Classification of Kuga fiber varieties 1214430642

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Journal of Algebra 612 (2022) 208–226

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Journal of Algebra www.elsevier.com/locate/jalgebra

Classification of Kuga fiber varieties Salman Abdulali Department of Mathematics, Mail Stop 561, East Carolina University, Greenville, NC 27858, USA

a r t i c l e

i n f o

Article history: Received 27 June 2021 Available online 9 September 2022 Communicated by Prakash Belkale MSC: primary 14G35, 14K10, 32M15

a b s t r a c t We complete Satake’s classification of Kuga fiber varieties by showing that if a representation ρ of a hermitian algebraic group satisfies Satake’s necessary conditions, then some multiple of ρ defines a Kuga fiber variety. © 2022 Elsevier Inc. All rights reserved.

Keywords: Kuga fiber variety Hodge group Mumford-Tate group

1. Introduction Kuga fiber varieties [20,23] are families of abelian varieties A → V, where V = Γ\G(R)0 /K is an arithmetic variety, and A is the pullback of the universal family of abelian varieties over a Siegel modular variety. Here, G is a semisimple algebraic group over Q such that G(R) is of hermitian type, K is a maximal compact subgroup of G(R)0 , and Γ is an arithmetic subgroup of G(Q). A Kuga fiber variety is constructed from a symplectic representation ρ : G → Sp(2n, Q) which is equivariant with a holomorphic map τ : X → Sn , where X = G(R)0 /K is the symmetric domain belonging to G, and E-mail address: [email protected]. https://doi.org/10.1016/j.jalgebra.2022.08.021 0021-8693/© 2022 Elsevier Inc. All rights reserved.

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Sn is the Siegel space of degree n. Kuga assumed that V is compact; we do not make this assumption. Kuga’s original motivation was to prove the Ramanujan conjecture, a goal achieved by Deligne [9,11]. Kuga fiber varieties, which include Shimura’s pel-families [42], have played a central role in the arithmetic theory of automorphic forms [26,31,32]. These varieties are also key to the study of algebraic cycles on abelian varieties and abelian schemes [4,5,14,15,18,21,30,43]; indeed, the concept of the Hodge group (or MumfordTate group) of an abelian variety arose in the context of Kuga fiber varieties [29]. Another area in which Kuga fiber varieties play a key role is in the study of K3-surfaces, via the Kuga-Satake construction of abelian varieties associated to K3-surfaces [25], as in Deligne’s proof of the Weil conjectures for these surfaces [10]. We consider the following problem in this paper: Given an arithmetic variety V, classify all Kuga fiber varieties over it. Equivalently, given the group G, find all representations of it into a symplectic group which are equivariant with holomorphic maps of the corresponding symmetric domains. From another point of view, this problem is equivalent to the classification (up to isogeny) of the semisimple parts of the Hodge groups of abelian varieties, together with their action on the first cohomology of the abelian variety. This problem was raised by Kuga [20] in the 1960’s, and partially answered by Satake [34–41]. Addington [6] completed Satake’s classification for Q-simple groups of type II (orthogonal groups) and type III (symplectic groups). This was extended to non-simple groups of type III by Abdulali [1,2]. Our results deal fully with groups which are not simple over Q. It is important to do so because the semisimple part of the Hodge group of a simple abelian variety need not be simple. We give an example of such an abelian variety in Section 7. Further examples may be found in Satake [37, Remark 2, p. 356], Kuga [22, §5], and Abdulali [2, §4], [3, §2.4]. Deligne [12, §1.3] and Milne [28, §10] considered this problem from a somewhat different point of view. Their results are similar to those of Satake, and are motivated by the problem of constructing canonical models of (connected) Shimura varieties. Given a connected Shimura variety associated to a semisimple group G, a canonical model can be constructed from an injective symplectic representation of G which defines a Kuga fiber variety. Such Shimura varieties are said to be of abelian type. (The existence of canonical models for all Shimura varieties is proved by Milne in [27].) Green, Griffiths, and Kerr [16,17] and Patrikis [33] have considered the more general problem of classifying the Hodge groups of Hodge structures of higher weight. They completely classify the reductive groups which are Mumford-Tate groups of polarizable Hodge structures of arbitrary weight; however the representations of the groups on the Hodge structures have not been classified. The key to our classification is a reduction to the rigid case, which is much easier. In the proof of our main theorem (Theorem 7) we reduce the general case to the rigid case, which is proved in Theorem 6. The inspiration for this strategy comes from the

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construction of families of families of abelian varieties by Kuga and Ihara [24], and the related concept of “sharing” in Kuga [22, §5, p. 277]. Acknowledgments. I am grateful to the referee for detailed comments and suggestions. This article has greatly benefited from them. Notations and conventions. All representations are finite-dimensional and algebraic. For a finite field extension E of a field F , we let ResE/F be the restriction of scalars functor, from schemes over E to schemes over F . For an algebraic or topological group G, we denote by G0 the connected component of the identity. 2. Kuga fiber varieties In this section we give an overview of the construction of Kuga fiber varieties. Our primary purpose is to fix the notations and terminology; for details we refer to Satake [40]. 2.1. Groups of hermitian type Let X be a bounded symmetric domain associated to a group G of hermitian type. Groups of hermitian type are assumed to be semisimple, but not necessarily connected. Choose a base point o ∈ X. Then K := {g ∈ G0 | g · o = o} is a maximal compact subgroup of G0 . Let g := Lie G be the Lie algebra of G, k := Lie K, and let g = k ⊕ p be the corresponding Cartan decomposition. Differentiating the map G0 → X given by g → g · o induces an isomorphism of p with To (X), the tangent space of X at o, and there exists a unique H0 in the center of k, called the H-element at o, such that ad H0 |p is the complex structure on To (X). 2.2. Equivariant holomorphic maps Let G1 and G2 be groups of hermitian type, with bounded symmetric domains X1 and X2 respectively. Let H0 and H0 be H-elements at base points o1 ∈ X1 and o2 ∈ X2 respectively. Let ρ : G1 → G2 be a homomorphism of Lie groups. We say that ρ satisfies the H1 -condition relative to the H-elements H0 and H0 if [dρ(H0 ) − H0 , dρ(g)] = 0

for all g ∈ g1 .

