j. differential geometry 66 (2004) 233-287

A CHARACTERIZATION OF CERTAIN SHIMURA CURVES IN THE MODULI STACK OF ABELIAN

VARIETIES

Eckart Viehweg & Kang Zuo

Abstract

Let f : X → Y be a semistable family of complex abelian varieties over a curve Y of genus g(Y), and smooth over the complement of s points. If ℓ denotes the rank of the nonflat (1, 0) part F^{1,0} of the corresponding variation of Hodge structures, the Arakelov inequality says that

2 · deg(F^{1,0}) ≤ ℓ · (2g(Y) − 2 + s).

The family reaches this bound if and only if the Higgs field of the variation of Hodge structures is an isomorphism. The latter is reflected in the existence of special Hodge cycles in the general fibre, and the base of such a family is a Shimura curve.

In particular, for s ≠ 0, such a family must be isogenous to the ℓ-fold product of a modular family of elliptic curves, and a constant abelian variety.

For s = 0, if the flat part of the variation of Hodge structures is defined over the rational numbers, one finds the family to be isogenous to the product of several copies of a family h : Z → Y, and a constant abelian variety. In this case, h : Z → Y is obtained from the corestriction of a quaternion algebra A, defined over a totally real numberfield F, and ramified over all infinite places but one.

In case the flat part is not defined over the rational numbers, one still can classify the corresponding variations of Hodge structures.

This work has been supported by the “DFG-Schwerpunktprogramm Globale Methoden in der Komplexen Geometrie”. The second named author was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 4239/01P).

Received 06/15/2003.

0. Introduction

Throughout this article, Y will denote a nonsingular complex projective curve, U an open dense subset, and X_{0} → U a smooth family of abelian varieties. We choose a projective nonsingular compactification X of X_{0} such that the family extends to a morphism f : X → Y, which we call again a family of abelian varieties although some of the fibres are singular. We write S = Y \ U, and Δ = f^{−1}(S). Consider the weight 1 variation of Hodge structures given by f : X_{0} → U, i.e., R^{1}f_{∗}ℤ_{X_{0}}. We will always assume that the monodromy of R^{1}f_{∗}ℤ_{X_{0}} around all points in S is unipotent. The Deligne extension of (R^{1}f_{∗}ℤ_{X_{0}}) ⊗ O_{U} to Y carries a Hodge filtration. Taking the graded sheaf one obtains the Higgs bundle

(E, θ) = (E^{1,0} ⊕ E^{0,1}, θ_{1,0})

with E^{1,0} = f_{∗}Ω_{X/Y}^{1} (log Δ) and E^{0,1} = R^{1}f_{∗}O_{X}. The Higgs field θ_{1,0} is given by the edge morphisms

f_{∗}Ω_{X/Y}^{1} (log Δ) ⟶ R^{1}f_{∗}O_{X} ⊗ Ω_{Y}^{1} (log S)

of the tautological sequence

0 → f^{∗}Ω_{Y}^{1} (log S) → Ω_{X}^{1}(log Δ) → Ω_{X/Y}^{1} (log Δ) → 0.

By [14] E can be decomposed as a direct sum F ⊕ N of Higgs bundles with E^{1,0} ∩ F ample and with N flat, hence for F^{i,j} = E^{i,j} ∩ F and N^{i,j} = E^{i,j} ∩ N the Higgs bundle E decomposes in

(F = F^{1,0} ⊕ F^{0,1}, θ_{1,0}|_{F^{1,0}}) and (N^{1,0} ⊕ N^{0,1}, 0). (0.0.1)

For g_{0} = rank(F^{1,0}) the Arakelov inequalities ([7], generalized in [20], [12]) say that

(0.0.2) 2 · deg(F^{1,0}) ≤ g_{0} · deg(Ω_{Y}^{1} (log S)).

In this note we will try to understand f : X → Y, for which (0.0.2) is an equality, or as we will say, of families reaching the Arakelov bound. By Proposition 1.2, this property is equivalent to the maximality of the Higgs field for F, saying that θ|_{F_{1,0}} : F^{1,0} → F^{0,1} ⊗ Ω_{Y}^{1} (log S) is an isomorphism.

As it will turn out, the base of a family of abelian varieties reaching the Arakelov bound is a Shimura curve, and the maximality of the Higgs field is reflected in the existence of special Hodge cycles on the general fibre. Before formulating a general result, let us consider two examples.

For families of elliptic curves, the maximality of the Higgs field just says that the family is modular (see Section 2).

Proposition 0.1. Let f : E → Y be a semistable family of elliptic curves, smooth over U ⊂ Y. If E → Y is nonisotrivial and reaching the Arakelov bound, E → Y is modular, i.e., U is the quotient of the upper half plane H by a subgroup of Sl_{2}(ℤ) of finite index, and the morphism U → ℂ = H/Sl_{2}(ℤ) is given by the j-invariant of the fibres.

For S ≠ ∅ the only families of abelian varieties reaching the Arakelov bound are built up from modular families of elliptic curves.

Theorem 0.2. Let f : X → Y be a family of abelian varieties smooth over U, and such that the local monodromies around s ∈ S are unipotent. If S ≠ ∅, and if f : X → Y reaches the Arakelov bound, then there exists an étale covering π : Y′ → Y such that f′ : X′ = X ×_{Y} Y′ → Y′ is isogenous over Y′ to a product

B × E ×_{Y′} ⋯ ×_{Y′} E,

where B is abelian variety defined over ℂ of dimension g−g_{0}, and where h : E → Y′ is a family of semistable elliptic curves reaching the Arakelov bound.

Results parallel to 0.2 have been obtained in [30] for families of K3surfaces, and the methods and results of [30] have been a motivation to study the case of abelian varieties.

As we will see in Section 4, Theorem 0.2 follows from the existence of too many endomorphisms of the general fibre of f : X → Y, which in turn implies the existence of too many cycles on the general fibre of X ×_{Y} X. We give an elementary proof of Theorem 0.2 in Section 4, although it is nothing but a first example for the relation between the maximality of Higgs fields, and the moduli of abelian varieties with a given special Mumford–Tate group Hg, constructed in [17] and [18] (see Section 2).

Proposition 0.3. Let f : X → Y be a family of g-dimensional abelian varieties reaching the Arakelov bound. Assume that g = g_{0}, or more generally that the largest unitary local subsystem 𝕌_{1} of R^{1}f_{∗}ℂ_{X_{0}} is defined over ℚ. Then there exists a finite cover Y′ → Y, étale over U, and a ℚ-algebraic subgroup Hg ⊂ Sp(2g, ℝ), such that pullback family f′ : X′ = X ×_{Y} Y′ → Y′ is a semistable compactification of the universal family of polarized abelian varieties with special Mumford–Tate group contained in Hg, and with a suitable level structure.

As a preparation for the proof of Proposition 0.3 we will show in Section 1, using Simpson’s correspondence between Higgs bundles and local systems, that the maximality of the Higgs field enforces a presentation of the local systems R^{1}f_{∗}ℂ_{X_{0}} and 𝔼nd(R^{1}f_{∗}ℂ_{X_{0}}) using direct sums and tensor products of one weight one complex variation of Hodge structures 𝕃 of rank two and several unitary local systems.

Proposition 0.3 relates families reaching the Arakelov bound to totally geodesic subvarieties of the moduli space of abelian varieties, as considered by Moonen in [16], or to the totally geodesic holomorphic embeddings, studied by Abdulali in [1] (see Remark 2.5 b). As in [21] one could use the classification of Shimura varieties due to Satake [22] to obtain a complete list of those families, and to characterize them in terms of properties of their variation of Hodge structures.

We choose a different approach, less relying on the theory of Shimura varieties, and more adapted to handle the remaining families of abelian varieties (see Remark 2.5 c), as well as some other families of varieties of Kodaira dimension zero (see [33]). We first show that the decompositions of R^{1}f_{∗}ℂ_{X_{0}} and 𝔼nd(R^{1}f_{∗}ℂ_{X_{0}}) mentioned above are defined over \overline{ℚ} ∩ ℝ. In case S ≠ ∅ it is then easy to see, that the unitary parts of the decompositions trivialize, after replacing Y by a finite étale cover Y′ (see 4.4).

For S = ∅, let us assume first that the assumptions made in 0.3 hold true. By [9] (see Proposition 6.3) they imply that the family is rigid, i.e., that the morphism from Y to the moduli stack of polarized abelian varieties has no nontrivial deformation, except those obtained by deforming a constant abelian subvariety.

Mumford gave in [18] countably many moduli functors of abelian fourfolds, where Hg is obtained via the corestriction of an quaternion algebra, defined over a totally real cubic number field F. Generalizing his construction one considers quaternion division algebras A defined over any totally real number field F, which are ramified at all infinite places except one. Choose an embedding

D = Cor_{F/ℚ}A ⊂ M(2^{m}, ℚ),

with m minimal. As we will see in Section 5 writing d = [F : ℚ] one finds m = d or m = d + 1. By 5.9 and 5.10 we get the following types of moduli functors of abelian varieties with special Mumford–Tate group

Hg = \{x ∈ D^{∗}; x\overline{x} = 1\}

and with a suitable level structure, which are represented by a smooth family Z_{A} → Y_{A} over a compact Shimura curve Y_{A}. Since we did not fix the level structure, Y_{A} is not uniquely determined by A. So it rather stands for a whole class of possible base curves, two of which have a common finite étale covering.

Example 0.4. Let Z_{η} denote the generic fibre of Z_{A} → Y_{A}. Then one of the following holds true:

i. 1 < m = d odd.

In this case dim(Z_{η}) = 2^{d−1} and End(Z_{η}) ⊗_{ℤ} ℚ = ℚ. ii. m = d + 1. Then dim(Z_{η}) = 2^{d} and:

a. For d odd, End(Z_{η})⊗_{ℤ}ℚ a totally indefinite quaternion algebra

over ℚ. b. For d even, End(Z_{η}) ⊗_{ℤ} ℚ a totally definite quaternion algebra

over ℚ.

Let us call the family Z_{A} → Y_{A} a family of Mumford type.

For d = 1 or 2 the examples in 0.4 include the only two Shimura curves of PEL-type, parameterizing:

• Moduli schemes of false elliptic curves, i.e., polarized abelian surfaces B with End(B) ⊗_{ℤ} ℚ a totally indefinite quaternion algebra over ℚ (see also [25]).

• Moduli schemes of abelian fourfolds B with End(B)⊗_{ℤ}ℚ a totally definite quaternion algebra over ℚ.

We will see in Section 6 that for g = g_{0}, up to powers and isogenies, the families of Mumford type are the only smooth families of abelian varieties over curves reaching the Arakelov bound.

Theorem 0.5. Let f : X → Y be a smooth family of abelian varieties. If the largest unitary local subsystem 𝕌_{1} of R^{1}f_{∗}ℂ_{X} is defined over ℚ and if f : X → Y reaches the Arakelov bound, then there exist:

a. a quaternion division algebra A, defined over a totally real number

field F, and ramified at all infinite places except one, b. an étale covering π : Y′ → Y, c. a family of Mumford type h : Z = Z_{A} → Y′ = Y_{A}, as in Exam-

ple 0.4, and an abelian variety B such that f′ : X′ = X×_{Y} Y′ → Y′ is isogenous to

B × Z ×_{Y′} ⋯ ×_{Y′} Z ⟶ Y′.

Things get more complicated if one drops the condition on the unitary local subsystem 𝕌_{1} of R^{1}f_{∗}ℂ_{X}. For one quaternion algebra A, there exist several nonisogenous families. Hence it will no longer be true, that up to a constant factor f : X → Y is isogenous to the product of one particular family.

Example 0.6 (See 5.11). Let A be a quaternion algebra defined over a totally real number field F, ramified at all infinite places but one, and let L be a subfield of F. Let β_{1}, …, β_{δ} : L → \overline{ℚ} denote the different embeddings of L. For μ = [F : L] + 1 (or may be μ = [F : ℚ] in case that L = ℚ and μ odd) there exists an embedding

Cor_{F/L}A ⊂ M(2^{μ}, L).

As is well-known (see Section 5) for some Shimura curve Y′ such an embedding gives rise to a representation of π_{1}(Y′, ∗) in M(2^{μ}, L), hence to a local L system 𝕍_{L}. Moreover there exists an irreducible ℚ-local system 𝕏_{ℚ} = 𝕏_{A,L;ℚ} for which 𝕏_{ℚ} ⊗ \overline{ℚ} is a direct sum of the local systems 𝕍_{L} ⊗_{L,β_{ν}} \overline{ℚ}.

There exist nonisotrivial families h : Z → Y′ with a geometrically simple generic fibre, such that R^{1}h_{∗}ℚ is a direct sum of ι copies of 𝕏_{ℚ}. Such examples, for g = 4 and 8 have been considered in [9]. Here F is a quadratic extension of ℚ, L = F and ι = 1 or 2. For g = 8, i.e., ι = 2, this gives the lowest-dimensional example of a nonrigid family of abelian varieties without a trivial subfamily [9]. A complete classification of such families is given in [21].

Theorem 0.7. Let f : X → Y be a smooth family of abelian varieties. If f : X → Y reaches the Arakelov bound, then there exist an étale covering π : Y′ → Y, a quaternion algebra A, defined over a totally real number field F and ramified at all of the infinite places except one, an abelian variety B, and ℓ families h_{i} : Z_{i} → Y′ of abelian varieties with geometrically simple generic fibre, such that:

i. f′ : X′ = X ×_{Y} Y′ → Y′ is isogenous to

B × Z_{1} ×_{Y′} ⋯ ×_{Y′} Z_{ℓ} ⟶ Y′.

ii. For each i ∈ {1, …, ℓ} there exists a subfield L_{i} of F such that the local system R^{1}h_{i∗}ℚ_{Z_{i}} is a direct sum of copies of the irreducible ℚ-local system 𝕏_{A,L_{i};ℚ} defined in Example 0.6.

iii. For each i ∈ {1, …, ℓ} the following conditions are equivalent: a. L_{i} = ℚ. b. h_{i} : Z_{i} → Y′ is a family of Mumford type, as defined in Exam-

ple 0.4. c. End(𝕏_{A,L_{i};ℚ})^{0,0} = End(𝕏_{A,L_{i};ℚ}). Moreover, if one of those conditions holds true, R^{1}h_{i∗}ℚ_{Z_{i}} is irreducible, hence R^{1}h_{i∗}ℚ_{Z_{i}} = 𝕏_{A,L_{i};ℚ}.

Here, contrary to 0.5, we do not claim that a component h_{i} : Z_{i} → Y′ is uniquely determined up to isogeny by 𝕏_{A,L_{i};ℚ} and by the rank of R^{1}h_{i∗}ℚ_{Z_{i}}.

We do not know for which g there are families of Jacobians among the families of abelian varieties considered in 0.2, 0.5 or 0.7, i.e., whether one can find a family φ : Z → Y of curves of genus g such that f : J(Z/Y) → Y reaches the Arakelov bound.

For Y = ℙ^{1} the Arakelov inequality (0.0.2) implies that #S ≥ 4. Our hope, that a family of abelian varieties with #S = 4 can not be a family of Jacobians, broke down when we found an example of a family of genus 2 curves over the modular curve X(3) in [13] (see also [37]), whose Jacobian is isogenous to the product of a fixed elliptic curve B with the modular curve E(3) → X(3) (see Section 7).

As mentioned already, this article owes a lot to the recent work of the second named author with Xiao-Tao Sun and Sheng-Li Tan. We thank Ernst Kani for explaining his beautiful construction in [13], and for sharing his view about higher genus analogs of families of curves with splitting Jacobians. It is also a pleasure to thank Ben Moonen, Hélène Esnault and Frans Oort for their interest and help, Ngaiming Mok, for explaining us differential geometric properties of base spaces of families and for pointing out Mumford’s construction in [18], Bruno Kahn and Claus Scheiderer for their help to understand quaternion algebras and their corestriction, and Martin Möller and the referee for pointing out several misprints and ambiguities.

This note grew out of discussions started when the first named author visited the Institute of Mathematical Science and the Department of Mathematics at the Chinese University of Hong Kong. His contributions to the final version (in particular to the proof of Theorems 0.5 and 0.7) were written during his visit to the I.H.E.S., Bures sur Yvette. He would like to thank the members of those three Institutes for their hospitality.

1. Splitting of ℂ-local systems

We will frequently use C. Simpson’s correspondence between polystable Higgs bundles of degree zero and representations of the fundamental group π_{1}(U, ∗).

Theorem 1.1 (C. Simpson [27]). There exists a natural equivalence between the category of direct sums of stable filtered regular Higgs bundles of degree zero, and of direct sums of stable filtered local systems of degree zero.

We will not recall the definition of a “filtered regular” Higgs bundle ([27], p. 717), just remark that for a Higgs bundle corresponding to a local system 𝕍 with unipotent monodromy around the points in S the filtration is trivial, and automatically deg(𝕍) = 0. By ([27], p.

720) the latter also holds true for local systems 𝕍 which are polarisable ℂ-variations of Hodge structures.

For example, 1.1 implies that the splitting of Higgs bundles (0.0.1) corresponds to a decomposition over ℂ

(R^{1}f_{∗}ℤ_{X_{0}}) ⊗ ℂ = 𝕍 ⊕ 𝕌_{1}

where 𝕍 corresponds to the Higgs bundle (F = F^{1,0} ⊕ F^{0,1}, τ = θ|_{F_{0,1}}) and 𝕌_{1} to (N = N^{1,0} ⊕ N^{0,1}, θ_{N} = θ|_{N} = 0). Let Θ(N, h) denote the curvature of the Hodge metric h on E^{1,0} ⊕ E^{0,1} restricted to N, then by [11], chapter II we have

Θ(N, h|_{N}) = −θ_{N} ∧ \overline{θ}_{N} − \overline{θ}_{N} ∧ θ_{N} = 0.

This means that h|_{N} is a flat metric. Hence, 𝕌_{1} is a unitary local system.

