Signature (n−2,2) CM Types and the Unitary Colmez Conjecture
aa r X i v : . [ m a t h . N T ] M a r SIGNATURE ( n − , CM TYPES AND THE UNITARY COLMEZCONJECTURE
SOLLY PARENTI
Abstract.
Colmez conjectured a formula relating the Faltings height of CM abelianvarieties to a certain linear combination of log derivatives of L -functions. In thispaper, we study the case of unitary CM fields and by studying the class functionsthat arise, we reduce the conjecture to a special case. Using the Galois action, weprove more cases of the Colmez Conjecture. Set Up and Theorem
The Colmez conjecture gives a formula for the Faltings height of a CM abelianvariety in terms of log derivatives of L -functions arising from the CM type. Thisconjecture has proven useful in giving bounds for the Faltings height of CM abelianvarieties (see [Col98] for the case of elliptic curves and [Tsi18] where a weaker formof the Colmez conjecture is used in the proof of the André-Oort conjecture for themoduli space of principally polarized abelian varieties). Definition 1.1.
A unitary CM field E is a CM field of the form E = kF , where F is a totally real number field and k ⊆ C is an imaginary quadratic field.Given a CM field E of degree n , a CM type of E consists of n embeddings of E into C such that no two of the embeddings differ by complex conjugation. Forunitary CM fields, we will stratify the CM types by signature. Definition 1.2.
Let E = kF be a unitary CM field. A CM type Φ ⊆ Hom( E, C ) has signature ( n − r, r ) if exactly n − r of the embeddings in Φ restrict to the identity k ֒ → C .The main theorem of this paper is that we can reduce the Colmez conjecture inthe unitary case to CM types of signature ( n − , . Theorem 1.3.
Let E = kF be a unitary CM field. Then, the Colmez conjectureholds for E if and only if it holds for all CM types of signature ( n − , . We spend the remainder of this section providing a brief description of the Colmezconjecture and refer the reader to [YY17] giving a more thorough background to theconjecture. Section 2 contains the proof of Theorem 1.3 and in Section 3 we applyTheorem 1.3 to obtain examples of CM fields where the Colmez conjecture holds.Let Φ be a CM type of a CM field E and identify Φ with its characteristic function Φ : Hom( E, C ) → { , } . If E c denotes the Galois closure of E (which is also a CMfield), then the restriction map Hom( E c , C ) → Hom( E, C ) can be used to extend Φ to Φ c , a CM type on E c . Mathematics Subject Classification.
Primary: 11G15.
Key words and phrases.
Colmez conjecture.
Choosing an identification of
Hom( E c , C ) with Gal( E c / Q ) , we obtain a function Φ c : Gal( E c / Q ) → { , } and we define the reflex CM type f Φ c : Gal( E c / Q ) → { , } by f Φ c ( g ) = Φ c ( g − ) .Let A Φ : Gal( E c / Q ) → C denote the function we obtain by taking a normalizedconvolution of Φ c and f Φ c . More concretely, A Φ ( g ) = 1 E c / Q ) X σ ∈ Gal( E c / Q ) Φ c ( σ ) f Φ c ( σ − g ) . To obtain a class function A , we take the average of A Φ among conjugates in Gal( E c / Q ) . That is to say, A ( g ) = 1 E c / Q ) X h ∈ Gal( E c / Q ) A Φ ( hgh − ) . As A is a class function, we may write A = X χ a χ χ, where a χ ∈ C and χ ranges through the irreducible representations of Gal( E c / Q ) .Then define the function Z ( s, A ) by Z ( s, A ) := X χ a χ Z ( s, χ ) , Z ( s, χ ) := L ′ ( s, χ ) L ( s, χ ) + 12 log f χ , where L ( s, χ ) is the Artin L -function of χ and f χ is the Artin conductor of χ .Let Q cm denote the compositum