Construction of Hecke Characters for Three-dimensional CM Abelian Varieties
aa r X i v : . [ m a t h . N T ] S e p CONSTRUCTION OF HECKE CHARACTERS FOR THREE-DIMENSIONAL CMABELIAN VARIETIES
ZHENGYUAN SHANG
Abstract.
It is well-known for an elliptic curve with complex multiplication that the existence of a Q -rational model is equivalent to its field of moduli being equal to Q , or its endomorphism ring being the ringof integers of 9 possible fields ( ∗ ). Murabayashi and Umegaki proved analogous results for abelian surfaces.For three dimensional CM abelian varieties with rational fields of moduli, Chun narrowed down to a list of37 possible CM fields. In this paper, we show that his list is exact. By constructing certain Hecke charactersthat satisfy a theorem of Shimura, we prove that precisely 28 isogeny classes of these abelian varieties have Q -models. Therefore the complete analogy to ( ∗ ) fails here. Introduction
For an elliptic curve E defined over C with its endomorphism ring isomorphic to the ring of integers of animaginary quadratic field K , it is a classical result that the followings are equivalent: i) the j -invariant of E is contained in Q (or equivalently, the field of moduli of E is Q ), ii) the class number h K of K is one, iii) E has a model defined over Q . The classification of class number one imaginary quadratic fields then enablesus to determine all the (nine) Q -rational points in the moduli space A represented by CM elliptic curves.Much effort has gone into generalizing this result to n -dimensional abelian varieties of CM-type for n ≥ i) ⇔ ii) for polarized abelian surfaces: precise criterions wereformulated for a surface to be simple and for its field of moduli to be Q . Then in [8] and [13], all Q -rationalCM points in A ( d ) for d ≥ iii) ⇒ i) is clear by the definition ofthe field of moduli. By constructing certain Hecke characters over the quartic field, Murabayashi [9] showedthat all such CM points have Q -rational models, finishing the analogue to ii) ⇒ iii) .Chun [2] gave necessary and sufficient conditions (Theorems 2.1 & 2.3) for three dimensional CM abelianvarieties to have rational fields of moduli, and narrowed down to 37 potential fields in Tables 1 through 4.In this paper, we first show that these 37 fields all satisfy Chun’s sufficient condition, so the abelian varietiescorresponding to them admit polarizations such that their fields of moduli coincide with Q . Then we provethat exactly 28 isogeny classes of them have Q -models, indicating the failure of an analogue to i) ⇔ iii) . Notation:
Unless otherwise specified, K is a CM field of degree six over Q such that G := Gal( K/ Q ) iscyclic with generator a σ , and ρ = σ is complex conjugation. Let F, F be the cubic and quadratic subfieldsof K . Let U = Q u O × K u and V = Q v O × F v , where u (resp. v ) runs through all finite places of K (resp. F ).For a number field k , we use O k , E k , W k , C k , I k , H k , P k , h k , f k , d k , k × A , k ×∞ , f to denote its ring of integers,group of units (in O k ), group of roots of unity, ideal class group, ideal group, Hilbert class field, definingpolynomial (over Q ), class number, conductor, discriminant, idele group, Archimedian part (of k × A ), and Date : September 29, 2020. onductor over F . For a place v of k , k v is the completion. For a fractional ideal I ∈ I k , cl ( I ), N ( I ) are itsclass of in C k and absolute norm resp. For x ∈ k × A , il ( x ) is the fractional ideal (in k ) associated to x . KF F Q Consider a structure P = ( A, C, θ ), where A is a three dimensional abelian variety over C with a polarization C and an injection θ : K → End ( A ) := End( A ) ⊗ Q such that θ − (End( A )) = O K . We always assume θ ( K )is stable under the involution of End ( A ) determined by C . Let Φ = { σ , σ , σ } be a CM type of K inducedby the representation of K, through θ , on the Lie algebra of A . Consider the isomorphism ˜Φ : K ⊗ Q R → C such that x ⊗ r ( rx σ , rx σ , rx σ ). For x ∈ K , let S Φ ( x ) be the diagonal matrix with x σ , x σ , x σ alongthe diagonal. By the theory of complex multiplication, there exists a fractional ideal J of K and an analyticisomorphism λ : C / ˜Φ( J ) → A such that ∀ x ∈ O K , the following diagram commutes C / ˜Φ( J ) A C / ˜Φ( J ) A λS Φ ( x ) θ ( x ) λ Take a basic polar divisor in C and consider its Riemann form E ( x, y ) on C with respect to λ . Then thereis an element η of K such that for x, y ∈ K , E ( ˜Φ( x ) , ˜Φ( y )) = Tr K/ Q ( ηxy ρ ), η ρ = − η , and Im( η σ j ) > ≤ j ≤
3. We say P is of type ( K, Φ; η, J ). Note that given ( K, Φ; η, J ) satisfying these conditions, we canalso reconstruct the corresponding ( A, C , θ ). Acknowledgment:
I want to thank my mentor Prof. Matthias Flach for introducing me to this fascinatingproblem and offering numerous helpful suggestions. This project would not be possible without his guidance.I also wish to thank Caltech Class of 1936 for their generous support through the SURF program.2.
Preliminaries
In this section we state explicitly the results mentioned in the introduction, starting with Chun’s criterions.
Theorem 2.1. [2, Theorem 1] Let P = ( A, C, θ ) be of type ( K, Φ; η, J ). If A is simple and the field ofmoduli of ( A, C ) is Q , then K is one of the fields in Tables 1 through 4, and Φ = { , σ, σ } up to G -action. Remark 2.2.
