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Complex Multiplication

J.S. Milne

April 7, 2006

These are preliminary notes^1 for a modern account of the theory of complex multipli- cation. A shortened (minimal) version will be included in my book on Shimura varieties, and a complete longer version may one day be published separately.

v0.00 April 7, 2006. First version posted on the web; 113 pages.

Please send comments and corrections to me at [email protected].

Available at http://www.jmilne.org/math

Copyright c^ 2006. J.S. Milne.

(^1) This should be taken seriously: there are omissions, repetitions, clumsy statements and proofs, and incon- sistencies in notation.

4 CONTENTS

78 ; The fundamental theorem in terms of id`eles 79 ; The fundamental theorem in terms of uniformizations 82 ; The fundamental theorem in terms of moduli 83 ; Alternative ap- proach using crystals (Deligne c1968) 85 10 The fundamental theorem of complex multiplication.............. 90 Statement of the Theorem 90 ; Definition of f˚ ./ 92 ; Proof of Theorem 10.2 up to an element of order 2 95 ; Completion of the proof (following Deligne) 97

III CM-motives 99

IV Applications 101

A Additional notes; solutions to the exercises 103

B Summary 105

Bibliography 109

Index of definitions 112

Preface

The theory of complex multiplication is not only the most beautiful part of mathematics but also of all science. D. Hilbert^2

Abelian varieties with complex multiplication^3 are special in that they have the largest possible endomorphism rings. For example, the endomorphism ring of an elliptic curve is usually Z, but when it is not, it is an order in an imaginary quadratic number field, and the elliptic curve is then said to have complex multiplication. Similarly, the endomorphism ring of a simple abelian variety of dimension g is usually Z, but, at the opposite extreme, it may be an order in a number field of degree 2g, in which case the abelian variety is said to have complex multiplication. Abelian varieties with complex multiplication correspond to special points on the moduli variety of abelian varieties, and their arithmetic is intimately related to that of the values of modular functions and modular forms at those points. The first important result in the subject, which goes back to Kronecker and Weber, states that the Hilbert class field (maximal abelian unramified extension) of an imaginary quadratic subfield E of C is generated by the special value j ./ of the j -function at any element  of E in the complex upper half plane generating the ring of integers in E. Here j is the holomorphic function on the complex upper half plane invariant under the action

of SL 2 .Z/, taking the values 0 and 1728 respectively at ^1 C

p 3 2 and^

p 1 , and having a simple pole at infinity. The statement is related to elliptic curves through the ideal class group of E, which acts naturally both on the Hilbert class field of E and on the set of isomorphism classes of elliptic curves with endomorphism ring OE. Generalizing this, Hilbert asked in the twelfth of his famous problems whether there ex- ist holomorphic functions whose special values generate the abelian extensions (in particu- lar, the class fields) of arbitrary number fields. For quadratic imaginary fields, the theory of elliptic curves with complex multiplication shows that elliptic modular functions have this property (Kronecker, Weber, Takagi, Hasse). Hecke began the study of abelian surfaces

(^2) As quoted by Olga Taussky in her obituary for Hilbert in Nature, 152 (1943), 182–183. The following is from a letter she sent to me in October 1990:

Yes it is true, Hilbert said this and I was in the audience when he said it and I was pleased he said it. It was at the Mathematiker Kongress Z¨urich 1932. Fueter ... had written an opus in 2 volumes: Vorlesungen ¨uber die singul¨aren Moduln und die komplexe Multiplikation der elliptischen Funktionen, Teubner, 1924, 1927. Hilbert presided at Fueter’s lecture. (^3) The name is both archaic and imprecise — the term “multiplication” is no longer used to denote an endomorphism, and “complex multiplication” is sometimes used to denote a more general class (Birkenhake and Lange 2004, p262) — but I know of no other.

5

Let A and B be sets and let  be an equivalence relation on A. If there exists a canonical surjection A! B whose fibres are the equivalence classes, then I say that B classifies the elements of A modulo  or that it classifies the -classes of elements of A.