In general, if 𝕌 is a local system, whose Higgs bundle is a direct sum of stable Higgs bundles of degree zero and with a trivial Higgs field, then 𝕌 is unitary.

As a typical application of Simpson’s correspondence one obtains the polystability of the components of certain Higgs bundles. We just formulate it in the weight one case.

Recall that F^{1,0} is polystable, if there exists a decomposition

F^{1,0} ≃ ⨁_{i} A_{i}

with A_{i} stable, and

\frac{deg A_{i}}{rankA_{i}} = \frac{deg F^{1,0}}{rankF^{1,0}} .

Proposition 1.2. Let 𝕍 be a direct sum of stable filtered local systems of degree zero with Higgs bundle (F = F^{1,0} ⊕F^{0,1}, τ). Assume that τ|_{F^{0,1}} = 0, that

τ_{1,0} = τ|_{F^{1,0}} : F^{1,0} ⟶ F^{0,1} ⊗ Ω_{Y}^{1} (log S) ⊂ F ⊗ Ω_{Y}^{1} (log S),

and that for g_{0} = rank(F^{1,0})

(1.2.1) 2 · deg(F^{1,0}) = g_{0} · deg(Ω_{Y}^{1} (log S)).

Then τ_{1,0} is an isomorphism, and the sheaf F^{1,0} is polystable.

Proof of 1.2. Let A ⊂ F^{1,0} be a subsheaf, and let B ⊗ Ω_{Y}^{1} (log S) be its image under τ_{1,0}. Then A ⊕ B is a Higgs subbundle of F^{1,0} ⊕ F^{0,1}, and applying 1.1 one finds deg(A) + deg(B) ≤ 0. Hence

deg(A) = deg(B) + rank(B) · deg(Ω_{Y}^{1} (log S))
≤ deg(B) + rank(A) · deg(Ω_{Y}^{1} (log S))
≤ − deg(A) + rank(A) · deg(Ω_{Y}^{1} (log S)).

The equality (1.2.1) implies that

\frac{deg(A)}{rank(A)} ≤ \frac{1}{2} deg(Ω_{Y}^{1} (log S)) = \frac{deg(F^{1,0})}{g_{0}},

and F^{1,0} is semistable. If

\frac{deg(A)}{rank(A)} = \frac{deg(F^{1,0})}{g_{0}},

rank(A) = rank(B) and deg(B) = − deg(A). The Higgs bundle (F^{1,0} ⊕ F^{0,1}, τ) splits by 1.1 as a direct sum of stable Higgs bundles of degree zero. Hence (A ⊕ B, τ|_{A⊕B}) is a direct factor of (F^{1,0} ⊕ F^{0,1}, τ). In particular, A is a direct factor of F^{1,0}. For A = F^{1,0} one also obtains that τ_{1,0} is injective and by (1.2.1) it must be an isomorphism. q.e.d.

The local system R^{1}f_{∗}ℚ_{X_{0}} on U = Y \ S is a ℚ-variation of Hodge structures with unipotent local monodromies around s ∈ S, obviously having a ℤ-form. By Deligne’s semisimplicity theorem [4] it decomposes as a direct sum of irreducible polarisable ℚ-variation of Hodge structures 𝕍_{iℚ}.

More generally, if 𝕍 is a polarized ℂ-variation of Hodge structures, and

𝕍 = ⨁_{i} 𝕍_{i},

a decomposition with 𝕍_{i} an irreducible ℂ-local system, then by [7] each 𝕍_{i} again is a polarisable ℂ-variation of Hodge structures.

In both cases, taking the grading of the Hodge filtration, one obtains a decomposition of the Higgs bundle

(E, θ) = (F^{1,0} ⊕ F^{0,1}, τ) ⊕ (N^{1,0} ⊕ N^{0,1}, 0)

as a direct sum of sub-Higgs bundles, as stated in 1.1. Obviously, each of the 𝕍_{iℚ} again reaches the Arakelov bound.

Our next constructions will not require the local system to be defined over ℚ. So by abuse of notations, we will make the following assumptions:

Assumption 1.3. For a number field L ⊂ ℂ consider a polarized L-variation of Hodge structures 𝕏_{L} of weight one over U = Y \ S with unipotent local monodromies around s ∈ S. Assume that the local system 𝕏 = 𝕏_{L} ⊗_{L} ℂ has a decomposition 𝕏 = 𝕍 ⊕ 𝕌_{1}, with 𝕌_{1} unitary, corresponding to the decomposition

(E, θ) = (F, τ_{1,0}) ⊕ (N, 0) = (F^{1,0} ⊕ F^{0,1}, τ_{1,0}) ⊕ (N^{1,0} ⊕ N^{0,1}, 0)

of Higgs fields. Assume that 𝕍 (or (F, τ)) has a maximal Higgs field, i.e., that

τ_{1,0} = τ|_{F^{1,0}} : F^{1,0} → F^{0,1} ⊗ Ω_{Y}^{1} (log S)

is an isomorphism. Obviously, for g_{0} = rank(F^{1,0}) this is equivalent to the equality (1.2.1). Hence we will also say, that 𝕏 (or (E, θ)) reaches the Arakelov bound.

Proposition 1.4. If deg Ω_{Y}^{1} (log S) is even there exists a tensor product decomposition of variations of Hodge structures

𝕍 ≃ 𝕃 ⊗_{ℂ} 𝕋,

with:

a. 𝕃 is a rank-2 local system. For some invertible sheaf L, with L^{2} = Ω_{Y}^{1} (log S) the Higgs bundle corresponding to 𝕃 is

(L ⊕ L^{−1}, τ′),

with τ′|_{L^{−1}} = 0 and τ′|_{L} given by an isomorphism

τ′_{1,0} : L ⟶ L^{−1} ⊗ Ω_{Y}^{1} (log S).

L has bidegree 1, 0, and L^{−1} has bidegree 0, 1. b. If g_{0} is odd, L^{g_{0}} = det(F^{1,0}) and L is uniquely determined. c. For g_{0} even, there exists some invertible sheaf N of order two in

Pic^{0}(Y) with L^{g_{0}} = det(F^{1,0}) ⊗ N. d. 𝕋 is a unitary local system and a variation of Hodge structures of

pure bidegree 0, 0. If (T , 0) denotes the corresponding Higgs field, then T = F^{1,0} ⊗ L^{−1} = F^{0,1} ⊗ L.

In Section 6 we will need a slightly stronger statement.

Addendum 1.5. If in 1.4, there exists a presentation 𝕍 = 𝕋_{1} ⊗_{ℂ}𝕍_{1} with 𝕋_{1} unitary and a variation of Hodge structures of pure bidegree 0, 0, then there exists a unitary local system 𝕋_{2} with 𝕋 = 𝕋_{1} ⊗_{ℂ} 𝕋_{2}.

In fact, write (T_{1}, 0) and (F_{1}^{1,0}⊕F_{1}^{0,1}, τ_{1}) for Higgs fields corresponding to 𝕋_{1} and 𝕍_{1}, respectively. Then deg(T_{1}) = 0 and

2 · deg(F_{1}^{1,0}) · rank(T ) = 2 · deg(F^{1,0})
= g_{0} · deg(Ω_{Y}^{1} (log S))
= rank(F_{1}^{1,0}) · rank(T ) · deg(Ω_{Y}^{1} (log S)).

So (F_{1}^{1,0} ⊕ F_{1}^{0,1}, τ_{1}) again satisfies the assumptions made in 1.4.

Proof of 1.4. Taking the determinant of

τ_{1,0} : F^{1,0} ⟶^{≃} F^{0,1} ⊗ Ω_{Y}^{1} (log S),

one obtains an isomorphism

det τ_{1,0} : det F^{1,0} ⟶^{≃} det F^{0,1} ⊗ Ω_{Y}^{1} (log S)^{g_{0}},

By assumption there exists an invertible sheaf L with L^{2} = Ω_{Y}^{1} (log S). Since F^{1,0} ≃ F^{0,1∨},

(det F^{1,0})^{2} ≃ Ω_{Y}^{1} (log S)^{g_{0}} = L^{2·g_{0}},

and det F^{1,0} ⊗ L^{−g_{0}} = N is of order two in Pic^{0}(Y).

If g_{0} is even, L is uniquely determined up to the tensor product with two torsion points in Pic^{0}(Y).

If g_{0} is odd, one replaces L by L ⊗ N and obtains det F^{1,0} = L^{g_{0}}. By 1.2 the sheaf

T = F^{1,0} ⊗ L^{−1}

is polystable of degree zero. 1.1 implies that the Higgs bundle (T , 0) corresponds to a local system 𝕋, necessarily unitary.

Choose 𝕃 to be the local system corresponding to the Higgs bundle

(L ⊕ L^{−1}, τ′), with τ′_{1,0} : L ⟶^{≃} L^{−1} ⊗ Ω_{Y}^{1} (log S).

The isomorphism

τ_{1,0} : T ⊗ L = F^{1,0} ⟶^{≃} F^{0,1} ⊗ Ω_{Y}^{1} (log S) ⟶^{≃} F^{0,1} ⊗ L^{2}

induces an isomorphism

ϕ : T ⊗ L^{−1} ⟶^{≃} F^{0,1},

such that τ_{1,0} = ϕ∘(id_{T} ⊗τ′_{1,0}). Hence the Higgs bundles (F^{1,0}⊕F^{0,1}, τ) and (T ⊗ (L ⊕ L^{−1}), id_{T} ⊗ τ′) are isomorphic, and 𝕍 ≃ 𝕋 ⊗_{ℂ} 𝕃. q.e.d.

Remark 1.6. i. If deg Ω_{Y}^{1} (log S) is odd, hence S ≠ ∅, and if the genus of Y is not

zero, one may replace Y by an étale covering, in order to be able to apply 1.4. Doing so one may also assume that the invertible sheaf N in 1.4 c), is trivial.

ii. For Y = ℙ^{1} and for 𝕏 reaching the Arakelov bound, #S is always even. This, together with the decomposition 1.4, for 𝕌 = ℂ^{g_{0}}, can easily obtained in the following way: by 1.2, F^{1,0} must be the direct sum of invertible sheaves L_{i}, all of the same degree, say ν. Since τ_{1,0} is an isomorphism, the image τ_{1,0}(L_{i}) is O_{ℙ^{1}}(2−s+ν)⊗ Ω_{ℙ^{1}}(log S). Since F^{0,1} is dual to F^{1,0} one obtains −ν = 2 − s + ν, and writing L_{i}^{−1} = τ_{1,0}(L_{i}),

(F^{1,0} ⊕ F^{0,1}, τ) ≃ (⨁_{i} O_{ℙ^{1}}(ν) ⊕ O_{ℙ^{1}}(−ν), ⨁_{i} τ) .

Consider now the local system of endomorphism 𝔼nd(𝕍) of 𝕍, which is a polarized weight zero L-variation of Hodge structures. The Higgs bundle

(F^{1,0} ⊕ F^{0,1}, τ)

for 𝕍 induces the Higgs bundle

(F^{1,−1} ⊕ F^{0,0} ⊕ F^{−1,1}, τ_{1,−1} ⊕ τ_{0,0})

corresponding to 𝔼nd(𝕍) = 𝕍 ⊗_{ℂ} 𝕍^{∨}, by choosing

F^{1,−1} = F^{1,0} ⊗ F^{0,1}^{∨}, F^{0,0} = F^{1,0} ⊗ F^{1,0}^{∨} ⊕ F^{0,1} ⊗ F^{0,1}^{∨}

and F^{−1,1} = F^{0,1} ⊗ F^{1,0}^{∨}.

The Higgs field is given by

τ_{1,−1} = (−id) ⊗ τ_{1,0}^{∨} ⊕ τ_{1,0} ⊗ id and τ_{0,0} = τ_{1,0} ⊗ id ⊕ (−id) ⊗ τ_{1,0}^{∨}.

Lemma 1.7. Assume as in 1.3 that 𝕏 reaches the Arakelov bound or equivalently that the Higgs field of 𝕍 is maximal. Let

F_{u}^{0,0} := Ker(τ_{0,0}) and F_{m}^{0,0} = Im(τ_{1,−1}).

Then there is a splitting of the Higgs bundle

(F^{1,−1} ⊕ F^{0,0} ⊕ F^{−1,1}, τ_{1,−1} ⊕ τ_{0,0})
= (F^{1,−1} ⊕ F_{m}^{0,0} ⊕ F^{−1,1}, τ_{1,−1} ⊕ τ_{0,0}) ⊕ (F_{u}^{0,0}, 0),

which corresponds to a splitting of the local system over ℂ

𝔼nd(𝕍) = 𝕎 ⊕ 𝕌.

𝕌 is unitary of rank g_{0}^{2} and a variation of Hodge structures concentrated in bidegree 0, 0, whereas 𝕎 is a ℂ-variation of Hodge structures of weight zero and rank 3g_{0}^{2}.

τ_{1,−1} : F^{1,−1} ⟶ F_{m}^{0,0} ⊗ Ω_{Y}^{1} (log S)  and
τ_{0,0} : F_{m}^{0,0} ⟶ F^{−1,1} ⊗ Ω_{Y}^{1} (log S)

are both isomorphisms.

Proof. By definition, (F_{u}^{0,0}, 0) is a sub-Higgs bundle of

(F^{1,−1} ⊕ F^{0,0} ⊕ F^{−1,1}, τ_{1,−1} ⊕ τ_{0,0}).

We have an exact sequence

0 ⟶ F_{u}^{0,0} ⟶ F^{0,0} ⟶^{τ_{0,0}} F^{−1,1} ⊗ Ω_{Y}^{1} (log S).

Since τ_{1,0} ⊗ id is surjective, τ_{0,0} is surjective, and

deg(F_{u}^{0,0}) = deg(F^{0,0}) − deg(F^{−1,1}) − rank(F^{−1,1}) · deg(Ω_{Y}^{1} (log S)).

By the Arakelov equality,

deg(F^{−1,1}) = g_{0} · deg(F^{0,1}) + g_{0} · deg(F^{1,0}^{∨})

= g_{0}^{2} · deg(Ω_{Y}^{1} (log S)) = rank(F^{−1,1}) · deg(Ω_{Y}^{1} (log S))

and one finds deg(F_{u}^{0,0}) = deg(F^{0,0}) = 0.

By 1.1 (F_{u}^{0,0}, 0), as a Higgs subbundle of degree zero with trivial Higgs field, corresponds to a unitary local subsystem 𝕌 of 𝔼nd(𝕍). The exact sequence

0 → F_{u}^{0,0} → F^{0,0} → F^{−1,1} ⊗ Ω_{Y}^{1} (log S) → 0

splits, and one obtains a direct sum decomposition of Higgs bundles

(F^{1,−1} ⊕ F^{0,0} ⊕ F^{−1,1}, τ) = (F^{1,−1} ⊕ F_{m}^{0,0} ⊕ F^{−1,1}, τ) ⊕ (F_{u}^{0,0}, 0),

which induces the splitting on 𝔼nd(𝕍) with the desired properties. q.e.d.

In 1.7 the local subsystem 𝕎 of 𝔼nd(𝕍) has a maximal Higgs field in the following sense:

Definition 1.8. Let 𝕎 be a ℂ-variation of Hodge structures of weight k, and let

(F, τ) = ( ⨁_{p+q=k} F^{p,q}, ⨁ τ_{p,q})

be the corresponding Higgs bundle. Recall that the width is defined as

width(𝕎) = Max{|p − q|; F^{p,q} ≠ 0}.

i. 𝕎 (or (F, τ)) has a generically maximal Higgs field if width(𝕎) > 0 and if: a. F^{p,k−p} ≠ 0 for all p with |2p − k| ≤ width(𝕎). b. τ_{p,k−p} : F^{p,k−p} → F^{p−1,k−p+1} ⊗ Ω_{Y}^{1} (log S) is generically an isomorphism for all p with |2p−k| ≤ width(𝕎) and |2p−2−k| ≤ width(𝕎).

ii. 𝕎 (or (F, τ)) has a maximal Higgs field if the τ_{p,k−p} in i) b. are all isomorphisms.

In particular, a variation of Hodge structures with a maximal Higgs field can not be unitary.

Properties 1.9.

a. If 𝕎 is a ℂ-variation of Hodge structures with a (generically) maximal Higgs field, and if 𝕎′ ⊂ 𝕎 is a direct factor, then width(𝕎′) = width(𝕎) and 𝕎′ has again a (generically) maximal Higgs field.

b. Let 𝕃 and 𝕋 be two variations of Hodge structures with 𝕃 ⊗ 𝕋 of weight 1 and width 1, and with a (generically) maximal Higgs field. Then, choosing the bidegrees for 𝕃 and 𝕋 in an appropriate way, either 𝕃 is a variation of Hodge structures concentrated in degree 0, 0, and 𝕋 is a variation of Hodge structures of weight one and width one with a (generically) maximal Higgs field, or vice versa.

Proof. For a) consider the Higgs field (⨁ F′^{p,q}, τ′_{p,q}) of 𝕎′, which is a direct factor of the one for 𝕎. Since the τ_{p,q} are (generically) isomorphisms, a) is obvious.

In b) denote the components of the Higgs fields of 𝕃 and 𝕋 by L^{p_{1},q_{1}} and T^{p_{2},q_{2}}, respectively. Shifting the bigrading one may assume that p_{1} = 0 and p_{2} = 0 are the smallest numbers with L^{p_{1},q_{1}} ≠ 0 and T^{p_{2},q_{2}} ≠ 0 and moreover that the corresponding q_{i} ≥ 0. Since q_{1} + q_{2} = 1, one of q_{i} must be zero, let us say the first one.

Then T^{p_{2},q_{2}} can only be nonzero, for (p_{2}, q_{2}) = (0, 1) or = (1, 0) and 𝕃 is concentrated in degree 0, 0.

Obviously this forces the Higgs field of 𝕃 to be zero. Then the Higgs field of 𝕃 ⊗ 𝕋 is the tensor product of the Higgs field

T^{1,0} → T^{0,1} ⊗ Ω_{Y}^{1} (log S)

with the identity on L^{0,0}, hence the first one has to be (generically) an isomorphism. q.e.d.