(2.2.1)

The stronger condition dρ(H0 ) = H0

(2.2.2)

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is called the H2 -condition. If either of these is satisfied, then there exists a unique holomorphic map τ : X1 → X2 such that τ (o1 ) = o2 , and the pair (ρ, τ ) is equivariant in the sense that τ (g · x) = ρ(g) · τ (x)

for all g ∈ G01 , x ∈ X1 .

In fact, Clozel [8] has shown that if G2 has no exceptional factors, then the H1 -condition is equivalent to the existence of an equivariant holomorphic map. 2.3. The Siegel space Let E be a nondegenerate alternating form on a finite-dimensional real vector space V . The symplectic group Sp(V, E) is a Lie group of hermitian type; the associated symmetric domain is the Siegel space S(V, E) = { J ∈ GL(V ) | J 2 = −I and E(x, Jy) is symmetric, positive definite }. Sp(V, E) acts on S(V, E) by conjugation. The H-element at J ∈ S(V, E) is J/2. Lemma 1. Let G be a group of hermitian type with symmetric domain X, and let E be a nondegenerate alternating form on a finite-dimensional real vector space V . Let ρ : G → Sp(V, E) satisfy the H2 -condition with respect to H-elements H0 and H0 = J0 /2 at base points o ∈ X and J0 ∈ S(V, E), respectively. Then J0 ∈ ρ(G). Proof. SinceJ0 is a complex structure on V , there exists a basis of V with respect to  0 In , where 2n = dim V . Then, using the H2 -condition, we calculate which J0 = −In 0 π that J0 = exp( 2 J0 ) = exp(πH0 ) = exp(dρ(πH0 )) = ρ(exp(πH0 )).  2.4. The fiber varieties We shall say that an algebraic group G over a subfield of R is of hermitian type if the Lie group G(R) is of hermitian type. Now let G be a connected, semisimple algebraic group of hermitian type over Q. Assume that G has no nontrivial, connected, normal subgroup H such that H(R) is compact. Let E be a nondegenerate alternating form on a finite-dimensional rational vector space V . The symplectic group Sp(V, E) is then a Q-algebraic group of hermitian type. We write S(V, E) for S(VR , ER ). Let ρ : G → Sp(V, E) be a representation defined over Q, which satisfies the H1 condition with respect to the H-elements H0 and H0 = J/2. Let τ : X → S(V, E) be the corresponding equivariant holomorphic map. Let Γ be a torsion-free arithmetic subgroup of G(Q), and L a lattice in V such that ρ(Γ)L = L. Then the natural map

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A = (Γ ρ L)\(X × VR ) −→ V := Γ\X is a morphism of smooth quasiprojective algebraic varieties (Borel [7, Theorem 3.10, p. 559] and Deligne [13, p. 74]), so that A is a fiber variety over V called a Kuga fiber variety. The fiber AP over any point P ∈ V is an abelian variety isomorphic to the torus VR /L with the complex structure τ (x), where x is a point in X lying over P . We say that a representation ρ : G → GL(V ) defines a Kuga fiber variety if ρ(G) is contained in a symplectic group Sp(V, E), and ρ satisfies the H1 -condition with respect to some H-elements. 3. Satake’s classification 3.1. Necessary conditions In a series of papers [34–41] Satake classified the H1 -representations of a given hermitian group into a symplectic group. We summarize his results below. Let G be a connected semisimple algebraic group over Q. Assume that G(R)0 is of hermitian type, and G has no nontrivial, connected, normal Q-subgroup H with H(R) compact. After replacing G by a finite covering, if necessary, we may write gR =

s 

gj ,

G(R) = G0 × G1 × · · · × Gs ,

j=0

where G0 is compact, each Gj is a noncompact absolutely simple Lie group for j > 0, and, each gj = Lie(Gj ). Suppose ρ : G → Sp(V, E) is a symplectic representation satisfying the H1 -condition. Then, (1) For j = 1, . . . s, we have that Gj is one of the following: (a) Type I: SU (p, q) with p ≥ q ≥ 1; (b) Type II: SU − (n, H) with n ≥ 5 (this is the group that Helgason [19, p. 445] calls SO (2n)); (c) Type III: Sp(2n, R) with n ≥ 1; (d) Type IV: Spin(p, 2) with p ≥ 1, p = 2. (2) Let ρ be a nontrivial C-irreducible subrepresentation of ρC . Then, for some index j (1 ≤ j ≤ s), we have that ρ is equivalent to ρ0 ⊗ ρj , where ρ0 is a representation of G0,C , and, ρj is a representation of Gj,C . We call this the stability condition (see §4.1 for the terminology). (3) Fix an index j with 1 ≤ j ≤ s. Let ρj be an irreducible subrepresentation of VR considered as a Gj -module. Then ρj is either trivial or given by one of the following: (a) If Gj = SU (p, q) with p ≥ q ≥ 2, then ρj,C is the direct sum of the standard representation of Gj,C = SLp+q (C) and its contragredient; it satisfies the H2 condition if and only if p = q.