REFERENCES

In addition to those listed at the end, I refer to the following of my course notes (available at www.jmilne.org/math/). AAG: Algebraic Groups and Arithmetic Groups, v1.0, May 22, 2005. AG: Algebraic Geometry, v4.0, October 30, 2003. ANT: Algebraic Number Theory, v2.1, August 31, 1998. CFT: Class Field Theory, v3.1, May 6, 1997. FT: Fields and Galois Theory, v3.0, August 31, 2003. LEC: Lectures on Etale Cohomology, v2.01, August 9, 1998. MF: Modular Functions and Modular Forms, v1.1, May 22, 1997.

PREREQUISITES

The reader is expected to have a good knowledge of basic algebraic number theory (e.g., ANT and parts of CFT), and basic algebraic geometry (e.g., AG and Hartshorne 1977, II) including abelian varieties (e.g., Milne 1986).

10 CHAPTER I. ANALYTIC THEORY

PROPOSITION 1.1 Let B be a simple Q -algebra with centre k. Let K be a field containing all conjugates of k and splitting B , and let V be a B ˝Q K -module. The characteristic polynomials det.T bjV / of all elements b of B acting on the K -vector space V have coefficients in Q if and only if V is isomorphic to a multiple of the reduced B ˝Q K - module (equivalently, V is free as a k ˝Q K -module).

PROOF. The isomorphism ˛W B ˝Q K!

Q

Wk!K Mn.K/^ is well-determined up to conju- gation by an element of

Q

Wk!K Mn.K/ , and hence the characteristic polynomial Pb .T /

of ˛.b ˝ 1 / for b 2 B is well-defined. It equals det.T bjV / where V is the reduced module. For any automorphism  of K, .Pb .T // D Pb .T /. Enlarging K to a Galois extension of Q doesn’t change Pb .T /, and so this shows that Pb .T / has coefficients in Q. Any other B ˝Q K-module M is isomorphic to a direct sum

L

 mV,^ m^ ^0 :^ As a 2 k acts on V as multiplication by .a/, the characteristic polynomial PM;a.T / of a

on

L

 mV^ is

Q

.T^ ^ a/ m

n

. When a generates k, this has coefficients in Q if and

only if the m are equal.^2

Let B be a semisimple k-algebra, and let B D

Q

Bi be its decomposition into a product of simple algebras Bi. The centre of each Bi is a field ki , and each degree ŒBi W ki ç is a square. The reduced degree of B over k is defined to be

ŒBW kçred D

X

i ŒBi W ki ç^1 =^2
Œki W kç:

For any field k^0 containing k,

ŒBW kç D ŒB ˝k k^0 W k^0 ç, and ŒBW kçred D ŒB ˝k k^0 W k^0 çred: (1)

PROPOSITION 1.2 Let B be a semisimple k -algebra. For any faithful B -module M ,

dimk M  ŒBW kç red;

and there exists a faithful module for which equality holds if and only if the simple factors of B are matrix algebras over their centres.

PROOF. Let B D

Q

Bi where Bi 

Q

Mni .Di / with Di a central division algebra over ki , and let Si D D in ibe a simple Bi -module. Then every B-module M is isomorphic to a sum

L

i mi^ Si^ , and^ M^ is faithful if and only if each^ mi^ >^0. Therefore, if^ M^ is faithful, dimk M D

X

i mi^
^ ni^
^ ŒDi^ W^ kç^
^ Œki^ W^ kç^ ^

X

i ni^
^ ŒDi^ W^ kç^
^ Œki^ W^ kç: On the other hand, ŒBW kçred D

X

i ni^
^ ŒDi^ W^ kç^

(^12)
Œki W kç:

The proposition is now obvious. (^2)

PROPOSITION 1.3 Let B be a semisimple k -algebra. Every maximal ´etale k -subalgebra of B has degree ŒBW kçred over k.

