Arithmetic Quotients of the Complex Ball

and a Conjecture of Lang

Mladen Dimitrov, Dinakar Ramakrishnan

Received: October 10, 2014 Revised: September 28, 2015

Communicated by Don Blasius

Abstract. We prove that various arithmetic quotients of the unit ball in ℂ^{n} are Mordellic, in the sense that they have only finitely many rational points over any finitely generated field extension of ℚ. In the previously known case of compact hyperbolic complex surfaces, we give a new proof using their Albanese in conjunction with some key results of Faltings, but without appealing to the Shafarevich conjecture. In higher dimension, our methods allow us to solve an alternative of Ullmo and Yafaev. Our strongest result uses in addition Rogawski’s theory and establishes the Mordellicity of the Baily-Borel compactifications of Picard modular surfaces of some precise levels related to the discriminant of the imaginary quadratic fields.

2010 Mathematics Subject Classification: 11G18, 14G05, 22E40. Keywords and Phrases: Rational points; Picard modular surfaces; Albanese varieties; Automorphic representations of unitary groups.

Introduction

Let F be a totally real number field of degree d and ring of integers 𝔬, and let M be a totally imaginary quadratic extension of F with ring of integers 𝔒. Let G be a unitary group over F defined by a hermitian form on M^{n+1} of signature (n, 1) at one infinite place ι and (n+1, 0) or (0, n+1) at the others. A subgroup Γ ⊂ G(F) is arithmetic if it is commensurable with G(𝔬), the stabilizer in G(F) of 𝔒^{n+1}, and we will denote by Y_{Γ} the quotient of the n-dimensional complex hyperbolic space by the natural action of ι(Γ) ⊂ G(F_{ι}) = U(n, 1). If F ≠ ℚ, then the hermitian form is anisotropic and Y_{Γ} is a projective variety defined over a number field (see Proposition 1.2). A projective variety X over ℂ is said to be Mordellic if it has only a finite number of rational points in every finitely generated field extension of ℚ over which X is defined. Lang conjectured in [L, Conjecture VIII.1.2] that X is Mordellic if and only if the corresponding analytic space X(ℂ) is hyperbolic, meaning that any holomorphic map ℂ → X(ℂ) is constant, which by Brody [B] is equivalent to requiring the Kobayashi semi-distance on X(ℂ) to be a metric. It is a consequence of a conjecture of Ullmo (see [U, Conjecture 2.1]) that a projective variety X defined over a number field k is Mordellic if it is arithmetically Mordellic, meaning that it has only a finite number of rational points in every finite extension of k. Our first result establishes that many arithmetic compact surfaces previously only known to be arithmetically Mordellic by [U, Théorème 3.2] are in fact Mordellic. To state it precisely we need to fix a Hecke character λ of M as in Definition 3.1. The existence of such characters is known (see Lemma 3.5). Denote by ℭ the conductor of λ and, if the extension M/F is everywhere unramified, multiply ℭ by any prime 𝔮 of F which does not split in M. Moreover, fix an auxiliary prime 𝔭 of F which splits in M and is relatively prime to ℭ. Finally, for every ideal 𝔑 ⊂ 𝔒 we consider the standard congruence subgroups Γ_{0}(𝔑), Γ_{1}(𝔑) and Γ(𝔑) of G(F) (see Definition 1.3).

Theorem 0.1. Let n = 2 and G over F ≠ ℚ as above. Then for every choice of (ℭ, 𝔭), and for any torsion free subgroup Γ ⊂ Γ_{1}(ℭ) ∩ Γ_{0}(𝔭) of finite index, Y_{Γ} is Mordellic.

