On the André-Oort conjecture for Hilbert modular surfaces. ^{∗}

Bas Edixhoven^{†}

April 5, 2000

1 Introduction.

In order to state the conjecture mentioned in the title, we need to recall some terminology and results on Shimura varieties; as a general reference for these, we use [19, Sections 1–2]. So let 𝕊 := Res_{ℂ/ℝ}𝔾_{m,ℂ} be the algebraic group over ℝ obtained by restriction of scalars from ℂ to ℝ of the multiplicative group. For V an ℝ-vector space, it is then equivalent to give an ℝ-Hodge structure or an action by 𝕊 on it. A Shimura datum is a pair (G, X), with G a connected reductive affine algebraic group over ℚ, and X a G(ℝ)-conjugacy class in the set of morphisms of algebraic groups Hom(𝕊, G_{ℝ}), satisfying the three conditions of [19, Def. 1.4] (i.e., the usual conditions (2.1.1–3) of [13]). These conditions imply that X has a natural complex structure (in fact, the connected components are hermitian symmetric domains), such that every representation of G on a ℚ-vector space defines a polarizable variation of Hodge structure on X. For (G, X) a Shimura datum, and K a compact open subgroup of G(𝔸_{f}), we let Sh_{K}(G, X)(ℂ) denote the complex analytic variety G(ℚ)\(X × G(𝔸_{f})/K), which has a natural structure of quasiprojective complex algebraic variety, denoted Sh_{K}(G, X)_{ℂ}; the projective limit Sh(G, X)_{ℂ} over all K of the Sh_{K}(G, X)_{ℂ} is a scheme on which G(𝔸_{f}) acts continuously. (The action being continuous means that the scheme has a cover by open affines U_{i} = Spec(A_{i}) such that each U_{i} is stabilized by some open subgroup K_{i} of G(𝔸_{f}) and each f in A_{i} has open stabilizer in K_{i}.) A morphism of Shimura data from (G_{1}, X_{1}) to (G_{2}, X_{2}) is a morphism f: G_{1} → G_{2} that maps X_{1} to X_{2}; for K_{1} and K_{2} compact open subgroups of G_{1}(𝔸_{f}) and G_{2}(𝔸_{f}) with f(K_{1}) contained in K_{2}, such an f induces a morphism Sh(f) from Sh_{K_{1}}(G_{1}, X_{1})_{ℂ} to Sh_{K_{2}}(G_{2}, X_{2})_{ℂ}.

∗A preprint version of this article together with an appendix that could be considered as the author’s scratch paper while working on this subject can be downloaded from the author’s home page. This appendix contains details that the author does not find interesting enough to publish, but that may be helpful for readers who got lost.

†partially supported by the Institut Universitaire de France, and by the European TMR Network Contract ERB FMRX 960006 “arithmetic algebraic geometry”.

1.1 Definition. Let (G, X) be a Shimura datum, K an open compact subgroup of G(𝔸_{f}), and Z an irreducible closed subvariety of Sh_{K}(G, X)_{ℂ}. Then Z is a subvariety of Hodge type if there is a Shimura datum (G′, X′), a morphism of Shimura data f: (G′, X′) → (G, X), and an element g of G(𝔸_{f}) such that Z is an irreducible component of the image of the map:

Sh(G′, X′)_{ℂ} ⟶^{Sh(f)} Sh(G, X)_{ℂ} ⟶^{·g} Sh(G, X)_{ℂ} ⟶ Sh_{K}(G, X)_{ℂ}.

This definition is equivalent to [19, 6.2], which uses only closed immersions f: G′ → G. In [20, Prop. 2.8] it is explained that the subvarieties of Hodge type are precisely the loci where certain given classes in certain variations of Hodge structures (obtained from representations of G) are Hodge classes; hence the terminology.

1.2 Definition. Let (G, X) be a Shimura datum. For h in X we let MT(h) be the Mumford-Tate group of h, i.e., the smallest algebraic subgroup H of G such that h factors through H_{ℝ}. A point h in X is called special if MT(h) is commutative (in which case it is a torus). For K a compact open subgroup of G(𝔸_{f}), a point in Sh_{K}(G, X)_{ℂ} is special if its preimages in Sh(G, X)_{ℂ} are of the form (h, g) with h in X special. Equivalently, the special points in Sh_{K}(G, X)_{ℂ} are the zero dimensional subvarieties of Hodge type.

1.3 Conjecture. (André-Oort) Let (G, X) be a Shimura datum. Let K be a compact open subgroup of G(𝔸_{f}) and let S be a set of special points in Sh_{K}(G, X)(ℂ). Then every irreducible component of the Zariski closure of S in Sh_{K}(G, X)_{ℂ} is a subvariety of Hodge type.

Some remarks are in order at this point. André stated this conjecture as a problem for curves containing infinitely many special points in general Shimura varieties in [2, X.4]. Independently, Oort raised the question for general subvarieties of the moduli spaces of principally polarized abelian varieties in [25]. In [2, X.4] André mentions the similarity with the Manin-Mumford conjecture (proved by Raynaud, see [1]), and [4] contains a version generalizing both the conjecture above and the Manin-Mumford conjecture; see also [19, 6.7.2].

Let us now discuss the results on the conjecture above that have been obtained until now. All of them deal with moduli spaces of abelian varieties. Moonen proved in his thesis (see [21, §5], in particular the equivalence between Conjectures 5.1 and 5.3, and [18, IV]) that the conjecture is true for sets S for which there exists a prime number p at which all s in S have an ordinary reduction of which they are the canonical lift. Needless to say, his methods use reduction modulo a prime number p. This gives a quite general result, but it has the disadvantage that one neglects most of the Galois action on the special points, and that one has to work with one Frobenius element simultaneously for all s in S.

In [14], the conjecture was proved for the moduli space of pairs of elliptic curves, assuming the generalized Riemann hypothesis (GRH) for imaginary quadratic fields. In a few words, the proof exploits the Galois action on CM-points and considers intersections of the subvarieties in question with images of them under suitable Hecke operators. In this approach, we work with a different Frobenius element for each s in S; GRH comes in via the existence of small primes with suitable properties. The same case of Conjecture 1.3 was proved unconditionally by André in [5]. He uses the Galois action on the CM-points, and a Diophantine approximation result of Masser on the j-function.

More recently, Yafaev has generalized the result in [14] to the case of products of two Shimura curves that are associated to quaternion algebras over ℚ, see [31], and B. Belhaj Dahman, a student of André, is working on the families of jacobians of the curves

y^{n} = x(x − 1)(x − λ).

The question about these families of jacobians is whether or not the various isogeny factors coming from the decomposition for the action of μ_{n}(ℂ) are simultaneously of CM type for infinitely many complex numbers λ.

Recently, Clozel and Ullmo have proved ([10]), for G among GSp_{2n} and GL_{n}, that sets of the form T_{p}x, with x in G(ℚ)\G(𝔸)/K and T_{p} certain Hecke operators with p tending to infinity, are equidistributed. The idea behind this is that one would like to imitate and apply the equidistribution results for Galois orbits of strict sequences of points of small height in abelian varieties as in [1]. A sequence of closed points of an algebraic variety is called strict if every proper closed subset contains only finitely many elements of the sequence. Phrased in this terminology, the André-Oort conjecture says that a sequence of special points is strict if no proper subvariety of Hodge type contains an infinite subsequence. Of course, to prove the André-Oort conjecture in this way, one has to replace T_{p}x by the Galois orbit of x, which seems to be a hard problem, and moreover, one has to deal with the fact that the heights of CM points tend to infinity and not to zero.

In this article, we prove the Conjecture 1.3, assuming GRH, for Hilbert modular surfaces. The method of proof is basically the same as in [14], but now we do use more advanced techniques. The two main results of the article are described in Section 2. The reason for which we state and prove Theorem 2.2 is that it has an interesting application to transcendence of special values of certain hypergeometric functions via work of Wolfart, Cohen and Wüstholz, see [11], without having to assume GRH.

Let us briefly describe the contents of this article. Section 2 introduces the Hilbert modular surfaces that we work with in terms of a Shimura datum, gives their interpretation as moduli spaces of abelian surfaces with multiplications by the ring of integers of a real quadratic field K, and states the main results.

Section 3, which is not so essential, discusses the difference between working with abelian surfaces with or without a given polarization. In group theoretical terms, the choice is between working with GL_{2}(K) or its subgroup GL_{2}(K)′ consisting of the elements of GL_{2}(K) whose determinant is in ℚ^{∗}. The reason for considering both cases is that with a polarization (and a suitable level structure), the variation of Hodge structure provided by the lattices of the abelian surfaces comes from a representation of the group in the Shimura datum, which is not true without given polarizations. We need variations of Hodge structure in Section 4. On the other hand, the size of Galois orbits of special points, studied in Section 6, is simpler to understand in terms of class groups when working without polarizations. We could have chosen to work throughout the article with GL_{2}(K)′, but we think that it is instructive to see the consequences of such a choice in the relatively easy case of Hilbert modular surfaces, before trying to treat general Shimura varieties completely in group theoretical terms.

In Section 4 we recall an important result of André, relating the generic Mumford-Tate group of a variation of Hodge structure to its algebraic monodromy group (i.e., the Zariski closure of the image of monodromy). We use it to prove that for a curve in a Hilbert modular surface that is not of Hodge type and that does contain a special point, the connected algebraic monodromy group is maximal, i.e., SL_{2,K}.

Section 5 introduces the Hecke correspondence T_{p} associated to a prime number p. We use a very powerful result of Nori in order to prove that for C a curve with maximal algebraic monodromy group, T_{p}C is irreducible if C is large enough.

The main result of Section 6 says that the size of the Galois orbit of a special point x grows at least as a positive power of the discriminant discr(R_{x}) of the ring of endomorphisms (commuting with the real multiplications) of the corresponding abelian variety. This section is quite long, and contains some messy computations, depending on the structure of the Galois group of the normal closure of the CM field in question. The problem is that one has to give a lower bound for the image under the reflex type norm of one class group in another.

Section 7 gives an upper bound for the number of points in intersections of the form Z_{1}∩T_{g}Z_{2}, with Z_{1} and Z_{2} fixed subvarieties of a general Shimura variety, and with T_{g} a varying Hecke correspondence.

Finally, Section 8 combines all these preliminary results as follows. One supposes that C is a curve in a Hilbert modular surface S, containing infinitely many special points, and not of Hodge type. If p is large enough (depending only on C), then T_{p}C is irreducible by Section 5.

Since the T_{p}-orbits in S are dense, one cannot have C = T_{p}C. Hence the intersections C ∩ T_{p}C are finite, and hence bounded above (Section 7) by a constant times p^{2}. Let now x be a special point on C. If p is a prime that is split in R_{x}, then C ∩ T_{p}C contains the Galois orbit of x, hence |C ∩ T_{p}C| grows at least as a positive power of |discr(R_{x})|. But this lower bound for primes that are split in R_{x} contradicts the conditional effective Chebotarev theorem (this is where GRH comes in). Hence, assuming GRH, one has proved that if C does contain infinitely many special points, then C is of Hodge type. The reason that one can prove Thm. 2.2 unconditionally is that in that case the CM field ℚ ⊗ R_{x} is independent of x, and hence Chebotarev’s theorem itself is sufficient.

In April 1999, we have proved Conjecture 1.3, assuming GRH, for arbitrary products of modular curves, extending the methods of [14]. A detailed proof, which is quite elementary, will be written up in the near future. One can hope that combining the techniques used for these last two results will make it possible to treat more general higher dimensional cases of Conjecture 1.3. Of course, eventually everything should be expressed in terms of “(G, X)-language”. In fact, in this article we could already have worked without mentioning abelian varieties.

Before we really start, let us first mention two obvious general principles. The first is that level structures don’t matter in Conjecture 1.3: for (G, X) a Shimura datum, K and K′ open compact in G(𝔸_{f}) with K ⊂ K′, an irreducible subvariety Z of Sh_{K}(G, X)_{ℂ} is of Hodge type if and only if its image in Sh_{K′}(G, X)_{ℂ} is. The second principle says that the irreducible components of intersections of subvarieties of Hodge type are again of Hodge type (this is clear from the interpretation of subvarieties of Hodge type given right after Definition 1.1).

2 The main results.

Let K be a real quadratic extension of ℚ, let O_{K} be its ring of integers, and let G be the ℤgroup scheme Res_{O_{K}/ℤ}(GL_{2,O_{K}}). After numbering the two embeddings of K in ℝ, we have ℝ ⊗ K = ℝ^{2}, and hence G(ℝ) = GL_{2}(ℝ)^{2}. We will study the Shimura variety:

S(ℂ) := G(ℚ)\(X × G(𝔸_{f})/G(ℤ̂)),

where X = (ℍ^{±})^{2}, and where ℍ^{±} is the usual GL_{2}(ℝ)-conjugacy class of morphisms from 𝕊 to GL_{2,ℝ}, i.e., the class of a + bi ↦ ( \begin{matrix}a & −b \\ b & a \end{matrix} ). The surface S_{ℂ}, called a Hilbert modular surface, is the coarse moduli space for pairs (A, α) with A an abelian surface and α a morphism from O_{K} to End(A) (see [30, Ch. X], and the end of Section 3 for the moduli interpretation for a closely related Shimura datum). This implies that the reflex field of (G, X) is ℚ and that the canonical model S_{ℚ} (see [19, Section 2] for this notion) is simply the coarse moduli space for pairs (A/S/ℚ, α) with S a ℚ-scheme, A/S an abelian scheme of relative dimension two, and α a morphism from O_{K} to End_{S}(A). The set of geometrically connected components of S_{ℚ} is K^{∗}\𝔸_{K}^{∗}/(ℝ ⊗ K)^{∗,+}O_{K}^{∧∗} = Pic(O_{K})^{+}, the group of isomorphism classes of invertible O_{K}- modules with orientations at the two infinite places, and has trivial action by Gal(\overline{ℚ}/ℚ) ([30, Ch. I, Cor. 7.3]). The main objective of this article is to prove the following two theorems.

2.1 Theorem. Assume GRH. Let C ⊂ S_{ℂ} be an irreducible closed curve containing infinitely many CM points. Then C is of Hodge type.

2.2 Theorem. Let C ⊂ S_{ℂ} be an irreducible closed curve containing infinitely many CM points corresponding to abelian varieties that lie in one isogeny class (the isogenies are not required to be compatible with the multiplications by O_{K}). Then C is of Hodge type.

Let us note immediately that these theorems apply in fact to all Hilbert modular surfaces, because the André-Oort conjecture is insensitive to level structure. Before proving the theorems we need to discuss some of the tools we will use in it.

3 Choosing a suitable Shimura variety.

For a variation of Hodge structure on a complex variety, one has the notions of generic Mumford-Tate group and that of monodromy. A relation between these two notions will be very useful for us. In order to get a suitable variation of Hodge structure on S(ℂ) as above, there is a little complication, and at least two options to get around it. The problem is that the tautological representation of G_{ℚ} on the ℚ-vector space K^{2} does not induce a variation of Hodge structure on Sh_{H}(G_{ℚ}, X)(ℂ), even if H is an arbitrary small open subgroup of G(𝔸_{f}); just consider the action of O_{K}^{∗} in G(ℚ) on X × G(𝔸_{f})/H (see [20, Section 2.3] for a general statement).

The first possible way out is to use an other representation, and have the monodromy take place in the image of G under this representation. For example, one can take the representation Sym^{2}(ρ_{0}) ⊗ det(ρ_{0})^{−1}, with ρ_{0} the tautological representation on O_{K}^{2}. This representation induces a faithful representation ρ of the quotient G^{ad} = Res_{O_{K}/ℤ}PGL_{2,O_{K}}. The morphism G → G^{ad} induces an isomorphism from X to a conjugacy class X^{ad} in Hom_{ℝ}(𝕊, G_{ℝ}^{ad}), and gives a morphism of Shimura data from (G, X) to (G^{ad}, X^{ad}). Let S^{ad}(ℂ) be the Shimura variety Sh_{G^{ad}(ℤ̂)}(G^{ad}, X^{ad})(ℂ). The natural morphism from S_{ℂ} to S_{ℂ}^{ad} is finite and surjective (this follows directly from the definition); one can show that it is the quotient for a faithful action of Pic(O_{K}), but that will not be used. Conjecture 1.3 is then true for S_{ℂ} if and only if it is true for S_{ℂ}^{ad}, and ρ induces a variation of Hodge structure on Sh_{H}(G^{ad}, X^{ad})(ℂ) for suitable H. The disadvantage of working with S_{ℂ}^{ad} is that it does not seem to have an interpretation as a moduli space of abelian varieties; this is not a real problem, but we prefer to work with Shimura varieties that are as simple as possible.