PROOF. When B is central simple, the proposition asserts that every maximal subfield of

B containing k has degree ŒBW kç

(^12)

. This case is proved in CFT, IV 3.5, and the general case follows easily. (^2)

(^2) Let ca.T / D Q .T a/ be the characteristic polynomial of a in the field extension k=Q. Because a generates k, ca.T / is irreducible. Any monic irreducible factor of PM;a.T / in QŒT ç shares a root with ca.T /, and therefore equals it. Hence, if PM;a.T / has coefficients in Q, it is a power of ca.T /.

1. CM-ALGEBRAS AND CM-TYPES 11

CM-algebras

A number field E is said to be totally real if its image under every homomorphism E ,! C is contained in R. When the image is never contained in R, the field is said to be totally imaginary. Equivalently, E is totally real if E ˝Q R  RŒEWQç^ and it is totally imaginary if E ˝Q R  CŒEWQç=^2. A number field QŒ˛ç ' QŒX ç=.f .X // is totally real if all the roots of f .X / are real and it is totally imaginary if none of the roots are real.

PROPOSITION 1.4 The following conditions on a number field E are equivalent: (a) E is a totally imaginary quadratic extension of a totally real number field; (b) there exists an automorphism E ¤ id of E such that  ı E D  ı  for all homomor- phisms W E ,! C ; (c) E D F Œ˛ç with F totally real, ˛^2 2 F , and .˛^2 / < 0 for all homomorphism W F ,! C.

PROOF. Assume (a), and let F be the totally real subfield. The unique nontrivial automor- phism of E fixing F has the property required for (b). Let ˛ generate E over F. After completing the square, we may suppose ˛^2 2 F , and then .˛^2 / < 0 for every embedding W F ,! C because E is totally imaginary. Assume (b). Then E has order 2 , because  ı ^2 E D  for any W E ,! C. Moreover, its fixed field F is totally real, and E is a totally imaginary quadratic extension of F. Assume (c). Certainly, the conditions imply that E is a totally imaginary quadratic extension of F. (^2)

Because they occur in the theory of complex multiplication, the fields satisfying these conditions are called CM-fields. Note that a number field E is CM if and only if it has exactly one complex conjugation (by (b)). Clearly, any field isomorphic to a CM-field is CM.

COROLLARY 1.5 A finite composite of CM-subfields of a field is CM; in particular, the Galois closure of a CM-field in any larger field is CM.

PROOF. Clearly, each complex embedding of the composite of two CM-fields will induce the same nontrivial complex conjugation on the field. (^2)

REMARK 1.6 Let K  Qal^ be a number field. If ^1 acts on K as  for every  2 Aut.Qal/, then K is totally real or is a CM-field according as  fixes E or not. It follows that

the union of all CM-subfields of Qal^ is the field fixed by the comutators Œ; ç def D ^1 ^1 of Gal.Qal=Q/, i.e., it is the subfield corresponding to the closure of the group generated by

fŒ; ç j  2 Gal.Qal=Q/g:

We denote this field by Qcm.

REMARK 1.7 Let K be a number field. Since a composite of totally real fields is totally real, K contains a largest totally real subfield F. Moreover, K contains at most one totally imaginary quadratic extension of F , because any such extension is of the form F Œ

p ˛ç with ˛ totally negative; if F Œ

p ˇç is a second such extension, then K contains the totally real field F Œ

p ˛ˇç, which must equal F , and this implies that F Œ

p ˛ç D F Œ

p ˇç. If K contains

a CM-field E, then K^0 def D E
F is the largest CM-subfield of K. It consists of all elements ˛ of K having a conjugate ˛^0 in K such that .˛^0 / D .˛/ for all embeddings W K! C. For any such embedding, K^0 D K \ Qcm.