It is worth noting that fields in Tables 1 through 4 are indeed cyclic over Q . We denote by A i the field in Table 1 with generator α i , and similarly the others by B i , Γ i , and Ω i . By degree consideration, | W K | = 2 , , , , or 18. Moreover, A = Q ( ζ ) (resp. A = Q ( ζ )) is the only field with | W k | = 14 (resp.18). Here ζ l := e πil for l ∈ Z + . By the uniqueness of the quadratic subfield of K , for fields containing i := √− ω := e πi ), | W K | = 4 (resp. 6). For the rest, we then have | W K | = 2 and W K = { , − } . Theorem 2.3. [2, Theorem 2] Let P = ( A, C, θ ) be of type ( K, Φ; η, J ). If one of the followings is satisfied: i) K belongs to Table 1, ii) K belongs to Table 2 and H F has four index 3 subfields, iii) K belongs to Table3 and J is principal, iv) K = Ω , cl ( J ) ∈ C F (see the remark below), and H F has four index 3 subfields,and if Φ = { , σ, σ } up to G -action, then A is simple and the field of moduli of ( A, C ) coincides with Q . emark 2.4. By [2, Lemma 2], the canonical map C F → C K is injective. Here C F is the image of C F .Moreover, we need the following theorem of Shimura, which is also crucial in Murabayashi’s proof in [9]. Let K ⊂ C be a degree 2 n CM field with F its maximal real subfield. Let Φ be a CM type of K and ( K ′ , Φ ′ ) be thereflex of ( K, Φ). For a number field k containing K ′ , consider f : k × → K × with f ( x ) = Q τ ∈ Φ ′ N k/K ′ ( x ) τ ,which can be extended to a continuous homomorphism k × A → K × A (also denoted f ). Suppose K, K ′ havesubfields D, D ′ respectively such that: i) D ⊂ F , D ′ ⊂ F ′ , ii) K, K ′ are cyclic over D, D ′ resp., iii) K = DS , where S/ Q is generated Tr Φ ′ ( a ) = P τ ∈ Φ ′ a τ for all a ∈ K ′ , and iv) there is an isomorphismGal( K ′ /D ′ ) → Gal(
K/D ) where σ [ σ ] such that (Tr Φ( a )) σ = Tr Φ( a [ σ ] ) for all a ∈ K . Theorem 2.5. [12, Theorem 5] Let A be an n -dimensional abelian variety over C with a polorization C and θ : O K ֒ −→ End( A ) such that the type of ( A, θ ) is ( K, Φ). Let k be an algebraic number field containingboth D ′ and the field of moduli of ( A, C , θ | D ). Let k = k K ′ and h = k F ′ . Suppose that K ′ and k arelinearly disjoint over D ′ and let χ : h × A → {± } be the quadratic character associated to the extension k/h .Suppose there exists a Hecke character ψ : k × A → C × satisfying a) x ∈ k ×∞ ⇒ ψ ( x ) = ( f ( x ) ∞ ) − , with ∞ K corresponding to K ֒ −→ C b) y ∈ h × A ⇒ ψ ( y ) = χ ( y ) | y | − h A , with | · | h A the volume of ideles c) x ∈ k × A , x ∞ = 1 ⇒ ψ ( x ) ∈ K × , ψ ( x ) ψ ( x ) ρ = | x | − k A , ( ψ ( x )) = il ( f ( x )) d) x ∈ k × A , x ∞ = 1 , σ ∈ Gal( k/k ) ⇒ ψ ( x ) [ σ ] = ψ ( x σ ), with Gal( k/k ) and Gal( K ′ /D ′ ) identified.Then there exists a structure ( A ′ , C ′ , θ ′ ) rational over k and isomorphic to ( A, C , θ ) such that correspondingHecke character is ψ , and ( A ′ , C ′ , θ ′ | D ) is rational over k . Remark 2.6.
Let ψ ∗ be the Hecke ideal character associated with ψ and c be the conductor of ψ . Then by[12, Remark 2], the conditions a) , b , c , d) are equivalent to a ′ ) ψ ∗ ( s O K ) = det Φ ′ ( N k/K ′ ( s )), if s ∈ k × and s ≡ c ). b ′ ) ψ ∗ ( t O K ) = (cid:16) k/h t (cid:17) N ( t ), for every ideal t of h prime to c . c ′ ) For every ideal I of k prime to c , ψ ∗ ( I ) ∈ K × , ψ ∗ ( I ) ψ ∗ ( I ) ρ = N ( I ), and ψ ∗ ( I ) O K = N k/K ′ ( I ) Φ ′ . d ′ ) For every ideal I of k prime to c and σ ∈ Gal( k/k ), ψ ∗ ( I ) [ σ ] = ψ ∗ ( I σ ).respectively. Here the equivalence c) ⇔ c ′ ) requires that the lattice J in K determined by A is a fractionalideal, which holds in our case as End( A ) = θ ( O K ).Finally, we have the following converse to Theorem 2.5 by Yoshida. Theorem 2.7. [14, Theorem 2, Corollary] Fix a structure ( A, C , θ ) as above. If in the isogeny class of ( A, θ ),there exists a structure ( A ′ , θ ′ ) such that ( A ′ , θ ′ | D ) is rational over k and End( A ′ ) = θ ′ ( O K ), then thereexists a Hecke character ψ of k × A that satisfies a) through d) .3. Main Results
Theorem 3.1.
Let (
A, θ ) be of type ( K, Φ). Then A is simple and the field of moduli of ( A, C ) is Q for somepolarization C if and only if K is one of the fields in Tables 1 through 4, and Φ = { , σ, σ } up to G -action. Proof.