Another way out is to replace the group G by its subgroup G′ given by the following Cartesian diagram:

G′ ↪  G
↓ □  ↓det
𝔾_{m,ℤ} ↪ Res_{O_{K}/ℤ}𝔾_{m,O_{K}}

Loosely speaking, G′ is the subgroup of G consisting of those elements whose determinant is in ℚ. As the morphism det in the diagram above is smooth, G′ is smooth over 𝔾_{m,ℤ}, hence over ℤ. It follows that G′ is the scheme-theoretic closure in G of its generic fibre. We note that G′(ℝ) is the subgroup of (x, y) in GL_{2}(ℝ)^{2} with det(x) = det(y). All h in X factor through G′_{ℝ}, but X consists of two G′(ℝ)-conjugacy classes. The conjugacy class X′ we work with is the disjoint union of (ℍ^{+})^{2} and (ℍ^{−})^{2}. This gives a morphism of Shimura data from (G′, X′) to (G, X), and a morphism of Shimura varieties S′_{ℂ} → S_{ℂ} with S′_{ℂ} = Sh_{G′(ℤ̂)}(G′, X′)_{ℂ}. One can prove that the Shimura variety S′_{ℂ} is connected, and that the morphism to its image in S_{ℂ} is the quotient by a faithful action of the finite group O_{K}^{∗,+}/O_{K}^{∗,2}, i.e., by the group of totally positive global units modulo squares of global units. We will only use that the morphism S′_{ℂ} → S_{ℂ} is finite and that its image is open and closed; these two facts follow directly from the definitions. It follows that Conjecture 1.3 is true for S_{ℂ} if and only if it is for S′_{ℂ}, and similarly for the two theorems above that we want to prove. Moreover, the tautological representation of G′ does induce a variation of Hodge structure on Sh_{H}(G′, X′)(ℂ) for H sufficiently small.

The option we choose is the last. The variety S′_{ℂ} is the (coarse) moduli space for triplets (A, α, λ) where:

(3.1)  \{ \begin{matrix}A is a complex abelian surface, \\ α: O_{K} → End(A) a morphism of rings, \\ and λ: A → A^{∗} a principal O_{K}-polarization, \end{matrix}

a notion that we will now explain. First of all, A^{∗} is the dual of A in the category of abelian varieties with O_{K}-action: A^{∗} := Ext^{1}(A, O_{K} ⊗ 𝔾_{m}). One verifies that A^{∗} = δ ⊗_{O_{K}} A^{t}, where δ is the different of the extension ℤ → O_{K}, and where A^{t} = Ext^{1}(A, 𝔾_{m}), the dual of A in the usual sense. The inclusion δ ⊂ O_{K} induces a morphism A^{∗} → A^{t}, which is an isogeny. A principal O_{K}-polarization is then an isomorphism λ: A → A^{∗} such that the induced morphism from A to A^{t} is a polarization. Interpreted in Hodge-theoretical terms, a triplet (A, α, λ) as in (3.1) corresponds to a triplet (V, h, ψ) with V a locally free O_{K}-module of rank two, h: 𝕊 → (GL_{ℤ}(V))_{ℝ} a Hodge structure of type (−1, 0), (0, −1), and ψ: V × V → O_{K} a perfect antisymmetric O_{K}- bilinear form such that tr∘ψ: V ×V → ℤ is a polarization. Note that for such a triplet (V, h, ψ), the pair (V, ψ) is isomorphic to the standard pair (O_{K} ⊕ O_{K}, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )). In order to prove that the set of isomorphism classes of (A, α, λ) as in (3.1) is S′(ℂ) one uses the following two facts: 1: G′(𝔸_{f})/G′(ℤ̂) is the set of O_{K}-lattices in K^{2} on which ψ = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) induces a perfect pairing of O_{K}-modules, up to a factor in ℚ^{∗}; and 2: X′ is the set of Hodge structures of type (−1, 0), (0, −1) on the K-vector space K^{2} such that, up to sign, tr ∘ ψ is a polarization. The moduli space over ℚ of triplets (A/S, α, λ) with S a ℚ-scheme, α: O_{K} → End_{S}(A) a morphism of rings, and λ a principal O_{K}-polarization, is then the canonical model S′_{ℚ} of S′_{ℂ} (see also [27, 1.27] and [12, 4.11]).

For n ≥ 1, let H_{n} be the kernel of the morphism G′(ℤ̂) → G′(ℤ/nℤ), and let S′_{ℚ,n} denote the Shimura variety Sh_{H_{n}}(G′, X′)_{ℚ}. Then S′_{ℚ,n} is the moduli space for 4tuples (A/S, α, λ, ϕ), with S a ℚ-scheme, (A/S, α, λ) an abelian scheme over S with multiplications by O_{K} and a principal O_{K}-polarization, and with ϕ an isomorphism of S-group schemes with O_{K}-action:

ϕ: (O_{K}/nO_{K})_{S}^{2} ⟶ A[n],

such that there exists a (necessarily unique) isomorphism \overline{ϕ}: (ℤ/nℤ)_{S} → μ_{n,S} making the diagram:

((O_{K}/nO_{K})_{S}^{2})^{2} ⟶^{ϕ}  A[n]^{2}
↓ ψ_{n}  ↓ e_{λ,n}
O_{K} ⊗ (ℤ/nℤ)_{S} ⟶^{id⊗\overline{ϕ}} O_{K} ⊗ μ_{n,S}

commutative. In this diagram, ψ_{n} is the pairing given by ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ), and e_{λ,n} is the perfect pairing on A[n] induced by λ. For n ≥ 3 the objects ((A/S, α, λ)) have no non-trivial automorphisms (see [22, IV, Thm. 5]), and S′_{ℚ,n} is a fine moduli space. (Representability by an algebraic space can be found in [27, §1.23]. Quasi-projectiveness follows from [7].) In particular, for n ≥ 3 we do have a polarized variation of ℤ-Hodge structure on S′_{n}(ℂ), given by the first homology groups of the fibers of the universal family.

4 Monodromy and generic Mumford-Tate groups.

We recall some results that can be found in [20, Sections 1.1–1.3], with references to [9] and [3].

The Mumford-Tate group MT(V) of a ℚ-Hodge structure V, given by h: 𝕊 → GL(V)_{ℝ}, is defined to be the smallest algebraic subgroup H of GL(V)_{ℚ} such that h factors through H_{ℝ}. Equivalently, MT(V) is the intersection in GL(V) of all stabilizers of all lines generated by Hodge classes (i.e., of some type (p, p)) in all ℚ-Hodge structures of the form ⊕_{i}V^{⊗n_{i}}⊗(V^{∗})^{⊗m_{i}}.

For S a smooth complex algebraic variety with a polarizable variation of ℚ-Hodge structure V on the associated analytic variety S(ℂ), there is a countable union Σ of proper algebraic subvarieties such that s ↦ MT(V_{s}) is locally constant outside Σ (this makes sense because V is a locally constant sheaf on S(ℂ)). The smallest such Σ is called the Hodge exceptional locus, and its complement the Hodge generic locus. For s in S(ℂ) and not in Σ, MT(V_{s}) ⊂ GL(V_{s}) is called the generic Mumford-Tate group (at s).

Assume now that S is connected, and that we have an element s of S(ℂ). Then the locally constant sheaf V corresponds to a representation ρ: π_{1}(S(ℂ), s) → GL(V_{s}), called the monodromy representation. The algebraic monodromy group is defined to be the smallest algebraic subgroup H of GL(V_{s}) such that ρ factors through H, i.e., it is the Zariski closure of the image of ρ; its connected component of identity is called the connected algebraic monodromy group, and denoted AM(V_{s}). With these hypotheses, we have the following theorem.

4.1 Theorem. (André) Assume moreover that V admits a ℤ-structure, that s in S(ℂ) is Hodge generic, and that there is a point t in S(ℂ) such that MT(V_{t}) is abelian (i.e., t is special). Then AM(V_{s}) is the derived subgroup MT(V_{s})^{der} of MT(V_{s}), i.e., the algebraic subgroup generated by commutators.

Let us now consider what this theorem implies for the variation of Hodge structure that we have on S′_{n}(ℂ) (n ≥ 3), and, more importantly, for its restrictions to subvarieties of S′_{n}(ℂ). The Hodge exceptional locus of S′_{n}(ℂ) is by construction the union of all lower dimensional subvarieties of Hodge type. The generic Mumford-Tate group on S′_{n}(ℂ) is G′ (use that it contains a subgroup of finite index of G′(ℤ), and that for all h = (h_{1}, h_{2}): ℂ^{∗} → GL_{2}(ℝ)^{2} in X′ one has det(h_{1}(z)) = zz̄ = det(h_{2}(z)) for all z).

4.2 Proposition. Let n ≥ 3. Let C be an irreducible curve in S′_{ℂ,n} (i.e., an irreducible closed subvariety of dimension one); let C^{nor} denote its normalization and C^{sm} its smooth locus. Then C is of Hodge type if and only if the generic Mumford-Tate group on C^{sm} is strictly smaller than G′_{ℚ}. If C is not of Hodge type and contains a special point, then the connected algebraic monodromy group on C^{nor} equals G′_{ℚ}^{der} = Res_{K/ℚ}SL_{2,K}.

Proof. Suppose that C is of Hodge type. Then some element in some tensor construction of the variation of Hodge structure on S′_{ℂ,n} is a Hodge class on C, but not on S′_{ℂ,n}. The interpretation of the Mumford-Tate group as stabilizer of lines generated by Hodge classes shows that the generic Mumford-Tate group on C^{sm} is strictly smaller than G′_{ℚ}. Now suppose that the generic Mumford-Tate group on C^{sm} is strictly smaller than G′_{ℚ}. Then C does carry an extra Hodge class. The locus where this class is a Hodge class is necessarily of dimension one, hence, C, being an irreducible component of it, is of Hodge type. The second statement follows now from André’s theorem above. □

5 Irreducibility of images under Hecke correspondences.

For (G, X) a Shimura datum, K_{1} and K_{2} open subgroups of G(𝔸_{f}), and g in G(𝔸_{f}), one has the so-called Hecke correspondence T_{g} that is defined as follows. Consider the diagram:

Sh_{K_{1}}(G, X)_{ℂ} ⟵^{π_{1}} Sh(G, X)_{ℂ} ⟶^{·g} Sh(G, X)_{ℂ} ⟶^{π_{2}} Sh_{K_{2}}(G, X)_{ℂ},

where π_{1} and π_{2} are the quotient maps for the actions by K_{1} and K_{2}, respectively. The morphism π_{2} ∘ ·g is the quotient for the action of gK_{2}g^{−1}, hence π_{1} and π_{2} ∘ ·g both factor through the quotient by K := K_{1} ∩ gK_{2}g^{−1}, and T_{g} is the correspondence:

Sh_{K_{1}}(G, X)_{ℂ} ⟵^{\overline{π_{1}}} Sh_{K}(G, X)_{ℂ} ⟶^{\overline{π_{2}∘·g}} Sh_{K_{2}}(G, X)_{ℂ}.

Of course, T_{g} exists already over the reflex field E of (G, X). In particular, for Z a closed subvariety of Sh_{K_{1}}(G, X)_{E}, its image T_{g}Z is a closed subvariety of Sh_{K_{2}}(G, X)_{E}.

We now specialize to our situation, i.e., to the Shimura datum (G′, X′) as above. For p a prime number, we let T_{p} be the Hecke correspondence on S′_{ℚ} given by the element g(p) in G′(𝔸_{f}) with g(p)_{p} = ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) and g(p)_{l} = 1 for l different from p. Note that g(p)^{−1} gives the same correspondence as g(p) does, because g(p) = ( \begin{matrix}p & 0 \\ 0 & p \end{matrix})( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )g(p)^{−1}( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). The modular interpretation of T_{p} is the following. Let [(A, λ)] in S′(ℂ) denote the isomorphism class of a complex abelian surface A with multiplications by O_{K} and with a principal O_{K}-polarization λ. Then, as a cycle, the image of [(A, λ)] is given by:

T_{p}[(A, λ)] = ∑_{H}[(A/H, \overline{pλ})],

where H ranges through the O_{K}/pO_{K}-submodules of A[p](ℂ) that are free of rank one, and where \overline{pλ} is the principal O_{K}-polarization induced by pλ on A/H. In order to see this, one uses, as in Section 3, that G′(𝔸_{f})/G′(ℤ̂) is the set of O_{K}-lattices in K^{2} on which ψ = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) induces a perfect pairing of O_{K}-modules, up to a factor in ℚ^{∗}, and that the correspondence on it induced by g(p)^{−1} sends such a lattice to the set of lattices containing it with quotient free of rank one over O_{K}/pO_{K}.

5.1 Proposition. Let C be an irreducible curve in S′_{ℂ}. Suppose that C is not of Hodge type and that it contains a special point. Then, for all primes p large enough, T_{p}C is irreducible.

Proof. Let n ≥ 3 be some integer, and let C_{n} be an irreducible component of the inverse image of C in S′_{ℂ,n}. Irreducibility of T_{p}C_{n} implies that of T_{p}C. Let V denote the polarized variation of ℤ-Hodge structure on S′_{n}(ℂ) that we considered before, let s be in C_{n}(ℂ). We choose an isomorphism of O_{K}-modules from O_{K}^{2} to V_{s}. Let ρ: π_{1}(C_{n}(ℂ), s) → SL_{2}(O_{K}) be the monodromy representation. Proposition 4.2 implies that the Zariski closure in G′ of ρ(π_{1}(C_{n}^{nor}(ℂ), s)) is the subgroup Res_{O_{K}/ℤ}SL_{2,O_{K}}. For p prime, the correspondence T_{p} on S′_{ℂ,n} is given by a diagram:

S′_{ℂ,n} ⟵^{π_{1}} S′_{ℂ,n,p} ⟶^{π_{2}} S′_{ℂ,n}.

For T_{p}C_{n} = π_{2}π_{1}^{−1}C_{n} to be irreducible, it suffices that C_{n,p} be irreducible, with C_{n,p} the covering of C_{n}^{nor} obtained from π_{1}. But this covering corresponds to the π_{1}(C_{n}^{nor}(ℂ), s)-set ℙ^{1}(O_{K}/pO_{K}) of O_{K}/pO_{K}-submodules of (O_{K}/pO_{K})^{2} that are free of rank one. Nori’s Theorem [24, Thm. 5.1] (Theorem 5.2 below) implies that for p large enough, the reduction map from π_{1}(C_{n}^{nor}(ℂ), s) to SL_{2}(O_{K}/pO_{K}) is surjective. Since SL_{2}(O_{K}/pO_{K}) acts transitively on ℙ^{1}(O_{K}/pO_{K}), irreducibility follows. □

5.2 Theorem. (Nori) Let π be a finitely generated subgroup of GL_{n}(ℤ), let H be the Zariski closure of π, and for p prime, let π(p) be the image of π in GL_{n}(𝔽_{p}). Then, for almost all p, π(p) contains the subgroup of H(𝔽_{p}) that is generated by the elements of order p.

6 Galois action.

The aim of this section is to show that the Galois orbits of special points in S′(\overline{ℚ}) are big, in a suitable sense. For A and B abelian surfaces (over some field) with O_{K}-action, we let Hom_{O_{K}}(A, B) be the O_{K}-module of morphisms from A to B that are compatible with the O_{K}- actions.

6.1 Lemma. Let x in S′(\overline{ℚ}) be a special point, corresponding to a triplet (A, α, λ) with A an abelian surface over \overline{ℚ}, α: O_{K} → End(A) and λ a principal O_{K}-polarization. Then End_{O_{K}}(A) is an order, containing O_{K}, of a totally imaginary quadratic extension of K.

Proof. Let R be the endomorphism algebra ℚ ⊗ End(A) of A. Then R is a semi-simple ℚalgebra containing a commutative semi-simple subalgebra of dimension 4. Suppose that A is simple. Then R is a division algebra. Since R acts faithfully on H_{1}(A(ℂ), ℚ), it has dimension dividing 4, hence R is a quadratic extension of K. Since ℝ ⊗ R has a complex structure commuting with the R-action, R is a totally imaginary. Suppose now that A is not simple. Then A is isogeneous to the product of two elliptic curves, B_{1} and B_{2}, say. These elliptic curves are in fact isogeneous to each other, because otherwise K does not admit a morphism to the endomorphism algebra of B_{1} × B_{2}. So A is isogeneous to B^{2}, with B some elliptic curve. Since A is of CMtype, ℚ ⊗ End(B) is an imaginary quadratic field E, and R = M_{2}(E). In this case End_{O_{K}}(A) is an order in the totally imaginary extension K ⊗ E of K. □

6.2 Theorem. There exist real numbers ε > 0 and c > 0 such that for (A, α, λ) corresponding to a special point x in S′(\overline{ℚ}) one has:

|Gal(\overline{ℚ}/ℚ)·x| > c |discr(R_{x})|^{ε},

where R_{x} = End_{O_{K}}(A).