1. CM-ALGEBRAS AND CM-TYPES 13

PROOF. Assume initially that E is Galois over Q, and define E 0 to be the fixed subfield of H D f 2 Gal.E=Q/ j ˚ D ˚g: (A) E 0 is a CM-subfield of E and ˚jE 0 is a CM-type on E 0. As E is a CM-field, ˚E D ˚ ¤ ˚;

and so E is not in H ; it therefore acts nontrivially on E 0. To show that E 0 is a CM- subfield, it remains to show that it is stable under E , i.e., that E a D E a for all  2 H and a 2 E 0. But, for  2 H ,

˚E E D ˚E D ˚E D ˚;

and so E E 2 H. This implies that E a D E a for all a 2 E 0. If ˚jE 0 is not a CM-type, then '^0 jE 0 D  ı 'jE 0 (4)

for distinct '; '^0 2 ˚. But (4) implies that  ı ' 2 '^0 H  ˚, which is a contradiction. (B) If E^0 is a CM-subfield of E and ˚jE^0 is a CM-type on E^0 , then E^0  E 0. The conditions imply that ˚ is the extension to E of the CM-type ˚^0 def D ˚jE^0 on E^0. Let  be an element of G fixing E^0. Then ˚^0  D ˚^0 , which implies that ˚ D ˚, and so  2 H. (A) and (B) prove the proposition when E is Galois over Q. In the general case, we can embed E in a CM-field E 1 Galois over Q and extend ˚ to a CM-type ˚ 1 on E 1. The preceding argument applied to .E 1 ; ˚ 1 / gives a smallest CM-field E 0  E such that ˚jE 0 is a CM-type on E 0. (^2)

COROLLARY 1.10 A CM-pair .E; ˚/ is primitive if and only if for some (hence all) CM- fields E 1 containing E and Galois over Q , the subgroup of Gal.E 1 =Q/ fixing E is

f 2 Gal.E 1 =Q/ j ˚ 1  D ˚ 1 g

where ˚ 1 is the extension of ˚ to E 1.

PROOF. Immediate from the proposition. (^2)

EXERCISE 1.11 (Shimura and Taniyama 1961, 8.2 = Shimura 1998, 8.2). Let E be a CM- field, and write E D F Œ˛ç with ˛^2 2 F and totally negative. The embeddings 'W E! C such that =.'.˛// > 0 form a CM-type ˚ on E. Show that .E; ˚/ is primitive if and only if (a) F Œ˛ç D QŒ˛ç, and (b) for any conjugate ˛^0 of ˛ over Q other than ˛ itself, ˛^0 =˛ is not totally positive.

EXERCISE 1.12 (ibid. 8.4). Let E D Q΍ where  is a primitive 13 th^ root of 1 in C. Of the 32 CM-types on E containing the given embedding of E into C, show that only 2 are nonprimitive, and that the remaining 30 CM-types fall into 6 orbits under the action of Gal.E=Q/, each with 5 elements.

DEFINITION 1.13 Let E be an ´etale Q-algebra, and let Q be an algebraic closure of Q. A CM-type on E with values in Q is a subset ˚ of HomQ-alg.E; Q/ such that

HomQ-alg.E; Q/ D ˚ t ˚

for all complex conjugations  on Q.

14 CHAPTER I. ANALYTIC THEORY

Note that when E is a CM-algebra and Q D Qal, this agrees with Definition 1.8.

EXERCISE 1.14 Let ˚ be a CM-type on a field E with values in Q. Show that there exists a CM-subfield E 0 of E such that no two elements of ˚ are complex conjugates on E 0 (and hence there is a CM-type ˚ 0 on E 0 such that ˚ D f'W E! Q j 'jE 0 2 ˚ 0 g).

EXERCISE 1.15 Rewrite this subsection replacing Qal^ and C with Q.

The reflex field of a CM-pair

If  is an automorphism of C (or Qal/ and ˚ is a CM-type on a CM-algebra E, then

def D f ı ' j ' 2 ˚g

is again a CM-type on E.^3

PROPOSITION 1.16 Let .E; ˚/ be a CM-pair. The following conditions on a subfield E of Qal are equivalent: (a)  2 Gal.Q^ al=Q/ fixes E if and only if ˚ D ˚ ; (b) E is the subfield of Q generated by the elements

P

' 2 ˚ '.a/^ ,^ a^2 E^.