Remark 1.10. The splitting in 1.7 can also be described by the tensor product decomposition 𝕍 = 𝕋 ⊗_{ℂ} 𝕃 in 1.4 with 𝕋 unitary and 𝕃 a rank two variation of Hodge structures of weight one and with a maximal Higgs field. For any local system 𝕄 one has a natural decomposition 𝔼nd(𝕄) = 𝔼nd_{0}(𝕄) ⊕ ℂ, where ℂ acts on 𝕄 by multiplication. Applying 1.7 to 𝕃 instead of 𝕍, gives exactly the decomposition 𝔼nd(𝕃) = 𝔼nd_{0}(𝕃) ⊕ ℂ. One obtains

𝔼nd(𝕍) = 𝕋 ⊗_{ℂ} 𝕋^{∨} ⊗_{ℂ} 𝕃 ⊗_{ℂ} 𝕃^{∨}
= (𝔼nd_{0}(𝕋) ⊕ ℂ) ⊗_{ℂ} (𝔼nd_{0}(𝕃) ⊕ ℂ)
= 𝔼nd_{0}(𝕋) ⊕ ℂ ⊕ 𝔼nd(𝕋) ⊗_{ℂ} 𝔼nd_{0}(𝕃).

Here 𝔼nd_{0}(𝕋) ⊕ ℂ is unitary and 𝕎 = 𝔼nd(𝕋) ⊗_{ℂ} 𝔼nd_{0}(𝕃) has again a maximal Higgs field.

Remark 1.11. If one replaces 𝔼nd(𝕍) by the isomorphic local system 𝕍 ⊗_{ℂ} 𝕍, one obtains the same decomposition. However, it is more natural to shift the weights by two, and to consider this as a variation of Hodge structures of weight 2.

A statement similar to 1.7 holds true for ∧^{2}(𝕍). Here the Higgs bundle is given by

F′^{2,0} = F^{1,0} ∧ F^{1,0}, F^{1,1} = F′^{1,0} ⊗ F^{0,1} and F′^{0,2} = F^{0,1} ∧ F^{0,1}.

2. Shimura curves and the special Mumford–Tate group

Lemma 2.1. Let 𝕃 be a real variation of Hodge structures of weight 1 dimension 2, with a nontrivial Higgs field. Let

γ_{𝕃} : π_{1}(U, ∗) → Sl(2, ℝ)

be the corresponding representation and let Γ_{𝕃} denote the image of γ_{𝕃}. Assume that the local monodromies around the points s ∈ S are unipotent. Then the Higgs field of 𝕃 is maximal if and only if

U = Y \ S ≃ H/Γ_{𝕃}.

Proof. Writing L for the (1, 0) part, we have an nontrivial map

τ_{1,0} : L ⟶ L^{−1} ⊗ Ω_{Y}^{1} (log S). (2.1.1)

Since L is ample, Ω_{Y}^{1} (log S) is ample, hence the universal covering Ũ of U = Y \ S is the upper half plane H. Let

Ũ = H \underrightarrow{φ̃} H

be the period map. The tangent sheaf of the period domain H is given by the sheaf of homomorphisms from the (1, 0) part to the (0, 1) part of the variation of Hodge structures. Therefore τ_{1,0} is an isomorphism if and only if φ̃ is a local diffeomorphism. Note that by Schmid [23] the Hodge metric on the Higgs bundle corresponding to 𝕃 has logarithmic growth at S and bounded curvature. By the remarks following [28], Propositions 9.8 and 9.1, τ_{1,0} is an isomorphism if and only if φ̃: Ũ → H is a covering map, hence an isomorphism.

Obviously the latter holds true in case Y \ S ≃ H/Γ_{𝕃}.

Assume that φ̃ is an isomorphism. Since φ̃ is an equivariant with respect to the π_{1}(U, ∗)-action on Ũ and the Pρ_{𝕃}(π_{1}(U, ∗))-action on H, the homomorphism

ρ_{𝕃_{ℤ}} : π_{1}(U, ∗) ⟶ Pρ_{𝕃_{ℤ}}(π_{1}(U, ∗)) ⊂ PSl_{2}(ℝ)

must be injective, hence an isomorphism. So φ̃ descend to an isomorphism

φ : Y \ S ≃ H/Γ_{𝕃}.  q.e.d.

Proof of Proposition 0.1. h : E → Y be the semistable family of elliptic curves, reaching the Arakelov bound, smooth over U. Hence 𝕃_{ℤ} = R^{1}h_{∗}ℤ_{E_{0}} is a ℤ-variation of Hodge structures of weight one and of rank two. Writing L for the (1, 0) part, we have an isomorphism

τ_{1,0} : L ⟶ L^{−1} ⊗ Ω_{Y}^{1} (log S). (2.1.2)

Since L is ample, Ω_{Y}^{1} (log S) is ample, hence the universal covering of U is the upper half plane H. One obtains a commutative diagram

H \underrightarrow{φ̃} H
ψ′↓  ^{ψ}↓
U ⟶^{j} ℂ

where j is given by the j-invariant of the fibres of E_{0} → U, where ψ is the quotient map H → H/Sl_{2}(ℤ), and where φ̃ is the period map. 2.1 implies that

φ : U ⟶ H/ρ_{𝕃_{ℤ}}(π_{1}(U, ∗))

is an isomorphism, hence ρ_{𝕃_{ℤ}}(π_{1}(U, ∗)) ⊂ Sl_{2}(ℤ) is of finite index, and E → Y is a semistable model of a modular curve. q.e.d.

Let us recall the description of wedge products of tensor products (see [10], p. 80). We will write λ = {λ_{1}, …, λ_{ν}} for the partition of g_{0} as g_{0} = λ_{1} + ⋯ + λ_{ν}. The partition λ defines a Young diagram and the Schur functor 𝕊_{λ}. Assuming as in 1.4 that 𝕃 is a local system of rank 2, and 𝕋 a local system of rank g_{0}, both with trivial determinant, one has

∧^{k}(𝕃 ⊗ 𝕋) = ⨁ 𝕊_{λ}(𝕃) ⊗ 𝕊_{λ′}(𝕋)

where the sum is taken over all partitions λ of k with at most 2 rows and at most g_{0} columns, and where λ′ is the partition conjugate to λ. Similarly,

S^{k}(𝕃 ⊗ 𝕋) = ⨁ 𝕊_{λ}(𝕃) ⊗ 𝕊_{λ}(𝕋)

where the sum is taken over all partitions λ of k with at most 2 rows.

The only possible λ are of the form \{k − a, a\}, for a ≤ \frac{k}{2}. By [10], 6.9 on p. 79,

𝕊_{\{k−a,a\}}(𝕃) = \{ \begin{matrix}𝕊_{\{k−2a\}}(𝕃) = S^{k−2a}(𝕃) ⊗ det(𝕃)^{a} & if & 2a < k \\ 𝕊_{\{a,a\}}(𝕃) = det(𝕃)^{a} & if & 2a = k. \end{matrix}

For k = g_{0} one obtains:

Lemma 2.2. Assume that det(𝕃) = ℂ and det(𝕋) = ℂ.

a. If g_{0} is odd, then for some partitions λ_{c},

⋀^{g_{0}}(𝕃 ⊗ 𝕋) = ⨁_{c=0}^{\frac{g_{0}−1}{2}} S^{2c+1}(𝕃) ⊗ 𝕊_{λ_{2c}}(𝕋).

In particular, for c = \frac{g_{0}−1}{2} one obtains

S^{g_{0}}(𝕃) ⊗ ⋀^{g_{0}}(𝕋) = S^{g_{0}}(𝕃)

as a direct factor. b. If g_{0} is even, then for some partitions λ_{c},

⋀^{g_{0}}(𝕃 ⊗ 𝕋) = S^{g_{0}}(𝕃) ⊕ 𝕊_{\{2,...,2\}}(𝕋) ⊕ ⨁_{c=1}^{\frac{g_{0}}{2} −1} S^{2c}(𝕃) ⊗ 𝕊_{λ_{2c}}(𝕋).

Lemma 2.3. Assume that 𝕃 and 𝕋 are variations of Hodge structures, with 𝕃 of weight one, width one and with a maximal Higgs field, and with 𝕋 pure of bidegree 0, 0. a. If k is odd,

H^{0} (Y, ⋀^{k}(𝕃 ⊗ 𝕋)) = 0.

b. If k is even, say k = 2c, then for some λ_{c}

H^{0} (Y, ⋀^{k}(𝕃 ⊗ 𝕋)) = H^{0}(Y, det(𝕃)^{c} ⊗ 𝕊_{λ_{c}}(𝕋)).

c. For k = 2 one has in b) λ_{1} = {2}, hence 𝕊_{λ_{1}}(𝕋) = ⋀^{2}(𝕋).

d. H^{0}(Y, S^{2}(𝕃 ⊗ 𝕋)) = H^{0}(Y, det(𝕃) ⊗ ⋀^{2}(𝕋)).

Proof. S^{ℓ}(𝕃) has a maximal Higgs field for ℓ > 0, whereas for all partitions λ′ the variation of Hodge structures 𝕊_{λ′}(𝕋) is again pure of bidegree 0, 0. By 1.9 a), S^{ℓ}(𝕃) ⊗ 𝕊_{λ′}(𝕋) has no global sections. Hence ⋀^{k}(𝕃 ⊗ 𝕋) can only have global sections for k even. In this case, the global sections lie in

det(𝕃)^{c} ⊗ 𝕊_{λ_{c}}(𝕋),

for some partition λ_{c}, and one obtains a) and b). For k = 2 one finds λ_{1} = {2}. For d) one just has the two partitions {1, 1} and {2}. Again, the direct factor S^{2}(𝕃) ⊗ S^{2}(𝕋), corresponding to the first one, has no global section. q.e.d.

Let us shortly recall Mumford’s definition of the Hodge group, or as one writes today, the special Mumford–Tate group (see [17], [18], and also [5] and [24]). Let B be an abelian variety and H^{1}(B, ℚ) and Q the polarization on V. The special Mumford–Tate group Hg(B) is defined in [17] as the smallest ℚ-algebraic subgroup of Sp(H^{1}(B, ℝ), Q), which contains the complex structure. Equivalently Hg(B) is the largest ℚalgebraic subgroup of Sp(H^{1}(B, ℚ), Q), which leaves all Hodge cycles of B × ⋯ × B invariant, hence all elements

η ∈ H^{2p}(B × ⋯ × B, ℚ)^{p,p} = [⋀^{2p}(H^{1}(B, ℚ) ⊕ ⋯ ⊕ H^{1}(B, ℚ))]^{p,p} .

For a smooth family of abelian varieties f : X_{0} → U with B = f^{−1}(y) for some y ∈ U, and for the corresponding ℚ-variation of polarized Hodge structures R^{1}f_{∗}ℚ_{X_{0}} consider Hodge cycles η on B which remain Hodge cycles under parallel transform. One defines the special Mumford– Tate group Hg(R^{1}f_{∗}ℚ_{X_{0}}) as the largest ℚ-subgroup of Sp(H^{1}(B, ℚ), Q) which leaves all those Hodge cycles invariant ([5], §7, or [24], 2.2).

Lemma 2.4.

a. For all y ∈ U the special Mumford–Tate group Hg(f^{−1}(y)) is a subgroup of Hg(R^{1}f_{∗}ℚ_{X_{0}}). For all y in the complement U′ of the union of countably many proper closed subsets it coincides with Hg(R^{1}f_{∗}ℚ_{X_{0}}).

b. Let G^{Mon} denote the smallest reductive ℚ-subgroup of

Sp(H^{1}(B, ℝ), Q),

containing the image Γ of the monodromy representation

γ : π_{0}(U) ⟶ Sp(H^{1}(B, ℝ), Q).

Then the connected component G_{0}^{Mon} of one in G^{Mon} is a subgroup of Hg(R^{1}f_{∗}ℚ_{X_{0}}).

c. If f : X → Y reaches the Arakelov bound, and if R^{1}f_{∗}ℂ_{X} has no unitary part, then G_{0}^{Mon} = Hg(R^{1}f_{∗}ℚ_{X_{0}}).

Proof. The first statement of a) has been verified in [24], 2.3., and the second in [16] 1.2. As explained in [5], §7, or [24], 2.4, the Mumford– Tate group contains a subgroup of Γ of finite index, hence b) holds true. It is easy to see, that the same holds true for the special Mumford–Tate group (called Hodge group in [17]) by using the same argument.

Since the special Mumford–Tate group of an abelian variety is reductive, a) implies that Hg(R^{1}f_{∗}ℚ_{X_{0}}) is reductive. So G_{0}^{Mon} ⊂Hg(R^{1}f_{∗}ℚ_{X_{0}}) is an inclusion of reductive groups. The proof of 3.1, (c), in [6] carries over to show that both groups are equal, if they leave the same tensors

η ∈ [⋀^{2p}(H^{1}(B, ℚ) ⊕ ⋯ ⊕ H^{1}(B, ℚ))]

invariant.

Let η ∈ H^{k}(B, ℚ) be invariant under Γ, and let η̃ be the corresponding global section of

⋀^{k}(R^{1}f_{∗}ℚ_{X_{0}}) = ⋀^{k}(𝕃 ⊗ 𝕋).

By 2.3 a) and b), one can only have global sections for k = 2c, and those lie in

det(𝕃)^{c} ⊗ 𝕊_{λ_{c}}(𝕋).

In particular they are of pure bidegree c, c.

The same argument holds true, if one replaces B and f : X → Y by any product, which implies c). q.e.d.

For the Hodge group Hg(R^{1}f_{∗}ℚ_{X}) = Hg ⊂ Sp(2g, ℚ), as in Lemma 2.4 Mumford considers the moduli functor M(Hg) of isomorphy classes of polarized abelian varieties with special Mumford–Tate group equal to a subgroup of Hg. He shows that M(Hg) admits a quasiprojective coarse moduli space M(Hg), which lies in the coarse moduli space of polarized abelian varieties A_{g}. By Mumford ([17], Section 3, [18], Sections 1–2)

M(Hg) = Γ\Hg(ℝ)/K

where K is a maximal compact subgroup of Hg(ℝ), and Γ an arithmetic subgroup of Hg(ℚ). The embedding M(Hg) ↪ A_{g} is a totally geodesic embedding, and M(Hg) is a Shimura variety of Hodge Type Hg.

Let f : X_{0} → U be a family of abelian varieties with the special Mumford–Tate group Hg(R^{1}f_{∗}ℚ_{X_{0}}) = Hg. By Lemma 2.4 a), f induces a morphism

U → M(Hg).

Proof of 0.3. By Proposition 1.4 the image of the monodromy representation of f lies in Sl_{2}(ℝ) × G, for some compact group G, and its Zariski closure is Sl_{2}(ℝ) × G. Hence, G_{0}^{Mon}(ℝ) is again the product of Sl_{2}(ℝ) with a compact group. Lemma 2.4 c), implies that

Hg(ℝ) = Sl_{2}(ℝ) × G′

for a compact group G′, hence Hg(ℝ)/K ≃ Sl_{2}(ℝ)/SO_{2} is the upper half plane H.

In particular, dim M(Hg) = 1. Since we assumed the family to be nonisotrivial and semistable, the morphism U → M(Hg) is surjective.

Consider the composition ϕ : U → M(Hg) → A_{g}. Replacing U by an étale covering, we may assume that X_{0} → U is the pullback of a universal family of abelian varieties, defined over an étale covering A′_{g} of A_{g}. The pull back of the tangent bundle on A_{g} via ϕ is just

ϕ^{∗}T_{A_{g}} = S^{2}E^{0,1} ⊂ E^{0,1}^{⊗2}.

The differential dϕ : T_{U} → ϕ^{∗}T_{A_{g}} ⊂ E^{0,1}^{⊗2} is induced by the Kodaira-Spencer map E^{1,0} ⊗ T_{U} → E^{0,1}. By Proposition 1.4

E^{1,0} ⊕ E^{0,1} = (L ⊕ L^{−1}) ⊗ 𝕋,

and the map dϕ : T_{U} → E^{0,1}^{⊗2} lies in the component

dϕ : T_{U} ≃ L^{⊗−2} ⊂ L^{⊗−2} ⊗ End(𝕋).

This implies that the differential of the map U → M(Hg) is no where vanishing, hence U → M(Hg) is étale. q.e.d.

Remarks 2.5.

a) As is well-known (see [17], [18]) the moduli space of abelian varieties with a given special Mumford–Tate group is necessarily a Satake holomorphic embedding. Hence the assumptions made in Proposition 0.3 imply in particular that the period map from U to the corresponding moduli space of abelian varieties with a fixed level structure is a Satake holomorphic embedding.

b) Presumably Proposition 0.3 can also be obtained using [1]. Using Proposition 1.4 the maximality of the Higgs field should imply that the period map from U = H/Γ to the Siegel upper half plane is a rigid, totally geodesic, and equivariant holomorphic map. Then [1], Theorem 3.4, implies that f : X → Y is a family of Mumford type, and as mentioned in the introduction one can finish the proof of Theorem 0.5 going through the classification of Shimura varieties.

c) Without the assumption of rigidity, hidden behind the one saying that the maximal unitary local subsystem is defined over ℚ, we do not see a way to show directly, that the families are Kuga fibre spaces. One needs a precise description of the ℤ-structure on the decompositions of the variation of Hodge structures. On the other hand, the latter will allow one to prove Theorems 0.5 and 0.7 directly.

d) Theorems 0.5 and 0.7 imply that all families f : X → Y with maximal Higgs fields are Kuga fibre spaces, and that the period map is again a Satake holomorphic embedding.