A consequence of this is that for any arithmetic subgroup Γ ⊂ G(F) there exists a finite explicit cover of Y_{Γ} which is Mordellic. Note also that even though the theorem only concerns arithmetic subgroups, because F and M can vary, it can be applied to infinitely many pairwise non-commensurable cocompact discrete subgroups in U(2, 1). In order to apply our method to the analogous case of a unitary group G′′ defined by a division algebra of dimension 9 over M with an involution of a second kind, one would need to find a (cocompact) arithmetic subgroup Γ ⊂ G′′(F) such that the Albanese of Y_{Γ} is non-zero. This is an open question for any G′′ since, in contrast to our case, it is known by Rogawski [R1] that the Albanese of Y_{Γ} is zero for any congruence subgroup Γ ⊂ G′′(F). While Ullmo’s approach uses the Shafarevich conjecture, ours is based instead on the Mordell-Lang conjecture proved by Faltings [F2] and on the key Proposition 3.6, which we hope is of independent interest. Consider now the case when the hermitian form is isotropic, which necessarily implies that F = ℚ and M is imaginary quadratic. Then Y_{Γ} is not compact and, for Γ arithmetic, we denote by Y_{Γ}^{∗} the Baily-Borel compactification which is a normal, projective variety of dimension n. A smooth toroidal compactification X_{Γ} of Y_{Γ} can be defined over a number field (see [F3]), and it is not hyperbolic even if Γ is torsion-free; for example, if n = 2, then X_{Γ} is a union of Y_{Γ} with a finite number of elliptic curves – one above each cusp of Y_{Γ}^{∗}. However, by a result of Tai and Mumford [Mu, §4], X_{Γ} is of general type for Γ sufficiently small. The Bombieri-Lang conjecture asserts then that the points of X_{Γ} over any finitely generated field extension of ℚ over which X_{Γ} is defined are not Zariski dense. We prove this in Proposition 4.3 which allows us to solve an alternative of Ullmo and Yafaev [UY] regarding the Lang locus of Y_{Γ}^{∗}.

Theorem 0.2. For all Γ ⊂ G(ℚ) arithmetic and sufficiently small, Y_{Γ}^{∗} is arithmetically Mordellic.

Keeping the assumption that M is imaginary quadratic, say of fundamental discriminant −D, let us now suppose in addition that n = 2. The corresponding locally symmetric spaces Y_{Γ} are called Picard modular surfaces. We state here our main theorem.

Theorem 0.3. Let 𝔇 = \begin{cases}3𝔒 & , if D = 3, \\ \sqrt{−D}𝔒 & , if D ≠ 3 is odd, \\ 2\sqrt{−D}𝔒 & , if 8 divides D, \\ \sqrt{−D}𝔓_{2} & , otherwise, where 𝔓_{2}^{2} = 2𝔒. \end{cases}

(i) Let Γ = \begin{cases}Γ(𝔇^{2}) & , if D ∈ \{3, 4, 7, 11, 19, 43, 67, 163\}, \\ Γ(𝔇) & , if D ∈ \{8, 15, 20, 23, 24, 31, 39, 47, 71\}, \\ Γ_{1}(𝔇) & , otherwise. \end{cases}

Then Y_{Γ}^{∗} is Mordellic, while X_{Γ} is a minimal surface of general type. (ii) Let N > 2 be a prime inert in M and not equal to 3 when D = 4. Then Y_{Γ(N)∩Γ_{1}(𝔇)}^{∗} is Mordellic, while X_{Γ(N)∩Γ_{1}(𝔇)} is a minimal surface of general type.

The fact that one can take Γ(𝔇) instead of Γ(𝔇^{2}) when D ∈ {15, 20, 23, 24, 31, 39, 47, 71} in Theorem 0.3(i) depends on a preprint of Džambić [D] considered for publication elsewhere (see the proof for details). At the heart of our proof stand some arithmetical computations using certain key theorems of Rogawski [R1, R2]. They yield, for each imaginary quadratic field M, an explicit congruence subgroup Γ such that the smooth compactification X_{Γ} does not admit a dominant map to its Albanese variety. A geometric ingredient of the proof is a result of Holzapfel et al that X_{Γ} is of general type, though not hyperbolic, implying by a theorem of Nadel [N] that any curve of genus ≤ 1 on it is contained in the compactifying divisor. If there is anything new in our approach, it lies in the systematic use of the modern theory of automorphic representations in Diophantine geometry.

Acknowledgments. We would like to thank Don Blasius, Jean-François Dat, Najmuddin Fakhruddin, Dick Gross, Barry Mazur, Matthew Stover and Shing-Tung Yau for helpful conversations. In fact it was Fakhruddin who suggested our use of the Mordell-Lang conjecture for abelian varieties. Needless to say, this Note owes much to the deep results of Faltings. In addition, we thank the referee and Blasius for their corrections and suggestions which led to an improvement of the presentation. Thanks are also due to Serge Lang (posthumously), and to John Tate, for getting one of us interested in the conjectural Mordellic property of hyperbolic varieties. Finally, we are also happy to acknowledge partial support from the following sources: the Agence Nationale de la Recherche grants ANR-10-BLAN-0114 and ANR-11-LABX-0007-01 for the first author, and the NSF grant DMS-1001916 for the second author.