6.3 Remark. The proof will show that one can take ε to be any number less than 1/4. (To get this, one also has to optimize Theorem 6.4, noting that we only apply Stark’s result to fields L of degree at least 4.) Assuming the generalized Riemann hypothesis at this point does not improve this exponent (this is caused by the case where ℚ ⊗ R_{x} is Galois over ℚ with group (ℤ/2ℤ)^{2}).

Proof. Let f: S′_{ℚ} → S_{ℚ} be the morphism induced by the closed immersion of the Shimura data (G′_{ℚ}, X′) → (G_{ℚ}, X). Since f is finite, and since the Hecke correspondences on S_{ℚ} permute the irreducible components transitively, the statement we want to prove is equivalent to its analog for S_{ℚ}. So we will show in fact that there are positive ε and c such that for x special in S(\overline{ℚ}) corresponding to (A, α), we have:

|Gal(\overline{ℚ}/ℚ)·x| > c |discr(R_{x})|^{ε}.

For x special in S(\overline{ℚ}), let L_{x} be ℚ ⊗ R_{x}, and let M_{x} be the Galois closure in \overline{ℚ} of L_{x}. Since M_{x} is of degree at most 8 over ℚ, the statement we want to prove is equivalent to the existence of positive ε and c such that for all special x in S(\overline{ℚ}):

|Gal(\overline{ℚ}/M_{x})·x| > c |discr(R_{x})|^{ε}.

So let now x be special in S(\overline{ℚ}), corresponding to some (A_{x}, α_{x}). To study the Gal(\overline{ℚ}/M_{x})-orbit of x, we construct a zero-dimensional subvariety of Hodge type over M_{x}, containing x, and we use the theory of Shimura-Taniyama on complex multiplication, rephrased in the language of Shimura varieties (see [19, Section 2.2]). Let H_{x} be H_{1}(A(ℂ), ℤ); it is an R_{x}-module that is locally free of rank two as O_{K}-module. Let f: K^{2} → ℚ ⊗ H_{x} be an isomorphism of K-vector spaces. The Hodge structure on H_{x} given by A_{x} gives an element h_{x} of X. The lattice f^{−1}H_{x} in K^{2} corresponds to an element \overline{g_{x}} of G(𝔸_{f})/G(ℤ̂). By construction, x is the image of (h_{x}, g_{x}). Let T_{x} := Res_{L_{x}/ℚ}𝔾_{m,L_{x}}. Then f gives a closed immersion T_{x} → G_{ℚ}. Since ℚ ⊗ H_{x} is a one-dimensional L_{x}-vector space, T_{x} is its own centralizer in G_{ℚ}. It follows that h_{x} factors through T_{x,ℝ}. Hence we have a closed immersion of Shimura data: (T_{x}, {h_{x}}) → (G_{ℚ}, X). The reflex field of (T_{x}, {h_{x}}) is contained in M_{x}, hence we have a canonical model Sh(T_{x}, {h_{x}})_{M_{x}} over M_{x}. We put U_{x} := T_{x}(𝔸_{f}) ∩ g_{x}G(ℤ̂)g_{x}^{−1}. Then one easily verifies that we have an injective morphism of Shimura varieties Sh_{U_{x}}(T_{x}, {h_{x}})_{M_{x}} → S_{M_{x}}, which, on ℂ-valued points, is given by \overline{t} ↦ \overline{(h_{x}, tg_{x})}. By construction, U_{x} is the stabilizer in T_{x}(𝔸_{f}) of the lattice f^{−1}H_{x}; it follows that U_{x} = R_{x}^{∧,∗}, hence:

Sh_{U_{x}}(T_{x}, \{h_{x}\})_{M_{x}}(\overline{ℚ}) = L_{x}^{∗}\(𝔸_{f} ⊗ L_{x})^{∗}/R_{x}^{∧,∗} = Pic(R_{x}).

Our next objective is to describe in sufficient detail the action of Gal(\overline{ℚ}/M_{x}) on Pic(R_{x}) induced by the above bijections. Class field theory gives a continuous surjection from M_{x}^{∗}\𝔸_{M_{x},f}^{∗} to Gal(\overline{ℚ}/M_{x})^{ab}, characterized by the following property. In a representation of Gal(\overline{ℚ}/M_{x})^{ab} that is unramified at a finite place v of M_{x}, the arithmetic Frobenius element is the image of the class of an idèle that is trivial at all places other than v, and the inverse of a uniformizer at v. Let μ: 𝔾_{m,ℂ} → 𝕊_{ℂ} be the cocharacter obtained by composing ℂ^{∗} → ℂ^{∗} × ℂ^{∗}, z ↦ (z, 1) with the inverse of the isomorphism 𝕊(ℂ) = (ℂ ⊗_{ℝ} ℂ)^{∗} → ℂ^{∗} × ℂ^{∗}, x ⊗ y ↦ (xy, x\overline{y}). Then h_{x} ∘ μ is defined over M_{x}, and one defines:

r_{x}: T′_{x} := Res_{M_{x}/ℚ}𝔾_{m,M_{x}} → T_{x}

to be the morphism Res_{M_{x}/ℚ}(h_{x} ∘ μ) composed with the norm map from Res_{M_{x}/ℚ}T_{x,M_{x}} to T_{x}. With these definitions, the quotient Gal(\overline{ℚ}/M_{x})^{ab} of T′_{x}(𝔸_{f}) acts on Pic(R_{x}) via the morphism r_{x}, where we view Pic(R_{x}) as T_{x}(ℚ)\T_{x}(𝔸_{f})/R_{x}^{∧,∗}. It follows that:

|Gal(\overline{ℚ}/M_{x})·x| = |image of r_{x}(T′_{x}(𝔸_{f})) in Pic(R_{x})|.

We will need a more explicit description of r_{x}, in terms of the CM type associated to h_{x}. The morphism h_{x}: ℂ^{∗} → (ℝ ⊗ L_{x})^{∗} extends to a morphism of ℝ-algebras h: ℂ → ℝ ⊗ L_{x}. Extending scalars from ℝ to ℂ gives a morphism of ℂ-algebras id ⊗ h: ℂ ⊗_{ℝ} ℂ → ℂ ⊗ L_{x}. Via the isomorphisms:

ℂ ⊗_{ℝ} ℂ → ℂ × ℂ, x ⊗ y ↦ (xy, x\overline{y}),

and

ℂ ⊗ L_{x} → ℂ^{Hom(L_{x},ℂ)}, x ⊗ y ↦ (ϕ ↦ xϕ(y)),

the idempotent (1, 0) of ℂ×ℂ gives an idempotent in ℂ^{Hom(L_{x},ℂ)}, i.e., a partition of Hom(L_{x}, ℂ) into two sets Φ_{x} and ιΦ_{x}, where ι is the complex conjugation on ℂ. The set Φ_{x} is the CM type corresponding to h_{x}. Since M_{x} is the Galois closure of L_{x} in ℂ, Hom(L_{x}, M_{x}) = Hom(L_{x}, ℂ). With these notations, we have, for any ℚ-algebra R:

r_{x}: (R ⊗ M_{x})^{∗} ⟶ ∏_{ϕ∈Φ_{x}}(R ⊗ L_{x})^{∗}, u ⟼ ∏_{ϕ∈Φ_{x}} Norm_{ϕ}(u),

where Norm_{ϕ} is the norm map of the extension ϕ: R ⊗ L_{x} → R ⊗ M_{x}. Finally, let ϕ_{0} be in Hom(L_{x}, M_{x}), and define Σ_{x,ϕ_{0}} := {g ∈ Gal(M_{x}/ℚ) | gϕ_{0} ∈ Φ_{x}}. Then we have:

ϕ_{0} ∘ r_{x}: T′_{x} ⟶ T_{x} ↪ T′_{x}, u ↦ ∏_{g∈Σ_{x,ϕ_{0}}} g^{−1}u,

for all ℚ-algebras R and all u in (R ⊗ M_{x})^{∗}. This is the description of r_{x} that we work with.

Since M_{x} is generated over K by the extension L_{x} and its conjugate, M_{x} has degree 4 or 8 over ℚ, and its Galois group Gal(M_{x}/ℚ) is isomorphic to ℤ/4ℤ, ℤ/2ℤ × ℤ/2ℤ, or D_{4}, the dihedral group of order 8. We define T to be Res_{K/ℚ}𝔾_{m,K}; note that T is a subtorus of T_{x}, equal to the center of G_{ℚ}. We will see below that r_{x} ∘ ϕ_{0} : T_{x} → T_{x} induces an endomorphism of T_{x}/T whose image, after passing to 𝔸_{f}-valued points, in Pic(R_{x})/Pic(O_{K}) is big enough for our purposes.

Suppose first that Gal(M_{x}/ℚ) is isomorphic to ℤ/4ℤ, say with generator σ. Then M_{x} = L_{x}, σ^{2} is the complex conjugation and K = L_{x}^{⟨σ^{2}⟩}. After changing ϕ_{0}, if necessary, one has that Σ_{x,ϕ} = {1, σ}. The formula above for ϕ_{0} ∘ r_{x} shows that r_{x} is simply given by the element 1 + σ^{−1} of ℤ[Gal(M_{x}/ℚ)]. Since σ^{2} acts as −1 on T_{x}/T, we have r_{x} ∘ (1 − σ^{−1}) = 2 on T_{x}/T. It follows that:

|Gal(\overline{ℚ}/M_{x})·x| ≥ |image of ·2: Pic(R_{x})/Pic(O_{K}) → Pic(R_{x})/Pic(O_{K})|.

Theorem 6.4 below finishes the proof in this case.

Suppose now that Gal(M_{x}/ℚ) is isomorphic to ℤ/2ℤ × ℤ/2ℤ. After changing ϕ_{0}, if necessary, one has Σ_{x,ϕ_{0}} = {1, σ}, with σ of order two and K_{x} := L_{x}^{⟨σ⟩} ≠ K. Let R′_{x} be the order O_{K_{x}} ∩ R_{x} of K_{x}. Since r_{x} is given by 1 + σ, the induced map T′_{x}(𝔸_{f}) → Pic(R_{x}) factors through Pic(R′_{x}) → Pic(R_{x}) induced by the inclusion R′_{x} → R_{x}. The fact that σ acts as 1 on T′′_{x} := Res_{K_{x}/ℚ}𝔾_{m,K_{x}} and as −1 on T_{x}/T′′_{x} implies that the kernel of the map Pic(R′_{x}) → Pic(R_{x}) is killed by multiplication by 2. Since 1 + σ acts as multiplication by 2 on T′′_{x}, we get:

|Gal(\overline{ℚ}/M_{x})·x| ≥ |image of ·4: Pic(R′_{x}) → Pic(R′_{x})|.

Since the order O_{K} ⊗ R′_{x} of L_{x} is contained in R_{x}, and has discriminant discr(R′_{x})^{2}discr(O_{K})^{2}, we have:

|discr(R′_{x})| ≥ |discr(O_{K})|^{−1}|discr(R_{x})|^{1/2}.

The proof in this case is finished by Theorem 6.4.

Suppose that Gal(M_{x}/ℚ) is isomorphic to D_{4}. Let τ and σ be generators of Gal(M_{x}/ℚ), with M_{x}^{⟨τ⟩} = L_{x}, and with σ of order 4. Then σ^{2} is the complex conjugation, and τσ^{−1} = στ. After changing ϕ_{0}, if necessary, we have Σ_{x,ϕ_{0}} = {1, τ, σ, στ}. It follows that ϕ_{0} ∘ r_{x} is given by the element t := 1 + τ + σ^{3} + στ of ℤ[Gal(M_{x}/ℚ)]. Using that Norm_{ϕ_{0}} = 1 + τ, a simple computation gives:

ϕ_{0} ∘ r_{x} ∘ ϕ_{0} ∘ Norm_{ϕ_{0}} = 2(1 + τ) + σNorm_{M_{x}/K}.

It follows that r_{x} ∘ ϕ_{0} acts as 2 on T_{x}/T. We conclude that:

|Gal(\overline{ℚ}/M_{x})·x| ≥ |image of ·2: Pic(R_{x})/Pic(O_{K}) → Pic(R_{x})/Pic(O_{K})|.

Theorem 6.4 finishes the proof in this last case. □

6.4 Theorem. Let K be a totally real number field. There exists c > 0 such that for all orders R, containing O_{K}, in totally complex quadratic extensions L of K, one has:

|image of ·4 on Pic(R)/Pic(O_{K})| ≥ c |discr(R)|^{1/8}.

If one assumes the generalized Riemann hypothesis, then one can replace the exponent 1/8 by any number less than 1/2.

Proof. We will use the following lower bound for class numbers:

let K be a totally real number field; there exists c > 0 such that for all totally complex quadratic extensions L of K, one has:

|Pic(O_{L})| ≥ c |discr(O_{L})|^{1/6}.

In order to prove this, one distinguishes two cases: K = ℚ and K ≠ ℚ, and one notes that the regulator Reg(O_{L}) is at most Reg(O_{K}). In the case K ≠ ℚ one uses the following consequence of Stark’s Theorem 2 in [29]:

for K a totally real number field, there exists c > 0 such that for all totally complex quadratic extensions L of K, one has:

|Pic(O_{L})| ≥ c |discr(O_{L})|^{1/2−1/[L:ℚ]}.

In the case K = ℚ one applies the Brauer-Siegel theorem (see for example [17, Ch. XVI]):

for N > 0 and ε > 0, there exists c > 0 such that for all Galois extensions L of ℚ of degree at most N one has:

|Pic(O_{L})|·Reg(O_{L}) ≥ c |discr(O_{L})|^{1/2−ε}.

Combining these two results, and using that [L : ℚ] ≥ 3 if K ≠ ℚ gives the inequality we want. We could replace the exponent 1/6 by 1/4 if we would just use that [L : ℚ] ≥ 4 if K ≠ ℚ.

Let now K, R and L be as in the theorem. Then R is the inverse image of a subring \overline{R} of some finite quotient \overline{O_{L}} of O_{L}. We have an exact sequence:

0 ⟶ R^{∗} ⟶ O_{L}^{∗} ⟶ \overline{O_{L}}^{∗}/\overline{R}^{∗} ⟶ Pic(R) ⟶ Pic(O_{L}) ⟶ 0.

The torsion of O_{L}^{∗} is bounded in terms of the degree of K, and by Dirichlet’s theorem on units the quotient O_{L}^{∗}/O_{K}^{∗} is finite. The long exact cohomology sequence obtained by taking Gal(L/K)- invariants of the short exact sequence:

0 ⟶ tors(O_{L}^{∗}) ⟶ O_{L}^{∗} ⟶ O_{L}^{∗}/tors(O_{L}^{∗}) ⟶ 0

gives an injection from (O_{L}^{∗}/tors(O_{L}^{∗}))/(O_{K}^{∗}/tors(O_{K}^{∗})) into H^{1}(Gal(L/K), tors(O_{L}^{∗})), showing that (O_{L}^{∗}/tors(O_{L}^{∗}))/(O_{K}^{∗}/tors(O_{K}^{∗})) is of order at most two. We conclude that there exists c > 0, depending only on the degree of K, such that:

|Pic(R)| ≥ c \frac{|\overline{O_{L}}^{∗}|}{|\overline{R}^{∗}|} |Pic(O_{L})|.

On the other hand, we have:

discr(R) = (\frac{|\overline{O_{L}}|}{|\overline{R}|})^{2} discr(O_{L}).

We claim that for every ε > 0 there exists c > 0, depending only on the degree of K, such that:

\frac{|\overline{O_{L}}^{∗}|}{|\overline{R}^{∗}|} ≥ c (\frac{|\overline{O_{L}}|}{|\overline{R}|})^{1−ε} .

To prove this claim, one notes that:

\frac{|\overline{R}|}{|\overline{R}^{∗}|} = ∏_{k res field of \overline{R}} \frac{|k|}{|k^{∗}|}, and \frac{|\overline{O_{L}}|}{|\overline{O_{L}}^{∗}|} = ∏_{k res field of \overline{O_{L}}} \frac{|k|}{|k^{∗}|}.