The ⇒ direction is a restatement of Theorem 2.1. For F in Tables 2 and 4, we have h F = 3, so H F is Galois over Q of degree 9. In particular, Gal( H F / Q ) is either Z / Z or Z / Z × Z / Z . Since H F is belian, it coincides with the genus field of F , which is the compositum of cubic fields by [3, Theorem].Hence Gal( H F / Q ) ∼ = Z / Z × Z / Z and the converse follows from Theorem 2.3. (cid:3) Now let K be one of the fields in Tables 1 through 4 and Φ = { , σ, σ } . Choose θ such that ( A, θ ) is of type( K, Φ). Then there exists a polarization C such that the field of moduli of ( A, C , θ ) is Q . Since ( K, Φ) is simple,( K ′ , Φ ′ ) = ( K, { , σ , σ } ). To apply Theorem 2.5, take D = D ′ = Q . Note that Q ⊂ F = F ′ and K = K ′ iscyclic over Q . Meanwhile, the field over Q generated by P τ ∈ Φ ′ a τ for all a ∈ K ′ is just K , and K = K Q . Forthe identity isomorphism σ → [ σ ] from Gal( K ′ /D ′ ) → Gal(
K/D ), we have Tr(Φ( a )) σ = Tr Φ( a [ σ ] ), ∀ a ∈ K .Put k = Q , which contains both D ′ and the field of moduli of ( A, C , θ D ). Then k = K and h = F . Notethat K ′ and k are linearly disjoint over D ′ . Furthermore, f : K × A → K × A is defined by x xx σ x σ . Proposition P.
There exists a Hecke character ψ of K × A → C × satisfying the conditions a) , b) , c) , d) ifand only if | W K | 6 = 6 (or equivalently, K / ∈ { A , A , A , A , B , B , B , Γ , Γ } ). Proof.
This follows from Propositions 4.1, 4.10, 4.11, 4.12, 4.13, and 4.17. (cid:3)
Remark 3.2.
In our case, a) , b) , c) , d) translate to a ′′ ) x ∈ K ×∞ ⇒ ψ ( x ) = ( xx σ x σ ) − ∞ b ′′ ) y ∈ F × A ⇒ ψ ( y ) = χ ( y ) | y | − F A c ′′ ) x ∈ K × A , x ∞ = 1 ⇒ ψ ( x ) ∈ K × , ψ ( x ) ψ ( x ) ρ = | x | − K A , ( ψ ( x )) = il ( xx σ x σ ) d ′′ ) x ∈ K × A , x ∞ = 1 , σ ∈ Gal( K/ Q ) ⇒ ψ ( x ) σ = ψ ( x σ ) Theorem 3.3.
Let (
A, θ ) be of type ( K, Φ). If K is one of the fields in Tables 1 through 4, and Φ = { , σ, σ } up to G -action, then ( A, θ ) has a Q -model if and only if | W K | 6 = 6. Proof.
This is immediate from Proposition P , Theorem 2.5, and Theorem 2.7. (cid:3) Remark 3.4. If | W K | 6 = 6, then the Q -model of ( A, θ ) has conductors as listed in Tables 1 through 4 byProposition 4.14. Otherwise, (
A, θ ) has a F -model by Proposition 4.18.4. Hecke Characters
We prove Proposition P in this section, starting with the two cyclotomic fields K with | W K | = 14 or 18. Proposition 4.1.
Proposition P holds for K = Q ( ζ ) , Q ( ζ ) (i.e. A , A ). Proof.
The case K = A = Q ( ζ ) follows immediately from Shimura’s construction in [12, Theorem 7]. If K = A = Q ( ζ ), we adopt a similar approach. Consider the prime ideal l = (1 − ζ ) O K above (3). Then3 O K = l . By the theory of complex multiplication (cf. [2, Lemma 5]), for an ideal I in K , I Φ ′ = II σ I σ = r O K for some r ∈ K such that rr ρ = N ( I ). If I is prime to 3, then N ( I ) ≡ r ≡ ± l O K l ).Note that 1 , ζ , ζ , ..., ζ ∈ l O K l project to distinct classes in (1 + l O K l ) / (1 + l O K l ). Moreover, 1 , ζ , ζ map to distinct classes in (1 + l O K l ) / (1 + l O K l ) and 1 , ζ , ζ to distinct classes in (1 + l O K l ) / (1 + l O K l ).If r ≡ l O K l ), suppose r = 1 + t (1 − ζ ) for some t ∈ O K l . Then since t ≡ t ρ (mod l O K l ), we have rr ρ ≡ (1 + t (1 − ζ ) )(1 + t ρ (1 − ζ ) ) ≡ (1 + t (1 − ζ ) )(1 + t (1 − ζ ) ) ≡ t (2 − ζ − ζ ) ≡ − t ( ζ − ≡ l O K l ). Thus t ≡ l O K l ) and r ≡ l O K l ). Consequently, there is a unique sign of ± andan integer 0 ≤ m ≤ ± ζ m r ≡ l O K l ). We then define ψ ∗ ( I ) = ± ζ m r . Since rr ρ = N ( I )and {± ζ m } are all the roots of unity, ψ ∗ is well-defined. ote that ψ ∗ is a homomorphism from the group I l of ideals of K prime to l to K ∗ . For ( s ) ∈ I l with s ≡ l ), since σ fixes l , we have ss σ s σ ≡ l ). Then ψ ∗ (( s )) = ss σ s σ , so ψ ∗ is a Heckecharacter. By construction, it satisfies a ′ ) , c ′ ) , and d ′ ) in Remark 2.6. For b ′ ) , it suffices to show that( ∗ ) ψ ∗ ( p O K ) = (cid:18) K/F p (cid:19) N ( p )for every prime ideal p of F prime to 3. Such p is unramified in K . Let ρ be the generator of Gal( K/F ).If p O K = qq ρ for some prime ideal q of K , then ψ ∗ ( p O K ) = ψ ∗ ( q ) ψ ∗ ( q ) ρ = N ( q ) = N ( p ), so ( ∗ ) holds.Otherwise, p remains prime in K . Then ( K/F p ) = ρ and N ( p ) ≡ − p O K ) Φ ′ = N ( p ) O K , so ψ ∗ ( p O K ) = − N ( p ) and ( ∗ ) also holds. This completes the proof. (cid:3) For the remaining fields from Tables 1 through 4, | W K | ∈ { , , } . We first show the existence of satisfactoryHecke characters ψ for the cases when | W K | = 2 or 4, where the following proposition is an essential step. Proposition 4.2.