of all CM number fields. This field is an infinitedegree Galois extension of Q with a well defined complex conjugation, which we willdenote by ρ . Via the quotient map Gal( Q cm / Q ) ։ Gal( E c / Q ) , we may consider A as a class function on Gal( Q cm / Q ) .The other side of the conjecture involves the Faltings height of CM abelian varieties.If Φ is a CM type of a CM field E , let X Φ be an abelian variety of dimension n with CM by ( O E , Φ) . We can find a number field L over which X Φ has everywheregood reduction. Denote by X Φ the Neron model of X Φ defined over O L and let ǫ : Spec( O L ) → X Φ be the zero section. Define ω X Φ as the following line bundle on Spec( O L ) , ω X Φ = ǫ ∗ (Ω n X Φ / O L ) , and for a non-zero section α of ω X Φ , we define the (stable) Faltings height of X Φ via h Fal ( X Φ ) := − L : Q ] X σ : L֒ → C log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z X σ Φ ( C ) α σ ∧ α σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + log | ω X Φ / O L / O L α | . This definition of stable Faltings height is independent of choice of α and choice of L over which X Φ obtains everywhere good reduction.In his 1993 paper [Col93], Colmez looks at CM , the Q vector space of class func-tions f : Gal( Q cm / Q ) → Q such that f ( g ) + f ( ρg ) is independent of g ∈ Gal( Q cm / Q ) .One can check that for every CM type Φ , the function A is an element of CM .Colmez defines a Q -linear height function ht : CM → R such that if X Φ is an abelianvariety with CM by ( O E , Φ) , then h Fal ( X Φ ) = − ht( A ) . IGNATURE ( n − , CM TYPES AND THE UNITARY COLMEZ CONJECTURE 3
Here, E is a CM field, Φ is a CM type of E , and h Fal denotes the Faltings heightof an abelian variety. The Q -linearity of Colmez’s ht will be important to us later.Colmez’s conjecture is the following alternate formula for ht( A ) . Conjecture 1.4 (Colmez) . For any CM type Φ , ht( A ) = Z (0 , A ) .2. Proof of Theorem
Before we start the proof of Theorem 1.3, let us introduce some notation. If F isa totally real number field of degree n , denote by F c the Galois closure of F . Let k be an imaginary quadratic field and let E := kF be a unitary CM field with complexconjugation ρ . We will denote the Galois closure of E by E c , and thus E c = kF c .Furthermore, let H := Gal( F c /F ) ≤ Gal( F c / Q ) =: G and suppose H = h . Then,we can identify the embeddings of F into C , which we will call { σ , . . . , σ n } , withcoset representatives for H \ G .An embedding E ֒ → C is uniquely determined by a pair of embeddings F ֒ → C and k ֒ → C . We denote by { , ρ } the two embeddings of k into C and for an embedding σ : F → C , we write ρ i σ for the embedding of E into C given by the pair { ρ i , σ } . If i = 0 , we simply write σ for σ .A CM type of E := kF consists of a choice of one of the embeddings k ֒ → C foreach embedding of F ֒ → C . Thus we can parametrize CM types of E via subsets of { , , . . . , n } . Given S ⊆ { , , . . . , n } , the corresponding CM type of E is given by Φ S = { ρ j i σ i : j i = 1 if i ∈ S, j i = 0 if i S } . Then, Φ S is a CM type of signature ( n − ǫ, ǫ ) , where ǫ = S . We will often writeCM types as sums, Φ S = X i ∈ S ρσ i + X i S σ i = tr E/k +( ρ − X i ∈ S σ i . The first step in the proof of the theorem is an explicit calculation of A S . Theorem 2.1.
Let S ⊆ { , , . . . n } be of size ǫ . Then, A S = 12 tr E c /k − ǫn (1 − ρ ) tr E c /k + ǫn (1 − ρ ) χ Ind GH ( χ ) + 1 hn (1 − ρ ) X g ∈ G g X i = j ∈ S σ i Hσ − j ! g − . Proof.