A simple computation then shows:

\frac{|\overline{O_{L}}^{∗}|}{|\overline{R}^{∗}|} ≥ n ∏_{p|n} (1 − \frac{1}{p})^{[L:ℚ]} ≥ n (\frac{1}{5 log(n)})^{[L:ℚ]} ,

where n = |\overline{O_{L}}|/|\overline{R}| is assumed to be at least 2. We conclude that there exists c > 0, depending only on K, such that:

|Pic(R)| ≥ c |discr(R)|^{1/7}.

In order to finish the proof of the theorem, it suffices to prove that for every ε > 0 there exists c > 0, depending only on K, such that |Pic(R)[2]| ≤ c |discr(R)|^{ε}. To do this, we proceed in the same way as we did in [14, Lemma 3.4]. As Pic(R) is a finite commutative group, the two 𝔽_{2}-vector spaces Pic(R)[2] and 𝔽_{2} ⊗ Pic(R) have the same dimension. The cover of Spec(R) by the disjoint union of Spec(ℤ_{2} ⊗ R) and Spec(R[1/2]) gives an exact sequence:

(ℚ_{2} ⊗ L)^{∗} ⟶ Pic(R) ⟶ Pic(R[1/2]) ⟶ 0.

It follows that dim_{𝔽_{2}} 𝔽_{2}⊗Pic(R) is bounded by dim_{𝔽_{2}} 𝔽_{2}⊗Pic(R[1/2]) plus a number depending only on the degree of K. We put S := Spec(R[1/2]) and T := Spec(O_{K}[1/2]). The long exact sequence associated to the multiplication by two on the sheaf 𝔾_{m} on the etale site S_{et} of S shows that dim_{𝔽_{2}} 𝔽_{2} ⊗ Pic(S) is at most dim_{𝔽_{2}} H^{1}(S_{et}, 𝔽_{2}). Let π: S → T be the morphism induced by the inclusion of O_{K} in R. Then H^{1}(S_{et}, 𝔽_{2}) is the same as H^{1}(T_{et}, π_{∗}𝔽_{2}), and we have a short exact sequence:

0 ⟶ 𝔽_{2,S_{et}} ⟶ π_{∗}π^{∗}𝔽_{2,T} ⟶ j_{!}𝔽_{2,U} ⟶ 0,

where j: U → T is the maximal open immersion over which π is etale. Let i: Z → T be closed immersion giving the complement of U, with Z reduced. Then the long exact sequences of cohomology groups associated to the exact sequence above and to the exact sequence:

0 ⟶ j_{!}𝔽_{2,U_{et}} ⟶ 𝔽_{2,S_{et}} ⟶ i_{∗}𝔽_{2,Z_{et}} ⟶ 0

show that there exists an integer c, depending only on K, such that:

dim_{𝔽_{2}} 𝔽_{2} ⊗ Pic(R) ≤ c + [K : ℚ] |{p prime dividing discr(R)}| .

As 2^{|{p|n}|} = n^{o(1)}, for n → ∞, we have proved the first statement of the theorem. If one assumes GRH, then the Brauer-Siegel theorem as stated above is true without the condition that the extension ℚ → L be Galois, see [17, XIII, §4]. □

7 Intersection numbers.

The aim of this section is to give a bound on intersections of subvarieties of Shimura varieties, provided that they are finite. In particular, we need to study the intersection of a subvariety with its images under Hecke correspondences. As our arguments work for general Shimura varieties, we give such a result in the general case. The main tool used in proving the result is the Baily-Borel compactification, together with its given ample line bundles. We start by recalling some properties of these line bundles, that follow immediately from the results in [7] (see also [6]).

7.1 Theorem. Let (G, X) be a Shimura datum. For K ⊂ G(𝔸_{f}) a compact open subgroup, let S_{K} := Sh_{K}(G, X)_{ℂ} the corresponding complex Shimura variety, and \overline{S}_{K} its Baily-Borel compactification. For every such K, and for every sufficiently divisible positive integer n, the nth power of the line bundle of holomorphic forms of maximal degree of X descends to S_{K}, and extends uniquely to a very ample line bundle L_{K,n} on \overline{S}_{K}, such that, at the generic points of the boundary components of codimension one, it is given by nth powers of forms with logarithmic poles. Let K_{1} and K_{2} be compact open subgroups of G(𝔸_{f}), and g in G(𝔸_{f}) such that K_{2} ⊂ gK_{1}g^{−1}. Then the morphism from S_{K_{2}} to S_{K_{1}} induced by g extends to a morphism f: \overline{S}_{K_{2}} → \overline{S}_{K_{1}}. If n is positive and sufficiently divisible so that L_{K_{1},n} exists, then L_{K_{2},n} exists, and is canonically isomorphic to f^{∗}L_{K_{1},n}.

Proof. Let us briefly recall how the compactification \overline{S}_{K} is defined. Let X^{+} be a connected component of X. Then each connected component of S_{K}(ℂ) is of the form Γ_{i}\X^{+}, with Γ_{i} an arithmetic subgroup of G^{ad}(ℚ) (G^{ad} being the quotient of G by its center). The compactification \overline{S}_{K}(ℂ) is then defined as the disjoint union of the Γ_{i}\\overline{X^{+}}, where \overline{X^{+}} is the union of X^{+} with its so-called rational boundary components, endowed with the Satake topology. It follows that we can write \overline{S}_{K}(ℂ) as G(ℚ)\(\overline{X} × G(𝔸_{f})/K), with \overline{X} the disjoint union of the \overline{X^{+}}.

Let X^{ad} be the G^{ad}(ℝ)-conjugacy class of morphisms from 𝕊 to G_{ℝ}^{ad} containing the image of X. Each connected component of X maps isomorphically to one of X^{ad} (see [19, 1.6.7]). We first prove the Theorem above for the Shimura datum (G^{ad}, X^{ad}). The group G^{ad} is a product of simple algebraic groups G_{j} over ℚ, and X^{ad} decomposes as a product of X_{j}’s. For compact open subgroups K, K_{1} and K_{2} that are products of compact open subgroups of the G_{j}(𝔸_{f}), the corresponding Shimura varieties decompose as a product, so that it suffices to treat the G_{j} separately. If (G_{j}, X_{j}) gives compact Shimura varieties, Kodaira’s theorem ([15, Section 1.4]) implies what we want, for compact open subgroups K_{j} that are sufficiently small; for arbitrary K_{j} one takes quotients by finite groups. Suppose now that (G_{j}, X_{j}) does give Shimura varieties that are not compact. If G_{j} is of dimension 3, then it is isomorphic to PGL_{2,ℚ}, and we are in the case of modular curves, where the Theorem we are proving is well known (the canonical line bundle with log poles at the cusps on the modular curve X(n), n ≥ 3, has degree > 0). Suppose now that G_{j} has dimension > 3. Then the boundary components are of codimension > 1, and the results we want are given in [7, Thm. 10.11].

The case of arbitrary open compact subgroups of G^{ad}(𝔸_{f}) follows by considering quotients by finite groups. The theorem for (G, X) itself follows from the fact that the connected components of the S_{K}(ℂ) are of the Γ\X^{+}, with Γ an arithmetic subgroup of G^{ad}(ℚ). □

7.2 Theorem. Let (G, X) be a Shimura datum, let K_{1} and K_{2} be compact open subgroups of G(𝔸_{f}), and let Z_{1} and Z_{2} be closed subvarieties of the Shimura varieties S_{1} := Sh_{K_{1}}(G, X)_{ℂ} and S_{2} := Sh_{K_{2}}(G, X)_{ℂ}, respectively. Suppose that Z_{1} or Z_{2} is of dimension at most one. Then there exists an integer c such that for all g in G(𝔸_{f}) for which T_{g}Z_{1} ∩ Z_{2} is finite, one has:

|T_{g}Z_{1} ∩ Z_{2}| ≤ c deg(\overline{π_{1}} : S_{g} → S_{1}),

where S_{g} = Sh_{K_{g}}(G, X)_{ℂ} with K_{g} = K_{1} ∩ gK_{2}g^{−1}, and with T_{g} and \overline{π_{1}} as in the beginning of Section 5.

Proof. We start with two reductions. First of all, writing Z_{1} and Z_{2} as the unions of their irreducible components, one sees that we may suppose that Z_{1} and Z_{2} are irreducible. Secondly, for g in G(𝔸_{f}), let p_{1,g} and p_{2,g} be the morphisms from S_{g} to S_{1} and S_{2}, respectively. Then one has:

T_{g}Z_{1} ∩ Z_{2} = p_{2,g} (p_{1,g}^{−1}Z_{1} ∩ p_{2,g}^{−1}Z_{2}) ,

which shows that T_{g}Z_{1} ∩ Z_{2} is finite if and only if p_{1,g}^{−1}Z_{1} ∩ p_{2,g}^{−1}Z_{2} is, and that |T_{g}Z_{1} ∩ Z_{2}| is at most |p_{1,g}^{−1}Z_{1} ∩ p_{2,g}^{−1}Z_{2}|. This also shows that we may replace K_{1} and K_{2} by smaller compact open subgroups. Hence we may suppose, by the previous theorem, that we have very ample line bundles L_{1} and L_{2} on the Baily-Borel compactifications \overline{S_{1}} and \overline{S_{2}} such that, for each g, \overline{p_{1,g}}^{∗}L_{1} and \overline{p_{2,g}}^{∗}L_{2} are isomorphic to the same line bundle L_{g} on \overline{S_{g}}.

We let \overline{Z_{1}} and \overline{Z_{2}} be the closures of Z_{1} and Z_{2} in \overline{S_{1}} and \overline{S_{2}}, respectively. Let m denote the degree of \overline{Z_{2}} with respect to L_{2}. Let g be in G(𝔸_{f}), such that T_{g}Z_{1} ∩ Z_{2} is finite. If the intersection is empty, there is nothing to prove, so we suppose that the intersection is not empty. Then the codimension of Z_{2} is at least the dimension d of Z_{1}, and we can choose f_{1}, …, f_{d} in H^{0}(\overline{S_{2}}, L_{2}^{⊗m}) such that \overline{Z_{2}} is contained in V_{S_{2}}(f_{1}, …, f_{d}), and \overline{T_{g}Z_{1}} ∩ V_{S_{2}}(f_{1}, …, f_{d}) is finite (because of our assumption on the dimensions of Z_{1} and Z_{2}, \overline{Z_{1}} ∩ \overline{Z_{2}} is finite). It then follows that |\overline{p_{1,g}^{−1}Z_{1}} ∩ \overline{p_{2,g}^{−1}Z_{2}}| is at most m^{d} times the degree of \overline{p_{1,g}^{−1}Z_{1}} with respect to L_{g}. But this degree is deg(p_{1,g}) times the degree of \overline{Z_{1}} with respect to L_{1}, hence we have:

|T_{g}Z_{1} ∩ Z_{2}| ≤ deg(p_{1,g})m^{d} deg_{L_{1}}(\overline{Z_{1}}).
□

8 Proof of the main results.

We will now prove Theorems 2.1 and 2.2. We first deal with Thm. 2.1. As we have already noticed, we may as well replace S_{ℂ} by S′_{ℂ}, so let C be an irreducible closed curve in S′_{ℂ} that contains infinitely many CM points. We have to show that C is of Hodge type.

Since C has infinitely many points in S′(\overline{ℚ}), it is, as a reduced closed subscheme of S′_{ℂ}, defined over \overline{ℚ}. To be precise, there is a unique closed subscheme C\overline{ℚ} of S′ ; \overline{ℚ} that gives C after base change from \overline{ℚ} to ℂ. But then C\overline{ℚ} has only finitely many conjugates under Gal(\overline{ℚ}/ℚ); we let C_{ℚ} be the reduced closed subscheme of S′_{ℚ} that, after base change to \overline{ℚ}, gives the union of these conjugates. In other words, we simply let C_{ℚ} be the image of C under the morphism of schemes S′_{ℂ} → S′_{ℚ}.

Let x in C(\overline{ℚ}) be a CM point, corresponding to a pair (A, λ) with A an abelian variety over \overline{ℚ} with multiplications by O_{K} and with λ a principal O_{K}-polarization. As before, we let R_{x} denote End_{O_{K}}(A), L_{x} := ℚ ⊗ R_{x} and M_{x} the Galois closure of L_{x} in ℂ. Let \overline{x} be the image of x in S(\overline{ℚ}). In the proof of Theorem 6.2 we have seen that the quotient Gal(\overline{ℚ}/M_{x})^{ab} of (𝔸_{f} ⊗M_{x})^{∗} acts on the subset Gal(\overline{ℚ}/M_{x})·\overline{x} of L_{x}^{∗}\(𝔸_{f} ⊗ L_{x})^{∗}/R_{x}^{∧,∗} via the morphism

r_{x}: (𝔸_{f} ⊗ M_{x})^{∗} ⟶ (𝔸_{f} ⊗ L_{x})^{∗}, u ↦ ∏_{ϕ∈Φ_{x}} Norm_{ϕ}(u),

where the subset Φ_{x} of Hom(L_{x}, M_{x}) is the CM type of x. Since Φ_{x} is a set of representatives for the action of Gal(L_{x}/K) on Hom(L_{x}, M_{x}), it follows that this map r_{x} factors through the subgroup of elements of (𝔸_{f} ⊗ L_{x})^{∗} whose norm to (𝔸_{f} ⊗ K)^{∗} is in 𝔸_{f}^{∗}. Hence, in the notation of the proof of Theorem 6.2, the morphism r_{x} factors through the intersection of the subtorus T_{x} of G_{ℚ} and G′_{ℚ}. It follows that the action of Gal(\overline{ℚ}/M_{x}) on Gal(\overline{ℚ}/M_{x})·x is given by the same morphism r_{x}, taking values in G′(𝔸_{f}).

8.1 Lemma. Suppose that p is a prime that is split in R_{x}, i.e., such that 𝔽_{p} ⊗ R_{x} is isomorphic to a product of copies of 𝔽_{p}. Then Gal(\overline{ℚ}/ℚ)x is contained in C_{ℚ}(\overline{ℚ}) ∩ (T_{p}C_{ℚ})(\overline{ℚ}).

Proof. The localization ℤ_{(p)} ⊗ R_{x} of R_{x} is the same as that of O_{K}. Hence, if we let H_{x} denote H_{1}(A(ℂ), ℤ), then ℤ_{(p)} ⊗ H_{x} is free of rank one over ℤ_{(p)} ⊗ R_{x}. It follows that we can choose the isomorphism f: K^{2} → ℚ ⊗ H_{x} to preserve the integral structures on both sides at p, i.e., such that it induces an isomorphism from (ℤ_{(p)} ⊗R_{x})^{2} to ℤ_{(p)} ⊗H_{x}. We note that p is split in M_{x} (i.e., 𝔽_{p} ⊗ O_{M_{x}} is a product of copies of 𝔽_{p}), because M_{x} is the Galois closure of L_{x}. Consider now an element u of (𝔸_{f} ⊗ M_{x})^{∗} that is equal to p at one place above p and equal to 1 at all other finite places of M_{x}. Then r_{x}(u), viewed as an element of (𝔸_{f} ⊗ L_{x})^{∗}, is equal to p at exactly two places of L_{x} above p that are not in the same Gal(L_{x}/K)-orbit, and equal to 1 at all other finite places of L_{x}. It follows that r_{x}(u) is conjugated in G′(𝔸_{f}), by some element in G′(ℤ̂), to the element g(p) that induces T_{p} (use that G′(ℤ_{p}) acts transitively on the set of free rank one ℤ_{p} ⊗ O_{K}-submodules of ℤ_{p} ⊗ O_{K}^{2}). We conclude that r_{x}(u)x is in T_{p}x. But since x is in C_{ℚ}(\overline{ℚ}), r_{x}(u)x is also in C_{ℚ}(\overline{ℚ}). It follows that Gal(\overline{ℚ}/ℚ)x is contained in C_{ℚ}(\overline{ℚ}) ∩ (T_{p}C_{ℚ})(\overline{ℚ}). □

Theorem 6.2 gives a lower bound for |Gal(\overline{ℚ}/ℚ)x|, whereas Theorem 7.2 gives an upper bound for |C_{ℚ}(\overline{ℚ}) ∩ (T_{p}C_{ℚ})(\overline{ℚ})|, assuming that the intersection is finite. What we want, of course, is to show that we can choose x and then p such that the lower bound exceeds the upper bound, and conclude that C_{ℚ} and T_{p}C_{ℚ} do not intersect properly. We note that if x varies over the infinite set of CM points of C(\overline{ℚ}), then |discr(R_{x})| tends to infinity because there are only finitely many orders of degree 2 over O_{K} with a given discriminant, and for each such order there are only finitely many x in S(\overline{ℚ}) with R_{x} isomorphic to that order. Since our lower bound for |Gal(\overline{ℚ}/ℚ)x| is a positive constant times a positive power of |discr(R_{x})|, and our upper bound for |C_{ℚ}(\overline{ℚ}) ∩ (T_{p}C_{ℚ})(\overline{ℚ})| is some fixed power of p, we get what we want if we can take, for |discr(R_{x})| big, p of size something polynomial in log |discr(R_{x})|. We note that |discr(O_{M_{x}})| ≤ |discr(R_{x})|^{4} because M_{x} is the composite of the extension L_{x} of K and its conjugate. We also note that the number of primes dividing discr(R_{x}) is at most log_{2}(|discr(R_{x})|).