Suppose | W K | = 2 or 4. If there exists a character e χ f : Q u | f O × K u → W K such that e χ f | Q v | f O × Fv = χ | Q v | f O × Fv e χ f ( x σ ) = e χ f ( x ) σ e χ f ( x ) = x, ∀ x ∈ W K where u and v are finite places of K and F . Then there exists a continuous homomorphism ψ : K × U K ×∞ → C × satisfying a ′′ ) , b ′′ ) , c ′′ ) , d ′′ ) with K × A , F × A replaced by K × U K ×∞ , F × V F ×∞ respectively. Proof.
Since F is the unique cubic subfield of K , f σ = f , so the Galois equivariance condition on e χ f iswell-defined. We adopt the construction of Yoshida in [14, Section 4]. For x ∈ F × A with x v ∈ O × F v for eachfinite place v of F not dividing f , we have χ ( x ) = Y v | f χ v ( x v ) sgn( x ∞ ) x sgn( x ∞ ) sgn( x ∞ )where sgn : R × → {± } is the sign map. Now consider ψ : K × U K ×∞ → C × defined by ψ ( xyz ) = e χ f ( y f )( f ( z )) − ∞ where x ∈ K × , y ∈ U, z ∈ K ×∞ , and y f is the Q u | f O × K u -component of y . Note that K × ∩ U K ×∞ = E K = W K E F by [2, Lemma 2]. For x ∈ E F , x ′ ∈ W K , we have e χ f ( xx ′ )( f ( xx ′ )) − ∞ = x ′ ( f ( x ′ )) − ∞ Y v | f χ v ( x ) N F/ Q ( x ) = χ ( x ) = 1Hence ψ is well-defined. Note that a ′′ ) is satisfied. For x ∈ F × , y ∈ V, z = ( z , z , z ) ∈ F ×∞ , we have ψ ( xyz ) = e χ f ( y f )( f ( z )) − ∞ = Y v | f χ v ( y v )( z z z ) − Meanwhile, χ ( xyz ) | xyz | − F A = Y v | f χ v ( y v ) sgn( z ) sgn( z ) sgn( z ) N ( il ( y )) | z z z | − = Y v | f χ v ( y v )( z z z ) − so b ′′ ) holds. For x ∈ K × , y ∈ U, z = ( z , z , z ) ∈ K ×∞ , if ( xyz ) ∞ = 1, then xz = x σ z = x σ z = 1, so ψ ( xyz ) = e χ f ( y f )( f ( z )) − ∞ = e χ f ( y f )( z z σ z σ ) − = e χ f ( y f ) xx σ x σ hus ψ ( xyz ) ψ ( xyz ) ρ = N K/ Q ( x ) = N ( il ( xyz )) = | xyz | − K A . Moreover, we have ( ψ ( xyz )) = ( xx σ x σ ) = il ( f ( xyz )), so c ′′ ) holds. Note that ψ (( xyz ) σ ) = e χ f (( y σ ) f )( f ( z σ )) − ∞ = e χ f (( y f ) σ )( z z σ z σ ) − = e χ f ( y f ) σ xx σ x σ = ( e χ f ( y f ) xx σ x σ ) σ = ( e χ f ( y f )( z z σ z σ ) − ) σ = ψ ( xyz ) σ Hence d ′′ ) also holds, which completes the proof. (cid:3) Lemma 4.3.
For fields K with | W K | = 2 and F = Q ( √− e χ f satisfying the conditions above. Proof.
Since the rational prime (2) is unramified in both F and F , it is unramified in K and (2 , f ) = 1. Fora prime p ⊂ F dividing f , let P ⊂ K be the prime above. Note that χ p | O × F p is the quadratic symbol mod p in O × F p . Let f χ P be the quadratic symbol mod P in O × K P . Then e χ f = Q P | f f χ P is the desired character. (cid:3) To construct e χ f for fields with F = Q ( √−
2) or | W K | = 4 (i.e. F = Q ( i )), we make the following observation. Lemma 4.4.
For K = A , A , A , A , A , the prime ideal (2) is inert in F , while for Γ , Γ , Γ , Ω , it splitscompletely. Let p ⊂ F be a prime above (2). Then p O K = P for some prime P ⊂ K , and f is a power of p or p p p . Meanwhile, the quadratic character χ p : F × p → {± } induced by K P /F p is trivial on 1 + p r ,where r = 2 for A , A , A , Γ , Γ , Ω and r = 3 for A , A , Γ (corresponding to F = Q ( i ) or Q ( √− Proof.
The first two claims follow from [5, Theorem 1]. Since F and F are cubic and quadratic respectively, d F = f F and d F = f F are coprime. As K/F is quadratic, f = d K/F , where d K/F is the relative discriminant.Therefore d F N F/ Q ( f ) = d K = d F d F and N F/ Q ( f ) = d F , which proves the last two claims. (cid:3) In Lemmas 4.5 through 4.8, we consider p , P , χ p as defined above. Lemma 4.5. [ F = Q ( √−
2) and (2) splits in F ] For K = Γ , there exists a homomorphism f χ P : O × K P →{± } such that f χ P | O × F p = χ p | O × F p and f χ P ( x ρ ) = f χ P ( x ) ρ , ∀ x ∈ O K × P . Proof.