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(b) If Gj = SU (p, 1), then ρj,C is one of the following: k p+1−k (i) ⊕ , for some k with 1 ≤ k < p+1 2 ; k (ii) with k = p+1 , and p ≡ 1 (mod 4); 2 k (iii) the direct sum of two copies of with k = p+1 2 , and p ≡ 3 (mod 4). p+1 The H2 -condition is satisfied if and only if k = 2 . (c) If Gj = SU − (n, H) with n ≥ 5, then ρj,C is the direct sum of two copies of the standard representation. The H2 -condition is satisfied in this case. (d) If Gj = Sp(2n, R), then ρj is the standard representation, and satisfies the H2 -condition. (e) If Gj = Spin(p, 2) with p ≥ 1 and p odd, then (i) ρj is the spin representation if p ≡ 1, 3 (mod 8); (ii) ρj,C is the direct sum of two copies of the spin representation if p ≡ 5, 7 (mod 8). In both cases, ρj satisfies the H2 -condition. (f) If Gj = Spin(p, 2) with p ≥ 4, and p even, then ρj is (i) one of the two spin representations if p ≡ 2 (mod 8); (ii) the direct sum of two copies of a spin representation if p ≡ 6 (mod 8); (iii) the direct sum of the two spin representations if p ≡ 0 (mod 4). In each case, ρj satisfies the H2 -condition. We note that the above conditions imply that ρ is self-dual. 3.2. A sufficient condition Satake showed that the necessary conditions listed in §3.1 are sufficient if we make an additional assumption.  Theorem 2 (Satake [39]). Let G be a Q-simple hermitian group such that GR = α∈S Gα where each Gα is an absolutely simple real algebraic group. Let ρ be a representation of G satisfying conditions (1)-(3) of §3.1. Assume further that each irreducible subrepresentation of ρC is nontrivial on Gα for exactly one α. Then some multiple of ρ defines a Kuga fiber variety. 3.3. More on type I We now take a closer look at the H1 -representation ρ : SU (p, q) → Sp(VR , E) given by item (3a) of the list in §3.1. We recall the matrix representation of this given by Satake. Let J0 ∈ S(V, E) be the base point. The eigenvalues of J0 on VC are i and  −i, and we iIm 0 take a basis of VC with respect to which the matrix of J0 is . The Lie 0 −iIm algebra of Sp(VR , E) with respect to this basis is given by

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 sp(VR , E) =

X1 X 12

X12 X1

   t X1 , X12 ∈t Mm (C)  X 1 = −X1 , X12 = X12 ,

i

 0 and the H-element is = (see Satake [36, p. 431]). 0 − 2i Im With respect to a suitable basis the Lie algebra of SU (p, q) is given by 2 Im

H0

 su(p, q) =

t

X1

X12

X 12

X2



  X1 ∈ Mp (C), X2 ∈ Mq (C) ∈ slp+q (C)  t , X j = −Xj (j = 1, 2)

and an H-element is given by

H0 =

qi p+q Ip



0

.

pi − p+q Iq

0

Then, dρ : su(p, q) → sp2p+2q is given by  t

X1 X 12

X12 X2



⎛

X2 ⎜ 0 ⎜ → ⎝ 0 X 12

0 X1 t X 12 0

t

0 X12 X2 0

⎞ X12 0 ⎟ ⎟. 0 ⎠ X1

(3.3.1)

The representation ρ, being the direct sum of the standard representation and its contragredient, is the restriction to SU (p, q) of a representation ρ¯ of U (p, q). We see from (3.3.1) that d¯ ρ maps u(p, q) into sl2p+2q , so ρ¯ maps U (p, q) into Sp(2p + 2q, R). Let ¯ p,q = H 0

i

2 Ip



0

∈ u(p, q).

− 2i Iq

0

(3.3.2)

¯ p,q ) = H  . It follows, as in the proof of Lemma 1, that J0 ∈ ρ¯(U (p, q)). Then d¯ ρ(H 0 0 Consider, next, the H1 -representation ρ : SU (p, 1) → Sp(2m, R) given in item (3(b)i) of the list in §3.1. We extend it to a representation ρ¯ : U (p, 1) → Sp(2m, R). Let 0 = H



i 2k Ip

0

i k  I   (H0 ) = 2 p 0



0 1−2k 2k i

0 − 2i Iq



.

(3.3.3)





¯ p ,q , =H 0

   p   0 ) = H  , the H-element of where p = kp and q  = k−1 . From this we see that d¯ ρ(H 0 Sp(2m, R). It follows, as in the proof of Lemma 1, that J0 ∈ ρ¯(U (p, 1)).