At this point we invoke the effective Chebotarev theorem of Lagarias, Montgomery and Odlyzko, assuming GRH, as stated in [28, Thm. 4] and the second remark following that theorem. A simple computation shows that this theorem implies the following result.

8.2 Proposition. For M a finite Galois extension of ℚ, let n_{M} denote its degree, d_{M} its absolute discriminant |discr(O_{M})|, and for x in ℝ, let π_{M,1}(x) be the number of primes p ≤ x that are unramified in M and such that the Frobenius conjugacy class Frob_{p} contains just the identity element of Gal(M/ℚ). Then for M a finite Galois extension of ℚ and x sufficiently big (i.e., bigger than some absolute constant), and bigger than 2(log(d_{M})^{2}(log(log(d_{M})))^{2}, one has:

π_{M,1}(x) ≥ \frac{x}{3n_{M} log(x)}.

This result shows that there exist infinitely many primes p such that C_{ℚ} and T_{p}C_{ℚ} do not intersect properly. Since C_{ℚ} is irreducible, it follows that, for such primes p, C_{ℚ} is contained in T_{p}C_{ℚ}.

Assume now that C is not of Hodge type. Then Proposition 5.1 tells us that for all primes p large enough, T_{p}C is irreducible. Since the correspondence T_{p} is defined over ℚ, i.e., is given by a correspondence on S′_{ℚ}, it follows that T_{p}C_{ℚ} is irreducible for p large enough. But then we see that there exist infinitely many prime numbers p such that C_{ℚ} is equal to T_{p}C_{ℚ}. But this is absurd, since by Lemma 8.3 below, for each x in S′(ℂ), the Hecke orbit ∪_{n≥0}T_{p}^{n}x is dense in S′(ℂ) if p is unramified in K. This finishes the proof of Theorem 2.1.

8.3 Lemma. Let x be in S′(ℂ) and let p be a prime number that is not ramified in K. Then the Hecke orbit ∪_{n≥0}T_{p}^{n}x is dense in S′(ℂ) for the archimedean topology.

Proof. By Lemma 8.4 below, g_{0} := ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) and G′(ℤ_{p}) generate G′(ℚ_{p}) (here we use that p is not ramified in K). Let now x be in S′(ℂ), and let (y, g) be a preimage of it in X × G′(𝔸_{f}) under the quotient map for the action by G(ℚ) × G(ℤ̂). The fact that T_{p} is then given by right multiplication on G′(𝔸_{f}) by the element g_{0} at the place p shows that the T_{p}-orbit of x is the image in S′(ℂ) of the G′(ℚ_{p})-orbit of (y, g). Let now Γ be the subgroup of G′(ℚ) consisting of γ such that γg is in gG′(ℤ̂)G′(ℚ_{p}). Then the T_{p}-orbit of x is the image in S′(ℂ) of the subset Γy of X × {g}. Now one notes that Γ contains a congruence subgroup of G′(ℤ[1/p]). It follows that the intersection of Γ with SL_{2}(O_{K}[1/p]) = G^{der}(ℤ[1/p]) is dense in G^{der}(ℝ) (for the archimedean topology) because G^{der} is generated by additive subgroups. Since G^{der}(ℝ) acts transitively on X^{+} = (ℍ^{+})^{2}, the lemma is proved. □

8.4 Lemma. Let p be a prime that is not ramified in K. Then g_{0} := ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) and G′(ℤ_{p}) generate G′(ℚ_{p}).

Proof. In order to minimize notation, let O_{K,p} denote ℤ_{p} ⊗ O_{K}, let K_{p} denote ℚ_{p} ⊗ O_{K}, and let H denote the subgroup of G′(ℚ_{p}) generated by g_{0} := ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) and G′(ℤ_{p}). Let Y be the set of O_{K,p}-lattices in K_{p}^{2} on which the K_{p}-bilinear form ψ given by ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) is a perfect pairing of O_{K,p}-modules, up to a factor in ℚ_{p}^{∗}. The map G′(ℚ_{p}) → Y, g ↦ gO_{K,p}^{2} induces a bijection from G′(ℚ_{p})/G′(ℤ_{p}) to Y. Hence, in order to prove our claim, it suffices to show that H acts transitively on Y. So let L be in Y. We note that O_{K,p} is either a product of two copies of ℤ_{p}, or the ring of integers ℤ_{p^{2}} in the unramified quadratic extension ℚ_{p^{2}} of ℚ_{p}; in both cases, O_{K,p} is a product of discrete valuation rings with uniformizer p. The theory of finitely generated modules over a discrete valuation ring says that there exists r in ℤ and d_{1} and d_{2} in O_{K,p} such that p^{r}L is contained in O_{K,p}^{2} and has an O_{K,p}-basis of the form (d_{1}e_{1}, d_{2}e_{2}), with (e_{1}, e_{2}) the standard basis of O_{K,p}^{2}. We note that conjugating g_{0} by suitable elements of G′(ℤ_{p}) shows that ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix}) and ( \begin{matrix}p & 0 \\ 0 & p \end{matrix}) are in H, and that, in the split case, (( \begin{matrix}1 & 0 \\ 0 & p \end{matrix}), ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )) is in H. Since the element d_{1}d_{2} of O_{K,p} is the factor by which ψ differs from a perfect pairing on p^{r}L, it is actually in ℤ_{p}. It follows that ( \begin{matrix}d_{1} & 0 \\ 0 & d_{2} \end{matrix} ) is in H. This finishes the proof that H is G′(ℚ_{p}). □

Let us now prove Theorem 2.2. We keep the notations of the proof Theorem 2.1, and we assume again that C is not of Hodge type. So now we may suppose moreover that C contains infinitely many CM points that have the same CM type. In particular, we have infinitely many CM points x such that L_{x} and Φ_{x} are constant, say L and Φ. Of course, the orders R_{x} are such that |discr(R_{x})| tends to infinity. The classical Chebotarev theorem (see for example [17, Ch. VIII, §4]) asserts that the set of primes p that are split in M has natural density 1/[M : ℚ] (actually, Dirichlet density is good enough here). Also, recall that the number of primes dividing some discr(R_{x}) is at most log_{2}(|discr(R_{x})|). Hence there do exist x and p such that p is split in M, split in R_{x}, sufficiently large so that T_{p}C_{ℚ} is irreducible, and such that the lower bound for |Gal(\overline{ℚ}/ℚ)x| of Theorem 6.2 exceeds the upper bound for the intersection C_{ℚ}(\overline{ℚ}) ∩ (T_{p}C_{ℚ})(\overline{ℚ}), if it is finite. Then we have C_{ℚ} = T_{p}C_{ℚ}, hence a contradiction because of Lemma 8.3.

Acknowledgements.

It is a pleasure to thank Rutger Noot, for teaching a very useful course on Shimura varieties, for answering my questions concerning them, and for his useful comments on this manuscript. Without the influence of Johan de Jong, Ben Moonen and Frans Oort I would not even have started working on this subject. I thank Paula Cohen for sending me a preliminary version of [11], and for pointing out to me which version of the André-Oort conjecture was exactly needed in it. Peter Stevenhagen gave me the reference to Stark’s results in [29]. I thank Andrei Yafaev for useful remarks on this manuscript; his numerous questions on the subject have caused me to learn a good deal about Shimura varieties. I thank the organizers of the Texel conference for their excellent work. The referee deserves much credit for pointing out a serious mistake in Section 7 of the preprint version, and for a long list of detailed comments. Last but not least I thank my wife Reinie Erné for her influence on this article via conversations both at work and at home, and for letting me sleep when I am too tired to fetch a bottle for Tom at five o’clock in the morning.

References

[1] A. Abbes. Hauteurs et discrétude (d’après L. Szpiro, E. Ullmo et S. Zhang). Séminaire Bourbaki, Vol. 1996/97. Astérisque No. 245, (1997), Exp. No. 825, 4, 141–166.

[2] Y. André. G-functions and geometry. Aspects of mathematics, Vol. E13, Vieweg, Braunschweig, 1989.

[3] Y. André. Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part. Compositio Mathematica 82 (1992), pp. 1–24.

[4] Y. André. Distribution des points CM sur les sous-variétés des variétés de modules de variétés abéliennes. Manuscript, April 1997.

[5] Y. André. Finitude de couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire. J. Reine Angew. Math. 505 (1998), pp. 203–208.

[6] W.L. Baily and A. Borel. On the compactification of arithmetically defined quotients of bounded symmetric domains. Bull. Amer. Soc. 70 (1964), 588–593.

[7] W.L. Baily and A. Borel. Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math. (2) 84 (1966), 442–528.

[8] J-L. Brylisnki and J-P. Labesse. Cohomologie d’intersection et fonctions L de certaines variétés de Shimura. Ann. Sci. E.N.S. 4ème série, 17 (1984), 361–412.

[9] E. Cattani, P. Deligne and A. Kaplan. On the locus of Hodge classes. J. Amer. Math. Soc. 8 (1995), pp. 483–506.

[10] L. Clozel and E. Ullmo. Talk at the Journées Arithmétiques, July 1999, Roma.

[11] P.B. Cohen and G. Wüstholz. Application of the André-Oort conjecture to some questions in transcendence. Preprint.

[12] P. Deligne. Travaux de Shimura. Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, pp. 123–165. Lecture Notes in Math., Vol. 244, Springer, Berlin, 1971.

[13] P. Deligne. Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, in Automorphic forms, representations, and L-functions, Part 2, A. Borel and W. Casselman, eds., Proc. of Symp. in Pure Math., Vol. 33, American Mathematical Society, 1979, pp. 247–290.

[14] S.J. Edixhoven. Special points on the product of two modular curves. Compositio Mathematica 114, 315–328, 1998.

[15] P. Griffiths and J. Harris. Principles of algebraic geometry. Pure and applied mathematics. A Wiley-Interscience publication, John Wiley and sons, New York, Chichester, Brisbane, Toronto, 1978.

[16] A. Grothendieck et. al. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux. North-Holland, Amsterdam (1968).

[17] S. Lang. Algebraic number theory. GTM 110, Springer-Verlag, second edition, 1994.

[18] B.J.J. Moonen. Special points and linearity properties of Shimura varieties. Thesis, Utrecht, September 1995.

[19] B.J.J. Moonen. Models of Shimura varieties in mixed characteristic. Pages 267–350 in “Galois representations in arithmetic algebraic geometry”, edited by A.J. Scholl and R.L. Taylor. Cambridge University Press 1998.

[20] B.J.J. Moonen. Linearity properties of Shimura varieties, I. Journal of Algebraic Geometry 7 (1998), 539–567.

[21] B.J.J. Moonen. Linearity properties of Shimura varieties, II. Compositio Math. 114 (1998), no. 1, 3–35.

[22] D. Mumford. Abelian varieties. Oxford University Press, 1970.

[23] D. Mumford. The topology of normal singularieties of an algebraic surface and a criterion for simplicity. Publications Mathématiques de l’IHES 9 (1961), 5–22.

[24] M.V. Nori. On subgroups of GL_{n}(𝔽_{p}). Invent. math. 88 (1987), pp. 257–275.

[25] F. Oort. Canonical lifts and dense sets of CM-points. Arithmetic Geometry, Proc. Cortona symposium 1994, F. Catanese, ed., Symposia Math., Vol. XXXVII, Cambridge University Press, 1997, pp. 228–234.

[26] V.P. Platonov and S. Rapinchuk. Algebraic groups and number theory. Moscow, 1991 (in Russian). English translation: Academic Press, 1993.

[27] M. Rapoport. Compactifications de l’espace de modules de Hilbert-Blumenthal. Compositio Mathematica 36 (1978), 255–335.

[28] J-P. Serre. Quelques applications du théorème de densité de Chebotarev. Publications Mathématiques de l’IHES 54 (1981), 123–202.

[29] H.M. Stark. Some effective cases of the Brauer-Siegel theorem. Inventiones math. 23, 135– 152 (1974).

[30] G. van der Geer. Hilbert modular surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 16. Springer-Verlag, 1988.

[31] A. Yafaev. Special points on products of two Shimura curves. To appear in Manuscripta Mathematica.

Bas Edixhoven IRMAR Campus de Beaulieu 35042 Rennes cedex France

A Abelian surfaces with real multiplication.

As above, K is a real quadratic field, and O_{K} is its ring of integers. Let us describe a bijection

S(ℂ) = G(ℚ)\((ℍ^{±})^{2} × G(𝔸_{f})/G(ℤ̂)) ⟶^{∼} {(A, α)}/ ≅,

where A is an abelian surface and α: O_{K} → End(A) a morphism of rings. By the way things have been set up, (ℍ^{±})^{2} is the set of Hodge structures of type {(−1, 0), (0, −1)} on the K-vector space K^{2}, i.e., Hodge structures for which K acts by endomorphisms.

The set G(𝔸_{f})/G(ℤ̂) is the set of O_{K}-lattices in K^{2}. (By an O_{K}-lattice in K^{2} we mean a sub-O_{K}-module M of finite type that generates K^{2} as a K-vector space). To see why the two sets are equal, we need an adelic description of the lattices. By an O_{K}^{∧}-lattices in 𝔸_{K,f}^{2} we mean a sub-O_{K}^{∧}-module of 𝔸_{K,f}^{2} that is free of rank two (and hence give the full 𝔸_{K,f}^{2} after tensoring with ℚ). (It is equivalent to consider sub-O_{K}^{∧}-modules of 𝔸_{K,f}^{2} that are of finite type and that generate 𝔸_{K,f}^{2} as K-vector space (or, equivalently, as 𝔸_{K,f}-module).) Now let G(𝔸_{f}) act on the set of O_{K}^{∧}-lattices in 𝔸_{K,f}^{2}. This action is transitive (use that each such a lattice is free of rank two), and the stabilizer of the standard lattice (O_{K}^{∧})^{2} is precisely G(ℤ̂). This means that G(𝔸_{f})/G(ℤ̂) is the set of O_{K}^{∧}-lattices in 𝔸_{K,f}^{2}.

Let us now see why the set of O_{K}-lattices in K^{2} is the set of O_{K}^{∧}-lattices in 𝔸_{K,f}^{2}. This is always the same story, but let me just write it down in this case (the “classical case” as far as I’m concerned is for ℤ). Let M be an O_{K}-lattice in K^{2}. To it, we associate the O_{K}^{∧}-lattice ℤ̂ ⊗ M in 𝔸_{f} ⊗ M = 𝔸_{K,f}^{2}. In the other direction, let N be a O_{K}^{∧}-lattice in 𝔸_{K,f}^{2}. To N, we simply associate N ∩ K^{2}. These two maps are inverses of each other.

We can now show that S(ℂ) is the set of (A, α) up to isomorphism. Let (A, α) be given. Choose an isomorphism of K-vector spaces between K^{2} and H_{1}(A, ℚ). Then we get a Hodge structure on K^{2} and an O_{K}-lattice in K^{2}, hence an element of (ℍ^{±})^{2} × G(𝔸_{f})/G(ℤ̂), defined up to the choice of isomorphism, i.e., up to G(ℚ). Conversely, an element of (ℍ^{±})^{2} × G(𝔸_{f})/G(ℤ̂) gives a pair (A, α), of which the isomorphism class depends only on the G(ℚ)-orbit. So, after all, one just has to view complex abelian varieties as given by a ℚ-Hodge structure and a lattice, and use the usual stuff regarding lattices.

One can of course do something fancy now with the category of abelian varieties up to isogeny, and interpret (ℍ^{±})^{2} × G(𝔸_{f}) as the set of isomorphism classes of (A, α, β, γ), with A an abelian surface up to isogeny, α an K-action on it, β an isomorphism of 𝔸_{K,f}-modules from 𝔸_{K,f}^{2} to H_{1}(A, 𝔸_{f}), and γ an isomorphism of K-vector spaces from K^{2} to H_{1}(A, ℚ).