Note that χ p (when restricted to O × F p ) factors through O × F p / (1 + p ) ∼ = ( O F p / p ) × , so it induces ahomomorphism χ p : ( O F p / p ) × → {± } . Since O F p ∩ P = p , there is a canonical inclusion ( O F p / p ) × → ( O K P / P ) × . We construct a homomorphism χ P : ( O K P / P ) × → {± } lifting χ p that is also compatiblewith the action of ρ . Let η = √−
2, then η is a uniformizer of O K P and η ρ = − η . Since (2) splits completely,we can take F p = Q and K P = Q ( √− ξ = 1 + η, ξ = 1 + η , ξ = 1 + η ∈ ( O K P / P ) × . Then( O K P / P ) × = h ξ i × h ξ i × h ξ i ∼ = Z / Z × Z / Z × Z / Z ( O F p / p ) × = h ξ i × h ξ i ∼ = Z / Z × Z / Z Let χ P ( ξ ) = 1, χ P ( ξ j ) = χ p ( ξ j ) for j = 2 ,
3. As N K P /F p (5 + 5 η ) ≡ (1 + η )(1 + η ) (mod p ), χ p ( ξ ξ ) = 1.Since ξ ρ = ξ ξ ξ , χ P ( ξ ρ ) = χ P ( ξ ) ρ . Then the composition of χ P with the projection satisfies the claim. (cid:3) Lemma 4.6. [ F = Q ( √−
2) and (2) is inert in F ] For K = A , A , there exists a homomorphism f χ P : O × K P → {± } such that f χ P | O × F p = χ p | O × F p and f χ P ( x σ ) = f χ P ( x ) σ , ∀ x ∈ O K × P . Proof.
Note that F p is the unramied cubic extension of Q and K P is the composition of F p and Q ( √− × p Gal( K P /F p ) Q × Gal( K P / Q )Gal( Q ( √− / Q ) χ p N F p / Q Let f χ P be the composition of N K P / Q ( √− with the homomorphism O × Q ( √− → {± } defined in Lemma4.5. Then f χ P satisfies our requirement. (cid:3) Lemma 4.7. [ F = Q ( i ) and (2) splits in F ] For K = Γ , Γ , Ω , there exists a homomorphism f χ P : O × K P →{± , ± i } such that f χ P | O × F p = χ p | O × F p , f χ P ( i ) = i , and f χ P ( x ρ ) = f χ P ( x ) ρ , ∀ x ∈ O K × P . Proof.
By the reasoning in Lemma 4.5, it suffices to contruct a suitable lift χ P of χ p for K = Γ . We cantake F p , K P as Q and Q ( i ) resp. Let η = 1 − i . Consider ξ = 1 + η, ξ = 1 + η ∈ ( O K P / P ) × . Then( O K P / P ) × = h ξ i × h ξ i ∼ = Z / Z × Z / Z ( O F p / p ) × = h ξ i ∼ = Z / Z Let χ P ( ξ ) = i and χ P ( ξ ) = χ p ( ξ ) = −
1, so χ P lifts χ p . Since ξ ρ = ξ ξ , χ P ( ξ ρ ) = χ P ( ξ ) ρ . Moreover, f χ P ( i ) = χ P ( ξ ) = i . This completes the proof. (cid:3) Lemma 4.8. [ F = Q ( i ) and (2) is inert in F ] For K = A , A , A , there exists a homomorphism f χ P : O × K P → {± , ± i } such that f χ P | O × F p = χ p | O × F p , f χ P ( i ) = i , and f χ P ( x σ ) = f χ P ( x ) σ , ∀ x ∈ O K × P . Proof.
Let f χ P be the composition of N K P / Q ( i ) ◦ ρ with the homomorphism O × Q ( i ) → {± , ± i } above. (cid:3) Remark 4.9.
When K = Γ , Γ , Ω , suppose 2 = p p p = P P P for prime ideals p j ⊂ F and P j ⊂ K such that σ P = P and σ P = P . Then σ induces isomorphisms K P → K P and K P → K P . Let ϕ : K P → Q ( i ) be an isomorphism such that ϕ ( i ) = i . Q ( i ) K P K P K P σ σϕ Let χ be the character of O × Q ( i ) defined in Lemma 4.7. Consider e χ f : Q j =1 O × K P j → {± , ± i } with e χ f ( y , y , y ) = χ ( ϕ ( y σ )) χ ( ϕ ( y σ )) χ ( ϕ ( y ρ ))Then e χ f restricts to Q j =1 χ p j on Q j =1 O × F p j , e χ f ( i, i, i ) = χ ( − i ) = i and e χ f (( y , y , y ) σ ) = e χ f ( y σ , y σ , y σ ) = e χ f ( y , y , y ) σ . Note that this is simply the composition χ ◦ N K/ Q ( i ) ◦ ρ on K ⊗ Q . Similarly, when K = Γ ,we can construct a e χ f from the character in Lemma 4.5 that satisfies the conditions of Proposition 4.2. Proposition 4.10.
Proposition P holds for fields K in Table 1 ( h K = h F = 1) with | W K | = 2 or 4. Proof.
By Lemmas 4.3, 4.6, 4.8 and Remark 4.9, there exists e χ f satisfying the conditions in Proposition 4.2.Since h K = h F = 1, K × A = K × U K ×∞ and F × A = F × V F ×∞ . Then ψ := ψ satisfies a ′′ ) through d ′′ ) . (cid:3) Proposition 4.11.