PROOF. If  2 Gal.Qal=Q/ permutes the '’s in ˚, then clearly it fixes all elements of the form

P

' 2 ˚ '.a/. Conversely, if P ' 2 ˚ '.a/^ D^

P

' 2 ˚ .^ ı^ '/.a/^ for all^ a^2 E

then f ı ' j ' 2 ˚g D ˚ by Dedekind’s theorem on the independence of characters (FT 5.14).^4 This shows that conditions (a) and (b) define the same field. (^2)

DEFINITION 1.17 The field satisfying the equivalent conditions in the proposition is called the reflex field E^ of .E; ˚/.

Note that, in contrast to E, which need not even be a field, E^ is a subfield of Qal.

PROPOSITION 1.18 Let .E; ˚/ be a CM-pair. (a) The reflex field E of .E; ˚/ is a CM-field. (b) If .E; ˚/ D

Q

1 im.Ei^ ; ˚i^ /^ , then^ E

 D E

1


^ E

 m: (c) The reflex field of any extension .E 1 ; ˚ 1 / of .E; ˚/ equals that of .E; ˚/. (^3) Note that, because E is CM,

 ı . ı '/ D . ı '/ ı E D  ı . ı '/I

therefore, if  ı . ı '/ D  ı '^0 , then  ı . ı '/ D  ı '^0 and  ı ' D '^0. Hence ˚ \ ˚ D ;, and it follows (by counting) that Hom.E; C/ D ˚ t ˚. (^4) In more detail, the equation says that X ' 2 ˚ '^ ^

X ' 2 ˚ '^ D^0 ;

and Dedekind’s theorem says that this is possible only if each ' in ˚ occurs exactly once in ˚.

16 CHAPTER I. ANALYTIC THEORY

PROPOSITION 1.21 Let .E; ˚/ be a CM-pair, and let k be a subfield of Q. There exists a finitely generated E ˝Q k -module V such that

Trk .ajV / D

P

' 2 ˚ '.a/;^ all^ a^2 E;^ (5)

if and only if k  E , in which case V is uniquely determined up to an E ˝Q k - isomorphism.

PROOF. If a acts k-linearly on V , then Trk .ajV / 2 k, and so, if there exists a k-linear action of E satisfying (5) on a k-vector space V , then certainly k  E. For the converse, we initially assume that k contains all the conjugates of E. There is then a canonical isomorphism

e ˝ a 7! .e
a/W E ˝Q k!

Y

WE!k k;

and so any E ˝Q k-module V is of the form

L

mk for unique nonnegative integers m, where k denotes a one-dimensional k-vector space on which e 2 E acts as .e/. Thus, up to isomorphism, there exists exactly one E ˝Q k-module satisfying (5), namely,

L

' 2 ˚ k'^. For a general k containing E, we use the following statement:

Let ˝ be a finite Galois extension of k with Galois group ; the functor V 7! ˝ ˝k V is an equivalence from the category of k-vector spaces to the category of ˝-vector spaces endowed with a semilinear action of (see AG 16.14; an action is semilinear if .av/ D a
v for 2 , a 2 ˝, v 2 V ).

Let ˝ be any finite Galois extension ˝ of k containing all conjugates of E. Consider the E ˝Q ˝-module

L

' 2 ˚ ˝'^ , where^ ˝'^ is a one-dimensional^ ˝-vector space on which e 2 E acts as '.e/. Because ˚ is stable under , we can define a semilinear action of on

L

' 2 ˚ ˝'^ by the rule

.: : : ; ' v; : : :/ D .: : : ; ı' v ; : : :/; 2 ;

and one checks that this is the only such action commuting with the action of E.^6 Any E ˝Q k-module satisfying (5) becomes isomorphic to

Q

' 2 ˚ ˝'^ over^ ˝, and so this shows that, up to isomorphism, there exists exactly one such E ˝Q k-module. (^2)

COROLLARY 1.22 The reflex field E is the smallest subfield of Q such that there exists an E ˝Q E -module V with

V ˝E Q '

M

' 2 ˚ Q'^ (as an^ E^ ˝Q^ Q^ -module)^ (6)

where Q' is a one-dimensional Q -vector space on which E acts through '.