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4. Chemistry and rigidity 4.1. Chemistry Addington [6] used a combinatorial structure called chemistry to classify representations of Q-simple groups of type II and type III which define Kuga fiber varieties. We give below a generalization of chemistry to arbitrary groups of hermitian type. A chemistry is a triple (G, S, S0 ), where G is a finite group acting on a finite set S, and S0 is a subset of S such that each G-orbit of S contains at least one member of S0 . A member of S is called an atom; a member of S0 is called a bad or radioactive atom. n A subset of S is called a molecule. A polymer is a formal linear combination i=1 ai Mi , n where each Mi is a molecule, and each ai is a positive integer. A polymer i=1 ai Mi is called stable if each Mi contains at most one radioactive atom. Since G acts on S, it acts on the set of all molecules and on the set of all polymers. We next describe how to associate a chemistry to a hermitian group G. We assume that G is a connected and semisimple algebraic group over Q such that G(R)0 is of hermitian type and has no nontrivial connected normal Q-subgroups H with H(R) compact. Write t G = j=1 Gj , where G is the universal covering of G, and each Gj is a simple group of hermitian type. Then there are totally real number fields Fj , and absolutely simple  j over Fj , such that Gj = ResF /Q G  j for 1 ≤ j ≤ t. Let Sj be the set of groups G j embeddings of Fj into R, and let S be the disjoint union of the Sj ’s. Let F be a finite Galois extension of Q contained in R which is totally real and contains all the images α(Fj ) for all j and for all α ∈ Sj . Let G = Gal(F/Q). For α ∈ S, we let j(α) be the unique index such that α ∈ Sj(α) ; then α is an embedding of Fj(α) into F . The group G acts on S by g · α = g ◦ α, and the orbits of this action are the sets Sj . For α ∈ S, we let  j(α) ⊗F ,α F , so that GF =  Gα = G j(α) α∈S Gα . Let S0 = {α ∈ S | Gα,R is not compact}, and Sj,0 = S0 ∩ Sj . Then (G, S, S0 ) is a chemistry, and so is (G, Sj , Sj,0 ) for each j. Next, let ρ be a representation of G. Lift ρ to the universal covering G of G, and n denote it again by ρ. Write ρC = i=1 ρi , where each ρi is an irreducible representation  n  of GC . Since GC = α∈S Gα,C , we can write ρC = i=1 α∈S ρi,α where each ρi,α is an irreducible representation of Gα,C . Then Mi = {α ∈ S | ρi,α is nontrivial}, n is the molecule associated to ρi , and P = i=1 Mi is the polymer associated to ρ. Since ρ is defined over Q, the polymer P is G-invariant, and Satake’s stability condition (item (2) of §3.1) is equivalent to P being a stable polymer. 4.2. The space of equivariant maps Let ρ : G → Sp(V, E) be a symplectic representation satisfying the H1 -condition with respect to H-elements H0 and J0 /2 at base points o ∈ X and J0 ∈ S(V, E), respectively.

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Let Xρ be the set of all J ∈ S(V, E) such that ρ satisfies the H1 -condition with respect to the H-elements H0 and J/2 at the base points o ∈ X and J ∈ S(V, E), respectively. Each J ∈ Xρ gives rise to a holomorphic map τJ : X → S(V, E) which is equivariant with ρ and such that τ (o) = J. Definition 3. In the above situation, we say that ρ is rigid if Xρ consists of the single point J0 . Let G1 be the centralizer of ρ(G) in Sp(V, E) and G1 the derived group of G1 . Then Satake [40, Proposition IV.4.1, p. 180] shows that G1 is a group of hermitian type with symmetric domain Xρ , and the inclusion maps G1 → Sp(V, E) and Xρ → S(V, E) are equivariant. The group G1 could have simple factors which are compact over R; let Gρ be the product of those simple factors of G1 which are not compact over R. Observe that ρ is rigid (i.e. Xρ reduces to a point) if and only if G1 (R) is compact. Since Gρ centralizes ρ(G), we see that the product G2 = ρ(G)Gρ is an almost direct product. It follows that G2 is a group of hermitian type with associated symmetric domain X × Xρ , and the inclusion ρ2 : G2 → Sp(V, E) satisfies the H1 -condition. The associated equivariant holomorphic map τ2 : X × Xρ → S(V, E) is given by τ2 (J, J  ) = τJ  (J). Now, let G3 be the centralizer of G2 in Sp(V, E), and G2 its derived group. Since G3 centralizes ρ(G), it is contained in G1 , and therefore G3 is contained in G1 . Since G2 also centralizes Gρ , we see that G3 (R) is compact. This shows that ρ2 is rigid. (This is Lemma 2.3.5 of [3].) Proposition 4. Let ρ be a symplectic representation defining a Kuga fiber variety, and let P be the associated polymer. Then ρ is rigid if and only if each molecule occurring in P contains exactly one radioactive atom. Proof. Let (G, S, S0 ) be the chemistry associated to ρ : G → Sp(V, E), where G = (Gal F/Q). Let (G  , S  , S0 ) be the chemistry associated to the inclusion ρ1 : Gρ → Sp(V, E) constructed above, where G  = Gal(F  /Q). After replacing F and F  by a common finite extension, if necessary, we may assume that F = F  and thus G = G  . The chemistry associated to the representation ρ2 : G2 → Sp(V, E) constructed above is then (G, T, T0 ), where T is the disjoint union of S and S  , and T0 is the disjoint union of S0 and S0 . Let P , Q, and R be the polymers associated to ρ, ρ1 , and ρ2 , respectively. Any molecule in R is of the form M ∪N where M is a molecule in P and N is a molecule in Q. Suppose every molecule in P contains a radioactive atom. Then the stability condition implies that no molecule in Q contains a radioactive atom, which implies that S0 is empty, so Gρ is compact, and therefore ρ is rigid. n To show the converse, write P = a M , where Mi = Mj when i = j. Then n i=1 i i we have a decomposition VR = i=1 Vi such that each C-irreducible subrepresentation of Vi,C has associated molecule Mi . This makes sense because an irreducible

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subrepresentation of ρC has the same molecule as its complex conjugate. Moreover each GR → Sp(Vi , Ei ) satisfies the H1 -condition, where Ei is the restriction of E to Vi × Vi . If some Mi contains no radioactive atom, then dρ = 0 on the space Vi . But then S(Vi , Ei ) is contained in Xρ , showing that ρ is not rigid.  This proposition motivates the following definition. Definition 5. In a chemistry, a polymer P = exactly one radioactive atom.