1. Basics: lattices, general type and neatness

For any integer n > 1, let H_{ℂ}^{n} be the n-dimensional complex hyperbolic space, represented by the unit ball in ℂ^{n} equipped with the Bergman metric of constant holomorphic sectional curvature −4/(n + 1), on which the real Lie group U(n, 1) acts in a natural way. Given a lattice Γ ⊂ U(n, 1) we denote by Γ̄ = Γ/Γ ∩ U(1) its image in the adjoint group PU(n, 1) = U(n, 1)/U(1), where U(1) is centrally embedded in U(n, 1). Conversely any lattice Γ̄ ⊂ PU(n, 1) = PSU(n, 1) is the image of a lattice in U(n, 1), namely the lattice U(1)Γ̄∩SU(n, 1). We consider the quotient Y_{Γ} = YΓ̄ = Γ̄\H_{ℂ}^{n}.

Lemma 1.1. Let Γ be a lattice in U(n, 1).

(i) The analytic variety Y_{Γ} is an orbifold and one has the following implications:

Γ neat ⇒ Γ torsion-free ⇒ Γ̄ torsion-free ⇒ Y_{Γ} hyperbolic manifold.

(ii) Assume that Γ̄ is torsion-free. Then the natural projection H_{ℂ}^{n} → Y_{Γ} is an etale covering with deck transformation group Γ̄.

Proof. The stabilizer in U(n, 1) of any point of H_{ℂ}^{n} is a compact group, hence its intersection with the discrete subgroup Γ is finite, showing that Y_{Γ} is an orbifold. Recall that Γ is neat if the subgroup of ℂ^{×} generated by the eigenvalues of any γ ∈ Γ is torsion-free. In particular Γ is torsion-free. Since Γ ∩ U(1) is finite, this implies that Γ̄ is torsion-free too. Under the latter assumption, Γ ∩ U(1) acts trivially on H_{ℂ}^{n}, and Γ̄ acts freely and properly discontinuously on it, hence Y_{Γ} is a manifold. Since H_{ℂ}^{n} is simply connected, it is a universal covering space of Y_{Γ} with group Γ̄. Hence any holomorphic map from ℂ to Y_{Γ} lifts to a holomorphic map from ℂ to H_{ℂ}^{n} which must be constant because H_{ℂ}^{n} has negative curvature. Thus Y_{Γ} is hyperbolic. □

Deligne’s classification [De] of Shimura varieties implies, when Γ is a congruence subgroup, that Y_{Γ} admits an embedding in a Shimura variety. Hence, by Shimura’s theory of canonical models, Y_{Γ} can be defined over a finite abelian extension of the reflex field M. We claim that this is also true for Γ arithmetic, when sufficiently small. Indeed any such Y_{Γ} is a finite unramified cover of a congruence quotient Y_{Γ′} which we have seen is defined over a number field. By Grothendieck, the finite index subgroup Γ̄ of the topological fundamental group Γ̄′ of Y_{Γ′}(ℂ) gives rise to a finite index subgroup of the algebraic fundamental group of Y_{Γ′}, yielding a finite algebraic (etale) map from a model of Y_{Γ} to Y_{Γ′}. In the cocompact case, this remains true even when Γ is not arithmetic.

Proposition 1.2. Assume that Γ̄ is cocompact and torsion-free. Then the projective variety YΓ̄ is of general type and can be defined over a number field.

Proof. The existence of the positive Bergman metric on H_{ℂ}^{n} implies by the Kodaira embedding theorem that any quotient by a free action such as YΓ̄ has ample canonical bundle, which results in YΓ̄ being of general type; it even implies that any subvariety of YΓ̄ is of general type. For surfaces one may alternately use the hyperbolicity of YΓ̄ to rule out all the cases in the Enriques-Kodaira classification where the Kodaira dimension is less than 2, thus showing that YΓ̄ is of general type. Calabi and Vesentini [CV] have proved that YΓ̄ is locally rigid, hence by Shimura [S1] it can be defined over a number field. In order to highlight the importance of rigidity of compact ball quotients, we provide a short second proof when n = 2 and Γ is arithmetic, based on Yau’s algebro-geometric characterization of compact Kähler surfaces covered by H_{ℂ}^{2}. Since YΓ̄ has an ample canonical bundle it can be embedded in some projective space, hence is algebraic over ℂ by Chow. Since YΓ̄ is uniformized by H_{ℂ}^{2}, the Chern numbers c_{1}, c_{2} of its complex tangent bundle satisfy the relation c_{1}^{2} = 3c_{2}. Since everything can be defined algebraically, for any automorphism σ of ℂ, the variety Y σ ; Γ̄ also has ample canonical bundle and c_{1}^{σ2} = 3c_{2}^{σ}. By a famous result of Yau [Y, Theorem 4], this is equivalent to the fact that Y σ ; Γ̄ may be realized as Γ̄^{σ}\H_{ℂ}^{2} for some cocompact torsion-free lattice Γ̄^{σ}. Since Γ̄ is arithmetic, it has infinite index in its commensurator in PU(2, 1), denoted by Comm(Γ̄). For every element g ∈ Comm(Γ̄) there is a Hecke correspondence