PROOF. Restatement of the proposition. (^2)

(^6) Use that ˝' D fx 2 V j a
x D '.a/x all a 2 Eg:

1. CM-ALGEBRAS AND CM-TYPES 17

Let V˚ be an E ˝Q k-module satisfying (5). An element a of k defines an endomor- phism of V˚ regarded as an E-vector space, whose determinant we denote by detE .ajV˚ /. If a 2 k, then detE .ajV˚ / 2 E, and so in this way we get a homomorphism

Nk;˚ W k^! E:

More generally, for any Q-algebra R and invertible element of a of k ˝Q R, we get an invertible element Nk;˚ .a/ D detE˝QR .ajV˚ ˝Q R/

of E ˝Q R. In this way, we get a homomorphism

Nk;˚ .R/W .k ˝Q R/^! .E ˝Q R/

which is functorial in R and independent of the choice of V˚. It is called the reflex norm from k to E (relative to ˚). When k D E, we drop it from the notation.

PROPOSITION 1.23 For any number field k with E^  k  Q ,

Nk;˚ D N˚ ı Nmk=E : (7)

PROOF. Choose an E ˝Q E-module V˚ satisfying (6), and let V 0 D k ˝E V˚. When we use V 0 to compute Nk;˚ , and V˚ to compute N˚ , we obtain (7). (^2)

REMARK 1.24 (a) For any isomorphism W E! E^0 ,

N˚ .a/ D N˚ .a/, all a 2 E,

where ˚ D f' ı  j ' 2 ˚g. (b) Let V˚ be an E ˝Q k-module satisfying (5). Then V˚ ˚ V˚ satisfies

Trk .ajV / D

X

WE!Q .a/, all a 2 E:

Therefore V˚ ˚ V˚ is a free E ˝Q k-module of rank 1 , and so

N˚ .a/
N˚ .a/

Therefore V˚ ˚ V˚ is free of rank 1 , and so

N˚ .a/
N˚ .a/ D Nmk=Q.a/, all a 2 k. (8)

Since N˚ .a/ D N˚E .a/ D E N˚ .a/, this can be rewritten as

N˚ .a/
E N˚ .a/ D Nmk=Q.a/, all a 2 k: (9)

More generally, for any Q-algebra R,

N˚ .a/
E N˚ .a/ D Nmk˝QR=R.a/, all a 2 .k ˝Q R/: (10)

1. CM-ALGEBRAS AND CM-TYPES 19

PROOF. We have Nk;˚ .a^0 /

. 7 / D N˚ .Nmk=E a/ D N˚ .aŒkWE ç /

and so (12) follows from (11). (^2)

Note that (12) determines N˚ as a homomorphism from the fractional ideals of E^ to the fractional ideals of E.

EXAMPLE 1.28 Consider a CM-pair .E; ˚/ with E a subfield of Q. The reflex CM-pair .E; ˚/ can be described as follows: choose a subfield L of Q containing E and Galois over Q (for example, L D Q) and let ˚L D f 2 Gal.L=Q/ j jE 2 ˚g; then

Gal.L=E/ D f 2 Gal.L=Q/ j ˚ D ˚gI

the set ˚L is stable under the left action of Gal.L=E/, and when we write

˚L 1 D

G

2 ˚ L^ Gal.L=E

/ (disjoint union),

˚^ D f jE^ j 2 ˚ Lg is the reflex CM-type on E. The map

a 7!

Y

2 ˚^ .a/W E^! L

factors through E^  L, and the resulting map E^! E^ is N˚ — this is a restatement of (1.26). Because it has this description, other authors write N˚^ where we write N˚.^7

Classification of the primitive CM-pairs

An isomorphism of CM-pairs .E; ˚/! .E^0 ; ˚^0 / is an isomorphism ˛W E! E^0 of Q- algebras such that ' ı ˛ 2 ˚ whenever ' 2 ˚^0. Let .E; ˚/ be a CM-pair, and let k be a CM subfield of Q Galois over Q and containing E. For W E! Q and  2 Gal.Q=Q/, define

./^ D

1 if  2 ˚ 0 otherwise.