Recall that Φ S = tr E/k +( ρ − X i ∈ S σ i . Extending Φ S to Φ cS , the CM type on E c , amounts to determining which embeddings E c ֒ → C when restricted to E are in Φ S . Since E is the fixed field in E c by thesubgroup H , Φ cS is given by Φ cS = tr E c /k +( ρ − X i ∈ S σ i H. When we write Φ cS as a sum in this manner, we are interpreting Φ cS as an elementof C [Gal( E c / Q )] which is isomorphic (as a ring) to the ring (under convolution) of S. PARENTI all maps from
Gal( E c / Q ) to C . Next we find the reflex type f Φ cS by inverting everyelement in Φ cS , f Φ cS = tr E c /k +( ρ − X j ∈ S Hσ − j . Then take the convolution of Φ cS and f Φ cS , A Φ S = 1[ E c : Q ] Φ c f Φ c = 12 hn tr E c /k +( ρ − X i ∈ S σ i H ! tr E c /k +( ρ − X j ∈ S Hσ − j ! = 12 tr E c /k − ǫn (1 − ρ ) tr E c /k + 1 n (1 − ρ ) X i,j ∈ S σ i Hσ − j . Finally, we need to project A Φ S onto the space of class functions to obtain A S ,(2.1) A S = 12 tr E c /k − ǫn (1 − ρ ) tr E c /k + 1 n (1 − ρ ) 1 hn X g ∈ G g X i,j ∈ S σ i Hσ − j ! g − . The main difficulty in (2.1) is the final term. We first look at the elements of thesum with i = j . hn X g ∈ G g X i ∈ S σ i Hσ − i ! g − = 1 hn X i ∈ S X g ∈ G gσ i Hσ − i g − ! (2.2) = 1 hn X i ∈ S X g ∈ G gHg − ! (2.3) = ǫhn X g ∈ G gHg − . (2.4)The following proposition simplifies (2.4) and combining the following Propositionwith equation (2.1) concludes the proof. (cid:3) Proposition 2.2.
Let χ : H → C be the trivial character. As functions G → C , wehave the relation X g ∈ G gHg − = hχ Ind GH ( χ ) . Proof.
This is proven on page 18 of [YY17], but we sketch a proof. Recall that therepresentation
Ind GH ( χ ) is given by Ind GH ( χ ) = { f : G → C : f ( xg ) = f ( g ) ∀ x ∈ H, g ∈ G } , where G acts by right translation. The space Ind GH ( χ ) consists exactly of the func-tions f : H \ G → C . Therefore a standard calculation shows that the representation Ind GH ( χ ) is isomorphic to the representation arising from the action of G on H \ G via g · Hσ := Hσg − . Recall that we have identified { σ , . . . , σ n } with coset repre-sentatives for H \ G . IGNATURE ( n − , CM TYPES AND THE UNITARY COLMEZ CONJECTURE 5
It is straightforward to compute the character of a permutation representation,namely it is the number of fixed points. That is to say, for σ ∈ G , χ Ind GH ( χ ) ( σ ) = { i ∈ { , . . . , n } : σ · Hσ i = Hσ i } = { i ∈ { , . . . , n } : σ ∈ σ i Hσ − i } . On the other hand, X g ∈ G gHg − ! ( σ ) = { g ∈ G : σ ∈ gHg − } . However, for a given i with σ ∈ σ i Hσ − i , then every g ∈ σ i H satisfies σ ∈ gHg − andsince H = h , we obtain X g ∈ G gHg − = hχ Ind GH ( χ ) . (cid:3) Let us record a few particular cases of Theorem 2.1 which will be of use: A { ∅ } = 12 tr E c /k ,A { i } = 12 tr E c /k − n (1 − ρ ) tr E c /k + 1 n (1 − ρ ) χ Ind GH ( χ ) ,A { i,j } = 12 tr E c /k − n (1 − ρ ) tr E c /k + 2 n (1 − ρ ) χ Ind GH ( χ ) + 1 hn (1 − ρ ) X g ∈ G (cid:16) gσ i Hσ − j g − + gσ j Hσ − i g − (cid:17) . Proposition 2.3.