Proposition P holds for fields K in Table 2 ( h K = h F = 3) with | W K | = 2 or 4. roof. By Chebotarev’s density theorem, there exists a non-principal prime ideal q ⊂ F that is inert in K ,which is equivalent to q being inert in both H F and K . (By [1], we can take q as the prime above (3)). Let Q ⊂ K be the prime above q and λ ∈ F such that v q ( λ ) = 1. Consider ι q ∈ F × A (resp. ι Q ∈ K × A ) whichequals to λ at the q -component (resp. Q ) and 1 elsewhere. Then since h K = h F = 3, K × A = h ι Q , K × U K ×∞ i and F × A = h ι q , F × V F ×∞ i . Consider ψ from Proposition 4.2. We now extend it to ψ : K × A → C × by setting ψ ( ι Q ) := χ ( ι q ) | ι q | − F A = χ ( ι q ) q Since ι q ∈ F × V F ×∞ on which b ′′ ) holds, we have ψ ( ι Q ) = ψ ( ι q ) = χ ( ι q ) | ι q | − F A = ψ ( ι Q ) , so this extensionis well-defined. By construction, a ′′ ) and b ′′ ) are satisfied. Note that ψ ( ι Q ) ψ ( ι Q ) ρ = q = | ι Q | − K A . Moreover,we have ( ψ ( ι Q )) = q = il ( f ( ι Q )), so c ′′ ) holds. Finally, note that q σ , q σ are not principal, so q σ = q δ and q σ = q δ in C F for some δ , δ ∈ { , } . Then as qq σ q σ is principal, δ = δ = 1. Since ι σ Q ι − Q ∈ F × V F ×∞ and ψ ( ι σ Q ι − Q ) = χ ( ι σ q ι − q ) | ι σ q ι − q | − F A = 1, ψ ( ι σ Q ) = ψ ( ι Q ) = ψ ( ι Q ) σ , so d ′′ ) remains valid. (cid:3) Proposition 4.12.
Proposition P holds for fields K in Table 3 ( h K = 4 , h F = 1) with | W K | = 2 or 4. Proof.
Let q ∈ Z + be the prime that ramifies in F . Then by [1], we have q = q q q = Q Q Q for primes q j ⊂ F and Q j ⊂ K . Let ( η, ǫ ) = (1 + i,
1) if K = Γ , Γ and ( √− q,
0) otherwise. For 1 ≤ j ≤
3, note that K Q j ∼ = Q q ( η ) and F q j ∼ = Q q . Consider ι Q ∈ K × A (resp. ι Q ) that equals to η (resp. η ρ ) at the Q -component(resp. Q ) and 1 elsewhere. By [2, Lemma 4], K × A = h ι Q , ι Q , K × U K ×∞ i and ι Q j ∈ K × U K ×∞ . We extend ψ from Proposition 4.2 to ψ : K × A → C × by setting ψ ( ι Q ) ρ = ψ ( ι Q ) := ( − ǫ η If K = Γ , Γ , ψ ( ι Q j ) = χ ( ι Q j ) | ι Q j | − F A = χ q j ( − q , since N K Q j /F q j ( η ) = q . If q = 2, since q ≡ χ q j ( −
1) = −
1. This remains true for q = 2 by the proof of Lemma 4.5. Thus ψ ( ι Q j ) = − q = ψ ( ι Q j ) . If K = Γ , Γ , then ψ ( ι Q ) = 2 e χ f ( i, ,
1) = − i = ψ ( ι Q ) , as N K Q j /F q j ( η ) = 2 (see Remark 4.9). Similarly, ψ ( ι Q ) = ψ ( ι Q ) . Hence this extension is well-defined. Note that a ′′ ) and b ′′ ) hold as before. Moreover,for j = 1 , ψ ( ι Q j ) ψ ( ι Q j ) ρ = q = | ι Q j | − K A and ( ψ ( ι Q j )) = ( η ) = Q Q Q = il ( f ( ι Q j )), so c ′′ ) is satisfied.Finally, ψ ( ι σ Q ) = ψ ( ι Q ) = ψ ( ι Q ) σ . Since ι Q ι Q ι σ Q ∈ K × U K ×∞ and in particular ι Q ι Q ι σ Q η − ∈ U K ×∞ , ψ ( ι Q ι Q ι σ Q ) = e χ f (1 , η ρ η , f ( η − ) ∞ ) − = e χ f (1 , η ρ η , ηη σ η σ = e χ f (1 , η ρ η , η η ρ If K = Γ , Γ , then η ρ η − = −
1, so ψ ( ι σ Q ) = − η = ψ ( ι Q ) σ . If K = Γ , Γ , η ρ η − = − i . As e χ f (1 , − i,
1) = i ,we have ψ ( ι σ Q ) = iη = ( − η ) σ = ψ ( ι Q ) σ . Hence d ′′ ) is valid. (cid:3) Proposition 4.13.
Proposition P holds for K = Ω in Table 4 ( h K = 12 , h F = 3). Proof.
Suppose 2 = q q q = Q Q Q for primes q j ⊂ F , Q j ⊂ K . Let r ⊂ F be a non-principal prime thatis inert in K and R ⊂ K the prime above it. Let η = 1+ i and λ ∈ F such that v r ( λ ) = 1. Consider ι Q ∈ K × A (resp. ι Q , ι R ) that equals to η (resp. η ρ , λ ) at the Q -component (resp. Q , R ) and 1 elsewhere. Note that K × A = h ι Q , ι Q , ι R , K × U K ×∞ i . We extend ψ from Proposition 4.2 by setting ψ ( ι Q ) ρ = ψ ( ι Q ) := − η and ψ ( ι R ) := χ ( ι r ) | ι r | − F A , where ι r is ι R considered as an idele in F × A . Then by the arguments in Propositions4.11 and 4.12, we have ψ is well-defined and satisfies a ′′ ) through d ′′ ) . (cid:3) Proposition 4.14. If | W K | 6 = 6, the Q -models of ( A, θ ) have conductors cond( A / Q ) as listed in Tables 1through 4. Moreover, the conductors of ψ are denoted cond( ψ ). roof. Let cond( A /K ) be the conductor of A over K . By [10, Theorem 12] and [6, Section 3, Corollary (b)],we have cond( A /K ) = cond( ψ ) N K/ Q (cond( A /K )) d K = cond( A / Q ) Then we can compute cond( A / Q ) from d K and cond( ψ ), the latter being immediate from our construction.In the tables, P , Q are primes in K above the first and second primes in the factorization of | d K | . (cid:3) Now we show that the desired Hecke characters don’t exists for fields with | W K | = 6. Remark 4.15.