n i=1

ai Mi is called rigid if each Mi contains

5. The rigid case 5.1. Statement of the theorem Let G be an algebraic group over Q of hermitian type, and ρ a representation of G satisfying Satake’s necessary conditions in §3.1. We begin by classifying the rigid representations which define Kuga fiber varieties. Theorem 6. Let G be a semisimple connected algebraic group over Q such that G(R)0 is of hermitian type and has no compact factors defined over Q. Let ρ be a representation of G satisfying conditions (1)-(3) of §3.1. If ρ is rigid, then some multiple of ρ defines a Kuga fiber variety. The remainder of this section is devoted to the proof of this theorem. 5.2. Beginning of the proof Without loss of generality we assume that G is simply connected, and ρ is nontrivial and a multiple of a Q-irreducible representation (see Satake [40, p. 189]). We freely use the notation introduced in §4.1, denoting the chemistry of G by (G, S, S0 ), and the chemistry of Gj by (G, Sj , Sj,0 ). Let S1 = S \ S0 and Sj,1 = Sj \ Sj,0 .  An H-element of GR is given by H0 = α∈S0 H0,α , where H0,α is an H-element of Gα,R . Let ρ0 be a C-irreducible subrepresentation of ρC . Let M be the molecule associated    to ρ0 . Write ρ0 = α∈M ρα , where ρα is an irreducible representation of Gα,C . Each   ρα = ρα ⊕ ρ¯α is defined over R. Replacing F by a finite extension, if necessary, we assume without loss of generality that each ρα is defined over F . Moreover, some multiple of ρ0 ⊕ ρ¯0 is defined over F .  For each α ∈ M , let ρˆα = σ∈G ρσα . Then ρˆα is a representation of Gj(α) satisfying the hypotheses of Theorem 2, so some multiple of it defines a Kuga fiber variety. By Satake’s construction (see [40, §IV.6, Theorems 6.1, 6.2, 6.3]), this representation is defined over Fj(α) in the sense that it is the restriction from Fj(α) to Q of a symplectic representation

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218

 j(α) → Sp(Vα , E α ). ρ˜α : G α is an Fj(α) -bilinear alternating form on Vα , and Eα = TrF /Q E α is a Gj(α) Here, E j(α)  invariant Q-bilinear alternating form on Vα = ResFj(α) /Q Vα . Then Vα ⊗ F = σ∈G Vασ , and Vασ is the representation space of ρσα . For each σ ∈ G such that σ(α) ∈ S0 , there (σ)  σ (x, Jα(σ) y) is a positive definite exists a complex structure Jα on V σ such that E α

α,R

σ  σ ) satisfies the H1 -condition with resymmetric form, and Gσ(α),R → Sp(Vα,R ,E α   (σ) (σ) 1 spect to the H-elements H0,σ(α) and 2 Jα . We have Jα = 2dρσα,R H0,σ(α) unless   (σ) p,q p,q Gσ(α),R = SU (p, q) with p = q, in which case Jα = 2d¯ ρσα,R H0,σ(α) is where H0,σ(α) σ σ given by (3.3.2) or (3.3.3), and ρ¯α,R is the extension of ρα,R to U (p, q) given in §3.3.

ρσα,R :

5.3. Construction of the complex structure 

 σ  = The representation space of ρ is contained in W = σ∈G V , where V  σ σ σ     α∈M (Vα ⊗Fj(α) ,α F ). We have VR = α∈M Vα,R . Define Jσ on VR by Jσ (⊗xα ) = (σ)

(σ)

(σ)

(σ)

⊗Iα xα , where Iα = Jα if σ(α) ∈ S0 , and Iα is the identity if σ(α) ∈ S1 . Then each  Jσ is a complex structure on VRσ , and therefore J = σ∈G Jσ is a complex structure on WR . 5.4. Construction of symmetric forms  j(α) We claim that for each α ∈ M there exists a nondegenerate, symmetric G β  invariant, Fj(α) -bilinear symmetric form γα on Vα such that γα is positive definite on Vα ⊗Fj(α) ,β R if β ∈ Sj(α),1 . We follow Satake [36, §8.1, pp. 265–267] for the classification of groups of hermitian type.  j(α) is of type I. Then G  j(α) = SU (V /D, T ), where D is a central Suppose, first, that G division algebra over a quadratic imaginary extension F  of Fj(α) , V = Dn , and T is a hermitian form on V . Suppose ρα is the standard representation so that we are in either  j(α),F  acts as the case (3a) or case (3b) with k = 1 of §3.1. Write VF  = U ⊕ U where G standard representation on U and as the contragredient on U. If β ∈ Sj(α),1 , then Gβ,R β

is either SU (m, 0) or SU (0, m), so T β is positive definite on one of U β , U and negative definite on the other. Take a ∈ Fj(α) such that for all β ∈ Sj(α),1 , we have α(a) > 0 if T is positive definite on U β and α(a) < 0 if T is negative definite on U β . Let T  = aT . Let β T  equal T  on U β and equal −T  on U . Let τ be the reduced trace from D to Fj(α) . Then γα (u, v) = τ (T  (u, v)) is the required symmetric form. k Next, suppose we are in case (3b) of §3.1 with k > 1, so that Gα,R acts as . Then k   j(α) -invariant hermitian form on k V , so we have a map T is a G  j(α) = SU (V, T ) → G

 (k) G j(α)

= SU

k 

V,

k 

 T) .