B Polarizations.

Why do we never have to discuss polarizability of our Hodge structures? Well, that’s because they are in a sense only of dimension two, just as in the case of elliptic curves. So what is in fact true is that every complex torus of dimension two, with an action by O_{K}, is automatically an abelian variety. Of course, this is very standard, but I just write it down for myself, so that I understand it, and so that I have the argument available electronically.

Consider the standard symplectic form on the K-vector space K^{2}:

ψ_{0} : K^{2} × K^{2} ⟶ K, (x, y) ↦ x^{t}Jy, J = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) .

Then we compose this ψ_{0} with an arbitrary non-zero ℚ-linear map l from K to ℚ in order to get a ℚ-bilinear anti-symmetric form:

ψ_{l}: K^{2} × K^{2} ⟶^{ψ_{0}} K ⟶^{l} ℚ.

In particular, one can take for l the trace map; in that case, we will denote ψ_{l} simply by ψ.

For all g in GL_{2}(K) and all x and y in K^{2}, one has ψ_{0}(gx, gy) = det(g)ψ_{0}(x, y). Hence if moreover det(g) is in ℚ, one has: ψ_{l}(gx, gy) = det(g)ψ_{l}(x, y). This means that such a ψ_{l} is a Hodge class of weight two (or is it −2?) for all h: 𝕊_{ℝ} → G_{ℝ} that are in (ℍ^{±})^{2}, since they factor through the subgroup of G of elements that have determinant in 𝔾_{m,ℚ}.

Let us now check that ψ is a polarization on (ℍ^{+})^{2}. So the only condition left to check is that x ↦ ψ(x, h(i)x) should be positive definite on ℝ ⊗ K^{2}. (One checks indeed that this makes sense, in the sense that (x, y) ↦ ψ(x, h(i)y) is symmetric.) The fact that ℝ ⊗ K = ℝ^{2} means that it suffices to check that x ↦ x^{t}h(i)x is positive definite on ℝ^{2} for every h in ℍ^{+}. But now note that for all h in ℍ^{+} and all non-zero x in ℝ^{2}, x and h(i)x are ℝ-linearly independent (interpret h as giving a structure of complex vector space on ℝ^{2}). Hence either x ↦ x^{t}h(i)x is positive definite, or negative definite, for all h in ℍ^{+} simultaneously. So let us check just what it is for the standard h, the one that sends a + bi to ( \begin{matrix}a & −b \\ b & a \end{matrix} ). In that case, one has h(i) = −J, hence x^{t}Jh(i)x = x^{t}x, which is the standard inner product.

On ℍ^{−}, x^{t}Jh(i)x is negative definite, since the standard h there has h(i) = J. So we have seen that ψ is a polarization on (ℍ^{+})^{2}, and −ψ one on (ℍ^{−})^{2}. On the other two components of (ℍ^{±})^{2} one gets polarizations by varying the map l, for example by taking the composition of l with multiplication by a suitable element of K, i.e., an element with the right signs at the two infinite places.

C Some stuff on group (schemes).

Let G denote the group scheme GL_{2} (over ℤ, that is). Let V denote its standard representation. I use in the text that the kernel of G acting on Sym^{2}(V) ⊗ det^{−1} is precisely the scalar subgroup 𝔾_{m} of G. This can be checked as follows. First of all, the pairing V ×V → det(V), (x, y) → xy, is perfect. Hence it gives us an isomorphism between V ⊗ det^{−1} and V^{∗}, the dual of V. Hence we may as well consider End(V) = V^{∗} ⊗ V as V ⊗ V ⊗ det^{−1}. Under this isomorphism, the quotient Sym_{2}(V) ⊗ det^{−1} of V ⊗ V ⊗ det^{−1} corresponds to the quotient of End(V) by the submodule of scalar matrices. So we test the question there. One computes:

(\begin{matrix}a & b \\ c & d \end{matrix}) (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) − (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}0 & −b \\ c & 0 \end{matrix}) ,
(\begin{matrix}a & b \\ c & d \end{matrix}) (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) − (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}b & 0 \\ d − a & b \end{matrix}) .

The condition that these two matrices are scalar give that a = d and b = c = 0.

D The Shimura datum for G^{ad}.

For π: G → G^{ad} as above, we claim that π induces an isomorphism (of real algebraic varieties) from (ℍ^{±})^{2} in Hom_{ℝ}(𝕊, G_{ℝ}) to a conjugacy class (that we will also denote by (ℍ^{±})^{2}) in Hom_{ℝ}(𝕊, G_{ℝ}^{ad}). Let us denote the kernel of π by Z (it is the center of G). Let h_{0} be our standard element in (ℍ^{±})^{2}. Let g be in G(ℝ), and suppose that π∘int_{g} ∘h_{0} = π∘h_{0}. We have to show that int_{g} ∘ h_{0} = h_{0}. Here is how that goes. Define a map (i.e., morphism of real algebraic varieties) z: 𝕊 → Z_{ℝ} by: z(s) = gh_{0}(s)g^{−1}·h_{0}(s)^{−1}. Then, because it goes to the center, z is actually a morphism of groups. All we have to show is that it is trivial. Well, it is trivial on the 𝔾_{m,ℝ} in 𝕊, since that one is mapped centrally in G_{ℝ}. Now the argument is finished by noting that Z_{ℝ} is a split torus, and 𝕊/𝔾_{m,ℝ} is not split.

E Comparing various groups.

It is not yet clear to me with which group I actually want to work. The possibilities are: GL_{2}(K), PGL_{2}(K) and G′ = {g ∈ GL_{2}(K) | det(g) ∈ ℚ}. Just to get some idea of what actually happens with these groups, and with the morphisms of Shimura data between them, I think it is a good idea to make some things explicit, such as the sets of connected components, and the finite maps between the various Shimura varieties.

So let us first think a bit about the π_{0}’s. Let’s first consider G as above. Then clearly we have:

π_{0}(S(ℂ)) = GL_{2}(K)^{+}\GL_{2}(𝔸_{K,f})/GL_{2}(O_{K}^{∧}).

But this set is the set of isomorphism classes of locally free rank 2 O_{K}-modules M with an orientation on det(M) = Λ_{O_{K}}^{2}M at the two infinite places. Now each locally free rank two module over O_{K} is isomorphic to one of the form O_{K} ⊕ L (show first that it is decomposable by choosing a one dimensional K-sub-vector space in ℚ ⊗ M; then show that M has a nowhere vanishing element). Of course, L is determined by M since one has det(M) = L. It follows that:

π_{0}(S(ℂ)) = Pic(O_{K})^{+}, the strict class group of K.

Let us now consider π_{0}(S^{ad}(ℂ)). We have:

π_{0}(S^{ad}(ℂ)) = PGL_{2}(K)^{+}\PGL_{2}(𝔸_{K,f})/PGL_{2}(O_{K}^{∧}).

This we recognize as the set of isomorphism classes of ℙ^{1}-bundles on S := Spec(O_{K}), locally trivial in the Zariski topology, with an orientation at the two infinite places (it does not seem a complete tautology, the correspondence with the Zariski ℙ^{1}-bundles, namely, it says more directly something as: trivial over K, and over every completion). Anyway, let us show that each ℙ^{1}-bundle X on S comes from a locally free rank two bundle on S. Just note that each element of X(K) extends to one in X(S). An element in X(S) gives an invertible O_{X}-module L that has degree one on each fibre, hence with p_{∗}L a rank two bundle on S. Then one checks that X is isomorphic, over S, to ℙ(p_{∗}L) (it is easy to see that X is the Grassmannian of locally free rank one quotients of p_{∗}L). This has an interpretation in the long exact sequence coming from the short exact sequence of Zariski sheaves on S:

1 ⟶ 𝔾_{m,S} ⟶ GL_{2,S} ⟶ PGL_{2,S} ⟶ 1.

What is quite nice in this situation is that S is of dimension one, hence that H^{2}(S, 𝔾_{m}) = 0, which explains the observation above. Now that we know that each element of π_{0}(S^{ad}(ℂ)) comes from a locally free rank two O_{K}-module, we want to know when two such modules give isomorphic ℙ^{1}-bundles. Well, the Grassmannian interpretation says that that happens if and only if the two modules are isomorphic up to twist by an invertible O_{K}-module. Hence:

𝔽_{2} ⊗ Pic(O_{K}) = {ℙ^{1}-bundles on Spec(O_{K})}_{/≅}.

Let us now consider orientations. Note that Aut_{O_{K}}(O_{K} ⊕ L) maps surjectively, under det, to Aut_{O_{K}}(L) = O_{K}^{∗}, and that doubles in Pic(O_{K}) have a canonical orientation. It follows that:

π_{0}(S^{ad}(ℂ)) = Pic(O_{K})^{+}/2Pic(O_{K}),

and that:

#π_{0}(S^{ad}(ℂ)) = \begin{cases}#𝔽_{2} ⊗ Pic(O_{K}) & if N(O_{K}^{∗}) = \{1, −1\}, \\ 2#𝔽_{2} ⊗ Pic(O_{K}) & if N(O_{K}^{∗}) = \{1\}. \end{cases}

Let us now say something about the map S(ℂ) → S^{ad}(ℂ). This makes it necessary to know things about the morphism of group schemes GL_{2,O_{K}} → PGL_{2,O_{K}}. We would like to know that this morphism is surjective for the Zariski topology. For this, it suffices to show that on a scheme S, an S-automorphism of ℙ_{S}^{1} is induced, locally on S, by an element of GL_{2}(S). Now use that for any scheme T, to give an element of ℙ^{1}(T) is to give an invertible O_{T} -module with two sections that generate it. Let g be an S-automorphism of ℙ_{S}^{1}. Then g^{∗}O(1) is of the form p^{∗}L ⊗ O(1) for some invertible O_{S}-module L. Since L is locally trivial, we get what we want.

Hence: the morphism GL_{2}(O_{K}^{∧}) → PGL_{2}(O_{K}^{∧}) is surjective (use that O_{K}^{∧} is the product of the completions at all finite places and that one has the surjectivity for each such completion). And: GL_{2}(𝔸_{K,f}) → PGL_{2}(𝔸_{K,f}) is surjective (just use what elements of 𝔸_{K,f} look like, or use that to give a point of a scheme with values in 𝔸_{f} is to give, for each p, a point with values in ℚ_{p}, such that for almost all p the point comes from a point with values in ℤ_{p}). The more difficult thing that remains now is the question of surjectivity of the morphism GL_{2}(O_{K}) → PGL_{2}(O_{K}).

It follows that S(ℂ) → S^{ad}(ℂ) is surjective. The stabilizer of (ℍ^{+})^{2} × {1} in GL_{2}(K) and PGL_{2}(K) are GL_{2}(O_{K})^{+} and PGL_{2}(O_{K})^{+}, respectively. So let us find out what the cokernel of GL_{2}(O_{K})^{+} → PGL_{2}(O_{K})^{+} is.

An automorphism of ℙ := ℙ_{S}^{1} is given by an invertible O_{ℙ}-module of degree one together with two generating sections. Such a module is of the form p^{∗}L ⊗ O(1). But then we have the condition that p_{∗}p^{∗}L ⊗ O(1) = L ⊕ L is generated by two global sections. This can be done if and only if L ⊕ L ≅ O ⊕ O, i.e., if and only if L^{⊗2} ≅ O. This explains that we have an exact sequence:

1 ⟶ O_{K}^{∗} ⟶ GL_{2}(O_{K}) ⟶ PGL_{2}(O_{K}) ⟶ Pic(O_{K})[2] ⟶ 0

Likewise, one gets:

1 ⟶ O_{K}^{∗} ⟶ GL_{2}(O_{K})^{+} ⟶ PGL_{2}(O_{K})^{+} ⟶ Pic(O_{K})[2] ⟶ 0.

We conclude that S(ℂ)^{0} → S^{ad}(ℂ)^{0} is the quotient for a faithful action by the group Pic(O_{K})[2], where S(ℂ)^{0} and S^{ad}(ℂ)^{0} are the standard irreducible components of S(ℂ) and S^{ad}(ℂ). One computes directly that the map S(ℂ) → S^{ad}(ℂ) is the quotient for a faithful action by the group K^{∗}\𝔸_{K}^{∗}/O_{K}^{∧,∗}, i.e., by Pic(O_{K}).

Let us now do some comparing between S and S′, with S′ coming from the Shimura datum with the group G′. We first remark that G′(ℝ) is the set of (g_{1}, g_{2}) in GL_{2}(ℝ)^{2} such that det(g_{1}) = det(g_{2}). This means that the G′(ℝ) conjugacy class of morphisms from 𝕊 to G′_{ℝ} that we deal with is:

X′ := (ℍ^{+})^{2} ∐(ℍ^{−})^{2} = (ℍ^{2})^{±}.

Hence we have:

S′(ℂ) = G′(ℚ)\(X′ × G′(𝔸_{f})/G′(ℤ̂)), π_{0}(S′(ℂ)) = G′(ℚ)^{+}\G′(𝔸_{f})/G′(ℤ̂).

This last set is the set of isomorphism classes of triplets (M, ϕ, α) with M a locally free O_{K}- module of rank two, ϕ: O_{K} → det(M) an isomorphism, and α an orientation on ℝ ⊗_{ℤ} det(M) that induces plus or minus the standard orientation on ℝ ⊗ O_{K} = ℝ × ℝ via ϕ. Since every such triplet is isomorphic to (O_{K}^{2}, id, (+, +)), we see:

S′(ℂ) is connected.

What about the map S′(ℂ) → S(ℂ)? It suffices to look at what happens on (ℍ^{2})^{+} × {1}. The stabilizer of this in G′(ℚ) is simply SL_{2}(O_{K}), and the stabilizer in G(ℚ) is GL_{2}(O_{K})^{+}. Hence the map S′(ℂ) → S(ℂ)^{+} is the quotient for the faithful action by the group O_{K}^{∗,+}/O_{K}^{∗,2}, i.e., totally positive global units modulo squares of global units.

F Some stuff on bilinear forms and field extensions.

Let k → K be a finite field extension, say of degree d. Let V be a finite dimensional K-vector space, say of dimension n. Let X denote the k-vector space of k-bilinear forms b: V × V → k such that b(ax, y) = b(x, ay) for all x and y in V and all a in K. (I.e., the maps x ↦ ax are required to be self-adjoint.) We want to relate X to the set Y of K-bilinear forms on V.

Let l: K → k be a surjective k-linear map (for example, one can take the trace map if k → K is separable). Then we have a map:

L: Y ⟶ X, b ↦ l ∘ b.

Indeed, for b in Y we have: (l ∘ b)(ax, y) = l(b(x, ay)) = (l ∘ b)(x, ay). The map L is injective, since, for b a K-bilinear form, the image of b is either 0 or K.

Let us now assume that k → K is separable. Then one computes that both X and Y are of dimension n^{2}d over k (of course, for Y this is clearly true without the separability assumption; for X, one uses this assumption in order to reduce to the case K = k^{d} via base change from k to some algebraic closure for example). Hence we conclude that the map L above is bijective. (I did not bother to check if this is still true without the separability.) So we have the following result.

F.1 Proposition. Let k be a field, and K a finite separable k-algebra. Let l: K → k be a surjective k-linear map (for example the trace map). Let V be a finitely generated projective K-module. Then for every k-bilinear form b: V × V → k such that b(ax, y) = b(x, ay) for all x and y in V and all a in K there exists a unique K-bilinear b′: V × V → K such that b = l ∘ b′. With this notation, b is symmetric (antisymmetric) if and only if b′ is so.

Proof. It only remains to prove that b is symmetric (antisymmetric) if and only if b′ is so. For b as above, let b^{t} denote its adjoint, i.e., b^{t}(x, y) = b(y, x); we will use the same notation for elements of Y. Then one has (b^{t})′ = (b′)^{t}. Now b is symmetric if and only if b^{t} = b, and b is antisymmetric if and only if b^{t} = −b. Hence the result. □

The next result gives a construction of the inverse of L, if one takes l to be the trace map.

F.2 Proposition. Let k be a field, and K a finite separable k-algebra. Let V be a finitely generated projective K-module, and b: V × V → k a k-bilinear map such that b(ax, y) = b(x, ay) for all x and y in V and all a in K. Because of the separability, we have a natural isomorphism of K-algebras: K ⊗_{k} K = K ×K′, where we view K ⊗_{k} K as a K-algebra via the first factor. This decomposition gives a decomposition of K-modules: K ⊗_{k} V = K ⊗_{k} K ⊗_{K} V = V ⊕ V′ with V′ = K′ ⊗_{K} V. Let b_{K} denote the K-bilinear form on K ⊗_{k} V obtained by extension of scalars. Then the decomposition of K ⊗_{k} V in V and V′ is orthogonal for b_{K}, and b′ is the restriction to V of B_{K}. In particular, one has b = tr ∘ b′.