For K = A , A , A , A , the prime ideal (3) is inert in F . For B , B , B , it ramifies, whilefor Γ , Γ , it splits completely. Let p ⊂ F be a prime above (3). Then p O K = P for some prime P ⊂ K ,and f = p or p p p . Let q ( = p ) be the rational prime dividing the conductor of F . Then q O F = q for aprime q ⊂ F and q splits in K . (These follow from direct computation with Magma [1]). Lemma 4.16.
For K = A , A , A , A , B , B , B and P as in Remark 4.15, if f χ P : O × K P → W K is ahomomorphism with f χ P ( x σ ) = f χ P ( x ) σ , then f χ P ( ω ) = 1. Proof.
Note that K P is a cubic extension of Q ( ω ). By Kummer theory, there exists a size three subgroup h α i of Q ( ω ) × / Q ( ω ) × generated by α such that K P = Q ( ω, √ α ). Since K P is cyclic over Q , h α i is stableunder the action of Gal( Q ( ω ) / Q ) = h ρ i , and ρ acts nontrivially on α , so α ρ = α in Q ( ω ) × / Q ( ω ) × . Thisimplies α ∈ O × Q ( ω ) , so √ α ∈ O × K P . Since f χ P ( ω √ α ) = f χ P ( √ α σ ) = f χ P ( √ α ) σ = f χ P ( √ α ), f χ P ( ω ) = 1 (cid:3) Proposition 4.17.
Proposition P holds for fields K in Tables 1 through 4 with | W K | = 6. Proof.
Suppose there is a Hecke character ψ of K × A satisfying a ′′ ) through d ′′ ) . Let p ⊂ F be a prime and P ⊂ K (resp. p ∈ Z + ) the prime above (resp. below) it. Let ψ P be the character on K × P induced by ψ .Now we show that Q u ψ u ( ω ) = 1, where u runs through all the finite places of K . There are three cases: i) p O K = P P . Note that K P = F p = K P . Since p ∤ f , we have ψ P ( ω ) ψ P ( ω ) = χ p ( ω ) = 1 by b ′′ ) . ii) p O K = P . If p O K = P P P , suppose σ P = σ P = P and ψ P ( ω ) = λ ∈ { , ω, ω } . Then by d ′′ ) , ψ P ( ω ) = ψ P ( ω ) σ = λ ρ and ψ P ( ω ) = λ σ = λ , so Q j =1 ψ P j ( ω ) = λ λ ρ = 1. Otherwise, since q splits in K by Remark 4.15, we have p O K = P . Note that the residual field of O K P has size p . Since 9 | p −
1, byHensel’s Lemma, there exists ζ ∈ O × K P , so ψ P ( ω ) = 1. iii) p O K = P . Then p = 3. If K = Γ , Γ , ∀ λ ∈ O × K P , we have ψ P ( λ ) ψ P ( λ ) ρ = 1 by c ′′ ) , so ψ P ( λ ) ∈ W K .Then ψ P ( ω ) = 1 by Lemma 4.16. If K = Γ , Γ , suppose 3 O K = P P P . Then the argument in the firsthalf of ii) shows that Q j =1 ψ P j ( ω ) = 1.Thus Q u ψ u ( ω ) = 1. Let ι ∈ K ×∞ be the idele equal to the ∞ -component of ω ∈ K × A . Then for ω ∈ K × , ψ ( ω ) = ψ ( ι ) Q u ψ u ( ω ) = ( f ( ι ) ∞ ) − = ( ωω σ ω σ ) − = ω = 1. Contradiction! Hence no such ψ exists. (cid:3) Proposition 4.18.
For fields K in Tables 1 through 4 with | W K | = 6, ( A, θ ) has a F -model. Proof.
Take D = D ′ = F = k in Theorem 2.5. It suffices to construct a Hecke character ψ : K × A → C × thatsatisfies a ′′ ) , b ′′ ) , c ′′ ) and Galois equivariance by ρ . The corresponding local character e χ f can be constructedexplicitly in a fashion similar to the proof of Lemmas 4.5 and 4.7. (cid:3) . Future Directions
An immediate direction for future research is to verify if the analogue to ( ∗ ) from the abstract holds for fourdimensional CM abelian varieties with rational fields of moduli. This will probably involve an even longer listof degree eight CM fields than Tables 1 through 4. As we can see from the proof above, a large | W K | is themajor obstruction to the existence of our desired Hecke charcters, which is made precise by an obstructionclass in H in [14, Proposition 1]. Thus, it is unlikely that these abelian varieties all have Q -models.Alternatively, we can fix an integer n ≥ n CM fields from which n -dimensional CM abelian varieties with fields of moduli = Q could arise. For n = 2 ,
3, the approaches byMurabayashi [7] and Chun [2] start with the identification of the relative class numbers of
K/F . Hence anaffirmative answer may hinge on the classification of extensions of a fixed relative class number as in [11].Finally, recall that Jacobi sum characters (discovered by Weil and formalized by Anderson) are a class ofHecke charcters systematically constructed for all abelian fields, with known connection to Fermat hypersur-faces. Since our K is abelian, it is of interest to figure out which Hecke characters constructed for Proposition P arise from Jacobi sums, which could improve our understanding of the corresponding abelian varieties. AppendixTable 1.