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This map satisfies the H2 -condition [36, §3.2, pp. 447–449]. The symmetric form γα  (k) will then serve as the required constructed above for the standard representation of G j(α)  j(α) . symmetric form for the k-th exterior algebra representation of G   Next, suppose Gj(α) is of type II. Then Gj(α) = SU (V /D, T ), where D is a quaternion division algebra over Fj(α) such that D ramifies at β ∈ Sj(α) if β ∈ Sj(α),0 and splits otherwise, V = Dn , and T is a nondegenerate D-valued skew-hermitian form on V such that the symmetric form B associated to T (see [36, §3.4, pp. 232–235, and p. 266] is definite at all β ∈ Sj(α),1 . Let a ∈ Fj(α) be such that β(a) > 0 if B β is positive definite and β(a) < 0 if B β is negative definite. We then take γα = aB. We next consider a group of type III. Such groups can be of type III.1 or III.2. A group  j(α) is of type of type III.1 is a symplectic group Sp(V, E) for which S0 is empty. Thus if G III.1 then the hypotheses of Theorem 2 are satisfied, and there is nothing more to prove.  j(α) is of type III.2. This means that G j(α) = SU (V /D, T ), We may thus assume that G where D is a quaternion algebra which splits at β ∈ Sj(α),0 , and ramifies at β ∈ Sj(α),1 , V = Dn , and T is a nondegenerate D-valued hermitian form on D such that T β is indefinite if β ∈ Sj,0 and definite otherwise. Let a ∈ Fj(α) be such that β(a) < 0 if T β is negative definite and β(a) > 0 otherwise. Then we let γα (u, v) = TrD/Fj(α) aT (u, v).  j(α) of type IV, so that a noncompact factor is Spin(p, 2) We next consider a group G for some p. If p is odd, so that we are in case (3e) of §3.1, then there is a group G1 of hermitian type, such that the symplectic representation ρ˜α factors through an H2  j(α) → G1 [36, §3.7, p. 458]. The group G1 is of type III if p ≡ 1 (mod 8) morphism G or p ≡ 3 (mod 8), and G1 is of type II if p ≡ 5 (mod 8) or p ≡ 7 (mod 8). We may then use for γα the symmetric form which we know exists for groups of type II and type III. Next consider a group of type IV with p even, so that we are in case (3f) of §3.1. If p ≡ 2 (mod 8) (respectively p ≡ 6 (mod 8)) there exists a hermitian group G1 of type  j(α) → G1 [36, III (respectively type II) such that ρ˜α factors through an H2 -morphism G §3.6, pp. 456–457]. We may then use for γα the symmetric form which we know exists for groups of type II and type III. It remains to consider a group of type IV with p ≡ 0 (mod 4). In this case there p p is a Gα -invariant hermitian form T on Vα such that T β has signature (2 2 −1 , 2 2 −1 )  j(α) → SU (Vα , T ) satisfies the H2 -condition, and ρ˜α for β ∈ Sj(α),0 , the natural map G factors through SU (Vα , T ) [36, §3.6, pp. 454–456]. We may then use for γα the symmetric form which we have constructed for the type I group SU (Vα , T ). This completes the proof of the claim in all cases. 5.5. The symmetric form and the complex structure We claim that   γασ Jα(σ) x, Jα(σ) y = γασ (x, y)

(5.5.1)

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whenever σ(α) ∈ Sj(α),0 . If ρσα,R satisfies the H2 -condition, then Lemma 1 shows that (σ)

Jα belongs to the image of Gσ(α) (R) under ρσα,R , so (5.5.1) is a consequence of γασ being Gσ(α) -invariant. If the H2 -condition is not satisfied then we are in either case (3a) or case (3(b)i) of §3.1. In both cases Gσ(α) is a special unitary group SU (W, h), and we have seen in §3.3 that we can extend the representation to the full unitary group U (W, h), (σ) so that Jα belongs to the image of U (W, h), and we can then argue as before to prove (5.5.1) in this situation. Observe that (5.5.1) is equivalent to     γασ x, Jα(σ) y = −γασ y, Jα(σ) x ,

(5.5.2)

  (σ) (σ) since Jα is a complex structure. In other words, γασ x, Jα y is alternating when σ(α) ∈ Sj(α),0 . 5.6. Construction of an alternating form  on V =   Define an F -bilinear alternating form E α∈M (Vα ⊗Fj(α) F ) by

 α xα , ⊗α yα ) = E(⊗



α (xα , yα ) E

α∈M



 γβ (xβ , yβ ) .

β∈M β=α

 is G-invariant,   = t G  Then E where G j=1 j . Next, we define a Q-bilinear alternating form E on V = ResF/Q V by  y). E(x, y) = TrF/Q E(x, Then E is G-invariant. 5.7. Properties of the complex structure  σ (x, Jσ y) is symmetric and positive definite for each σ ∈ G. Let We next show that E α(σ) be the unique member of M σ ∩ S0 . For x = ⊗xα , y = ⊗yα ∈ VRσ , we have

σ

E (x, Jσ y) =

 α∈M

ασ (xα , Iα(σ) yα ) E

 β∈M β=α

 (σ) γβσ (xβ , Iβ yβ )

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   (σ) σ =E x , J y γβσ (xβ , yβ ) α(σ) α(σ) α(σ) α(σ)

+

β∈M β=α(σ)





ασ (xα , yα )γ σ (xα(σ) , J (σ) yα(σ) ) E α(σ) α(σ)

α∈M α=α(σ)

+



γβσ (yβ , xβ )

β∈M β=α(σ)



 σ (yα , xα )γ σ (yα(σ) , J (σ) xα(σ) ) E α α(σ) α(σ)

α∈M α=α(σ)

=



 γβσ (xβ , yβ )

β∈M β=α β=α(σ)

 σ (yα(σ) , J (σ) xα(σ) ) =E α(σ) α(σ) 



 σ (yα , I (σ) xα ) E α α

α∈M







 γβσ (yβ , xβ )

β∈M β=α β=α(σ)

(σ) γβσ (yβ , Iβ xβ )