(1)  YΓ̄ ← Y_{Γ̄∩g^{−1}Γ̄g} ⟶_{g·}^{∼} Y_{gΓ̄g^{−1}∩Γ̄} → YΓ̄

and the correspondences for g and g′ differ by an isomorphism Y_{gΓ̄g^{−1}∩Γ̄} ⟶^{∼} Y_{g′Γ̄g^{′−1}∩Γ̄} over YΓ̄ if and only if g′ ∈ Γ̄g. By Chow (1) is defined algebraically, hence yields a correspondence on Y σ ; Γ̄ = YΓ̄^{σ}:

YΓ̄^{σ} ← YΓ̄_{1} ⟶^{∼} YΓ̄_{2} → YΓ̄^{σ},

for some finite index subgroups Γ̄_{1} and Γ̄_{2} of Γ̄^{σ}. By the universal property of the covering space H_{ℂ}^{2}, the middle isomorphism is given by an element of g_{σ} ∈ PU(2, 1) ≃ Aut(H_{ℂ}^{2}). Since Aut(H_{ℂ}^{2}/YΓ̄_{i}) = Γ̄_{i} (i = 1, 2), it easily follows that Γ̄_{2} = g_{σ}Γ̄_{1}g_{σ}^{−1}, and by applying σ^{−1} one sees that Γ̄_{1} = Γ̄^{σ} ∩ g_{σ}^{−1}Γ̄^{σ}g_{σ}. It follows that g_{σ} ∈ Comm_{G}(Γ^{σ}) and one can check that g′_{σ} ∈ Γ̄^{σ}g_{σ} if and only if g′ ∈ Γ̄g. Therefore Comm(Γ̄^{σ})/Γ̄^{σ} ≃ Comm(Γ̄)/Γ̄ is infinite too, which by a major theorem of Margulis implies that Γ̄^{σ} is arithmetic, providing an alternative proof of a result of Kazhdan. Thus Aut(ℂ) acts on the set of isomorphism classes of cocompact arithmetic quotients YΓ̄, or equivalently, on the set of equivalence classes of cocompact arithmetic subgroups Γ̄ (up to conjugation by an element of PU(2, 1)). The latter set is countable for the following reason. The group U(2, 1) has only countably many ℚ-forms, classified by central simple algebras of dimension 9 over M, endowed with an involution of a second kind and verifying some conditions at infinity (see [PR, pp. 87-88]). Moreover, there are only countably many arithmetic subgroups for a given ℚ-form, since those are all finitely generated and contained in their common commensurator, which is countable. Finally, by [G, Corollary 2.13], the fact that YΓ̄ has a countable orbit under the action of Aut(ℂ) is equivalent to YΓ̄ being defined over a number field. □

It is a well known fact that any orbifold admits a finite cover which is a manifold. In view of Lemma 1.1, the two lemmas below provide such covers explicitly for arithmetic quotients.

Definition 1.3. For every ideal 𝔑 ⊂ 𝔒 we define the congruence subgroup Γ(𝔑) (resp. Γ_{0}(𝔑), resp. Γ_{1}(𝔑)) as the kernel (resp. the inverse image of upper triangular, resp. upper unipotent, matrices) of the composite homomorphism:

G(𝔬) ↪ GL(n + 1, 𝔒) → GL(n + 1, 𝔒/𝔑).

The following lemma is well-known (see [H1, Lemma 4.3]).

Lemma 1.4. For any integer N > 2 the group Γ(N) is neat.