3. Splitting over \overline{ℚ}

Up to now, we considered local systems of ℂ-vector spaces induced by the family of abelian varieties. We say that a ℂ-local system 𝕄 is defined over a subring R of ℂ, if there exists a local system 𝕄_{R} of torsion-free R-modules with 𝕄 = 𝕄_{R} ⊗_{R} ℂ. In different terms, the representation

γ_{𝕄} : π_{0}(U, ∗) ⟶ Gl(μ, ℂ)

is conjugate to one factoring like

γ_{𝕄} : π_{0}(U, ∗) ⟶ Gl(μ, R) ⟶ Gl(μ, ℂ).

If 𝕄 is defined over R, and if σ : R → R′ is a ring isomorphism, we will write 𝕄_{R}^{σ} for the local system defined by

γ_{𝕄} : π_{0}(U, ∗) ⟶ Gl(μ, R) ⟶^{σ} Gl(μ, R′),

and 𝕄^{σ} = 𝕄_{R}^{σ}⊗_{R′} ℂ. In this section we want to show, that the splittings 𝕏 = 𝕍 ⊕ 𝕌_{1} and 𝔼nd(𝕍) = 𝕎 ⊕ 𝕌 considered in the last section are defined over \overline{ℚ}, i.e., that there exists a number field K containing the field of definition for 𝕏 and local K-subsystems

𝕍_{K} ⊂ 𝕏_{K}, 𝕌_{1K} ⊂ 𝕏_{K}, 𝕎_{K} ⊂ 𝔼nd(𝕏_{K}) and 𝕌_{K} ⊂ 𝔼nd(𝕏_{K})

with

𝕏_{K} = 𝕏_{L} ⊗_{L} K = 𝕍_{K} ⊕ 𝕌_{1K}, 𝕍_{K} = 𝕎_{K} ⊕ 𝕌_{K}, and with 𝕍 = 𝕍_{K} ⊗_{K} ℂ, 𝕌_{1} = 𝕌_{1K} ⊗_{K} ℂ 𝕎 = 𝕎_{K} ⊗_{K} ℂ, 𝕌 = 𝕌_{K} ⊗_{K} ℂ.

We start with a simple observation. Suppose that 𝕄 is a local system defined over a number field L. The local system 𝕄_{L} is given by a representation ρ : π_{1}(U, ∗) → Gl(M_{L}) for the fibre M_{L} of 𝕄_{L} over the base point ∗.

Fixing a positive integer r, let G(r, 𝕄) denote the set of all rank-r local subsystems of 𝕄 and let Grass(r, M_{L}) be the Grassmann variety of r-dimensional subspaces. Then G(r, 𝕄) is the subvariety of

Grass(r, M_{L}) ×_{Spec(L)} Spec(ℂ)

consisting of the π_{1}(U, ∗)-invariant points. In particular, it is a projective variety defined over L. A K-valued point of G(r, 𝕄) corresponds to a local subsystem of 𝕄_{K} = 𝕄_{L} ⊗_{L} K. One obtains the following well-known property:

Lemma 3.1. If [𝕎] ∈ G(r, 𝕄) is an isolated point, then 𝕎 is defined over \overline{ℚ}.

In the proof of 3.7 we will also need:

Lemma 3.2. Let 𝕄 be a variation of Hodge structures defined over L, and let 𝕎 ⊂ 𝕄 be an irreducible local subsystem of rank r defined over ℂ,. Then 𝕎 can be deformed to a local subsystem 𝕎_{t} ⊂ 𝕄, which is isomorphic to 𝕎 and which is defined over a finite extension of L.

Proof. By [4] 𝕄 is completely reducible over ℂ. Hence we have a decomposition 𝕄 = 𝕎 ⊕ 𝕎′.

The space G(r, 𝕄) of rank r local subsystems of 𝕄 is defined over L and the subset

{𝕎_{t} ∈ G(r, 𝕄); the composit 𝕎_{t} ⊂ 𝕎 ⊕ 𝕎′ ⟶^{pr_{1}} 𝕎 is nonzero}

forms a Zariski open subset. So there exists some 𝕎_{t} in this subset, which is defined over some finite extension of L. Since p : 𝕎_{t} → 𝕎 is nonzero, rank(𝕎_{t}) = rank(𝕎), and since 𝕎 is irreducible, p is an isomorphism. q.e.d.

Lemma 3.3. Let 𝕄 be the underlying local system of a variation of Hodge structures of weight k defined over a number field L. Assume that there is a decomposition

𝕄 = 𝕌 ⊕ ⨁_{i=1}^{ℓ} 𝕄_{i} (3.3.1)

in subvariations of Hodge structures, and let

(E, θ) = (N, 0) ⊕ ⨁_{i=1}^{ℓ}(F_{i}, τ_{i} = θ|_{F_{i}}) (3.3.2)

be the induced decomposition of the Higgs field. Assume that width(𝕄_{i}) = i, and that the 𝕄_{i} have all generically maximal Higgs fields. Then the decomposition (3.3.1) is defined over \overline{ℚ}. If L is real, it is defined over \overline{ℚ} ∩ ℝ. If 𝕄 is polarized, then the decomposition (3.3.1) can be chosen to be orthogonal with respect to the polarization.

Proof. Consider a family {𝕎_{t}}_{t∈Δ} of local subsystems of 𝕄 defined over a disk Δ with 𝕎_{0} = 𝕄_{ℓ}. For t ∈ Δ let (F_{𝕎_{t}}, τ_{t}) denote the Higgs bundle corresponding to 𝕎_{t}. Hence (F_{𝕎_{t}}, τ_{t}) is obtained by restricting the F -filtration of 𝕄⊗O_{U} to 𝕎_{t} ⊗O_{U} and by taking the corresponding graded sheaf. So the Higgs map

τ_{p,k−p} : F_{t}^{p,k−p} ⟶ F_{t}^{p−1,k−p+1} ⊗ Ω_{Y}^{1} (log S)

will again be generically isomorphic for t sufficiently closed to 0 and

|2p − k| ≤ ℓ and |2p − 2 − k| ≤ ℓ.

If the projection

ρ : 𝕎_{t} ⟶ 𝕄 = 𝕌 ⊕ ⨁_{i=1}^{ℓ} 𝕄_{i} ⟶ 𝕌 ⊕ ⨁_{i=1}^{ℓ−1} 𝕄_{i}

is nonzero, the complete reducibility of local systems coming from variations of Hodge structures (see [4]) implies that 𝕎_{t} contains an irreducible nontrivial direct factor, say 𝕎′_{t} which is isomorphic to a direct factor of 𝕌 or of one of the local systems 𝕄_{i}, for i < ℓ.

Restricting again the F filtration and passing to the corresponding graded sheaf, we obtain a Higgs bundle (F_{𝕎′_{t}}, τ′_{t}) with trivial Higgs field, or whose width is strictly smaller than ℓ. On the other hand, (F_{𝕎′_{t}}, τ′_{t}) is a sub-Higgs bundle of the Higgs bundle (F_{𝕎_{t}}, τ_{t}) of width ℓ, a contradiction. So ρ is zero and 𝕎_{t} = 𝕄_{ℓ}.

Thus 𝕄_{ℓ} is rigid as a local subsystem of 𝕄, and by Lemma 3.1 𝕄_{ℓ} is defined over \overline{ℚ}.

Assume now that L is real, hence 𝕄 = 𝕄_{ℝ} ⊗ ℂ. The complex conjugation defines an involution ι on 𝕄. Let \overline{𝕄}_{ℓ} denote the image of 𝕄_{ℓ} under ι. Then \overline{𝕄}_{ℓ} has again generically isomorphic Higgs maps τ_{p,k−p}, for

|2p − k| ≤ ℓ and |2p − 2 − k| ≤ ℓ.

If \overline{𝕄}_{ℓ} ≠ 𝕄_{ℓ}, repeating the argument used above, one obtains a map

\overline{𝕄}_{ℓ} ⟶ 𝕌 ⊕ ⨁_{i=1}^{ℓ−1} 𝕄_{i},

from a Higgs bundle of width ℓ and with a maximal Higgs field to one with trivial Higgs field or of lower width. Again such a morphism must be zero, hence 𝕄_{ℓ} = \overline{𝕄}_{ℓ} in this case.

So we can find a number field K, real in case L is real, and a local system 𝕄_{ℓ,K} ⊂ 𝕄_{K} with 𝕄_{ℓ} = 𝕄_{ℓ,K} ⊗_{K} ℂ. The polarization on 𝕄 restricts to a nondegenerated intersection form on 𝕄_{K}. Choosing for 𝕄_{ℓ,K}^{⊥} the orthogonal complement of 𝕄_{ℓ,K} in 𝕄_{K} we obtain a splitting

𝕄_{K} = 𝕄_{ℓ,K} ⊕ 𝕄_{ℓ,K}^{⊥}

inducing over ℂ the splitting of the factor 𝕄_{ℓ} in (3.3.1). By induction on ℓ we obtain 3.3. q.e.d.

For a reductive algebraic group G and for a finitely generated group Γ let M(Γ, G) denote the moduli space of reductive representations of Γ in G.

Theorem 3.4 (Simpson, [29], Corollary 9.18). Suppose Γ is a finitely generated group. Suppose ϕ : G → H is a homomorphism of reductive algebraic groups with finite kernel. Then the resulting morphism of moduli spaces

ϕ : M(Γ, G) ⟶ M(Γ, H)

is finite.

Corollary 3.5. Let Γ be π_{1}(Y, ∗) of a projective manifold, and γ : Γ → G be a reductive representation. If ϕγ ∈ M(Γ, H) comes from a ℂ-variation of Hodge structures, then γ comes from a ℂ-variation of Hodge structures as well.

Proof. By Simpson a reductive local system is coming from a variation of Hodge structures if and only if the isomorphism class of the corresponding Higgs bundle is a fix point of the ℂ^{∗}-action. Since the ℂ^{∗}-action contains the identity and since it is compatible with ϕ, the finiteness of the preimage ϕ^{−1}ϕ(γ) implies that the isomorphism class of the Higgs bundle corresponding to γ is fixed by the ℂ^{∗}-action, as well.

q.e.d.

Definition 3.6. Let 𝕄 be a local system of rank r, and defined over \overline{ℚ}. Let γ_{𝕄} : π_{1}(U, ∗) → Sl(2, \overline{ℚ}) be the corresponding representation of the fundamental group. For η ∈ π_{1}(U, ∗) we write tr(γ_{𝕄}(η)) ∈ \overline{ℚ} for the trace of η and

tr(𝕄) = {tr(γ_{𝕄}(η)); η ∈ π_{1}(U, ∗)}.

Corollary 3.7. Under the assumptions made in 1.3:

i. The splitting 𝕏 = 𝕍 ⊕ 𝕌_{1} is defined over \overline{ℚ}, and over \overline{ℚ} ∩ ℝ in

case L is real. If 𝕏 is polarized, it can be chosen to be orthogonal. ii. The splitting 𝔼nd(𝕍) = 𝕎⊕𝕌 constructed in Lemma 1.7 is defined

over \overline{ℚ}, and over \overline{ℚ} ∩ ℝ in case L is real. If 𝕏 is polarized, it can be chosen to be orthogonal.

iii. Replacing Y by an étale covering Y′, one can choose the decomposition 𝕍 ≃ 𝕃 ⊗ 𝕋 in 1.4 such that: a. 𝕃 and 𝕋 are defined over a number field K, real if L is real. b. One has an isomorphism 𝕍\overline{ℚ} ≃ 𝕃\overline{ℚ} ⊗\overline{ℚ} 𝕋\overline{ℚ}. c. tr(𝕃) is a subset of the ring of integers O_{K} of K.

Proof. i) and ii) are direct consequences of 3.3. For iii) let us first remark that for L real, passing to an étale covering 𝕃 and 𝕋 can both be assumed to be defined over ℝ. In fact, the local system \overline{𝕃} has a maximal Higgs field, hence its Higgs field is of the form (L′ ⊕ L′^{−1}, τ′) where L′ is a theta characteristic. Hence it differs from L at most by the tensor product with a two torsion point in Pic^{0}(Y). Replacing Y by an étale covering, we may assume 𝕃 = \overline{𝕃}. From 1.4 d), we obtain \overline{𝕋} = 𝕋.

Consider the isomorphism of local systems ϕ : 𝕃 ⊗ 𝕋 ⟶^{≃} 𝕍 and the induced isomorphism

ϕ^{2} : 𝔼nd_{0}(𝕃⊗𝕋) = 𝔼nd_{0}(𝕃)⊕𝔼nd_{0}(𝕋)⊗𝔼nd_{0}(𝕃)⊕𝔼nd_{0}(𝕋) ⟶ 𝔼nd(𝕍).

Since ϕ^{2}𝔼nd_{0}(𝕋) is the unitary part of this decomposition, by 3.3 it is defined over \overline{ℚ} ∩ ℝ, as well as ϕ^{2}(𝔼nd_{0}(𝕃) ⊕ 𝔼nd_{0}(𝕋) ⊗ 𝔼nd_{0}(𝕃)). The 1, −1 part of the Higgs field corresponding to ϕ^{2}End_{0}(𝕃) has rank one, and its Higgs field is maximal. Hence ϕ^{2}End_{0}(𝕃) is irreducible, and by 3.2 it is isomorphic to a local system, defined over \overline{ℚ}. Hence 𝕋 ⊗ 𝕋 ≃ 𝔼nd(𝕋) and 𝕃 ⊗ 𝕃 ≃ 𝔼nd(𝕃) are both isomorphic to local systems defined over some real number field K′. An O_{K′}-structure can be defined by

ϕ^{2}(𝔼nd(𝕃))_{O_{K′}} = ϕ^{2}(𝔼nd(𝕃))_{K} ∩ 𝕍_{O_{K′}} .

Consider for ν = 2 or ν = g_{0} the moduli space M(U, Sl(ν^{2})) of reductive representations of π(U, ∗) into Sl(ν^{2}). It is a quasi-projective variety defined over ℚ. The fact that 𝕃 ⊗ 𝕃 (or 𝕋 ⊗ 𝕋) is defined over \overline{ℚ} implies that its isomorphy class in M(U, Sl(ν^{2})) is a \overline{ℚ} valued point.

Consider the morphism induced by the second tensor product

ρ : M(U, Sl(ν)) ⟶ M(U, Sl(ν^{2}))

which is clearly defined over ℚ. By Theorem 3.4, ρ is finite, hence the fibre ρ^{−1}([𝕃 ⊗ 𝕃]) (or ρ^{−1}([𝕋 ⊗ 𝕋])) consists of finitely many \overline{ℚ}-valued points, hence 𝕃 and 𝕋 can be defined over a number field K. If L is real, as already remarked above, we may choose K to be real.

Obviously, for ρ ∈ π_{1}(Y, ∗) one has

tr(γ_{𝕃}(ρ))^{2} = tr(γ_{𝕃⊗𝕃}(ρ)).

In fact, one may assume that γ_{𝕃}(ρ) is a diagonal matrix with entries a and b on the diagonal. Then tr(γ_{𝕃⊗𝕃}(ρ)) has a^{2}, b^{2}, ab and ba as diagonal elements. Since tr(γ_{𝕃⊗𝕃}(ρ)) ∈ O_{K′} we find tr(γ_{𝕃}(ρ)) ∈ O_{K}.

q.e.d.

4. Splitting over ℚ for S ≠ ∅ and isogenies

In this section, we will consider the case L = ℚ and 𝕏_{ℚ} = R^{1}f_{∗}ℚ_{X_{0}}, where f : X → Y is a family of abelian varieties, S = Y \ U ≠ ∅, and where the restriction X_{0} → U of f is a smooth family.

Lemma 4.1. Assume that S ≠ ∅ and let 𝕄_{ℚ} be a ℚ-variation of Hodge structures of weight k and with unipotent monodromy around all points s ∈ S. Assume that over some number field K there exists a splitting

𝕄_{K} = 𝕄_{ℚ} ⊗_{ℚ} K = 𝕎_{K} ⊕ 𝕌_{K}

where 𝕌 = 𝕌_{K} ⊗_{K} ℂ is unitary and where the Higgs field of 𝕎 = 𝕎_{K} ⊗_{K} ℂ is maximal. Then 𝕎, 𝕌 and the decomposition 𝕄 = 𝕎 ⊕ 𝕌 are defined over ℚ. Moreover, 𝕌 extends to a local system over Y.

Proof. Let 𝕋 be a local subsystem of 𝕎. Writing

( ⨁_{p+q=k} F_{𝕋}^{p,q}, ⨁_{p+q=k} τ_{p,q}) ,

for the Higgs bundle corresponding to 𝕋, the maximality of the Higgs field for 𝕎 implies that the Higgs field for 𝕋 is maximal, as well. In particular, for all s ∈ S and for p > 0 the residue maps

res_{s}(τ_{p,q}) : F_{𝕋,s}^{p,q} ⟶ F_{𝕋,s}^{p−1,q+1}

are isomorphisms. By [27] the residues of the Higgs field at s are defined by the nilpotent part of the local monodromy matrix around s. Hence if γ is a small loop around s in Y, and if ρ_{𝕋}(γ) denotes the image of γ under a representation of the fundamental group, defining 𝕋, the nilpotent part N(ρ_{𝕋}(γ)) = log ρ_{𝕋}(γ) of ρ_{𝕋}(γ) has to be nontrivial.

We may assume that K is a Galois extension of ℚ. Recall that for σ ∈ Gal(K/ℚ) we denote the local systems obtained by composing the representation with σ by an upper index σ. Consider the composite

p : 𝕌_{K}^{σ} ⟶ 𝕄_{K} = 𝕎_{K} ⊕ 𝕌_{K} ⟶ 𝕎_{K},

and the induced map 𝕌^{σ} = 𝕌_{K}^{σ} ⊗_{K} ℂ → 𝕎.

Let γ be a small loop around s ∈ S, and let ρ_{𝕌}(γ) and ρ_{𝕌^{σ}}(γ) be the images of γ under the representations defining 𝕌 and 𝕌^{σ} respectively. Since 𝕌 is unitary and unipotent, the nilpotent part of the monodromy matrix N(ρ_{𝕌}(γ)) = 0. This being invariant under conjugation, N(ρ_{𝕌^{σ}}(γ)) is zero, as well as N(ρ_{p(𝕌^{σ})}(γ)).