Let us now note the special case where V is of dimension two. In that case, the K-vector space Y of antisymmetric K-bilinear forms is of dimension one, hence one gets the following corollary, which is of interest for Hilbert modular varieties.

F.3 Corollary. Let k be a field, and k → K a finite separable k-algebra. Let V be a free Kmodule of rank two. Let ψ_{0} : V × V → K be a non-degenerate alternating K-bilinear form. Then for every alternating k-bilinear form ψ: V × V → k such that ψ(ax, y) = ψ(x, ay) for all x and y in V and all a in K, there exists a unique b in K such that ψ(x, y) = tr(bψ_{0}(x, y)) for all x and y in V.

G Moduli interpretation for the symplectic group.

For details, see [12, Sections 1, 4]. Just in this section, let G denote the group of symplectic similitudes of rank 2n. More precisely, let n ≥ 0 be an integer, and let G denote the group of automorphisms of the ℤ-module ℤ^{2n} that preserve, up to scalar multiple, the standard symplectic form, i.e., the form given by the matrix ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). Let X := ℍ_{n}^{±} the set of h: 𝕊 → G_{ℝ} that are Hodge structures of weight −1 such that ψ is a polarization up to a sign. Then this X is one G(ℝ)-conjugacy class and it is called the Siegel double space. Let us consider:

A_{n}(ℂ) := G(ℚ)\(X × G(𝔸_{f})/G(ℤ̂)).

What we want to show is that A_{n}(ℂ) is the set of isomorphisms classes of pairs (A, λ) of principally polarized abelian varieties of dimension n. We already know what the interpretation of X is: it is the set of Hodge structures of weight −1 such that ψ is a polarization up to a sign. Let us now interpret G(𝔸_{f})/G(ℤ̂). Consider the action of G(𝔸_{f}) on the set of lattices in 𝔸_{f}^{2}. The stabilizer of the standard lattice ℤ̂^{2} is G(ℤ̂). Hence G(𝔸_{f})/G(ℤ̂) is the set of lattices of the form xℤ̂^{2}, with x in G(𝔸_{f}). We claim that this is the set of lattices L on which a suitable multiple of ψ induces a perfect pairing. For x in G(𝔸_{f}) we have: ψ(xu, xv) = μ(x)ψ(u, v), which proves that μ(x)^{−1}ψ is a perfect pairing on xℤ̂^{2}. On the other hand, let L be a lattice and a in 𝔸_{f}^{∗} be such that aψ is a perfect pairing on L. Then take a ℤ̂-basis l_{1}, …, l_{2n} of L such that aψ is in standard form, i.e., given by the matrix ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). Then the element x of GL_{2n}(𝔸_{f}) with xe_{i} = l_{i} is in G(𝔸_{f}). This finishes the proof of the fact that G(𝔸_{f})/G(ℤ̂) is the set of lattices on which a multiple of ψ is perfect.

Let us now describe the constructions that give A_{n}(ℂ) the interpretation as the set of isomorphism classes of abelian varieties of dimension n, with a principal polarization.

Suppose (A, λ) is given. Then choose an isomorphism f: ℚ^{2n} → H_{1}(A, ℚ) such that ψ corresponds to a multiple of λ (such an f is unique up to an element of G(ℚ)). Let x be the element of X that is given by the Hodge structure on ℚ^{2n} induced from A via f. Let L in G(𝔸_{f})/G(ℤ̂) be the lattice corresponding to ℤ^{2n} via f. The class of (x, L) modulo G(ℚ) depends only on the isomorphism class of (A, λ).

Suppose now that we have (x, L) in X × G(𝔸_{f})/G(ℤ̂). Then let A be (ℝ ⊗ L)/L with the complex structure given by the Hodge structure corresponding to x. Let a be the element of ℚ^{∗} such that aψ is perfect on L (this fixes a up to sign) and is a polarization λ on A (this fixes the sign). For g in G(ℚ), multiplication by g gives an isomorphism from (A, λ) to the (A′, λ′) obtained from (gx, gL).

Let us end with a remark which is just a reminder to myself.

G.1 Remark. Let V be a free finitely generated ℤ-module, with h: 𝕊 → GL(V)_{ℝ} a Hodge structure of type (−1, 0), (0, −1). Let A := (ℝ ⊗ V)/V be the associated complex torus. Then the dual complex torus corresponds to the Hodge structure t ↦ (h(t)^{∨})^{−1}N(t) = h(v̄)^{∨} on V^{∨}. In other words, the dual of A is (ℝ⊗V^{∨})/V^{∨}, with the complex structure on ℝ⊗V^{∨} = (ℝ⊗V)^{∨_{ℝ}} such that z in ℂ acts as z̄^{∨}. In order to prove this, one notes that the tangent space of A^{t} is H^{1}(A, O_{A}), which is naturally ℂ-anti-linearly isomorphic to H^{0}(A, Ω_{A}^{1}), which is the dual of the tangent space of A at zero.

H Moduli interpretation of S′(ℂ).

Let us recall:

S′(ℂ) = G′(ℚ)\(X′ × G′(𝔸_{f})/G′(ℤ̂)).

H.1 Proposition. The Shimura variety S′_{ℚ} is the moduli space of triplets (A, α, λ)with A an abelian surface, α: O_{K} → End(A) a ring morphism, and λ: A → A^{∗} a principal O_{K}-polariza-tion.

First of all, we have to explain what A^{∗} is, and what we call a principal O_{K}-polarization. Let us begin with A^{∗}: it is the dual of A in the category of abelian varieties with O_{K}-action. More precisely, since for A an abelian variety the dual is defined to be A^{t} := Ext^{1}(A, 𝔾_{m}), we put:

A^{∗} := Ext_{O_{K}}^{1}(A, O_{K} ⊗ 𝔾_{m}) = δ ⊗_{O_{K}} Ext^{1}(A, 𝔾_{m}) = δ ⊗_{O_{K}} A^{t},

with

δ := Hom_{O_{K}}(Hom_{ℤ}(O_{K}, ℤ), O_{K})

the different of the extension ℤ → O_{K}. In order to prove the above equalities, it is useful to note that for A → B a morphism of rings, for M a B-module and N and A-module, one has the adjunction:

Hom_{B}(M, Hom_{A}(B, N)) = Hom_{A}(_{A}M, N),

where _{A}M denotes the A-module given by M. Then one uses that for B locally free of finite rank as A-module one has Hom_{A}(B, N) = B^{∨_{A}} ⊗_{A} N, with B^{∨_{A}} = Hom_{A}(B, A) the A-dual of B. And then one uses that for P a finitely generated locally free B-module one has:

Hom_{B}(M, P ⊗_{A} N) = P ⊗_{B} Hom_{B}(M, B ⊗_{A} N).

This establishes:

Hom_{B}(M, B ⊗_{A} N) = δ ⊗_{B} Hom_{A}(M, N),

with δ = (B^{∨_{A}})^{∨_{B}}. Deriving with respect to N then gives:

Ext_{B}^{i}(M, B ⊗_{A} N) = δ ⊗_{B} Ext_{A}^{i}(M, N).

This explains the reason that A^{∗} occurs in this context. In our context, A = ℤ and B = O_{K}, so that δ is an ideal in O_{K}, since the trace map tr: O_{K} → ℤ gives an injective morphism O_{K} → (O_{K})^{∨_{ℤ}} (in fact, the trace map from B to A always gives a morphism δ → B, but it might be zero). This gives an isogeny:

A^{∗} = δ ⊗_{O_{K}} A^{t} ⟶ A^{t}.

We define an O_{K}-polarization to be an O_{K}-morphism λ: A → A^{∗} such that the induced morphism from A to A^{t} is a polarization; λ is called principal if it is an isomorphism (note that this means that the induced morphism A → A^{t} is not an isomorphism, since O_{K} is ramified over ℤ (we do suppose that K is a field, after all).

Let us now turn to the proof of the proposition above that gives the moduli interpretation of S′_{ℚ}. So we want to show that S′(ℂ) is the set of isomorphism classes of triplets (A, α, λ) over ℂ. In Hodge theoretical terms, such triplets are given by triplets (V, h, ψ) with V a locally free O_{K}-module of rank two, h: 𝕊 → (GL_{ℤ}(V))_{ℝ} a Hodge structure of type (−1, 0), (0, −1), and ψ: V × V → O_{K} a perfect antisymmetric O_{K}-bilinear form such that tr ∘ ψ: V × V → ℤ is a polarization. Note that for such a triplet (V, h, ψ), the pair (V, ψ) is isomorphic to the standard pair (O_{K} ⊕ O_{K}, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )). The proof of the proposition can now be easily described. As in the last section, one shows that G′(𝔸_{f})/G′(ℤ̂) is the O_{K}-lattices in K^{2} on which ψ = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) induces a perfect pairing of O_{K}-modules, up to a factor in ℚ^{∗}. The space X′ is the set of Hodge structures. As in the last section, one shows that G′(𝔸_{f})/G′(ℤ̂) is the O_{K}-lattices in K^{2} on which ψ = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) induces a perfect pairing of O_{K}-modules, up to a factor in 𝔸_{f}^{∗}. The space X′ is the set of Hodge structures of type (−1, 0), (0, −1) on the K-vector space K^{2} such that, up to sign, tr ∘ ψ is a polarization. f type (−1, 0), (0, −1) on the K-vector space K^{2} such that, up to sign, tr ∘ ψ is a polarization. After these remarks one simply follows the lines of the proof above of the modular interpretation for the symplectic group. Anyway, for details, one can consult [12, 4.11].

Let us end by stating that the multiplier character μ: G′ → 𝔾_{m} is the determinant (view G′ as a subgroup of Res_{O_{K}/ℤ}GL_{2,O_{K}} and 𝔾_{m} as a subgroup of Res_{O_{K}/ℤ}𝔾_{m,O_{K}}. More precisely, for all g in G′(ℚ) and all x and y in K^{2} we have (tr ∘ ψ)(gx, gy) = det(g)(tr ∘ ψ)(x, y).

I A remark on Mumford-Tate groups.

What I want to say is that to an isomorphism class of ℚ-Hodge structures one can associate its Mumford-Tate group. Namely, if V and V′ are isomorphic ℚ-Hodge structures, and if f and f′ are isomorphisms from V to V′, then f′ = fg with g an automorphism of V. But then g centralizes the Mumford-Tate group in GL(V). Hence f and f′ induce the same isomorphism from MT(V) to MT(V′). For example, the functor V ↦ Aut(V) does not have this property.

The same argument shows that a point P on a Shimura variety Sh_{K}(G, X)(ℂ) defines an algebraic group MT(P), with a given G(ℚ)-conjugacy class of embeddings in G.

J On computing the generic Mumford-Tate group on S′(ℂ).

First note that for all h = (h_{1}, h_{2}): ℂ^{∗} → GL_{2}(ℝ)^{2} in X′ one has det(h_{1}(z)) = det(h_{2}(z)) for all z. This shows that MT is contained in G′_{ℚ}. The locally constant sheaf V becomes constant on X′. Hence MT_{ℝ} contains all h(ℂ^{∗}) ⊂ GL_{2}(ℝ)^{2} for the h in X′. In particular, it contains all conjugates under G′(ℝ) of those images. but then it contains all (x, x), all (yxy^{−1}, x), hence all (xyx^{−1}y^{−1}, 1), etc. It follows that MT = G′_{ℚ}.

K Other remarks on Mumford-Tate groups.

We have defined the Mumford-Tate group MT(V) of a ℚ-Hodge structure V given by a morphism h: 𝕊 → GL(V)_{ℝ} to be the smallest algebraic subgroup H of GL(V) such that h factors through H_{ℝ}. This is not the usual definition, perhaps. The usual definition is to take MT′(V), the smallest subgroup H of GL(V) × 𝔾_{m} such that h′: 𝕊 → GL(V)_{ℝ} × 𝔾_{m,ℝ} factors through H_{ℝ}, where 𝕊 → 𝔾_{m,ℝ} corresponds to ℚ(1). The difference between the two choices is that MT′(V) keeps track of weights, whereas MT(V) doesn’t. The Tannakian interpretation of MT(V) is that it is the automorphism functor of the fibre functor “forget Hodge structure” on the tensor category generated by V. For MT′(V), one considers the tensor category generated by V and ℚ(1). Yet another (of course related) characterization is that MT′(V) seems to be the biggest subgroup of GL(V) × 𝔾_{m} that fixes all elements of type (0, 0) in ℚ-Hodge structures of the form V^{⊗n} ⊗ (V^{∗})^{⊗m} ⊗ ℚ(p). For this, see Deligne-Milne-Ogus-Shih. In the same way, MT(V) is characterized by the fact that it stabilizes all lines generated by Hodge classes (i.e., classes of some type (p, p)) in ℚ-Hodge structures of the form ⊕_{i}V^{⊗n_{i}} ⊗ (V^{∗})^{⊗m_{i}}.

Since I did not find this explicitly written (but I haven’t looked very much, I should say), let me write a proof. So let H be the intersection of the stabilizers of such lines. Let us first prove that MT(V) ⊂ H. So let t in some T = ⊕_{i}V^{⊗n_{i}} ⊗ (V^{∗})^{⊗m_{i}} be of some type (p, p). Then ℝt ⊂ T_{ℝ} is fixed by 𝕊, hence by MT(V). This proves that MT(V) ⊂ H. Let’s now prove that H ⊂ MT(V). Now we use Chevalley’s result: every subgroup of GL(V) is the stabilizer of a line in some finite dimensional representation of GL(V), plus the fact that each finite dimensional representation of GL(V) is contained in a representation of the form ⊕_{i}V^{⊗n_{i}} ⊗ (V^{∗})^{⊗m_{i}} (I will even give proofs for these two facts below, since I do not like the proof given in DMOS). Anyway, let t in some T be such that MT(V) is the stabilizer of ℚt. Then ℝt is fixed by 𝕊, hence t is of some type (p, p).

(Use for example that the norm 𝕊 → 𝔾_{m,ℝ} generates Hom(𝕊, 𝔾_{m,ℝ}).)

As I said, I do not like the proof of parts (a) and (b) of Proposition 3.1 in Chapter I of DMOS. So I give one.

K.1 Theorem. Let G be an affine algebraic group over a field k ⊃ ℚ. Let H be an algebraic subgroup of G, and V a finite dimensional faithful representation of G. Then there exists a line L in some representation of G of the form ⊕_{i}V^{⊗n_{i}} ⊗ (V^{∗})^{⊗m_{i}}, such that H is the stabilizer of L.

Proof. First of all, we may and do suppose that G = GL(V). The idea is now the following: let G act on itself by right translation; then G acts on k[G], and H is the stabilizer of the ideal I_{H}; then use that I_{H} is finitely generated, and that k[G] is locally finite. Let us first write down what k[G] is, as a G-module via right translation on G. Well,

k[G] = k[End(V)][1/ det] = Sym_{k}(End(V)^{∗})[1/ det] = Sym_{k}(V^{d})[1/ det],

where the last equality comes from the fact that End(V)^{∗}, as G-module given by right translation on End(V), is simply V^{d}, where d is of course the dimension of V (note that the G-action on End(V)^{∗} extends to an End(V)-action). Also, note that det is in Sym^{d}(End(V)^{∗}), and that we have g· det = det(g) det. The scalar subgroup 𝔾_{m} of G induces a ℤ-grading on k[G]. We have:

k[G]_{i} = ⋃_{j} k[End(V)]_{i+dj}det^{−j},

and:

k[End(V)]_{i} = Sym^{i}(End(V)^{∗}) = Sym^{i}(V^{d}) ⊂ (V^{d})^{⊗i} = (V^{⊗i})^{d^{i}},
k· det = Λ^{d}V ⊂ V^{⊗d},
k·det^{−1} = (Λ^{d}V)^{∗} ⊂ (V^{∗})^{⊗d}.

This describes k[G] as G-module. Let f_{1}, …, f_{r} be a finite set of generators of the ideal I_{H} of k[G]. Let W ⊂ k[G] be a finite dimensional sub-G-module containing the f_{i}. Then H is the stabilizer of the subspace I_{H} ∩ W of W, hence of the line Λ^{n}(I_{H} ∩ W) ⊂ Λ^{n}(W), with n = dim(I_{H} ∩ W). Now note that W is a subrepresentation of a representation of the form ⊕_{i}V^{⊗n_{i}} ⊗ (V^{∗})^{⊗m_{i}}. □

K.2 Remark. If we allow subquotients of the ⊕_{i}V^{⊗n_{i}} ⊗ (V^{∗})^{⊗m_{i}}, then we can drop the hypothesis that k is of characteristic zero.