β∈M β=α

 σ (y, Jσ x), =E   (σ) (σ) σ because E u, J v and γασ are symmetric, Iβ is the identity for β = α(σ), and α(σ) α(σ)  (σ)  σ (u, v) and γ σ σ E α α(σ) u, Jα(σ) v are alternating. Thus E (x, Jσ y) is a symmetric form on V σ . R

If x = y we have 

 σ (x, Jσ x) = E



α∈M

     (σ) σ  σ xα , I (σ) xα γ , I x E x β β α α β β



β∈M β=α

   (σ) σ x , J x γβσ (xβ , xβ ), =E α(σ) α(σ) α(σ) α(σ) β∈M β=α(σ)

 σ (x, Jσ y) is positive definite. so E 5.8. Conclusion of the proof  σ (x, Jσ y) is symmetric and positive definite, so is E(x, Jy). Thus J ∈ Since each E  S(V, E), so an H-element for Sp(V, E) is given by 12 J. Since H0 = α∈S0 H0,α is an Helement of G, and ρα satisfies the H1 -condition with respect to H0,α and 12 Jα whenever α ∈ S0 , it follows from our construction that ρ satisfies the H1 -condition, and therefore defines a Kuga fiber variety.

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6. The general case We next derive the general case from the rigid case. Theorem 7. Let ρ be a representation of G satisfying conditions (1)-(3) of §3.1. Then some multiple of ρ defines a Kuga fiber variety. Proof. Without loss of generality we assume that ρ is a primary representation, i.e., a multiple of an irreducible representation. Let (G, S, S0 ) be the chemistry associated to G, and let P be the polymer associated to ρ. We will freely use the notation from §4.1. Let ρ0 be an irreducible representation of ρC . Then, after replacing ρ by a multiple if  necessary, we have ρC = n(ρ0 ⊕ ρ¯0 )σ for some positive integer n. If M is the  σ∈G molecule for ρ0 , then P = σ∈G 2nM σ . For each j = 1, . . . , t, let Bj be a quaternion algebra over Fj such that Bj splits at an infinite place α if and only if α ∈ S0 . Let SL1 (Bj ) be the group of norm 1 units of Bj , and t let Lj = ResFj /Q SL1 (Bj ). Let L = j=1 Lj . Then L is a group of hermitian type with the same chemistry (G, S, S0 ) as G. Since P is a stable G-invariant polymer, a multiple of P is associated to a representation ξ : L → Sp(V, E) of L which defines a Kuga fiber variety (Addington [6, Theorem 2.1, p. 71] and Abdulali [2, Theorem 4.1, p. 341]). Let ξ0  be an irreducible representation of ξC with molecule M . Then ξC = σ∈G n(ξ0 ⊕ ξ¯0 )σ , where ξ0 is an irreducible subrepresentation of ξC with molecule M . Let Lρ be the group constructed from the centralizer of ξ(L) in Sp(V, E) as in §4.2.  Let ι : Lρ → Sp(V, E) be the inclusion, and let Q = σ∈G mN σ be the polymer for ι. Proposition 4 shows that the inclusion μ : ξ(L)Lρ → Sp(V, E) defines a Kuga fiber variety which is rigid. Let μ1 : ξ(L)Lρ → Sp(V, E) be a primary subrepresentation of  μ. Let R be the polymer for μ1 . We can write R = σ∈G m(M ∪ N )σ , where N is a molecule for the chemistry of Lρ , and m is a positive integer. Then μ1,C =



m((ξ0 ⊗ ι0 ) ⊕ (ξ¯0 ⊗ ¯ι0 ))σ ,

σ∈G

where ι0 is an irreducible subrepresentation of ιC with molecule N . Let μ ˜ be the representation of G × Lρ given by μ ˜C =



m((ρ0 ⊗ ι0 ) ⊕ (¯ ρ0 ⊗ ¯ι0 ))σ .

σ∈G

The polymer of μ ˜ is R which is rigid, so by Theorem 6, kμ ˜ defines a Kuga fiber variety for some positive integer k. Since the restriction of kμ ˜ to G is a multiple of ρ, this completes the proof of the theorem. 

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7. An example √ Let F = Q( 3). Let F1 and F2 be fields isomorphic to F . Let α, β be the embeddings of F1 into R, and let γ, δ be the embeddings of F2 into R. Let B be a quaternion algebra  1 be the group of norm 1 units in B, over F1 which splits at α and ramifies at β. Let G    2 ⊗F ,γ R = SU (3, 1) and and G1 = ResF1 /Q G1 . Let G2 be a group over F2 such that G 2  2 ⊗F ,δ R = SU (4, 0). Let G2 = ResF /Q G  2 . Let G = G1 × G2 . Then G 2 2 G(R) = SL2 (R) × SU (2) × SU (3, 1) × SU (4, 0), and G(C) = SL2 (C) × SL2 (C) × SL4 (C) × SL4 (C). Using the notations of §4.1, we have S = {α, β, γ, δ} and S0 = {α, γ}. The Galois group G of F over Q is of order 2. The nontrivial element of G acts on S by transposing α and β, and also transposing γ and δ. Let  1 ⊗F ,α C ∼ pα : G(C) → G = SL2 (C), 1  1 ⊗F ,β C ∼ pβ : G(C) → G = SL2 (C), 1

