proof. See Eiji's
Q talk. We only note here that explicitly t is dened to be the F E -submodule of Gal(C=E )invariants in 2 C .
We now dene a map N : ResE=Q Gm ! ResF=Q Gm which for any Q-algebra A is given by
N (A) : (E A) ! (F A) ;
via s 7! detF Q A (s : t Q A ! t Q A).
Example 2.4. Let F be a quadratic imaginary eld. Then = f : F ,! C g is a one-element set, hence
E = (F ). In this case t is a one-dimensional F -vector space and N (Q) : E ! F is given by 1 .
The following proposition shows that N coincides with the reex norm in the sense of Denition 2.2.
Proposition 2.5. Let : T ,! ResF=Q Gm be the inclusion map. Then r = N.
Remark 2.6. It is important that the target group of the reex norm is the torus T , and not merely the
full group ResF=Q Gm . The reason which makes this fact signicant for us is (as will be proved later, using
results in Nick's talk) that the Q-points of T are discrete in T (Af ), where Af denotes the nite adeles of Q.
This makes the quotient space T (Q) n T (Af ) Hausdor. This is not the case if we replace T with ResF=Q Gm
unless the group of intgral units of F is nite, which is to say F is imaginary quadratic.
Proof of Proposition 2:5. First note that E C by denition. To ease notation we will denote by E the set
HomQ (E; C ). Also for each 2 E we choose ~ 2 Gal(C=Q) satisfying ~ jE = . Then ~ makes sense for
every 2 .
We will show that the following diagram commutes:
(E C )
E C -action
ResE=Q ;E
/
(F E C )
Q
AutF C (t C )
›
detF C /
(F
4
2E []()
C ) = Q :,!C C
›
(6)
where the top map is on C -points and [](f e z ) = f (e)z as an F C -algebra map from F E C
to F C . By Remark 2.1 the composite of the right vertical arrow with the top arrow is r (C ), while the
composite of the other two arrows is N (C ). We will rst describe the map N (C ) = detF C (E C )-action.
We have an F E C -linear decomposition
t C = t E E C =
Y
2E
t E C =
Y
2E
YY
t E C C C =
2E 2
C C C =
YY
2E 2
C;
where the second equality from the right follows from the denition of t in the proof of Lemma 2.3;
the notation C; means C made into an E -algebra and F -algebra via and ~ respectively.
Q CMoreover
via
the
diagonal
action
of
C
on
E C = Q2E C acts on t C = Q2E Q2 CQ
; Q
2 ; . To
C
according
to
the
action
of F C .
compute detF C we need to rearrange the factors
in
2E 2 ; Q
Q
As F C -modules we have t C = :F,!C V , where V = (;)2(; ) C; , with (; ) :=
f(; ) 2 E j ~ = g. For 2 E , let 0 : E C ! C be the C -algebra map induced by e z 7! (e)z .
On each V an element 2 E C acts by multiplication
by the diagonal matrix diag (0 ( ))(;)2(; ) , so
Q
the C -determinant of this C -linear action equals (;)2(; ) 0 ( ) 2 C for each : F ,! C .
After we prove that diagram (6) commutes, this computation will provide a practical way to calculate the
reex norm (cf. Remark 2.7).
We will now describe the composite of the top and right maps in diagram (6). We rst consider the top
map; the diculty is that over C is concrete but ;E is \abstract". Note that given two Q-schemes X
f
YE , with E a subeld of C , we have a commutative diagram:
and Y , and an E -map XE !
ResE=Q f
ResE=Q XE (C ) = XE (E C )
o
›
ResE
C=C XE
C=C
Q
(C )
:E,!C ResC =C XC (C )
Q
ResC =C fC
Q
o
:E,!C XC (C )
o
/
Q
›
ResE=Q YE (C ) = YE (E C )
ResE
C=C fC
o
›
/
Q
›
/
(7)
ResE
C=C YE
C=C (C )
o
:E,!C ResC =C YC (C )
›
Q
/
fC
o
:E,!C YC (C )
›
where fC = ~ (fC ) : XC ! YC . Hence if X = Gm , Y = ResF=Q Gm , and f = ;E then since
=C = ~ ( ) = ~ (by Lemma 1.5), we have that ResE=Q ;E on C -points is identied with the map
Y
:E,!C
C
Q
~ !
Y
:E,!C
Y
ResF=Q Gm C (C ) =
:E,!C
(F C ) :
(8)
For any : E ,! C , we have the following commutative diagram of rings
C R C
o
C C
›
Q
p
2P (~ p ) 1 1 / F
(z ;z )7!(z ;z )p
C = Qp2P Fp R C
Q
/
(fp z )p; 7!(~p (f~ p )z ;~ p (f~ p )z )p
2P C ; C ;
›
p
and a commutative diagram of unit groups
5
p
p
(9)
jC
C SSSS
/ (C
SSS
SSS
SS
z 7!(z ;1)SSSSS)
R C )
(10)
o
C C
Note that the composite of the top map in diagram (9)Qwith jC is the map ~ . Hence if z 2 C ,
then ~ (z ) = (z ; 1)p if we view it as an element of p2P C ; C ; via the right vertical arrow
Q
Q
in diagram (9). If we identify p2P C ; C ; with :F,!C C; , then (z ; 1)p is identied with the
element (z; ); , where z; := z if 2 and z; := 1 otherwise.
Now, for every 0 : E ,! C we have the following commuative diagram of rings:
›
p
p
p
[0 ]()
E
F C
Q
Q
Q
Q
/F
C
(11)
o
:E,!C F
›
:E,!C
:E,!C
p
Q
Q
C
o
p2P Fp R C
›
o
(fp; zp; )p; 7!(fp;0 zp;0 )p
o
/
Q
p2P Fp R C
o
Q
p 2 Cp ; Cp ;
›
›
2P C
›
p
o
p
C
p
o
(z; ); 7!(z0 ;0 )~ 0 0 =
Q›
Q › C
/
:E,!C :F,!C C;
:F,!C Q
Our goal is to compute 2A [] ResE=Q ;E ( ), where 2 ResE=Q GmQ (C ) = (E C ) , but for
0
z := ( ) by using diagrams (7) and (11) this is the same as rst computing
(z; ); 2
Y Y
:E,!C :F,!C
:E,!C ~ (z ), which is
;
C;
Q
then composing it with the bottom arrow in diagram (11) giving (z0 ;0 )~0 0 = 2 :F,!C C , and nally
taking the product over all 0 : E ,! C , yielding
0
1
Y
B@
z0;0 C
A
0 ;0
~0 0 =
1
0
Y Y
C :
z A 2
=@
2(; )
:F,!C
This is the same element we obtained when we calculated the composite of the other two maps in diagram
(6).
Remark 2.7. The proof of Proposition 2.5 gives a concrete expression
Q for the reex norm of an element
1
e 2 E . Indeed, x : F ,! C . Then the reex norm r (e) is 2(; ) (e) . Note that this is in
fact independent of the choice of , as it should be. Moreover, we have r (e)r (e) = NE=Q (e).
6