Lemma 1.5. Suppose that n = 2 and that M is an imaginary quadratic field of fundamental discriminant −D ∉ \{−3, −4, −7, −8, −24\}. Then Γ_{1}(\sqrt{−D}𝔒) is neat.

Proof. Suppose that the subgroup of ℂ^{×} generated by the eigenvalues of some γ ∈ Γ_{1}(\sqrt{−D}𝔒) contains a non-trivial root of unity. Note first that det(γ) ∈ 𝔒^{×} ∩ (1 + \sqrt{−D}𝔒) = \{1\}. If γ is elliptic then it is necessarily of finite order. Otherwise γ fixes a boundary point of H_{ℂ}^{2} ⊂ ℙ^{2}(ℂ) and is therefore conjugated in GL(3, ℂ) to a matrix of

the form ( \begin{matrix}ᾱ & ∗ & ∗ \\ 0 & β & ∗ \\ 0 & 0 & α^{−1} \end{matrix} ), where β is necessarily a root of unity. If β = 1, then

det(γ) = 1 implies that α ∈ ℝ, leading to α = −1. Hence, in all cases, one may assume γ has a non-trivial root of unity ζ as an eigenvalue. By the Cayley-Hamilton theorem we have [M(ζ) : M] ≤ 3 and since D ≠ 7 we may assume (after possibly raising γ to some power) that ζ has order 2 or 3. By the congruence condition, each prime p dividing D has to divide also the norm of ζ − 1, hence D can be divisible only by the primes 2 or 3. Thus D ∈ {3, 4, 8, 24}, leading to a contradiction. □

2. Irregularity of arithmetic varieties

Let z ↦ z̄ be the non-trivial automorphism of M/F and let ω be the quadratic character of M/F, viewed as a Hecke character of F. Put M^{1} = {z ∈ M^{×} | zz̄ = 1}, which we will view as an algebraic torus over F and denote by 𝔸_{M}^{1} its 𝔸_{F} -points. We denote by q(X) the irregularity of X, given by the dimension of H^{0}(X, Ω_{X}^{1}).

2.1. Automorphic forms contributing to the irregularity. Fix a maximal compact subgroup K_{∞} ≃ (U(n) × U(1)) × U(n + 1)^{d−1} of the real linear Lie group G_{∞} = G(F ⊗_{ℚ} ℝ) ≃ U(n, 1) × U(n + 1)^{d−1}. Let Γ ⊂ G(F) be a lattice such that Γ̄ is torsion free.

Since Y_{Γ} is the Eilenberg-MacLane space of \overline{Γ}, there is a decomposition:

(2)  H^{1}(Y_{Γ}, ℂ) ≃ H^{1}(Γ̄, ℂ) ≃ ⨁_{π_{∞}} H^{1}(Lie(G_{∞}), K_{∞}; π_{∞})^{⊕m(π_{∞},Γ)},

where π_{∞} runs over irreducible unitary representations of G_{∞} occurring in the discrete spectrum of L^{2}(Γ\G_{∞}) with multiplicity m(π_{∞}, Γ), and H^{∗}(Lie(G_{∞}), K_{∞}; π_{∞}) is the relative Lie algebra cohomology. When Γ is cocompact, the entire L^{2}-spectrum is discrete and this decomposition follows from [BW, XIII]. When Γ is non-cocompact, one gets by [BC, §4.4-4.5] such a decomposition, but only for the L^{2}-cohomology of Y_{Γ}. However, one knows (see [MR, §1]) that H^{1}(Y_{Γ}, ℂ) is isomorphic to the middle intersection cohomology (in degree 1) of Y_{Γ}^{∗}, which is in turn isomorphic to the L^{2}-cohomology (in degree 1) of Y_{Γ}. By [BW, VI.4.11] there are exactly two irreducible non-tempered unitary representations of SU(n, 1) with trivial central character, denoted J_{1,0} and J_{0,1}, each of whose relative Lie algebra cohomology in degree 1 does not vanish and is in fact one dimensional. Since U(n, 1) is the product of its center with SU(n, 1), J_{1,0} and J_{0,1} can be uniquely extended to representations π^{+} and π^{−}, say, of U(n, 1) with trivial central characters (when n = 2 those are the representations J^{+} and J^{−} from [R1, p.178]). It follows that at the distinguished Archimedean place ι, where G(F_{ι}) = U(n, 1), we have

H^{1}(Lie(U(n, 1)), U(n) × U(1); π_{ι}) = \begin{cases}ℂ , if π_{ι} = π^{+} or π^{−}, \\ 0 , otherwise. \end{cases}