Therefore p(𝕌^{σ}) = 0, hence 𝕌^{σ} = 𝕌, and 𝕌 is defined over ℚ. Taking again the orthogonal complement, one obtains the ℚ-splitting asked for in 4.1.

Since N(ρ_{𝕌}(γ)) = 0, the residues of 𝕌 are zero in all points s ∈ S, hence 𝕌 extends to a local system on Y. q.e.d.

Corollary 4.2. Suppose that S ≠ ∅. Then the splittings in Corollary 3.7 i) and ii), can be defined over ℚ.

Lemma 4.3. Let 𝕄 be a local system, defined over ℤ, and let 𝕄_{ℚ} = 𝕎_{ℚ} ⊕ 𝕌_{ℚ} be a decomposition, defined over ℚ. Then there exist local systems 𝕌_{ℤ} and 𝕎_{ℤ}, defined over ℤ with

𝕌_{ℚ} = 𝕌_{ℤ} ⊗ ℚ, 𝕎_{ℚ} = 𝕎_{ℤ} ⊗ ℚ, and 𝕄_{ℤ} ⊃ 𝕎_{ℤ} ⊕ 𝕌_{ℤ}. (4.3.1)

Moreover, if 𝕌_{ℚ} is unitary with trivial local monodromies around S, then there exists an étale covering π : Y′ → Y such that π^{∗}𝕌_{ℚ} is trivial.

Proof. Defining a ℤ-structure on 𝕎_{ℚ} and 𝕌_{ℚ} by

𝕎_{ℤ} = 𝕎_{ℚ} ∩ 𝕄_{ℤ} and 𝕌_{ℤ} = 𝕌_{ℚ} ∩ 𝕄_{ℤ}

(4.3.1) obviously holds true.

Since the integer elements of the unitary group form a finite group, the representation defining 𝕌 factors through a finite quotient of the fundamental group π_{1}(U, ∗) ↦ G. The condition on the local monodromies implies that this quotient factors through π_{1}(Y, ∗), and we may choose Y′ to be the corresponding étale covering. q.e.d.

By 4.2 we obtain decompositions

R^{1}f_{∗}ℚ_{X_{0}} = 𝕍_{ℚ} ⊕ 𝕌_{1ℚ} and 𝔼nd(𝕍_{ℚ}) = 𝕎_{ℚ} ⊕ 𝕌_{ℚ}.

By 4.1 the local monodromies of the unitary parts 𝕌_{1} and 𝕌 are trivial. Moreover, 𝕌 is a subvariation of Hodge structures of weight 0, 0. Summing up, we obtain:

Corollary 4.4. Let f : X → Y be a family of abelian varieties with unipotent local monodromies around s ∈ S, and reaching the Arakelov bound. If S ≠ ∅ there exists a finite étale cover π : Y′ → Y with:

i. For a ℤ-variation of Hodge structures 𝕍′_{ℤ} of weight 1 with maximal Higgs field, we have

π^{∗}(R^{1}f_{∗}(ℤ_{X_{0}})) ⊃ 𝕍′_{ℤ} ⊕ ℤ^{2(g−g_{0})}, and
π^{∗}(R^{1}f_{∗}(ℤ_{X_{0}})) ⊗ ℚ = (𝕍′_{ℤ} ⊕ ℤ^{2(g−g_{0})}) ⊗ ℚ.

ii. 𝔼nd(𝕍′_{ℤ}) ⊃ 𝕎′_{ℤ} ⊕ ℤ^{g_{0}^{2}}, 𝔼nd(𝕍′_{ℤ}) ⊗ ℚ = (𝕎′_{ℤ} ⊕ ℤ^{g_{0}^{2}}) ⊗ ℚ, where 𝕎′_{ℤ} is an ℤ-variation of Hodge structures of weight 0 with maximal Higgs field, and where ℤ^{g_{0}^{2}} is a local ℤ-subsystem of type (0, 0).

Proof of Theorem 0.2. Let Y′ be the étale covering constructed in 4.4 ii). So using the notations introduced there,

R^{1}f′_{∗}(ℤ_{X′_{0}}) ⊗ ℚ = 𝕍′_{ℚ} ⊕ ℤ^{2(g−g_{0})} and 𝔼nd(𝕍′_{ℚ}) = 𝕎′_{ℚ} ⊕ ℤ^{g_{0}^{2}}. (4.4.1)

The left-hand side of (4.4.1) implies that f′ : X′ → Y′ is isogenous to a product of a family of g_{0}-dimensional abelian varieties with a constant abelian variety B of dimension g − g_{0}. By abuse of notations we will assume from now on, that B is trivial, hence g = g_{0} and R^{1}f′_{∗}(ℤ_{X′_{0}})⊗ℚ = 𝕍′_{ℚ}, and we will show that under this assumption f′ : X′ → Y′ is isogenous to a g-fold product of a modular family of elliptic curves.

Let us write

End(∗) = H^{0}(Y′, 𝔼nd(∗))

for the global endomorphisms. End(𝕍′_{ℚ}) = ℚ^{g^{2}} is a ℚ Hodge structure of weight zero, in our case the Hodge filtration is trivial, i.e.,

End(𝕍′_{ℚ})^{0,0} = End(𝕍′_{ℚ}).

If X_{η} = X′ ×_{Y′} Spec(\overline{ℂ(Y′)}) denotes the general fibre of f′, one obtains from [4], 4.4.6,

End(X_{η}) ⊗ ℚ = End(𝕍′_{ℚ})^{0,0} = End(𝕍′_{ℚ}).

By the complete reducibility of abelian varieties, there exists simple abelian varieties B_{1}, …, B_{r} of dimension g_{i}, respectively, which are pairwise nonisogenous, and such that X_{η} is isogenous to the product

B_{1}^{×ν_{1}} × ⋯ × B_{r}^{×ν_{r}}.

Moreover, since 𝕍 has no flat part, none of the B_{i} can be defined over ℂ. Let us assume that g_{i} = 1 for i = 1, …, r′ and g_{i} > 1 for i = r′+1, …, r.

By [19], p. 201, D_{i} = End(B_{i}) ⊗ ℚ is a division algebra of finite rank over ℚ with center K_{i}. Let us write

d_{i}^{2} = dim_{K_{i}}(D_{i}) and e_{i} = [K_{i} : ℚ].

Hence e_{i} · d_{i}^{2} = dim_{ℚ}(D_{i}).

By [19], p. 202, or by [15], p. 141, either d_{i} ≤ 2 and e_{i} · d_{i} divides g_{i}, or else e_{i} · d_{i}^{2} divides 2 · g_{i}. In both cases, the rank e_{i} · d_{i}^{2} is smaller than or equal to 2 · g_{i}. If i ≤ r′, hence if B_{i} is an elliptic curve, not defined over ℂ, we have e_{i} = d_{i} = 1.

Writing M_{ν_{i}}(D_{i}) for the ν_{i} × ν_{i} matrices over D_{i}, one finds ([19], p. 174)

End(X_{η}) ⊗ ℚ = M_{ν_{1}}(D_{1}) ⊕ ⋯ ⊕ M_{ν_{r}}(D_{r})

hence

g^{2} ≤ dim_{ℚ}(End(X_{η}) ⊗ ℚ)

= (∑_{i=1}^{r} ν_{i} · g_{i})^{2} = ∑_{i=1}^{r}(e_{i} · d_{i}^{2}) · ν_{i}^{2}

≤ ∑_{i=1}^{r′} ν_{i}^{2} + ∑_{i=r′+1}^{r} ν_{i}^{2} · 2 · g_{i} ≤ ∑_{i=1}^{r} ν_{i}^{2} · g_{i}^{2}.

Obviously this implies that r = 1 and that g_{1} ≤ 2. If g_{1} = 1, we are done. In fact, the isogeny extends all over Y′ \ S′ and, since we assumed the monodromies to be unipotent, B_{1} is the general fibre of a semistable family of elliptic curves. The Higgs field for this family is again maximal, and 0.2 follows from 0.1.

It remains to exclude the case that g_{1} = 2, and that e_{1} · d_{1}^{2} = 4. If the center K_{1} is not a totally real number field, e_{1} must be lager than 1 and one finds:

I. d_{1} = 1 and D_{1} = K_{1} is a quadratic imaginary extension of a real quadratic extension of ℚ.

If K_{1} is a real number field, looking again to the classification of endomorphisms of simple abelian varieties in [19] or [15], one finds that e_{1} divides g_{1}, hence the only possible case is:

II. d_{1} = 2 and e_{1} = 1, and D_{1} is a quaternion algebra over ℚ.

The abelian surface B_{1} over Spec(ℂ(Y′)) extends to a nonisotrivial family of abelian varieties B′ → Y′, smooth outside of S and with unipotent monodromies for all s ∈ S. This family again has a maximal Higgs field, and thereby the local monodromies in s ∈ S are nontrivial. As we will see below, in both cases, I and II, the moduli scheme of abelian surfaces with the corresponding type of endomorphisms turns out to be a compact subvariety of the moduli scheme of polarized abelian varieties, a contradiction.

I. By [15], Example 6.6 in Chapter 9, there are only finitely many g_{1}-dimensional abelian varieties with a given type of complex multiplication, i.e., with D_{1} a quadratic imaginary extension of a real number field of degree g_{1} over ℚ.

II. By [15], Exercise (1) in Chapter 9, there is no abelian surface for which D_{1} is a totally definite quaternion algebra. If D_{1} = End(B)⊗ℚ is totally indefinite, B is a false elliptic curve, as considered in Example 0.4, ii, for d = 1. Such abelian surfaces have been studied in [25], and their moduli scheme is a compact Shimura curve. The latter follows from Shimura’s construction of the moduli scheme as a quotient of the upper half plane H (see [15], §8 in Chapter 9, for example) and from [26], Chapter 9. q.e.d.

5. Quaternion algebras and Fuchsian groups

Let A denote a quaternion algebra over a totally real algebraic number field F with d distinct embeddings

σ_{1} = id, σ_{2}, …, σ_{d} : F → ℝ,

which satisfies the following extra condition: for 1 ≤ i ≤ d there exists ℝ-isomorphism

ρ_{1} : A^{σ_{1}} ⊗ ℝ ≃ M(2, ℝ), ρ_{i} : A^{σ_{i}} ⊗ ℝ ≃ ℍ, 2 ≤ i ≤ d,

where ℍ is the quaternion algebra over ℝ. An order O ⊂ A over F is a subring of A containing 1 which is a finitely generated O_{F} -module generating the algebra A over F. The group of units in O of reduced norm 1 is defined as

O^{1} = {x ∈ O; Nrd(x) = 1}.

As Shimura has shown ρ_{1}(O^{1}) ⊂ Sl_{2}(ℝ) is a discrete subgroup and for a torsion-free subgroup Γ ⊂ O^{1} of finite index H/ρ_{1}(Γ) is a quasiprojective curve, called Shimura curve. Furthermore, if A is a division algebra H/ρ_{1}(Γ) is projective (see [26], Chapter 9).

Remark 5.1. We will say that over some field extension F′ of F the quaternion algebra splits, if A_{F′} = A ⊗_{F} F′ ≃ M(2, F′). If F′ = F_{v} is the completion of F with respect to a place v of F, one says that F is ramified at v, if A_{v} = A_{F_{v}} does not split. As well-known, there exists some a ∈ F for which A_{F (\sqrt{a})} splits. As explained in [34], for example, we can choose such a ∈ F in the following way:

Fix one nonarchimedian prime p^{0} of ℚ, such that A is unramified over all places of F lying over p^{0}. Then choose a such that for all places v of F not lying over p^{0} the quaternion algebra A ramifies at v if and only if F_{v}(\sqrt{a}) ≠ F_{v}. Moreover one may assume, that the product over all conjugates of a is not a square in ℚ.

Definition 5.2. If Γ̃ ∈ PSl_{2}(ℝ) is a subgroup of finite index of some P ρ_{1}(O^{1}), then we call Γ̃ a Fuchsian group derived from a quaternion algebra A.

Theorem 5.3 (Takeuchi [31]). Let Γ̃ ⊂ PSl_{2}(ℝ) be a discrete subgroup such that H/Γ̃ is quasi-projective. Then Γ̃ is derived from a quaternion algebra A over a totally real number field F with d distinct embeddings

σ_{1} = id, σ_{2}, …, σ_{d} : F ⟶ ℝ,

with

ρ_{1} : A^{σ_{1}} ⊗ ℝ ≃ M(2, ℝ), ρ_{i} : A^{σ_{i}} ⊗ ℝ ≃ ℍ, 2 ≤ i ≤ d

if and only if Γ̃ satisfies the following conditions:

(I) Let k be the field generated by the set tr(Γ̃) over ℚ. Then k is an algebraic number field of finite degree, and tr(Γ̃) is contained in the ring of integers of k, O_{k}.

(II) Let σ be any embedding of k into ℂ such that σ ≠ id_{k}. Then σ(tr(Γ̃)) is bounded in ℂ.

In the proof of Theorem 5.3 one gets, in fact, k = F. If A is a division algebra, for example if d > 1, then Y = H/Γ̃ is projective, and it is determined by A, and by the choice of the order O ⊂ A up to finite étale coverings.

Assumption 5.4. Let 𝕏_{ℚ} be an irreducible ℚ-variation of Hodge structures of weight one and width one, and with a maximal Higgs field. Assume moreover, that 𝕏_{ℚ} is polarized. There are isomorphisms

ψ : 𝕏 = 𝕏_{ℚ} ⊗_{ℚ} ℂ ⟶^{≃} 𝕍 ⊕ 𝕌_{1} and ϕ : 𝕍 ⟶^{≃} 𝕃 ⊗ 𝕋

where 𝕌_{1} and 𝕋 are both unitary, and where 𝕃 is a rank two variation of Hodge structures of weight one and width one, with a maximal Higgs field. Moreover 𝕍, 𝕌_{1}, 𝕃, 𝕋 and ψ are defined over some real number field K, and ϕ over some number field K′. We fix an embedding of K′ into ℂ and denote by k ⊂ K ⊂ K′ ⊂ ℂ the field spanned by tr(𝕃) over ℚ.

Proposition 5.5. Keeping the notations and assumptions made in 5.4, replacing Y by an étale covering, one may assume that:

i. Γ_{𝕃} is derived from a quaternion algebra A over a totally real number field F with d distinct embeddings

σ_{1} = id, σ_{2}, …, σ_{d} : F ⟶ ℝ.

ii. For 1 ≤ i ≤ d there exists ℝ-isomorphism

ρ_{1} : A^{σ_{1}} ⊗ ℝ ≃ M(2, ℝ), and ρ_{i} : A^{σ_{i}} ⊗ ℝ ≃ ℍ, for 2 ≤ i ≤ d.

iii. The representation γ_{𝕃} : π_{1}(Y, ∗) → Sl(2, ℝ) defining the local system 𝕃 factors as

π_{1}(Y, ∗) ⟶^{≃} Γ ⊂ ρ_{1}(O^{1}) ⟶ Sl(2, ℝ ∩ \overline{ℚ}) ⊂ Sl(2, ℝ),

and Y ≃ H/Γ. iv. For a as in 5.1 F (\sqrt{a}) is a field of definition for 𝕃. v. If τ_{i}, 1 ≤ i ≤ d are extension of σ_{i} to F (\sqrt{a}), and if 𝕃_{i} denotes

the local system defined by

π_{1}(Y, ∗) ⟶ Sl(2, F (\sqrt{a})) ⟶^{τ_{i}} Sl(2, \overline{ℚ}),

then 𝕃_{i} is a unitary local system, for i > 1, and 𝕃_{1} ≃ 𝕃. vi. Up to isomorphism, 𝕃_{i} does not depend on the extension τ_{i} chosen.

Proof. i) and ii): By Corollary 3.7 iii), Γ_{𝕃} satisfies Condition (I) in Theorem 5.3. So, we only have to verify Condition (II) for Γ_{𝕃}. Let σ be an embedding of k into ℂ which is not the identity, and let σ̃ : K′ → ℂ be any extension of σ.

By 3.5 ψ^{−1}𝕍σ̃ is a subvariation of Hodge structures of 𝕏, hence of width zero or one. On the other hand, 𝕍σ̃ is isomorphic to 𝕃σ̃ ⊗ 𝕋σ̃. Both factors are variations of Hodge structures, hence at least one of them has a trivial Higgs field.

Assume both have a trivial Higgs field, hence 𝕍σ̃ as well. By 1.9 a), the composite

ψ^{−1}𝕍σ̃ ⟶ 𝕏 ⟶^{ψ} 𝕍 ⊕ 𝕌_{1} ⟶ 𝕍

has to be zero. Hence 𝕍σ̃ is a sublocal system of the unitary system 𝕌_{1}, hence unitary itself. The \overline{ℚ}-isomorphism ϕ : 𝕍 ⟶^{≃} 𝕃 ⊗ 𝕋 induces an isomorphism

ϕ^{⊗} : ⨂^{g_{0}} 𝕍σ̃ ⟶ (⨂^{g_{0}} 𝕃σ̃) ⊗ (⨂^{g_{0}} 𝕋σ̃) .

The right-hand side contains S^{g_{0}}(𝕃σ̃) as a direct factor, hence S^{g_{0}}(𝕃σ̃) is unitary, as well as 𝕃σ̃. So tr(𝕃σ̃) = σ(tr(𝕃)) is bounded in this case.

If the Higgs field of 𝕍σ̃ is nontrivial, it is generically maximal. This implies that the composite

𝕍σ̃ ⟶ 𝕏 ⟶ 𝕌_{1}

is zero. Hence 𝕍σ̃ ≃ 𝕍. If the Higgs field of 𝕃σ̃ is an isomorphism, by 1.4 replacing Y by an étale covering, 𝕃σ̃ ≃ 𝕃. Hence up to conjugation the representations γ_{𝕃σ̃} and γ_{𝕃} coincide and for all η ∈ π_{1}(Y, ∗)

tr(γ_{𝕃σ̃}(η)) = tr(γ_{𝕃}(η)).