In other words, (^) ./ D .^1 ı / where  is the characteristic function of ˚.

LEMMA 1.29 For each  , the number (^) ./ depends only on the restriction of  to E and the map  7! (^) ./W Hom.k; Q/! f 0 ; 1 g

is a CM-type on k:

PROOF. If ^0 jE^ D jE, then ^0 D  ı  for some  fixing E,^8 and so

^0 ˚ D ˚ D ˚I

hence (^) .^0 / D (^) ./. As ˚ is a CM-type on E,  lies in exactly one of ˚ or ˚, and so ./^ C^ .^ ı^ /^ D^1 :^2 (^7) For us, the reflex CM-type plays almost no role. (^8) Think of  and  (^0) as automorphisms of k, and take  D  (^1) ı  (^0).

20 CHAPTER I. ANALYTIC THEORY

For any  2 Gal.Q=Q/,

ı./^ D^ .^1 ı^ ^ ı^ /^ D^ .^1 ı^ /^ D^ .^ /./;

and so, as  runs over the embeddings E ,! Q, (^)  runs over a Gal.Q=Q/-orbit of CM- types on k.

PROPOSITION 1.30 The map .E; ˚/ 7! f (^) g defines a bijection from the set of isomor- phism classes of primitive CM-pairs .E; ˚/ whose reflex field is contained in k to the set of Gal.Q=Q/ -orbits of CM-types on k.

PROOF. We construct an inverse. For a CM-type on k, let .E ; ˚ / be the reflex CM- pair of .k; / (see 1.19). By definition, .E ; ˚ / is the primitive subpair of .k; ^1 /. Its isomorphism class depends only on the Gal.Qal=Q/-orbit of ,^9 and the map 7! .E ; ˚ / provides the required inverse.^10

Let k be a composite of CM-subfields of Q (e.g., k could be the composite Qcm^ of all CM-subfields of Q). We define a CM-type on k to be a locally constant map W Hom.k; Q/! f 0 ; 1 g such that ./ C . ı / D 1 for all . For example, the CM-types on Qcm^ are the extensions to Qcm^ of a CM-type on some CM-subfield of Q.

COROLLARY 1.31 The map .E; ˚/ 7! f (^) g defines a bijection from the set of isomor- phism classes of primitive CM-pairs .E; ˚/ to the set of Gal.Q=Q/ -orbits of CM-types on Qcm.

PROOF. Pass to the limit over all CM-subfields of Q in the proposition. (^2)

EXAMPLE 1.32 From (1.12) we can read off the list of isomorphism classes of primitive CM-pairs whose reflex field is contained in QŒe^2 ^ i=^13 ç.

REMARK 1.33 Let .E; ˚/ be a CM-pair with reflex field contained in k, and let (^)  be the CM-type on k defined by an embedding W E ,! Q. For any Q-algebra R and a ˝ r 2 .k ˝Q R/,

N.a ˝ r / def D

Y

Wk!Qal^ .a^ ˝^ r^ /^

 ./

is independent of , and equals N˚ .a ˝ r /.

REMARK 1.34 As the above discussion makes clear, attached to a CM-pair .E; ˚/ there is only an orbit of CM-types on the reflex field E. However, when E is a subfield of Qal, there is a well-defined CM-type ˚^ on E^ corresponding to the given embedding of E, called the reflex of ˚ (see 1.19).

EXERCISE 1.35 Rewrite this section for a k that is not necessarily Galois over Q.

(^9) For  2 Gal.Qal=Q/, E D E and W E! E is an isomorphism .E ; ˚ /. (^10) Consider, for example, a CM-pair .E; ˚/ and a fixed embedding of E into Qal. The composite

.E; ˚/ 7! 7! .E ; ˚ /

sends .E; ˚/ to the reflex .E; ˚/ of its reflex .E; ˚/. It is obvious from the definition of the reflex CM-pair, that .E; ˚/ is a primitive CM-subpair of .E; ˚/, and therefore equals it if .E; ˚/ is primitive.