Moreover the only irreducible unitary representation with non-zero relative Lie algebra cohomology in degree 0 is the trivial representation 𝟙, which does not contribute in degree 1; in particular π^{+} ≠ 𝟙 and π^{−} ≠ 𝟙 . This allows us to deduce from (2) the following formula

(3)  dim_{ℂ} H^{1}(Y_{Γ}, ℂ) = m(π^{+} ⊗ 𝟙^{⊗d−1}, Γ) + m(π^{−} ⊗ 𝟙^{⊗d−1}, Γ),

where π^{+} and π^{−} are viewed as representations of G(F_{ι}) = U(n, 1) and 𝟙^{⊗d−1} denotes the trivial representation of U(n + 1)^{d−1}. By [MR, §1], H^{1}(Y_{Γ}, ℂ) is isomorphic to H^{1}(X_{Γ}, ℂ), hence admits a pure Hodge structure of weight 1 and its dimension is given by 2q(X_{Γ}). In particular, the natural map H^{0}(X_{Γ}, Ω_{X_{Γ}}^{1}) → H^{0}(Y_{Γ}, Ω_{Y_{Γ}}^{1}) is an isomorphism, i.e.,

(4)  q(Y_{Γ}) = q(X_{Γ}).

It is known that π^{+} ⊗ 𝟙^{⊗d−1} (resp. π^{−} ⊗ 𝟙^{⊗d−1}) contributes to H^{0}(Y_{Γ}, Ω_{Y_{Γ}}^{1}) (resp. H^{1}(Y_{Γ}, Ω_{Y_{Γ}}^{0})). Since the latter two groups have the same dimension, it follows from (3) that

(5)  q(Y_{Γ}) = m(π^{+} ⊗ 𝟙^{⊗d−1}, Γ) = m(π^{−} ⊗ 𝟙^{⊗d−1}, Γ).

We will now focus on the case when Γ is a congruence subgroup and switch to the adelic setting which is better suited for computing the irregularity. For any open compact subgroup K of G(𝔸_{F,f}), where 𝔸_{F,f} denotes the ring of finite adeles of F, we consider the adelic quotient

(6)  Y_{K} = G(F)\G(𝔸_{F})/KK_{∞}.

Let G^{1} = ker(det : G → M^{1}) be the derived group of G. Since G^{1} is simply connected and G_{∞}^{1} is non-compact, G^{1}(F) is dense in G^{1}(𝔸_{F,f}) by strong approximation (see [PR, Theorem 7.12]). It follows that the group of connected components of Y_{K} is isomorphic to the idele class group:

(7)  π_{0}(Y_{K}) ≃ 𝔸_{M}^{1}/M^{1} det(K)M_{∞}^{1}.

To describe each connected component of Y_{K}, choose t_{i} ∈ G(𝔸_{F,f}), 1 ≤ i ≤ h, such that (det(t_{i}))_{1≤i≤h} forms a complete set of representatives of 𝔸_{M}^{1}/M^{1} det(K)M_{∞}^{1}, and let Γ_{i} = G(F) ∩ t_{i}Kt_{i}^{−1}G_{∞}. Then

(8)  G(F)\G(𝔸_{F})/K = ∐_{i=1}^{h} Γ_{i}\G_{∞} and Y_{K} = ∐_{i=1}^{h} Y_{Γ_{i}}.

Therefore (2) and (5) can be rewritten as:

(9) H^{1}(Y_{K}, ℂ) ≃ ⨁_{π=π_{∞}⊗π_{f}} (H^{1}(Lie(G_{∞}), K_{∞}; π_{∞}) ⊗ π_{f}^{K})^{⊕m(π)} and

(10) q(Y_{K}) = ∑_{\begin{matrix}π=π_{∞}⊗π_{f} \\ π_{∞}=π_{ι}⊗𝟙^{⊗d−1},π_{ι}≃π^{+} \end{matrix}} m(π) dim(π_{f}^{K}) = ∑_{\begin{matrix}π=π_{∞}⊗π_{f} \\ π_{∞}=π_{ι}⊗𝟙^{⊗d−1},π_{ι}≃π^{−} \end{matrix}} m(π) dim(π_{f}^{K}),

where π runs over all automorphic representation of G(𝔸_{F}) occurring discretely, with multiplicity m(π), in L^{2}(G(F)\G(𝔸_{F})).