For any subset S ⊆ { , , . . . , n } of size ǫ , we have A S = X { i,j }⊆ S A { i,j } − ( ǫ − X i ∈ S A { i } + ( ǫ − ǫ − A { ∅ } . Proof.
This proposition is clear when ǫ = 0 , , where we interpret an empty sum as0. For ǫ ≥ , we have X { i,j }⊆ S A { i,j } = X { i,j }⊆ S
12 tr E c /k − n (1 − ρ ) tr E c /k + 2 n (1 − ρ ) χ Ind GH ( χ ) + 1 hn (1 − ρ ) X g ∈ G (cid:16) gσ i Hσ − j g − + gσ j Hσ − i g − (cid:17)! = ǫ ( ǫ − E c /k − ǫ ( ǫ − n (1 − ρ ) tr E c /k + ǫ ( ǫ − n (1 − ρ ) χ Ind GH ( χ ) + 1 hn (1 − ρ ) X g ∈ G g (cid:16) X i = j ∈ S σ i Hσ − j (cid:17) g − . S. PARENTI
From Theorem 2.1, we can conclude that A S − X { i,j }⊆ S A { i,j } = − ( ǫ + 1)( ǫ − E c /k + ǫ ( ǫ − n (1 − ρ ) tr E c /k − ǫ ( ǫ − n (1 − ρ ) χ Ind GH ( χ ) = − ( ǫ − X i ∈ S A { i } + ( ǫ − ǫ − A { ∅ } . (cid:3) For a given CM type Φ S , we have written A S in terms of the correspondingfunction A for CM types of signatures ( n, , ( n − , , and ( n − , . To finish offthe proof, we will use Colmez’s ht function as well as recent work of Yang and Yin[YY17] proving the Colmez conjecture for CM types of signature ( n, and ( n − , . Theorem 2.4.
Suppose the Colmez conjecture holds for all CM types of E of signa-ture ( n − , . Then the Colmez conjecture holds for all CM types of E .Proof. Recall that Colmez defined a Q -linear height function ht : CM → R andconjectured that ht( A ) = Z (0 , A ) for any CM type Φ of a CM field. We note that Z (0 , · ) is also a Q -linear function on CM .Let S ⊆ { , , . . . , n } be a subset of size ǫ . In [YY17], Yang and Yin show that theColmez conjecture holds for all CM types of signature ( n, and ( n − , . Then,supposing that ht( A { i,j } ) = Z (0 , A { i,j } ) for any i, j , we obtain ht( A S ) = ht X { i,j }⊆ S A { i,j } − ( ǫ − X i ∈ S A { i } + ( ǫ − ǫ − A { ∅ } ! = X { i,j }⊆ S ht( A { i,j } ) − ( ǫ − X i ∈ S ht( A { i } ) + ( ǫ − ǫ − A { ∅ } )= X { i,j }⊆ S Z (0 , A { i,j } ) − ( ǫ − X i ∈ S Z (0 , A { i } ) + ( ǫ − ǫ − Z (0 , A { ∅ } )= Z , X { i,j }⊆ S A { i,j } − ( ǫ − X i ∈ S A { i } + ( ǫ − ǫ − A { ∅ } ! = Z (0 , A S ) (cid:3) Galois Action on CM Types
There is an action of Galois on the set of CM types. Namely, g ∈ Gal( E c / Q ) actson a CM type Φ by g · Φ := { gσ : σ ∈ Φ } . It is well known and straightforward tocheck that if Φ and Φ are two CM types that are equivalent under this action, then A = A .As in Section 2, let F be a totally real number field with Galois closure F c , with G := Gal( F c / Q ) and H := Gal( F c /F ) . Let k be an imaginary quadratic field andconsider the unitary CM field E := kF . We can describe the action of Gal( E c / Q ) ∼ = G × Z / Z on the set of CM types. The Z / Z component acts as complex conjugation, IGNATURE ( n − , CM TYPES AND THE UNITARY COLMEZ CONJECTURE 7 taking a CM type of signature ( n − ǫ, ǫ ) to a CM type of signature ( ǫ, n − ǫ ) . Theaction of Gal( E c /k ) fixes the signature of a CM type and this action on CM types ofsignature ( n − ǫ, ǫ ) is isomorphic to the action of G on the set of subsets of G/H ofsize ǫ .One of the main results of [YY17] is that the Colmez conjecture holds if we averageamongst CM types of a given signature. That is to say, if Φ( E ) ǫ denotes all CM typesof E of signature ( n − ǫ, ǫ ) , then X Φ ∈ Φ( E ) ǫ ht( A ) = X Φ ∈ Φ( E ) ǫ Z (0 , A ) . If there is only one equivalence class of CM types in a given signature, then theirresult immediately implies that the Colmez conjecture holds for the CM type of thatsignature. This is the idea behind Yang and Yin’s proof that the Colmez conjectureholds for CM types of signature ( n, and ( n − , . In particular, combining theseideas with our Theorem 2.4 gives the following theorem. Theorem 3.1.