 2 ⊗F ,γ C ∼ pγ : G(C) → G = SL4 (C), 2  1 ⊗F ,δ C ∼ pδ : G(C) → G = SL4 (C), 2

be the projections. Complex conjugation acts on the projections by p¯α = pα , p¯β = pβ , 3 3 p¯γ = pγ , and p¯δ = pδ . With the above notations, the representations of G defining Kuga fiber varieties are equivalent over C to direct sums of multiples of the following: = pα ⊕ pβ , = pα ⊗ pβ , 3 3 = pγ ⊕ pδ ⊕ p ⊕ pδ , 2 2 γ = 2  pγ ⊕ 2 pδ  j k 4−j 4−k  (5) ρj,k pγ ⊗ pδ ⊕ pγ ⊗ pδ , 5 =  2   2  (6) ρ6 = pα ⊗ pδ ⊕ pβ ⊗ pγ ,  3   3  (7) ρ7 = (pα ⊗ pδ ) ⊕ (pβ ⊗ pγ ) ⊕ pα ⊗ pδ ⊕ pβ ⊗ pγ . (1) (2) (3) (4)

ρ1 ρ2 ρ3 ρ4

Each of these representations is stable under complex conjugation, hence some multiple is defined over R. Since each one is also stable under the nontrivial member of G, some multiple of it is defined over Q. Since Satake’s necessary conditions are satisfied in each case, Theorem 7 implies that some multiple of each one defines a Kuga fiber variety.

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The representations ρ1 and ρ2 are representations of G1 alone, with ρ2 being rigid, and ρ1 non-rigid. The representations ρ3 , ρ4 , and ρ5 are representations of G2 alone; ρj,k 5 is rigid, but ρ3 and ρ4 are not. ρ6 and ρ7 are representations of the product group in an essential manner. They are both rigid, and the general fibers of the corresponding Kuga fiber varieties are simple abelian varieties. Since ρ6 satisfies the H2 -condition, the Hodge group of a general fiber is semisimple and equals the image of G1 × G2 [4, Proposition 2.2, p. 1124]. For ρ7 , the image of G1 × G2 is the semisimple part of the Hodge group of a general fiber. References [1] Salman Abdulali, Absolute Hodge Cycles in Kuga Fiber Varieties, Thesis, State University of New York, Stony Brook, 1980, MR2634784. [2] Salman Abdulali, Zeta functions of Kuga fiber varieties, Duke Math. J. 57 (1988) 333–345, https:// doi.org/10.1215/S0012-7094-88-05715-8, MR952238. [3] Salman Abdulali, Conjugates of strongly equivariant maps, Pac. J. Math. 165 (1994) 207–216, https://doi.org/10.2140/pjm.1994.165.207, MR1300831. [4] Salman Abdulali, Algebraic cycles in families of abelian varieties, Can. J. Math. 46 (1994) 1121–1134, https://doi.org/10.4153/CJM-1994-063-0, MR1304336. [5] Salman Abdulali, Hodge structures on abelian varieties of type III, Ann. Math. (2) 155 (2002) 915–928, https://doi.org/10.2307/3062136, MR1923969. [6] Susan L. Addington, Equivariant holomorphic maps of symmetric domains, Duke Math. J. 55 (1987) 65–88, https://doi.org/10.1215/S0012-7094-87-05504-9, MR883663. [7] Armand Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Differ. Geom. 6 (1972) 543–560, https://doi.org/10.4310/jdg/1214430642, MR0338456. [8] Laurent Clozel, Equivariant embeddings of Hermitian symmetric spaces, Proc. Indian Acad. Sci. Math. Sci. 117 (2007) 317–323, https://doi.org/10.1007/s12044-007-0027-8, MR2352051. [9] Pierre Deligne, Formes modulaires et représentations -adiques, in: Séminaire Bourbaki, 21e Année (1968/69), Exp. No. 355, in: Lecture Notes in Math., vol. 179, Springer-Verlag, Berlin-HeidelbergNew York, 1971, pp. 139–172, MR3077124. [10] Pierre Deligne, La conjecture de Weil pour les surfaces K3, Invent. Math. 15 (1972) 206–226, https://doi.org/10.1007/BF01404126, MR0296076. [11] Pierre Deligne, La conjecture de Weil. I, Publ. Math. Inst. Hautes Études Sci. (43) (1974) 273–307, https://doi.org/10.1007/BF02684373, MR0340258. [12] Pierre Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, in: Automorphic Forms, Representations and L-Functions, Part 2, Corvallis, Ore., 1977, in: Proc. Sympos. Pure Math., vol. 33, Amer. Math. Soc., Providence, R.I., 1979, pp. 247–289, MR546620. [13] Pierre Deligne, (notes by J.S. Milne) Hodge cycles on abelian varieties, in: Hodge Cycles, Motives, and Shimura Varieties, in: Lecture Notes in Math., vol. 900, Springer-Verlag, Berlin, 1982, pp. 9–100, corrected 2nd printing, 1989, MR654325. [14] B. Brent Gordon, Topological and algebraic cycles in Kuga-Shimura varieties, Math. Ann. 279 (1988) 395–402, https://doi.org/10.1007/BF01456276, MR922423. [15] B. Brent Gordon, Algebraic cycles and the Hodge structure of a Kuga fiber variety, Trans. Am. Math. Soc. 336 (1993) 933–947, https://doi.org/10.1090/S0002-9947-1993-1097167-2, MR1097167. [16] Mark Green, Phillip Griffiths, Matt Kerr, Mumford-Tate Groups and Domains: Their Geometry and Arithmetic, Annals of Mathematics Studies, vol. 183, Princeton Univ. Press, Princeton, NJ, 2012, MR2918237. [17] Mark Green, Phillip Griffiths, Matt Kerr, Hodge Theory, Complex Geometry, and Representation Theory, CBMS Regional Conference Series in Mathematics, vol. 118, Published for the Conference Board of the Mathematical Sciences by the Amer. Math. Soc., Providence, RI, 2013, MR3115136. [18] Rita Hall, Michio Kuga, Algebraic cycles in a fiber variety, Sci. Pap. Coll. Gen. Educ. Univ. Tokyo 25 (1975) 1–6, MR0469919.

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