2.2. Irregularity growth. Non-vanishing of q(Y_{Γ}) for sufficiently small congruence subgroups is known by a theorem of Shimura [S2, Theorem 8.1], extending earlier works of Kazhdan and Borel-Wallach [BW, VIII]. Our Diophantine results require however the stronger assumption that q(Y_{Γ}) > n, which we establish as a corollary of the next proposition.

Proposition 2.1. For every open compact subgroup K ⊂ G(𝔸_{F,f}) such that q(Y_{K}) ≠ 0 there exist infinitely many primes 𝔭 of F for which one can find an explicit finite index subgroup K′ ⊂ K differing form K only at 𝔭, such that π_{0}(Y_{K′}) = π_{0}(Y_{K}) and q(Y_{K′}) > q(Y_{K}).

Proof. Since q(Y_{K}) ≠ 0 by assumption, formula (10) implies that there exists an automorphic representation π with π_{∞} = π_{ι} ⊗ 𝟙^{⊗d−1}, π_{ι} ≃ π^{+}, such that m(π) ≠ 0 and π_{f}^{K} ≠ 0. Let 𝔭 be a prime of F which splits in M, so that G(F_{𝔭}) = GL(n + 1, F_{𝔭}). Assume that K = K_{𝔭}^{0} × K^{(𝔭)} where K_{𝔭}^{0} = GL(n + 1, 𝔬_{𝔭}) is the standard maximal compact subgroup of G(F_{𝔭}) and K^{(𝔭)} is the part of K away from 𝔭. In particular π_{𝔭} is unramified. Moreover π_{𝔭} is a unitary representation, since it is a local component of an automorphic representation. By the main result of [T], π_{𝔭} is then the full induced representation of GL(n + 1, F_{𝔭}) from an unramified character μ of a parabolic subgroup P (F_{𝔭}). We claim that, in our case, P is a proper parabolic subgroup. Otherwise π_{𝔭} will be one dimensional, hence G^{1}(F_{𝔭}) will act trivially. Since by strong approximation G^{1}(F)G^{1}(F_{𝔭}) is dense in G^{1}(𝔸_{F}), the latter will act trivially on any smooth vector in π, contradicting the fact that π_{ι} ≄ 𝟙. Let 𝔽_{q} = 𝔬/𝔭 be the residue field of F_{𝔭} and denote by P (𝔽_{q}) the corresponding parabolic subgroup of G(𝔽_{q}). Let K_{0,P} (𝔭) is the inverse image of P (𝔽_{q}) under the reduction modulo 𝔭 homomorphism GL(n + 1, 𝔬_{𝔭}) → GL(n + 1, 𝔽_{q}). Consider K′ = K_{0,P} (𝔭)K^{(𝔭)}. Since det(K′) = det(K), (7) implies that π_{0}(Y_{K′}) = π_{0}(Y_{K}). Moreover, formula (10) implies that

q(Y_{K′}) ≥ q(Y_{K}) + dim(π_{𝔭}^{K_{0,P} (𝔭)}) − dim(π_{𝔭}^{K_{𝔭}^{0}}) = q(Y_{K}) + dim(π_{𝔭}^{K_{0,P} (𝔭)}) − 1,

hence it suffices to show that the K_{0,P} (𝔭)-invariants in π_{𝔭} form at least a 2dimensional space. We claim that we even have

(11)  dim(π_{𝔭}^{K_{0,P} (𝔭)}) ≥ n + 1.

Indeed, since μ is unramified, its restriction to P ∩ K_{𝔭}^{0} is trivial. Therefore by the Iwasawa decomposition G(F_{𝔭}) = P (F_{𝔭}) · K_{𝔭}^{0} the restriction of π_{𝔭} to K_{𝔭}^{0} is

isomorphic to Ind_{P (F_{𝔭})∩K_{𝔭}^{0}}^{K_{𝔭}^{0}}(𝟙).

The subspace of K_{0,P} (𝔭)-invariant vectors in Ind_{P (F_{𝔭})∩K_{𝔭}^{0}}^{K_{𝔭}^{0}}(𝟙) identifies naturally with the space of ℂ-valued functions on the set:

(P (F_{𝔭}) ∩ K_{𝔭}^{0}) \K_{𝔭}^{0}/K_{0,P} (𝔭) ≃ P (𝔽_{q})\G(𝔽_{q})/P (𝔽_{q}).

and the number of such double cosets is the number of double cosets of the Weyl group of G relative to the subgroup attached to P. We may assume, for getting a lower bound, that P is maximal (and proper). The smallest number appears for P of type (n, 1) and it is n + 1. The claim follows. □

Corollary 2.2. For every arithmetic subgroup Γ ⊂ G(F) there exists an explicit subgroup Γ′ of finite index in Γ such that q(Y_{Γ′}) > q(Y_{Γ}).