So σ is the identity, a contradiction.

It remains to consider the case that 𝕍σ̃ ≃ 𝕍 and that 𝕃σ̃ is concentrated in degree 0, 0.

For g_{0} even, one has a \overline{ℚ}-isomorphism

∧^{g_{0}}ϕ : ∧^{g_{0}}𝕍 ≃ S^{g_{0}}(𝕃) ⊕ 𝕊_{\{2,...,2\}}(𝕋) ⊕ ⨁_{c=1}^{\frac{g_{0}}{2} −1} S^{2c}(𝕃) ⊗ 𝕊_{λ_{2c}}(𝕋),

where 𝕊_{{2,...,2}}(𝕋) is of width zero, where S^{g_{0}}𝕃 has a maximal Higgs field of width g_{0}, and where all other factors have a maximal Higgs field of width between 2 and g_{0} − 2. Let K denote the field of definition Γ_{𝕃}. Then K ⊃ k is a finite extension of k. Let σ be an embedding of k into ℂ which is not identity, and let σ̃ : K → ℂ be an extension of σ. Via the isomorphisms ∧^{g_{0}}ϕ and ∧^{g_{0}}ϕσ̃ we obtain an embedding

S^{g_{0}}𝕃σ̃ ⟶ S^{g_{0}}(𝕃) ⊕ 𝕊_{\{2,...,2\}}(𝕋) ⊕ ⨁_{c=1}^{\frac{g_{0}}{2} −1} S^{2c}(𝕃) ⊗ 𝕊_{λ_{2c}}(𝕋).

The projection of S^{g_{0}}𝕃σ̃ into S^{g_{0}}𝕃 must be zero, for otherwise, we would get an isomorphism S^{g_{0}}𝕃σ̃ ≃ S^{g_{0}}𝕃. By Corollary 3.5 𝕃σ̃ is a subvariation of Hodge structures, hence it has a maximal Higgs field.

The projection

S^{g_{0}}𝕃σ̃ ⟶ ⨁_{c=1}^{\frac{g_{0}}{2} −1} S^{2c}(𝕃) ⊗ 𝕊_{λ_{2c}}(𝕋)

must be also zero, for otherwise, by applying again Corollary 3.5 to S^{g_{0}}𝕃σ̃ one would find 𝕃σ̃ to have a maximal Higgs field, hence S^{g_{0}}𝕃σ̃ to have a maximal Higgs field of width g_{0}. But, then it can not be embedded in a local system of width < g_{0}.

Thus, the projection

S^{g_{0}}𝕃σ̃ ⟶ S_{{2,2,...,2}}𝕋

is an embedding. This implies that 𝕃σ̃ is unitary. In particular, again tr(𝕃σ̃) = σ(tr(𝕃)) is bounded in ℂ.

Finally, the assumption 𝕍σ̃ ≃ 𝕍 and 𝕃σ̃ unitary does not allow g_{0} = rank(𝕍) to be odd:

The \overline{ℚ}-isomorphism ϕ : 𝕍 ≃ 𝕃 ⊗ 𝕋, induces a \overline{ℚ}-isomorphism

∧^{g_{0}}ϕ : ∧^{g_{0}}𝕍 ≃ ⨁_{c=0}^{\frac{g_{0}−1}{2}} S^{2c+1}(𝕃) ⊗ 𝕊_{λ_{2c}}(𝕋)

(see 2.2). The left-hand side contains a local subsystem isomorphic to S^{g_{0}}(𝕃σ̃), hence with a trivial Higgs field, whereas the right-hand side only contains factors of width > 0, with a maximal Higgs field, a contradiction.

Applying 5.3 we obtain a quaternion algebra A satisfying i), ii) and the first part of iii). By 2.1 one has Y ≃ H/Γ.

For iv) we recall that by the choice of a the quaternion algebra A splits over F (\sqrt{a}). So v) follows from i) and ii).

To see that 𝕃_{i} is independent of the extension of σ_{i} to τ_{i} : F (\sqrt{a}) → \overline{ℚ} it is sufficient to show vi) for i = 1. Let \overline{𝕃} denote the local system obtained by composing the representation with the involution on F (\sqrt{a}). Then both, 𝕃 and \overline{𝕃} have a maximal Higgs field, hence by 1.4 c), their Higgs fields differ at most by the product with a two torsion element in Pic^{0}(Y). Replacing Y by an étale covering, we may assume both to be isomorphic. q.e.d.

Given a quaternion algebra A as in 5.5 i) and ii) allows us to construct certain families of abelian varieties. To this aim we need some wellknown properties of quaternion algebras A defined over number fields F. Let us fix a subfield L of F.

Notations 5.6. Let us write δ = [L : ℚ], δ′ = [F : L] and

β_{1} = id_{L}, β_{2}, …, β_{δ} : L ⟶ ℂ

for the different embeddings. We renumber the embeddings σ_{i} : F → ℂ in such a way, that

σ_{i}|_{L} = β_{ν} for (ν − 1)δ′ < i ≤ νδ′.

Recall, that the corestriction Cor_{F/L}(A) is defined (see [34], p. 10) as the subalgebra of Gal(\overline{ℚ}/L)-invariant elements of

⨂_{i=1}^{δ′} A^{σ_{i}} = ⨂_{i=1}^{δ′} A ⊗_{F,σ_{i}} \overline{ℚ}.

Lemma 5.7. Let A be a quaternion division algebra defined over a totally real number field F, of degree d over ℚ. Assume that A is ramified at all infinite places of F except one. For some subfield L of F let D_{L} = Cor_{F/L}(A) be the corestriction of A to L. Finally let a ∈ F be an element, as defined in 5.1, and

b = a · σ_{2}(a) · · · · · σ_{δ′}(a) ∈ L.

a. If L = ℚ, i.e., if d = δ′, then either i. D_{ℚ} ≃ M(2^{d}, ℚ), and d is odd, or ii. D_{ℚ} ≄ M(2^{d}, ℚ). Then

D_{ℚ} ≃ M(2^{d}, ℚ(\sqrt{b})).

ℚ(\sqrt{b}) is a quadratic extension of ℚ, real if and only if d is odd.

b. If L ≠ ℚ, then D_{L} ≄ M(2^{δ′}, L), and: i. L(\sqrt{b}) is an imaginary quadratic extension of L. ii. D_{L} ⊗_{L} L(\sqrt{b}) ≃ M(2^{δ′}, L(\sqrt{b})).

In a) ii), or in b), choosing an embedding L(\sqrt{b}) → M(2, L), one obtains an embedding

D_{L} ⟶ M(2^{d+1}, L).

Proof. For δ = [L : ℚ] ≥ 1, choose δ different embeddings β_{ν} : L → \overline{ℚ}, corresponding to infinite places v_{1}, …, v_{δ}. We may assume that β_{1} extends to the embedding σ_{1} of F. Hence A is ramified over δ′ − 1 extensions of v_{1} to F, and over all δ′ extensions of v_{ν} to F, for ν ≠ 2. Writing L_{v} for the completion of L at v, one has

D_{v_{ν}} = Cor_{F/L}A ⊗_{L} L_{v_{ν}} = \{ \begin{matrix}M(2, ℝ) ⊗ ⨂^{δ′−1} ℍ & for ν = 1 \\ ⨂^{δ′} ℍ & for ν ≠ 1. \end{matrix}

Recall that the r-fold tensor product of ℍ is isomorphic to M(2^{r}, ℝ) if and only if r is even. By our choice of a and b this holds true, if and only if L_{v}(\sqrt{b}) = L_{v}. In fact, the image of b in L_{v_{ν}} has the sign (−1)^{δ′−1}, for ν = 1 and (−1)^{δ′} otherwise.

In particular D_{L} ≃ M(2^{δ′}, L) can only hold true for L = ℚ and d = δ′ odd.

For L = ℚ, one also finds b > 0 if and only if d is odd.

For all but finitely many nonarchimedian places v of L, in particular for those dominating the prime p^{0} in 5.1, and for the completion L_{v} with respect to v, one has

D_{v} = Cor_{F/L}A ⊗_{L} L_{v} = M(2^{δ′}, L_{v}).

If this is not the case, consider the extension L′_{v} = L(\sqrt{b}) ⊗_{L} L_{v} of L_{v}. One finds

D_{v} ⊗_{L_{v}} L′_{v} = M(2^{δ′}, L′_{v}).

In fact, let v_{1}, …, v_{ℓ} be the places of F, lying over v, and let F_{1}, …, F_{ℓ} be the corresponding local fields. Then

D_{v} = ⨂_{i=1}^{ℓ} Cor_{F_{i}/L_{v}}(A ⊗_{F} F_{i}),

and it is sufficient to show that D_{i} = Cor_{F_{i}/L_{v}}(A ⊗_{F} F_{i}) splits over L′_{v}. If L′_{v} is a subfield of F_{i}

D_{i} ⊗ L′_{v} = Cor_{F_{i}/L_{v}}(A ⊗_{F} F_{i})^{⊗2}

splits, since (A ⊗_{F} F_{i})^{⊗2} does. The same holds true, if L′_{v} is not a field.

If L′_{v} is a field, not contained in F_{i}, then F′_{i} = F_{i} ⊗_{L_{v}} L′_{v} is a field extension of F_{i} of degree two, and

D_{i} ⊗ L′_{v} = Cor_{F′_{i}/L_{v}}(A ⊗_{F} F′_{i})

splits again, since (A ⊗_{F} F′_{i}) does. By [35], Chapter XI, §2, Theorem 2 (p. 206),

D_{L(\sqrt{b})} = D ⊗_{L} L(\sqrt{b}) = M(2^{δ′}, L(\sqrt{b})).  q.e.d.

Choose again an order O in A, and let O^{1} be the group of units in O of reduced norm 1. For any discrete torsion-free subgroup Γ̃ ⊂ P ρ_{1}(O^{1}) with preimage Γ in O^{1} ⊂ Sl_{2}(ℝ) the diagonal embedding

Γ ⟶ O^{1} ⟶ ⨂_{i=1}^{δ′} A^{σ_{i}}

induces an embedding

Γ ⟶ O^{1} ⟶ D_{L} = Cor_{F/L}A. (5.7.1)

Construction 5.8. For L = ℚ the morphism (5.7.1) and 5.7 a), give a morphism

Γ ⊂ D = Cor_{F/ℚ}A ⊂ D ⊗_{ℚ} ℚ(\sqrt{b}) = M(2^{d}, ℚ(\sqrt{b})) ⊂ M(2^{d+ϵ}, ℚ) for ϵ = 0 or 1, where b ∈ ℚ is either a square, or as defined in 5.5. One obtains a representation

η : Γ ⟶ Gl(2^{d}, ℚ(\sqrt{b})) ⟶ Gl(2^{d+ϵ}, ℚ).

If ϵ = 0, the degree d must be odd. Over ℝ one has

D ⊗_{ℚ} ℝ ≃ M(2, ℝ) ⊗ ℍ ⊗ ⋯ ⊗ ℍ. (5.8.1)

The ℚ-algebraic group G := \{x ∈ D^{∗}; Nrd(x) = x\overline{x} = 1\} is ℚ-simple and by (5.8.1) it is a ℚ-form of the ℝ-algebraic group

G(ℝ) ≃ Sl(2, ℝ) × SU(2) × ⋯ × SU(2).

Projection to the first factor, gives a representation of Γ in Sl(2, ℝ), hence a quotient Y = H/Γ with Γ = π_{1}(Y, ∗).

Let us denote by 𝕍_{ℚ} or by 𝕏_{ℚ} the ℚ-local system on Y induced by η. If we want to underline, that the local systems are determined by A we also write 𝕍_{Aℚ} and 𝕏_{Aℚ}, respectively.

Lemma 5.9. Keeping the assumptions and notations from 5.8 one finds:

a.

dim(End(X_{A,ℚ})) = \{ \begin{matrix}1 & for ϵ = 0 \\ 4 & for ϵ = 1. \end{matrix}

b. For ϵ = 1 one has

dim (H^{0} (Y, ⋀^{2}(X_{A,ℚ}))) = \{ \begin{matrix}3 & for d odd \\ 1 & for d even. \end{matrix}

Proof. Consider for ϵ′ = 2^{ϵ}

𝕏 = 𝕏_{Aℚ} ⊗ ℂ = 𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{d} ⊗ ℂ^{ϵ′}

where for σ̃ ∈ Gal(\overline{ℚ}/ℚ) the local system 𝕃σ̃ ; i has a maximal Higgs field if and only if σ̃|_{F} = σ_{i}^{−1}. Otherwise this local system is unitary and of pure bidegree 0, 0.

The determinant of each 𝕃_{i} is ℂ, hence End(𝕏) contains ℂ^{ϵ′} ⊗ ℂ^{ϵ′} as a direct factor. Then

(5.9.1) dim_{ℚ}(End(𝕏_{Aℚ})) = dim_{ℂ}(End(𝕏)) ≥ 4^{ϵ}.

One has

End(𝕏_{ℚ}) = H^{0}(Y, 𝔼nd(𝕏_{ℚ})) ≃ H^{0} (Y, ⋀^{2}(𝕏_{ℚ})) ⊕ H^{0}(Y, S^{2}(𝕏_{ℚ})).

By 2.3

H^{0} (Y, ⋀^{2}(𝕏)) = H^{0}(Y, S^{2}(𝕃_{2} ⊗ ⋯ ⊗ 𝕃_{d} ⊗ ℂ^{ϵ′}))  and

H^{0}(Y, S^{2}(𝕏)) = H^{0} (Y, ⋀^{2}(𝕃_{2} ⊗ ⋯ ⊗ 𝕃_{d} ⊗ ℂ^{ϵ′})) .

Since End(𝕏_{ℚ}) is invariant under Gal(\overline{ℚ}/ℚ), for σ̃ with σ̃|_{F} = σ_{2} it is for d > 1 contained in the direct sum of

H^{0}(Y, S^{2}(𝕃σ̃ ; 2 ⊗ ⋯ ⊗ 𝕃σ̃ ; d ⊗ ℂ^{ϵ′})) = H^{0} (Y, ⋀^{2}(𝕃σ̃ ; 3 ⊗ ⋯ ⊗ 𝕃σ̃ ; d ⊗ ℂ^{ϵ′}))

and

H^{0} (Y, ⋀^{2}(𝕃σ̃ ; 2 ⊗ ⋯ ⊗ 𝕃σ̃ ; d ⊗ ℂ^{ϵ′})) = H^{0}(Y, S^{2}(𝕃σ̃ ; 3 ⊗ ⋯ ⊗ 𝕃σ̃ ; d ⊗ ℂ^{ϵ′})).

Repeating this game we find

(5.9.2) H^{0} (Y, ⋀^{2}(𝕏_{ℚ})) ⊂ \{ \begin{matrix}H^{0}(Y, S^{2}(ℂ^{ϵ′})) & for d odd \\ H^{0}(Y, ⋀^{2}(ℂ^{ϵ′})) & for d even \end{matrix}

(5.9.3) H^{0}(Y, S^{2}(𝕏_{ℚ})) ⊂ \{ \begin{matrix}H^{0}(Y, ⋀^{2}(ℂ^{ϵ′})) & for d odd \\ H^{0}(Y, S^{2}(ℂ^{ϵ′})) & for d even. \end{matrix}

For ϵ′ = 1 we obtain that End(𝕏_{ℚ}) is a most one-dimensional and for ϵ′ = 2 we find

dim_{ℚ} (H^{0} (Y, ⋀^{2}(𝕏_{ℚ}))) ≤ 3 and dim_{ℚ}(H^{0}(Y, S^{2}(𝕏_{ℚ}))) ≤ 1

or vice versa. Comparing this with (5.9.1) one obtains 5.9 a) and b). q.e.d.

Lemma 5.10. Given a quaternion division algebra A, as in 5.5 i) and ii), there exists a smooth family of abelian varieties f : X_{A} → Y with R^{1}f_{∗}ℚ_{X_{A}} = 𝕏_{Aℚ}. Moreover, the special Mumford–Tate group Hg of the general fibre of f is the same as the group G in 5.8.

Proof. (See [18].) The group G in 5.8 and the representation

G ⟶ D^{∗} → Gl(2^{d+ϵ}, ℚ)

are ℚ forms of an ℝ-representation

Sl(2, ℝ) × SU(2)^{×(d−1)} ⟶ Sl(2, ℝ) × SO(2^{d−1}) ⟶ Gl(2^{d+ϵ}, ℝ).

The group in the middle acts on ℝ^{2} ×ℝ^{2^{d−1}}. Over ℝ, this representation leaves a unique nondegenerate symplectic form ⟨ , ⟩ on ℝ^{2^{d}} invariant, the tensor product of the Sl(2, ℝ)-invariant symplectic form on ℝ^{2} with the SO(2^{d−1})-invariant Hermitian form.

Hence for ϵ = 0 and V = ℚ^{2^{d}} there is a unique symplectic form Q on V, invariant under Γ ⊂ G.

For ϵ = 1, one chooses V = ℚ(\sqrt{b})^{2^{d}}. Again one has a unique ℚ(\sqrt{b}) valued symplectic form on V. Regarding V as a ℚ vector space, the trace ℚ(\sqrt{b}) → ℚ gives a ℚ valued symplectic form Q, again invariant under Γ ⊂ G.

Note that Γ is the group of units of an order O in A. Hence Γ leaves a ℤ-module L ⊂ V of rank dim V invariant. For some submodule H ⊂ L of the form H = mL, for m ≫ 0, one has Q(H × H) ⊂ ℤ. Obviously Γ leaves H again invariant. So one obtains a representation

Γ ⟶ Sp(H, Q) ⊗ ℚ.

Finally let

ϕ_{0} : T = {z ∈ ℂ; |z| = 1} ⟶ Sl(2, ℝ) × SO(2^{d−1}) ⊂ Sp(H, Q) ⊗ ℝ

be the homomorphism defined by

e^{iθ} ↦ [ \begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix} ] × I_{2^{d−1}}.