Let k be an imaginary quadratic field and let F be a totally realnumber field with Galois closure F c . Let H := Gal( F c /F ) ≤ Gal( F c / Q ) =: G . If G acts 2-transitively on G/H , then the Colmez conjecture holds for every CM type ofthe unitary CM field E := kF . We note that there are examples of such a ( G, H ) . In particular, PSL n ( F q ) and PGL n ( F q ) both act 2-transitively on P n − ( F q ) , and so we can take H to be thestabilizer of a point. More precisely, we may take H := a a . . . a n a . . . a n ... . . . a n . . . a nn In the case that n = 2 , H is the Borel subgroup of upper triangular matrices and werecover the result from [Par17].For small values of n, q , the groups PSL n ( F q ) and PGL n ( F q ) can be realized asthe Galois groups of totally real number fields over Q . In particular, consulting theLMFDB [LMF17] shows that for q ≤ , the groups PSL ( F q ) and PGL ( F q ) appearas the Galois groups of totally real fields.Furthermore, the fact that G is the Galois group of the Galois closure of F , and H is the subgroup that fixes the field F implies that the action of G on G/H inducesan embedding
G ֒ → Sym(
G/H ) of G into the symmetric group on the set of cosetsof G by H .Thus, we may apply Theorem 3.1 to any G which is a doubly transitive subgroupof a symmetric group. As a corollary of the classification of finite simple groups,a classification of doubly transitive groups is known. There are infinite families ofexamples and sporadic examples. We list the other doubly transitive groups andrefer the reader to [DM96] and [BHR13] for further details on these groups and theirdoubly transitive actions.The alternating and symmetric groups A n and S n are doubly transitive subgroupsof S n for any n . For these groups, we take H to be A n − and S n − respectively. TheColmez conjecture was already discussed in this case by Yang and Yin [YY17].Another family of examples is the symplectic groups Sp m ( F ) for m ≥ . For thesegroups, two choices of subgroups induce doubly transitive actions. This discussion S. PARENTI involves quadratic forms, symmetric bilinear forms, and alternating bilinear forms incharacteristic 2, so we will explain in detail. Let m and m denote the m × m zeroand identity matrix respectively and define the matrix J by J := (cid:20) m m − m m (cid:21) . If we denote x T to be the transpose of a matrix x , we define the symplectic group asfollows, Sp m ( F ) := { x ∈ GL m ( F ) : x T J x = J } . We can also view Sp m ( F ) as the set of linear transformations which preserve thefollowing alternating bilinear form ψ . ψ : F m × F m → F (cid:20) u u (cid:21) , (cid:20) v v (cid:21) ! u · v − u · v That is to say, Sp m ( F ) := { x ∈ GL m ( F ) : ψ ( xu, xv ) = ψ ( u, v ) ∀ u, v ∈ F m } . However, ψ is also a symmetric form over F and the doubly transitive action weare interested in involves orthogonal groups of associated quadratic forms. Definition 3.2. If V is a vector space over F , then a quadratic form on V is afunction q : V → F such that(1) q ( λv ) = λ q ( v ) for all λ ∈ F , v ∈ V ;(2) The function f ( u, v ) := q ( u + v ) − q ( u ) − q ( v ) is a symmetric bilinear form.Over F , a quadratic form determines a symmetric bilinear form, but many qua-dratic forms can determine the same symmetric bilinear form. Up to isometry, thereare two quadratic forms on F m , given by Q + ( v , . . . , v m ) = v v m +1 + · · · + v m v m ,Q − ( v , . . . , v m ) = v v m +1 + · · · + v m v m + v m + v m . We define two orthogonal groups, GO +2 m ( F ) and GO − m ( F ) , as the isometry groupsof these quadratic forms, GO ± m ( F ) = { x ∈ GL m ( F ) : Q ± ( xv ) = Q ± ( v ) ∀ v ∈ F m } . Both Q + and Q − determine the same bilinear symmetric form, ψ , and thus wehave that GO +2 m ( F ) ⊆ Sp m ( F ) and GO − m ( F ) ⊆ Sp m ( F ) . Furthermore, Sp m ( F ) acts doubly transitively on the cosets by either of the aforementioned subgroups.Applying Theorem 3.1 to this situation shows that if F is a totally real numberfield whose Galois closure F c has Galois group Sp m ( F ) and F is the fixed field byeither GO +2 m ( F ) or GO − m ( F ) , then the Colmez conjecture holds for E := kF .Another class of examples is the unitary groups PSU ( F q ) and PGU ( F q ) where q is a prime power. To describe these groups and the relevant subgroups, we will follow IGNATURE ( n − , CM TYPES AND THE UNITARY COLMEZ CONJECTURE 9 the convention of [DM96]. Let ϕ be the following bilinear form on F q . ϕ : F q × F q → F q u u u , v v v ! u v q + u v q + u v q We define GU ( F q ) to be the matrices which preserve this form, GU ( F q ) := { A ∈ GL ( F q ) : ϕ ( Au, Av ) = ϕ ( u, v ) ∀ u, v ∈ F q } . The action of GU ( F q ) on the 1-dimensional subspaces of F q defines the projec-tive group PGU ( F q ) and taking those matrices of determinant 1 defines the group PSU ( F q ) .An isotropic vector v ∈ F q is a vector such that ϕ ( v, v ) = 0 . The groups PGU ( F q ) and PSU ( F q ) both act doubly transitively on the set of 1-dimensional isotropic sub-spaces of F q , so therefore we may take H to be the stabilizer of any 1-dimensionalisotropic subspace. A quick calculation shows that the stabilizer in PSU ( F q ) and PGU ( F q ) of the subspace spanned by is exactly the upper triangular matricesof the respective groupApplying Theorem 3.1 to this group action shows that the Colmez conjecture holdsfor E := kF where k is an imaginary quadratic field, F is a totally real number fieldsuch that Gal( F c / Q ) ∼ = PSU ( F q ) or PGU ( F q ) and Gal( F c /F ) is the subgroup ofupper triangular matrices.There are a two more infinite families and a few more sporadic examples which welist here. • The Suzuki groups
Sz( q ) for q an odd power of 2 is a doubly transitive sub-group of S q +1 . • The Ree groups R( q ) for q an odd power of 3 is a doubly transitive subgroupof S q +1 . • The Mathieu groups M , M , M , M , M are doubly transitive subgroupsof S , S , S , S , and S respectively. • The Mathieu group M is a doubly transitive subgroup of S and PSL ( F ) is a transitive subgroup of S . • The alternating group A is a transitive subgroup of S . • The Higman-Sims group HS is a doubly transitive subgroup of S . • The Conway group Co is a doubly transitive subgroup of S .4. Acknowledgements
The author would like to thank Tonghai Yang for his assistance throughout workon this paper. The author would also like to thank Moisés Herradón Cueto for hishelp with doubly transitive groups. This work was done with the partial support ofNational Science Foundation grant DMS-1502553.
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University of Wisconsin-Madison, Department of Mathematics, Madison WI 53703
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