Degree six CM fields K with h K = 1 and h F = 1 K f K f F f F P F | d K | cond( ψ ) cond( A / Q ) Q ( α , √−
7) 7 7 7 x − x + 7 7 P Q ( α , √−
3) 9 9 3 x − x − P Q ( α , √−
19) 19 19 19 x − x + 133 19 P Q ( α , √−
3) 21 7 3 x − x + 7 3 · Q ( α , √−
1) 28 7 4 x − x + 7 2 · P · Q ( α , √−
1) 36 9 4 x − x − · P · Q ( α , √−
3) 39 13 3 x − x −
65 3 · Q ( α , √−
43) 43 43 43 x − x −
344 43 P Q ( α , √−
2) 56 7 8 x − x + 7 2 · P · Q ( α , √−
7) 63 9 7 x − x − · Q · Q ( α , √−
67) 67 67 67 x − x −
335 67 P Q ( α , √−
1) 76 19 4 x − x + 133 2 · P · Q ( α , √−
11) 77 7 11 x − x + 7 7 · Q · Q ( α , √−
7) 91 13 7 x − x −
65 7 · P · Q ( α , √−
3) 93 31 3 x − x + 124 3 · Q ( α , √−
2) 104 13 8 x − x −
65 2 · P · Q ( α , √−
3) 129 43 3 x − x −
344 3 · Table 2.
Degree six CM fields K with h K = 3 and h F = 3 K f K f F f F P F | d K | cond( ψ ) cond( A / Q ) Q ( β , √−
3) 63 63 3 x − x −
28 3 · Q ( β , √−
3) 63 63 3 x − x + 35 3 · Q ( β , √−
7) 63 63 7 x − x + 35 3 · Q · Q ( β , √−
7) 91 91 7 x − x − x −
27 7 · P · f K f F f F P F | d K | cond( ψ ) cond( A / Q ) Q ( β , √−
3) 117 117 3 x − x + 26 3 · Q ( β , √−
7) 133 133 7 x − x − x −
69 7 · P · Q ( β , √−
19) 171 171 19 x − x −
152 3 · Q · Q ( β , √−
7) 217 217 7 x − x − x + 225 7 · P · Q ( β , √−
19) 247 247 19 x − x − x + 64 13 · Q · Table 3.
Degree six CM fields K with h K = 4 and h F = 1 K f K f F f F P F | d K | cond( ψ ) cond( A / Q ) Q ( γ , √−
1) 124 31 4 x − x + 124 2 · P P P · Q ( γ , √−
7) 133 19 7 x − x + 133 7 · P P P · Q ( γ , √−
19) 171 9 19 x − x − · Q Q Q · Q ( γ , √−
1) 172 43 4 x − x −
344 2 · P P P · Q ( γ , √−
3) 183 61 3 x − x + 61 3 · Q ( γ , √−
3) 201 67 3 x − x −
335 3 · Q ( γ , √−
11) 209 19 11 x − x + 133 11 · P P P · Q ( γ , √−
2) 248 31 8 x − x + 124 2 · P P P · Q ( γ , √−
11) 473 43 11 x − x −
344 11 · P P P · Q ( γ , √−
7) 511 73 7 x − x + 511 7 · P P P · Table 4.
Degree six CM fields K with h K = 12 and h F = 3 K f K f F f F P F | d K | cond( ψ ) cond( A / Q ) Q ( ω , √−
1) 252 63 4 x − x −
28 2 · · P P P · · References [1] W. Bosma, J. Cannon, and C. Playoust,
The Magma algebra system. I. The user language , J. Symbolic Comput., 1997.[2] D. Chun,
The Field of Moduli of 3-dimensional Abelian Varieties of CM-type , Preprint (available here).[3] M. Ishida,
On the Genus Field of an Algebraic Number Field of Odd Prime Degree , J. Math. Soc. Japan, 1975[4] J. Jones, D. Roberts,
A Database of Local Fields , J. Symb. Comput., 2006.[5] P. Llorente, E. Nart,
Effective Determination of the Decomposition of the Rational Primes in a Cubic Field , Proc. Am.Math. Soc., 1983.[6] J. Milne,
On the Arithmetic of Abelian Varieties , Invent. Math., 1972.[7] N. Murabayashi,
The Field of Moduli of Abelian Surfaces with Complex Multiplication , J. Reine. Angew. Math., 1996.[8] N. Murabayashi, A. Umegaki,
Determination of All Q -rational CM Points in the Moduli Space of Principally PolarizedAbelian Surfaces , J. Algebra, 2001.[9] N. Murabayashi, Determination of Simple CM Abelian Surfaces Defined over Q , Math. Ann., 2008.[10] J. Serre, J. Tate, Good Reduction of Abelian Varieties , Ann. Math., 1968.[11] Y. Park, S. Kwon,
Determination of All Imaginary Abelian Sextic Number Fields with Class Number ≤ , Acta Arith.,1997.[12] G. Shimura, Models of an Abelian Variety with Complex Multiplication over Small Fields , J. Number Theory, 1982.[13] A. Umegaki,
Determination of All Q -rational CM Points in the Moduli Spaces of Polarized Abelian Surfaces , AnalyticNumber Theory, Dev. Math., 2002.[14] H. Yoshida, Hecke Characters and Models of Abelian Varieties with Complex Multiplication , J. Fac. Univ. Tokyo. Sect. IAMath., 1981.