K.3 Remark. If H contains the scalars in G = GL(V), then one can take L to be in some representation of the form (V^{⊗n})^{m}. To prove this, consider the Zariski closure \overline{H} of H in End(V), and use that it is a cone.

Just for fun, let us look at some examples in G := GL_{2}. The Borel subgroup B := \{( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix})\} is the stabilizer of the line generated by (1, 0) in V := k^{2}. The subgroup \{( \begin{matrix}1 & ∗ \\ 0 & ∗ \end{matrix})\} is the stabilizer of k(1, (1, 0)) in k ⊕ V. The subgroup \{( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} )\} is the stabilizer of k(1, (0, 1)^{∗}) in k ⊕ V^{∗}. The subgroup \{( \begin{matrix}t & 0 \\ 0 & t \end{matrix})\} is the stabilizer of k((1, 0), (0, 1)) in V ⊕ V. The subgroup \{( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix})\} is the stabilizer of the two-dimensional subspace of the ((x, 0), (0, y)) in V ⊕V; note that the proof above gives the same result. Finally, the trivial subgroup \{( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )\} is the stabilizer of (1, (1, 0), (0, 1)) in k ⊕ V ⊕ V.

L Modular interpretation of T_{p}.

Let A be a complex abelian surface with multiplication by O_{K} and with a principal O_{K}-polariza-tion λ: A → A^{∗}. Let H be an O_{K}-submodule of A[p](ℂ) that is free of rank one. Then we claim that pλ induces a principal O_{K}-polarization on A/H. So how does this work? Write pλ = π_{2}π_{1}, with π_{1} : A → B the quotient by H. Then (A/H)^{∗} is the quotient of A by ker(π_{2}^{∗}). So we have to see that ker(π_{2}^{∗}) = ker(π_{1}). Since both have the same number of elements, it suffices to see that one is contained in the other. Since ker(π_{1}) is maximal isotropic for the pairing e_{λ,p} that λ induces on A[p], it suffices to see that ker(π_{2}^{∗}) and ker(π_{1}) are orthogonal for that pairing. That results from standard things about such pairings coming from expressions like A^{∗} = Ext^{1}(A, O_{K} ⊗ 𝔾_{m}).

The general statement is this: let f: A → B and g: B → C be isogenies of abelian varieties with multiplications by O_{K}. Let h := gf. Then we have a short exact sequence:

0 ⟶ ker(f) ⟶ ker(h) ⟶ ker(g) ⟶ 0.

Applying Hom(·, O_{K} ⊗ 𝔾_{m}) gives an isomorphism of short exact sequences:

0 ⟶ ker(g^{∗}) ⟶ ker(h^{∗}) ⟶ ker(f^{∗}) ⟶ 0
↓ ↓ ↓ ↓ ↓
0 ⟶ ker(g)^{∗} ⟶ ker(h)^{∗} ⟶ ker(f)^{∗} ⟶ 0.

The fact that the map from ker(g^{∗}) to ker(f)^{∗} is zero means that ker(f) and ker(g^{∗}) are orthogonal for the paring induced by h between ker(h) and ker(h^{∗}).

M Some stuff on orders in finite separable ℚ-algebras.

I need lower bounds for orders of Picard groups of certain orders in certain CM fields. Therefore, some general theory should be quite useful.

Let ℚ → K be a finite separable ℚ-algebra. Then K is a finite product of number fields, say K = K_{1} × ⋯ × K_{m}, and the integral closure of ℤ in K is then the product of the maximal orders of the K_{i}. Let R ⊂ K be an order in K, i.e., a subring of K with ℚ ⊗ R = K and which is finitely generated as a ℤ-module. Then R is contained in O_{K} since the elements of R are integral over ℤ, and O_{K}/R is a finite additive group, since R and O_{K} are free ℤ-modules of the same finite rank. Consider ideals of R that are also O_{K}-ideals. Clearly a lot of such ideals do exist: for every n in ℤ that annihilates O_{K}/R, we have the example nO_{K}. The sum of a family of such ideals is again one such, hence there exists a unique maximal such ideal, called the conductor of R (relative to O_{K}). I don’t think that we will use this conductor so much, since we want estimates in terms of the discriminant of R.

Let I ⊂ R be a non-zero ideal that is also an O_{K}-ideal. Then R is the inverse image in O_{K} of the subring R/I of the quotient O_{K}/I of O_{K}). Actually, the diagram:

R ⟶ R/I
↓ ↓
O_{K} ⟶ O_{K}/I

is both Cartesian and co-Cartesian. For us, the most important is that every order of K is obtained as follows: take the inverse image in O_{K} of a subring of a finite quotient of O_{K}.

M.1 Discriminants.

Recall that discr(O_{K}) is the discriminant of the trace form on O_{K}. To be precise: for M a free ℤ-module of finite rank and b a bilinear form on M, let discr(M, b) be the integer defined by: let m be a basis of M, then discr(M, b) is the determinant of the matrix of b relative to m. In more intrinsic terms, one can use that b induces a bilinear form on the maximal exterior power of M, and use the integer coming from there. The whole thing does not depend on the basis because changing the basis changes it by the square of a unit. Over more general rings, and for projective modules, one obtains an ideal, locally principal, with some extra structure due to the squares of units that intervene. In fact, one sees that if the local generators of the ideal are non zero divisors, then the ideal is, as invertible module, the square of ΛM. In our case, we use the bilinear form (x, y) ↦ tr(xy). The separability of ℚ → K shows that discr(O_{K}) ≠ 0. Choosing a basis of O_{K} adapted to R shows that:

discr(R) = discr(O_{K}) |O_{K}/R|^{2}.

M.2 Theorem. Let ζ_{R} denote the zeta function of the order R, i.e., the zeta function of Spec(R) in the usual sense. Then:

Res_{1}(ζ_{R}) := lim_{s→1}(s − 1)^{m}ζ_{R}(s) = \frac{2^{r_{1}}(2π)^{r_{2}}|Pic(R)| Reg(R)}{|tors(R^{∗})| |discr(R)|^{1/2}},

with ℝ ⊗ K ≅ ℝ^{r_{1}} × ℂ^{r_{2}} and Reg(R) the regulator of R (see in the proof for the definition). (Recall that K is the product of m number fields.)

Proof. For O_{K}, see for example Lang’s “Algebraic number theory”, 2nd edition, VIII, §2. In fact, Lang gives the proof when K is a field, but for O_{K} in a product of number fields everything decomposes into products. Let us digress a little bit on the regulator. I find that the regulator Reg′(R) should be defined as follows: one considers

O_{K}^{∗} ⟶ (ℝ ⊗ O_{K})^{∗} ⟶^{log ∥ ∥} ℝ^{r_{1}+r_{2}},

and puts:

Reg′(R) := Vol(ℝ^{r_{1}+r_{2},+=0}/image of O_{K}^{∗}),

with the volume measured with respect to the volume form coming from the standard inner product on ℝ^{r_{1}+r_{2}}, and where log ∥ ∥ is taking log of absolute value at every factor of ℝ ⊗ O_{K}, with ∥x∥ being the factor by which the Haar measures change (|x| for a real place, |x|^{2} for a complex one). But this does not give the usual definition, as given in Lang. There one omits any one of the infinite places in order to get a square matrix of which one takes absolute value of the determinant. One easily proves that Reg′(R) = 2^{−r_{2}}(r_{1} + 2r_{2})(r_{1} + r_{2})^{−1/2}Reg(R), which actually makes my definition a bit ugly.

Anyway, let’s proceed. Since we know the theorem for O_{K}, all we have to do is to compare our R to O_{K}. Let X := Spec(O_{K}), Y := Spec(R), and N: X → Y the morphism induced by the inclusion of R in O_{K}. Then we have a short exact sequence of sheaves on X:

0 ⟶ O_{Y}^{∗} ⟶ N_{∗}O_{X}^{∗} ⟶ Q ⟶ 0,

with Q a skyscraper sheaf given by \overline{O_{K}}^{∗}/\overline{R}^{∗} in case R is given by the subring \overline{R} of the finite quotient \overline{O_{K}} of O_{K}. This gives a long exact sequence:

0 ⟶ R^{∗} ⟶ O_{K}^{∗} ⟶ \overline{O_{K}}^{∗}/\overline{R}^{∗} ⟶ Pic(R) ⟶ Pic(O_{K}) ⟶ 0.

Let A be the cokernel of R^{∗} → O_{K}^{∗}, and B the kernel of Pic(R) → Pic(O_{K}). Then one gets:

|Pic(R)| = |B| |Pic(O_{K})|, |\overline{O_{K}}^{∗}/\overline{R}^{∗}| = |A| |B|, |A| = \frac{|tors(O_{K}^{∗})|}{|tors(R^{∗})|} |\frac{O_{K}^{∗}/tors}{R^{∗}/tors}|

Reg(R) = |\frac{O_{K}^{∗}/tors}{R^{∗}/tors}| Reg(O_{K}), |discr(R)|^{1/2} = \frac{|\overline{O_{K}}|}{|\overline{R}|}|discr(O_{K})|^{1/2}.

Putting this all together shows that the right hand side of the equality we want to prove changes by the factor |\overline{O_{K}}^{∗}| |\overline{R}^{∗}|^{−1} |\overline{R}| |\overline{O_{K}}|^{−1} when going from O_{K} to R. So all that we have to do now is to show that the left hand side changes by the same factor. But then note:

\frac{|\overline{R}|}{|\overline{R}^{∗}|} = ∏_{k res field of \overline{R}} \frac{|k|}{|k^{∗}|} = ∏_{k} \frac{1}{1 − |k|^{−1}} ,

which is clearly the contribution to Res_{1}(ζ_{R}) of those residue fields. □

M.3 Theorem. Let N > 0. Then there exists a real number c > 0 such that for every order R in a separable ℚ-algebra K of degree at most N, one has:

|Pic(R)| Reg(R) ≥ c |discr(R)|^{1/7}.

M.4 Remark. As the proof will show, we can actually get 1/6 − ε as exponent, instead of 1/7, with a c depending on ε, for every ε > 0. If one assumes the generalized Riemann hypothesis, then one can get 1/2 − ε as exponent, for every ε > 0, with again c depending on ε. In that case, one uses Siegel’s theorem that one finds in [17, Ch. XIII, §4].

Proof. We will first prove this for maximal orders in number fields of bounded degree, then for maximal orders in finite separable ℚ-algebras of bounded degree, and then for arbitrary orders of bounded degree.

In the case of a maximal order of a number field of bounded degree, we just apply two theorems. The first one is the Brauer-Siegel theorem (see for example [17, Ch. XVI]), that states that:

for N > 0 and ε > 0, there exists c > 0 such that:

|Pic(O_{K})| Reg(O_{K}) ≥ c |discr(O_{K})|^{1/2−ε}

for all Galois extensions K of ℚ of degree at most N.

The second theorem is one of Stark ([29, Thm. 1]):

let N > 0. There exists c > 0 such that for all number fields K of degree at most N over ℚ, one has: |Pic(O_{K})| Reg(O_{K}) ≥ c |discr(O_{K})|^{1/2−1/[K:ℚ]}.

Together, these two results show:

let N > 0. There exists c > 0 such that for every number field K of degree at most N over ℚ one has: |Pic(O_{K})| Reg(O_{K}) ≥ c |discr(O_{K})|^{1/6}.

This settles the case where the ℚ-algebra K is a field. The case for a maximal order in a finite separable ℚ-algebra of degree at most N then follows, because everything decomposes into a product of at most N factors, for which one has the result already.

So let now K be a finite separable ℚ-algebra of degree at most N, and let R be an order in it, given by the subring \overline{R} of some finite quotient \overline{O_{K}} of O_{K}. We have already seen that:

|Pic(R)| Reg(R) = \frac{|\overline{O_{K}}^{∗}|}{|\overline{R}^{∗}|}|Pic(O_{K})| Reg(O_{K})\frac{|tors(R^{∗})|}{|tors(O_{K}^{∗})|},

|discr(R)| = (\frac{|\overline{O_{K}}|}{|\overline{R}|})^{2} |discr(O_{K})|.

We note that the quotient |tors(R^{∗})| |tors(O_{K}^{∗})|^{−1} and its inverse are bounded in terms of N only. Hence the theorem follows from the following claim:

for N > 0 and ε > 0 there exists c > 0 such that for R an order in a finite separable ℚ-algebra K of degree at most N, one has:

\frac{|\overline{O_{K}}^{∗}|}{|\overline{R}^{∗}|} ≥ c (\frac{|\overline{O_{K}}|}{|\overline{R}|})^{1−ε} ,

where R is the inverse image of the subring \overline{R} of the finite quotient \overline{O_{K}} of O_{K}.

We now prove this claim. Let n denote |\overline{O_{K}}/\overline{R}|. We may and do assume that n > 1. Localizing at the maximal ideals of \overline{R}, followed by a simple computation, shows that:

\frac{|\overline{O_{K}}^{∗}|}{|\overline{R}^{∗}|} ≥ n ∏_{p|n} (1 − \frac{1}{p})^{N} ≥ n (\frac{1}{5 log(n)})^{N} .

Since log(n) = n^{o(1)}, this shows our claim, and hence finishes the proof of the theorem. □

N On effective Chebotarev.

As usual, let Li(x) := ∫_{2}^{x} dt/ log(t). If one assumes GRH, then the effective Chebotarev theorem of Lagarias, Montgomery and Odlyzko, stated as in [28, Thm. 4] and the second remark following that theorem, says:

for M a finite Galois extension of ℚ, let n_{M} denote its degree, d_{M} its absolute discriminant |discr(O_{M})|, and for x in ℝ, let π_{M,1}(x) be the number of primes p ≤ x that are unramified in M and such that the Frobenius conjugacy class Frob_{p} contains just the identity element of Gal(M/ℚ). Then one has, for all sufficiently large x and all finite Galois extensions M of ℚ:

|π_{M,1}(x) − \frac{1}{n_{M}} Li(x)| ≤ \frac{1}{3n_{M}} x^{1/2} (log(d_{M}) + n_{M} log(x)) .

This result shows that for all x sufficiently large, and all finite Galois extensions M of ℚ, one has:

π_{M,1}(x) ≥ \frac{x}{n_{M} log(x)} (Li(x)\frac{log(x)}{x} − \frac{log(x)}{3x^{1/2}} (log(d_{M}) + n_{M} log(x))) .

If x tends to infinity, Li(x) log(x)/x tends to 1 and log(x)^{2}/x^{1/2} tends to 0. Some computation (that we will do below) shows that if x is sufficiently big (i.e., bigger than some absolute constant), and bigger than 2(log(d_{M})^{2}(log(log(d_{M})))^{2}, then log(x) log(d_{M})/3x^{1/2} < 1/2, and hence:

π_{M,1}(x) ≥ \frac{x}{3n_{M} log(x)}.

Here is the computation that I promised. Put a := log(d_{M})/3. We want to find a lower bound for x that implies that a log(x)/\sqrt{x} < 1/2. We put x := y^{2} (with y > 0, of course). Then what we want is a lower bound for y such that b log(y) < y, with b = 4a. we put y = zb. Then what we want is a lower bound for z such that z − log(z) > c, with c = log(b). Now write z = (1 + u)c. Then what we want is: uc − log(1 + u) − log(c) > 0. Since log(1 + u) ≤ u, it suffices that uc − u − log(c) > 0, i.e., that u > log(c)/(c − 1) (by the way, since we are willing to let x be sufficiently large, we may take care of small d_{M} by that, and suppose that c is sufficiently large). For c sufficiently large, for any ε > 0, u > ε is good enough. Translating this back to x and log(d_{M}), one gets that x > (1 + ε)^{2}(4a log(4a))^{2} is good. Then one uses that 2 > 16/9.

O Real approximation.

It is known that for G an affine algebraic group over ℚ one has G(ℚ) dense in G(ℝ). This is what Deligne calls real approximation. To prove it, one reduces to the case of tori. But even that case is not so trivial to me. Of course, tori are unirational (they are images of tori that are products of multiplicative groups of number fields), but that is not enough: that only gives that the rational points of G are dense in the connected components of G(ℝ) that do contain a rational point. Anyway, for a detailed proof I would refer to the book [26] of Platonov and Rapinchuk.