J_{0} = ϕ_{0}(i) defines a complex structure on H ⊗ ℝ, and

Q(x, J_{0}x) > 0, for all x ∈ H.

The image of G in Sp(H, Q) ⊗ ℝ is normalized by ϕ_{0}(T), i.e., for all g ∈ G one has

gϕ_{0}(T)g^{−1} = ϕ_{0}(T).

So 𝕏_{Aℚ} defines a smooth family of abelian varieties f : X_{A} → Y = H/Γ.

By the construction this family reaches the Arakelov bound and 𝕏_{Aℚ} has no unitary part. By Lemma 2.4 c), one knows that

G_{0}^{Mon} = Hg(R^{1}f_{∗}ℚ_{X_{A}}).

On the other hand, G_{0}^{Mon} is contained in the image of G in Sp(H, ℚ)⊗ℚ. Since

𝕏_{Aℂ} = 𝕃_{1ℂ} ⊗ 𝕃_{2ℂ} ⊗ ⋯ ⊗ 𝕃_{dℂ} ⊗ ℂ^{2^{ϵ}}

and since all factors are Zariski dense in Sl(2, ℂ) one finds that

G_{0ℂ}^{Mon} = Sl(2, ℂ)^{×d} = G_{ℂ},

hence

G_{0}^{Mon} = Hg(R^{1}f_{∗}ℚ_{X_{A}}) = G.  q.e.d.

Let us remark, that in the proof of Theorem 0.5 in Section 6 we will see, that the families f : X_{A} → Y in 5.10 are unique up to isogenies, and up to replacing Y by étale coverings, and that they belong to one of the examples described in 0.4.

Construction 5.11. If L ≠ ℚ choose b as in 5.7. The morphism (5.7.1) and 5.7 b), give a map

Γ ⊂ D_{L} = Cor_{F/L}A ⊂ D_{L} ⊗_{L} L(\sqrt{b}) = M(2^{δ′}, L(\sqrt{b})) ⊂ M(2^{δ′+1}, L),

inducing a representation Γ → Gl(2^{δ′+1}, L), hence an L-local system 𝕍_{L} on Y = H/Γ.

An embedding L ⊂ M(δ, ℚ) gives rise to

Γ ⊂ D_{L} = Cor_{F/L}A ⊂ M(2^{δ′+1}, L) ⊂ M(δ2^{δ′+1}, ℚ),

hence to a ℚ-local system 𝕏_{ℚ} = 𝕏_{A,L;ℚ}.

In different terms, choose extensions β̃_{ν} of β_{ν} to \overline{ℚ}. For 𝕍\overline{ℚ} = 𝕍_{L}⊗_{L}\overline{ℚ}, the \overline{ℚ}-local system

X\overline{ℚ} = 𝕏_{A,L;\overline{ℚ}} = 𝕍\overline{ℚ} ⊕ 𝕍β̃_{2} ; \overline{ℚ} ⊕ ⋯ ⊕ 𝕍β̃_{δ} ; \overline{ℚ}

is invariant under Gal(\overline{ℚ}/ℚ), hence defined over ℚ.

Remark 5.12. Consider any family X → Y of abelian varieties, with a geometrically simple generic fibre. If 𝕏_{A,L;ℚ} is an irreducible component of R^{1}f_{∗}ℚ_{X}, all irreducible components of R^{1}f_{∗}ℚ_{X} are isomorphic to 𝕏_{A,L;ℚ}. As in [4], p. 55, for Δ = End(𝕏_{A,L;ℚ}) one finds

R^{1}f_{∗}ℚ_{X} ≃ 𝕏_{A,L;ℚ} ⊗_{Δ} Hom(𝕏_{A,L;ℚ}, R^{1}f_{∗}ℚ_{X}),

and for some m

End(R^{1}f_{∗}ℚ_{X}) ≃ M(m, Δ).

In [21], Section 9, one finds examples showing that all m > 0 occur.

6. The proof of Theorems 0.5 and 0.7

In order to prove Theorems 0.5 and 0.7 we will show, that the local subsystem 𝕏_{ℚ} in 5.4 is for some L ⊂ F isomorphic to the one constructed in 5.8 or 5.11.

Let us consider the subgroup H of Gal(\overline{ℚ}/ℚ) of all β with (ψ^{−1}𝕍)^{β} = ψ^{−1}𝕍, and let L denote the field of invariants under H. So 𝕍 = 𝕍_{L}⊗_{L}ℂ.

Proposition 6.1. Let us keep the assumptions made in 5.4 and use the notations introduced in 5.5. Replacing Y by a finite étale covering, the field of invariants L under H is a subfield of F. Using the notations introduced in 5.6 for such a subfield, there exists a decomposition

𝕍_{L} ≃ 𝕃_{1L} ⊗ ⋯ ⊗ 𝕃_{δ′L} ⊗ 𝕋′_{L}

with:

i. For β ∈ Gal(\overline{ℚ}/L) and i ≤ δ′ one has 𝕃^{β} ≃ 𝕃_{i}, if and only if

β|_{F} = σ_{i}. ii. For β ∈ Gal(\overline{ℚ}/L) the Higgs field of

(𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′})^{β}

is maximal. iii. For β ∈ Gal(\overline{ℚ}/ℚ) with β|_{L} ≠ id_{L} the local system

(𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′})^{β}

is unitary. iv. For β ∈ Gal(\overline{ℚ}/L) the local system 𝕋′^{β} is a unitary.

Proof. Replacing Y by an étale covering, we are allowed to apply 5.5. In particular we have the rank 2 local systems 𝕃_{1}, …, 𝕃_{d}, defined there. Consider any decomposition 𝕍 ≃ 𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{r} ⊗ 𝕋_{r} with:

i′. If for β ∈ Gal(\overline{ℚ}/ℚ) one has β|_{F} = σ_{i}, with i ∈ \{1, …, r\}, then

𝕃^{β} ≃ 𝕃_{i}. For r = 1, 1.4 gives a decomposition 𝕍 = 𝕃 ⊗ 𝕋. Write again 𝕃_{1} = 𝕃 and 𝕋_{1} = 𝕋. By 5.5 iv), the local system is defined over F (\sqrt{a}) and by 5.5 vi), 𝕃_{1}^{β} ≃ 𝕃_{1} if the restriction of β to F is σ_{1} = id_{F}. Hence i′) holds true for this decomposition.

Consider for r ≥ 1 a decomposition satisfying i′).

Step 1. If for some β′ ∈ Gal(\overline{ℚ}/ℚ) and for i ∈ \{1, …, r\} one has 𝕃^{β′} ≃ 𝕃_{i}, then necessarily β′|_{F} = σ_{i}.

In fact, let β ∈ Gal(\overline{ℚ}) be an automorphism with β_{F} = σ_{i}. Then 𝕃^{β^{−1}∘β′} ≃ 𝕃, and 5.5 v), implies that β^{−1} ∘ β′|_{F} = id_{F}.

Step 2. There exists no τ ∈ Gal(\overline{ℚ}/ℚ) with 𝕃_{1}^{τ} ⊗ ⋯ ⊗ 𝕃_{r}^{τ} not unitary and with τ|_{F} ≠ σ_{i} for i = 1, …, r.

Assume the contrary. Renumbering the embeddings of F → ℝ one may assume that τ|_{F} = σ_{r+1}. Recall that by 3.5 𝕃_{i}^{τ} is a variation of Hodge structures of rank 2. It either is of width zero, hence unitary, or of width one, hence with maximal Higgs field. By assumption there exists some i < r + 1 for which 𝕃_{i}^{τ} has a maximal Higgs field. Choose β ∈ Gal(\overline{ℚ}/ℚ) with β|_{F} = σ_{i}. Then 𝕃^{β∘τ} = 𝕃_{i}^{τ} has a maximal Higgs field. 5.5 v), implies β ∘ τ|_{F} = id_{F}, a contradiction.

Step 3. Assume there exists some τ ∈ Gal(\overline{ℚ}/ℚ) with τ|_{F} ≠ σ_{i} for i = 1, …, r, with 𝕍^{τ} not unitary, but with 𝕃_{1}^{τ} ⊗ ⋯ ⊗ 𝕃_{r}^{τ} unitary. Then (renumbering the embeddings F → ℝ, if necessary) one finds a decomposition with r + 1 factors, again satisfying i′).

𝕃_{i}^{τ} is unitary for i = 1, …, r. By 1.4 and by 1.5 over some étale covering of Y we find a splitting 𝕋_{r}^{τ} ≃ 𝕃 ⊗ 𝕋′′, with

𝕍^{τ} ≃ (𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{r})^{β} ⊗ 𝕃 ⊗ 𝕋′′.

Apply τ^{−1}. Then one has

𝕍 ≃ 𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{r} ⊗ 𝕃^{τ^{−1}} ⊗ 𝕋_{r+1}.

Since 𝕃_{1} has maximal Higgs field, 𝕃_{r+1} := 𝕃^{τ^{−1}} must be unitary, as well as 𝕋_{r+1}. Applying any extension τ_{i} of σ_{i}^{−1} for i ≤ r, one finds 𝕃_{r+1}^{τ_{i}} to be unitary, since otherwise there would be two factors with a maximal Higgs field, 𝕃_{i}^{τ_{i}} and 𝕃_{r+1}^{τ_{i}}.

So τ|_{F} must be one of the remaining σ_{j}, and renumbering we may assume τ|_{F} = σ_{r+1}.

Step 4. Assume we have found a decomposition as in i′), and of maximal possible length. Then for all τ ∈ Gal(\overline{ℚ}/ℚ) with τ|_{F} ≠ σ_{i} for i = 1, …, r the local system

𝕍^{τ} ≃ 𝕃_{1}^{τ} ⊗ ⋯ ⊗ 𝕃_{r}^{τ} ⊗ 𝕋_{r}^{τ}

is unitary. For those τ one has (ψ^{−1}𝕍)^{τ} ≠ ψ^{−1}𝕍. On the other hand, for all β with β|_{F} = σ_{i} with 1 ≤ i ≤ r the local system 𝕍^{β} has a maximal Higgs field, hence (ψ^{−1}𝕍)^{β} = ψ^{−1}𝕍. So

H = \{β ∈ Gal(\overline{ℚ}/ℚ); β|_{F} = σ_{i} with 1 ≤ i ≤ r\}

and L as the field of invariants under H is contained in F. Using the notations introduced in 5.6 for such subfields, one finds r = δ′ and 𝕃_{1}^{β} ⊗ ⋯ ⊗ 𝕃_{δ′}^{β} has a maximal Higgs field, for all β ∈ H. This in turn implies that 𝕋′^{β} is unitary for those β. q.e.d.

Theorem 6.2. Let us keep the assumption made in 5.4 and use the notations introduced in 5.6. Replacing Y by an étale covering, there exists some ϵ′ > 0 and a decomposition

ψ : 𝕏 ⟶^{≃} ⨁_{ν=1}^{δ} ( ⨂_{i=(ν−1)δ′}^{νδ′} 𝕃_{i})^{⊕ϵ′} (6.2.1)

such that:

a. For β ∈ Gal(\overline{ℚ}/ℚ) the local system 𝕃_{i}^{β^{−1}} has a maximal Higgs field

if and only if β|_{F} = σ_{i}. Moreover 𝕃^{β} = 𝕃_{i} in this case. b. The direct sum in (6.2.1) is orthogonal with respect to the polar-

ization. c. If the local subsystems ψ^{−1}𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′} of 𝕏 are defined over L

then ϵ′ = 1, L = ℚ and [F : ℚ] is odd. d. If ψ^{−1}𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′} ⊂ 𝕏 is not defined over L choose b to be the

element defined in 5.5 and ι ∈ Gal(\overline{ℚ}/L) with ι(\sqrt{b}) = −\sqrt{b}. Then ϵ′ = 2, the direct factor ψ^{−1}𝕃_{1} ⊗⋯⊗𝕃_{δ′} ⊗ℂ^{2} in (6.2.1) is defined over L and it decomposes over L(\sqrt{b}) like

ψ^{−1}𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′} ⊕ (ψ^{−1}𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′})^{ι} ⊂ 𝕏.

e. 𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′} is irreducible as a ℂ-local system.

Proof. Using the notations from 6.1 let us define 𝕃_{i} = 𝕃σ̃_{i}, where σ̃_{i} is any extension of σ_{i} to \overline{ℚ}. Obviously, fixing any extension β̃_{ν} of β_{ν} one has

𝕍β̃_{ν} = 𝕃_{(ν−1)δ′+1} ⊗ ⋯ ⊗ 𝕃_{νδ′} ⊗ 𝕋′β̃_{ν}.

𝕍 has a maximal Higgs field, whereas ⨁_{ν=2}^{δ} 𝕍β̃_{ν} is unitary. Hence their intersection is zero. Applying β̃_{ν} one obtains the same for the intersection of 𝕍β̃_{ν} and ⨁_{μ=1,μ≠ν}^{δ} 𝕍β̃_{μ}. So

ψ^{−1} (⨁_{ν=1}^{δ} 𝕍β̃_{ν})

is a local subsystem of 𝕏, defined over ℚ. By assumption both must be equal. One obtains

ψ : 𝕏 ⟶^{≃} ⨁_{ν=1}^{δ} ( ⨂_{(ν−1)δ′+1}^{νδ′} 𝕃_{i}) ⊗ 𝕋′β̃_{ν}. (6.2.2)

Let us show next, that 𝕋′ is a trivial local system. The \overline{ℚ}-isomorphism in (6.2.2) induces an isomorphism

𝔼nd(𝕏) ⟶^{≃} 𝔼nd (⨁_{ν=1}^{δ} 𝕍β̃_{ν}) .

Since β ∈ Gal(\overline{ℚ}/ℚ) permutes the direct factors 𝕍β̃_{ν} of 𝕏,

⨁_{ν=1}^{δ} 𝔼nd(𝕍β̃_{ν})

is a local subsystem, defined over ℚ. So ϕ^{−1} induces an embedding

ϕ′ : ⨁_{ν=1}^{δ} 𝔼nd(𝕃_{(ν−1)δ′+1}) ⊗ ⋯ ⊗ 𝔼nd(𝕃_{νδ′}) ⊗ 𝔼nd(𝕋′β̃_{ν}) ⟶ 𝔼nd(𝕏),

Writing 𝔼nd(𝕃_{i}) = ℂ ⊕ 𝔼nd_{0}(𝕃_{i}) we obtain a decomposition of the left-hand side in direct factors, all of the form

𝔼nd_{0}(𝕃_{j_{1}}) ⊗ ⋯ ⊗ 𝔼nd_{0}(𝕃_{j_{ℓ}}) ⊗ 𝔼nd(𝕋′β̃_{ν}),

for some (ν − 1)δ′ + 1 ≤ j_{1} < ⋯ < j_{ℓ} ≤ νδ′.

The only ones without any 𝔼nd_{0}(𝕃_{i}) are the 𝔼nd(𝕋′β̃_{ν}). We claim that

ϕ′ (⨁_{ν=1}^{δ} 𝔼nd (𝕋′β̃_{ν}))^{β} = ϕ′ (⨁_{ν=1}^{δ} 𝔼nd (𝕋′β̃_{ν})) ,

for all β ∈ Gal(\overline{ℚ}/ℚ). Otherwise, we would get a nonzero projection from ϕ′(⨁_{ν=1}^{δ} 𝔼nd(𝕋′β̃_{ν})) to an irreducible local system 𝔼, containing at least one of the 𝔼nd_{0}(𝕃_{i}). By construction, there exists an β_{i} ∈ Gal(\overline{ℚ}/ℚ), such that 𝕃_{i}^{β_{i}} has a maximal Higgs field. Hence 𝔼^{β_{i}} has a maximal Higgs field.

Applying β_{i} we obtain a nonzero map

ϕ′ (⨁_{ν=1}^{δ} 𝔼nd (𝕋′β̃_{ν}))^{ββ_{i}} ⟶ 𝔼^{β_{i}}.

The right-hand side has a maximal Higgs field induced by the one on 𝔼nd_{0}(𝕃_{i}^{β_{i}}), whereas the left-hand side is unitary, a contradiction.

So ϕ′(⨁_{ν=1}^{δ} 𝔼nd(𝕋′β̃_{ν})) is Gal(\overline{ℚ}/ℚ)-invariant, hence a unitary local

system admitting a ℤ-structure. This implies that ϕ′(⨁_{ν=1}^{δ} 𝔼nd(𝕋′β̃_{ν})) is trivial, after replacing Y by a finite étale cover. So the same holds true for 𝔼nd(𝕋′), hence for 𝕋′ as well. Let us write 𝕋′ = ℂ^{⊕ϵ′}. Hence for some ϵ′ one has the decomposition (6.2.1), and a) holds true by construction.

Recall that the local system 𝕃 is defined over F (\sqrt{a}) for a as in 5.1. Hence 𝕃_{i} is defined over σ_{i}(F)(\sqrt{σ_{i}(a)}), and 𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′} is defined over the compositum F′ of those fields, for i = 1, …, δ′.

By 5.5 Cor_{F/L}A can only split if L = ℚ and if [F : ℚ] is odd. Let us write L′ = ℚ in this case. Otherwise it splits over the subfield L′ = L(\sqrt{b}) of F′, where b is given in 5.7 b). In both cases one finds

(Cor_{F/L}A) ⊗_{L} L′ ≃ M(2^{δ′}, L′)

and correspondingly 𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′} is defined over L′.

If L′ = ℚ, this is a local subsystem of 𝕏_{ℚ}. Since it is a ℚ-variation of Hodge structures, and since we assumed 𝕏_{ℚ} to be irreducible, both coincide.

If L′ ≠ L consider the L′-local subsystem (𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′})_{L′} of 𝕍_{L′}. For ι as in d),

𝕍′ = (𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′}) ⊕ (𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′})^{ι},

is a local subsystem of 𝕍, defined over L, and of rank 2^{δ′+1}. Then

⨁_{nu=1}^{δ} ψ^{−1}(𝕍′)β̃_{ν}

is a local subsystem of rank δ·2^{δ′+1} of 𝕏, defined over ℚ. It is also a subvariation of Hodge structures. Since we assumed 𝕏_{ℚ} to be irreducible, both must coincide and ϵ′ is equal to two.

It remains to verify e). Assume that 𝕄 is a direct factor of 𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′}. By 3.2 we may assume that 𝕄 is defined over \overline{ℚ}.

1.9 a), implies that 𝕄 has a maximal Higgs field. By 1.4 𝕄 = 𝕃′ ⊗𝕋′_{1}, and replacing Y by an étale covering we may assume that 𝕃′ = 𝕃_{1} = 𝕃, and that 𝕋′ is a direct factor of 𝕃_{2} ⊗ ⋯ ⊗ 𝕃_{δ′}. Using the notations introduced in 5.6, let σ̃_{i} ∈ Gal(\overline{ℚ}/L) be an extension of σ_{i}, for i = 1, …, δ′. For those i by 6.1

(𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′})σ̃_{i}

has again a maximal Higgs field. Applying 1.9 a), one obtains the same for

𝕄σ̃_{i} = 𝕃σ̃_{i} ; 1 ⊗ 𝕋′_{1}σ̃_{i}.

For i = 2, the first factor is unitary, hence the second has again a maximal Higgs field. 1.5 tell us, that replacing Y again by some étale covering,

𝕋′_{1}σ̃_{i} = 𝕃 ⊗ 𝕋′′,

hence for 𝕋′_{2} = 𝕋′′^{σ_{i}^{−1}}

𝕄 = 𝕃_{1} ⊗ 𝕃_{2} ⊗ 𝕋′_{2}.

Repeating this construction one finds

𝕄 = 𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′} ⊗ 𝕋′_{δ′},

necessarily with 𝕋′_{δ′} = ℂ. q.e.d.

Proposition 6.3. Let f : X → Y be a family of abelian varieties with general fibre X_{η}, and reaching the Arakelov bound. Then:

i. For a generic fibre X_{η} of f

End(X_{η}) ⊗ ℚ ≃ End_{Y} (X) ⊗ ℚ ≃ End(R^{1}f_{∗}ℚ_{X})^{0,0}.

ii. If R^{1}f_{∗}ℂ_{X} has no unitary part then: a. End(R^{1}f_{∗}ℚ_{X})^{0,0} = End(R^{1}f_{∗}ℚ_{X}). b. If X_{η} is geometrically simple, R^{1}f_{∗}ℚ_{X} is irreducible. c. f : X → Y is rigid, i.e., the morphism from Y to the moduli scheme of polarized abelian varieties has no nontrivial deformation.

Proof. i) is a special case of [4], 4.4.6.

If R^{1}f_{∗}ℂ_{X} has no unitary part, for 𝕍 = R^{1}f_{∗}ℂ_{X} 1.7 gives a decomposition 𝔼nd(𝕍) = 𝕎 ⊕ 𝕌 where 𝕎 has a maximal Higgs field, and where 𝕌 is concentrated in bidegree 0, 0. Since 1.9 a), implies that 𝕎 has no global section, one gets a).

For X_{η} geometrically simple End(X_{η}) ⊗ ℚ = End(R^{1}f_{∗}ℚ_{X})^{0,0} is a skew field, hence a) implies that R^{1}f_{∗}ℚ_{X} is irreducible.

ii) c), follows from [9] (see also [21]). q.e.d.

Proposition 6.4. Let f : X → Y be a family of abelian varieties, with a geometrically simple generic fibre X_{η} and reaching the Arakelov bound. Assume that (replacing Y by an étale covering, if needed) one has the decomposition (6.2.1) in 6.2. Then R^{1}f_{∗}ℂ_{X} has no unitary part if and only if

End(R^{1}f_{∗}ℚ_{X})^{0,0} = End(R^{1}f_{∗}ℚ_{X}). (6.4.1)

Proof. By 6.3 ii), a) and b), if 𝕏_{ℚ} = R^{1}f_{∗}ℚ_{X} has no unitary part, 𝕏_{ℚ} is irreducible, and (6.4.1) holds true.

If on the other hand, R^{1}f_{∗}ℂ_{X} has a unitary part, the same holds true for 𝕏. Let us write again 𝕌_{1} for the unitary part of 𝕏. So the field L in 5.4 can not be ℚ. Recall that the Higgs field of 𝕌_{1} splits in two components, one of bidegree 1, 0, the other of bidegree 0, 1, both with a trivial Higgs field. Correspondingly 𝕌_{1} is the direct sum of two subsystems, say 𝕌^{1,0} and 𝕌^{0,1}.

By 6.2 𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{δ′} is an irreducible ℂ-local system. Let us choose one element of ℂ^{ϵ′} and the corresponding local subsystem

𝕄 = ψ^{−1}(L_{1} ⊗ ⋯ ⊗ 𝕃_{δ′})

of 𝕏. There exists some β ∈ Gal(\overline{ℚ}/ℚ) with 𝕄^{β} and \overline{𝕄}^{β} unitary. Replacing 𝕄 by \overline{𝕄}, if necessary we may assume that 𝕄^{β} lies in 𝕌^{1,0} and \overline{𝕄}^{β} in 𝕌^{0,1}. Then

𝕄^{β} ⊗ \overline{𝕄}^{β^{∨}} ⊂ 𝕌^{1,0} ⊗ 𝕌^{0,1}^{∨} ⊂ 𝔼nd(R^{1}f_{∗}ℂ_{X})^{1,−1}.

In 5.5 v), we have seen that 𝕃_{i} ≃ \overline{𝕃}_{i} for all i. Hence

\overline{𝕄} ≃ \overline{𝕃}_{1} ⊗ ⋯ ⊗ \overline{𝕃}_{δ′} ≃ 𝕄,

and 𝕄^{β} and \overline{𝕄}^{β} are isomorphic. One obtains

End(R^{1}f_{∗}ℂ_{X})^{1,−1} ≠ 0.  q.e.d.

Proof of 0.5. Replacing Y by an étale covering, we may assume that Rf_{∗}ℂ_{X} has no unitary part as all. 1.4 provides us with a local system 𝕃, independent of all choices, again after replacing Y by some étale covering.

Hence it is sufficient to consider the case that the generic fibre of f : X → Y is geometrically simple. By 6.3 i), the local system 𝕏_{ℚ} = R^{1}f_{∗}ℚ_{X} is irreducible. In 6.2 the nonexistence of a unitary part implies that δ = 1, hence L = ℚ, and

𝕏 = 𝕍 = (𝕃_{1} ⊗ ⋯ ⊗ 𝕃_{d})^{⊕ϵ′}.

For ϵ′ = 1, the ℚ-local system 𝕏_{ℚ} is given by the representation

η : π_{1}(Y, ∗) ⟶ D^{∗} = (Cor_{F/ℚ}A)^{∗} = Gl(2^{d}, ℚ).

By 2.1 π_{1}(Y, ∗) → Γ = η(π_{1}(Y, ∗)) is an isomorphism and Y = H/Γ. Hence 𝕏_{ℚ} is isomorphic to the local system 𝕏_{Aℚ} constructed in 5.8. In particular, d = [F : ℚ] is odd, and by 6.3 i), 6.4, and 5.9

End(X_{η}) = End(𝕏_{ℚ}) = ℚ and H^{0}(Y, 𝕏_{ℚ} ⊗ 𝕏_{ℚ}) = ℚ.

The second equality implies that the polarization of 𝕏_{ℚ} is unique, up to multiplication with constants, hence 𝕏_{ℚ} and 𝕏_{Aℚ} are isomorphic as polarized variations of Hodge structures. For some ℤ-structure on 𝕏_{Aℚ} we constructed in 5.10 a smooth family of abelian varieties X_{A} → Y, and this family is isogenous to f : X → Y. Both satisfy the properties, stated in Example 0.4 i).

For ϵ′ = 2 and for b as in 5.5, 𝕏_{ℚ} is given by

π_{1}(Y, ∗) ⟶ D^{∗} = (Cor_{F/ℚ}A)^{∗} ⊂ (D ⊗_{ℚ} ℚ(\sqrt{b}))^{∗}
= Gl(2^{d}, ℚ(\sqrt{b})) ⊂ Gl(2^{d+1}, ℚ),

hence again X_{ℚ} is isomorphic to the local system 𝕏_{Aℚ} constructed in 5.8. By 6.3 i), 6.4 and 5.9 a), one finds that

End(X_{η}) = End(𝕏_{ℚ})^{0,0} = End(𝕏_{ℚ}),

is of dimension 4.

For b as in 5.5, consider the local system

𝕃_{1ℚ(\sqrt{b})} ⊗ ⋯ ⊗ 𝕃_{dℚ(\sqrt{b})}

defined by the representation π_{1}(Y, ∗) → Gl(2^{d}, ℚ(\sqrt{b})), together with a embedding into 𝕏_{ℚ(\sqrt{b})}. Restricting the polarization, one obtains a polarization Q′ on 𝕃_{1ℚ(\sqrt{b})} ⊗ ⋯ ⊗ 𝕃_{dℚ(\sqrt{b})}, unique up to multiplication with constants. Regarding this local system as a ℚ-local system, the inclusion

Gl(2^{d}, ℚ(\sqrt{b})) ⊂ Gl(2^{d+1}, ℚ)

defines an isomorphism

𝕃_{1ℚ(\sqrt{b})} ⊗ ⋯ ⊗ 𝕃_{dℚ(\sqrt{b})} ⟶ 𝕏_{ℚ}

and the restriction of the polarization of 𝕏_{ℚ} is the composite of Q′ with the trace on ℚ(\sqrt{b}). In particular, the polarization is uniquely determined, and the family f : X → Y is isogenous to the family X_{A} → Y_{A} = Y constructed in 5.10.

Since, up to a shift in the bidegrees,

ℝ^{2}f_{∗}ℚ_{X} = ⋀^{2} 𝕏_{ℚ}

is a subvariation of Hodge structures of 𝔼nd(𝕏_{ℚ}) one obtains the first equality in

dim(H^{0}(Y, R^{2}f_{∗}ℚ)^{1,1}) = dim(H^{0}(Y, R^{2}f_{∗}ℚ)) = \{ \begin{matrix}3 & for d odd \\ 1 & for d even, \end{matrix}

whereas the second one has been verified in 5.9 b). The dimension of H^{0}(Y, R^{2}f_{∗}ℚ)^{1,1} is the Picard number of a general fibre of f : X → Y. In fact, the Neron–Severi group of a general fibre is invariant under the special Mumford–Tate group of the fibre, hence by 2.4 a), it coincides with dim(H^{0}(Y, R^{2}f_{∗}ℚ)^{1,1}).

Looking to the list of possible Picard numbers and to the structure of the corresponding endomorphism algebras for simple abelian varieties (for example in [15], p. 141), one finds that End(X_{η})⊗ℚ is a quaternion algebra over ℚ, totally indefinite for d odd, and totally definite otherwise. Hence f : X → Y satisfies the properties stated in Example 0.4 ii).

q.e.d.

Proof of 0.7. Again we may assume that R^{1}f_{∗}ℂ_{X} has no nontrivial unitary subbundle defined over ℚ. Let 𝕍 ⊕ 𝕌_{1} be the decomposition of R^{1}f_{∗}ℂ_{X} in a part with a maximal Higgs field and a unitary bundle. By 1.4 one can write 𝕍 = 𝕃 ⊗ 𝕋, where after replacing Y by a finite covering, 𝕃 only depends on Y. If h : Z → Y is a subfamily of f : X → Y with a geometrically simple generic fibre, then repeating this construction with g instead of f, we obtain the same local system 𝕃, hence by 5.3 the same quaternion algebra A. Hence we may assume that f : X → Y has a geometrically simple generic fibre, and we have to show, that f : X → Y is one of the families in Example 0.6.

By [4], §4, R^{1}f_{∗}ℚ_{X} is a direct sum of the same irreducible ℚ-local system 𝕏_{ℚ}. From 1.4 and 5.3 we obtain 𝕃 and a quaternion algebra A, defined over a totally real number field F. By 6.1, 𝕏 contains a local system 𝕍, defined over a subfield L of F. which satisfies the conditions stated there. By 6.2, for b as in 5.7, 𝕍 is given by the representation π_{1}(Y, ∗) → Gl(2^{δ′+1}, L) induced by

π_{1}(Y, ∗) ⟶ D_{L} = Cor_{F/L}A ⊂ D_{L} ⊗_{L} L(\sqrt{b})
= M(2^{δ′}, L(\sqrt{b})) ⊂ M(2^{δ′+1}, L),

hence it is isomorphic to the local system in 5.11. Then the decomposition of 𝕏 in direct factors in 6.2 coincides with the one in 5.11, and f : X → Y is one of the families in Example 0.6.

In iii), the condition b) implies a) and vice versa. On the other hand, L_{i} = ℚ if and only if R^{1}h_{i∗}ℂ_{Z_{i}} has no unitary part, which by 6.4 is equivalent to c). q.e.d.

7. Families of curves and Jacobians

Let us shortly discuss the relation between Theorems 0.2 and 0.5 and the number of singular fibres for semistable families of curves.

Let Y be a curve, let h : C → Y be a semistable nonisotrivial family of curves of genus g > 1, smooth over V, and let f : J(C/Y) → Y be a compactification of the Neron model of the Jacobian of h^{−1}(V) → V. Let us write S for the points in Y − V with f^{−1}(y) singular, and Υ for the other points in Y \ V, i.e., for the points y with h^{−1}(y) singular but f^{−1}(y) smooth. Let g(Y) be the genus of Y and U = Y \ S.

The Arakelov inequality for nonisotrivial families of curves says that

0 < 2 · deg(F^{1,0}) ≤ g_{0} · (2 · g(Y) − 2 + #S + #Υ), (7.0.2)

whereas the Arakelov inequality for f : J(C/Y) → Y gives the stronger bound

0 < 2 · deg(F^{1,0}) ≤ g_{0} · (2 · g(Y) − 2 + #S). (7.0.3)

Hence for a family of curves, the right-hand side of (7.0.2) can only be an equality, if Υ is empty. On the other hand, if both, S and Υ are empty, the Miyaoka-Yau inequality for the smooth surface C implies that

deg(h_{∗}ω_{C/Y}) ≤ \frac{g − 1}{6}(2 · g(Y) − 2).

Hence if h : C → Y is smooth and if h_{∗}ℂ_{C} has no unitary part, the inequalities (7.0.2) and (7.0.3) have both to be strict.

Let us consider the case g(Y) = 0, i.e., families of curves over ℙ^{1}. S.-L. Tan [32] has shown that h : C → ℙ^{1} must have at least 5 singular fibres, hence that #S +#Υ ≥ 5, and (7.0.2) is strict in this special case.

Moreover, he and Beauville [2] gave examples of families with exactly 5 singular fibres for all g > 1. In those examples one has Υ = ∅.

On the other hand, (7.0.3) implies that #S ≥ 4. For #S = 4, the family f : J(C/Y) → Y reaches the Arakelov bound, hence by 0.2 it is isogenous to a product of a constant abelian variety with a product of modular elliptic curves, again with 4 singular fibres. By [3] there are just 6 types of such families, among them the universal family E(3) → X(3) of elliptic curves with a level 3-structures.

Being optimistic one could hope, that those families can not occur as families of Jacobians, hence that there is no family of curves h : C → ℙ^{1} with #S = 4. However, a counterexample has been constructed in [13]. In fact, the same family had been obtained before by G. Xiao in [37].

Example 7.1. Let B be a fixed elliptic curve, defined over ℂ. Consider the Hurwitz functor H_{B,N} defined in [13], i.e., the functor from the category of complex schemes to the category of sets with

H_{B,N}(T) = {f : C ⟶ B×T; f is a normalized covering of degree N and C a smooth family of curves of genus 2 over T }.

The main result of [13] says that for N ≥ 3 this functor is represented by an open subscheme V = H_{B,N} of the modular curve X(N) parameterizing elliptic curves with a level N-structure.

The universal curve C → H_{B,N} extends to a semistable curve C → X(N) whose Jacobian is isogenous to B × E(N). Hence writing S for the cusps, J(C/X(N)) is smooth outside of S, whereas C → X(N) has singular semistable fibres outside of H_{B,N}. Theorem 6.2 in [13] gives an explicit formula for the number of points in Υ = X(N) \ (H_{B,N} ∪ S).

Evaluating this formula for N = 3 one finds #Υ = 3. For N = 3 the modular curve X(3) is isomorphic to ℙ^{1} with 4 cusps. So the number of singular fibres is 4 for J(C/ℙ^{1}) → ℙ^{1} and 7 for C → ℙ^{1}.

We do not know whether similar examples exist for g > 2. For g > 7 the constant part B in Theorem 0.2 can not be of codimension one. In fact, the irregularity q(C) of the total space of a family of curves of genus g over a curve of genus q satisfies by [36], p. 461, the inequality

q(C) ≤ \frac{5 · g + 1}{6} + g(Y).

If J(C/Y) → Y reaches the Arakelov bound, hence if it is isogenous to a product

B × E ×_{Y} ⋯ ×_{Y} E,

one finds

dim(B) ≤ \frac{5 · g + 1}{6}.

As explained in [8] it is not known, whether for g ≫ 2 there are any curves C over ℂ whose Jacobian is isogenous to the product of elliptic curves. Here we are even asking for families of curves whose Jacobian is isogenous to the product of the same nonisotrivial family of elliptic curve, up to a constant factor.

For the smooth families of abelian varieties, considered in 0.5 or 0.7 we do not know of any example, where such a family is a family of Jacobians.

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Universität Duisburg-Essen Mathematik, Campus Essen

45117 Essen, Germany

E-mail address: viehweg@uni-essen.de

Universität Mainz

Fachbereich 17, Mathematik

55099 Mainz, Germany

E-mail address: kzuo@informatik.uni-mainz.de