Proof. By [S2, Theorem 8.1] there exists an open compact subgroup K of G(𝔸_{F,f}) such that q(Y_{K}) ≠ 0 (one might take the principal congruence subgroup of level 2N, which is included in the index 4 subgroup of the principal congruence subgroup of level N considered by Shimura). Denote by h the cardinality of π_{0}(Y_{K}). Applying recursively Proposition 2.1 yields a finite index subgroup K′ ⊂ K, such that π_{0}(Y_{K′}) = π_{0}(Y_{K}) and

q(Y_{K′}) > h · q(Y_{Γ}).

Write Y_{K′} = ∐_{i=1}^{h} Y_{Γ′_{i}} as in (8), and let Γ′ = ∩_{i=1}^{h}Γ′_{i}. Since the irregularity cannot decrease by going to a finite cover, one has:

q(Y_{Γ′∩Γ}) ≥ q(Y_{Γ′}) ≥ max_{1≤i≤h}q(Y_{Γ′_{i}}) ≥ \frac{1}{h} ∑_{i=1}^{h} q(Y_{Γ′_{i}}) = \frac{1}{h}q(Y_{K′}) > q(Y_{Γ}).

□

One can simplify the final step of the proof above and use any Γ′_{i} instead of ∩_{i=1}^{h}Γ′_{i}, since Shimura’s theory of canonical models implies that the connected components of Y_{K′} are all Galois conjugates, hence share the same irregularity.

3. Irregularity of arithmetic surfaces

The positivity of q(Y_{Γ}) is an essential ingredient in the proof of our Diophantine results. The starting point for the arithmetic application of this paper was our knowledge that Rogawski’s classification [R1, R2] of cohomological automorphic forms on G, combined with some local representation theory, would allow us to compute q(Y_{Γ}) precisely and show that it does not vanish for some explicit congruence subgroups Γ. Marshall [Ma] gives sharp asymptotic bounds for q(Y_{Γ}) when Γ shrinks, also by using Rogawski’s theory. In this section we assume that n = 2.

3.1. Rogawski’s theory. Rogawski [R1, R2] gives an explicit description, in terms of global Arthur packets, of the automorphic representations π of G(𝔸_{F}) occurring in (10), which we will now present. Let T denote the maximal torus of the standard upper-triangular Borel subgroup B of G. Let G′ denote the quasi-split unitary group associated to M/F, so that G is an inner form of G′. Note that G_{v} ≃ G′_{v} for any finite place v and that G ≃ G′ only for d = 1. Let λ be a unitary Hecke character of M whose restriction to F is ω, and let ν be a unitary character of 𝔸_{M}^{1}/M^{1}. At a place v of F which does not split in M, which includes any Archimedean v, the local Arthur packet Π′(λ_{v}, ν_{v}) consists of a square-integrable representation π_{s}(λ_{v}, ν_{v}) and a non-tempered representation π_{n}(λ_{v}, ν_{v}) of G′(F_{v}). These constituents of the packet can be described (see [R1, §12.2]) as the unique subrepresentation and the corresponding (Langlands) quotient representation of the induction of the character of B(F_{v}) which is trivial on the unipotent subgroup and given on T (F_{v}) by:

(12)  (ᾱ, β, α^{−1}) ↦ λ_{v}(ᾱ)|α|_{M_{v}}^{3/2}ν_{v}(β), where α ∈ M_{v}^{×}, β ∈ M_{v}^{1}.

If one considers unitary induction, then one has to divide the above character by the square root of the modular character of B(F_{v}), that is to say by (ᾱ, β, α^{−1}) ↦ |α|_{M_{v}}. At any finite place v of F which splits in M, G_{v} ≃ G′_{v} also splits and is isomorphic to GL(3, F_{v}). The local Arthur packet Π′(λ_{v}, ν_{v}) has a unique element π_{n}(λ_{v}, ν_{v}) which is induced from the character:

( \begin{matrix}h_{2} & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & h_{1} \end{matrix} ) ↦ λ_{v}(det(h_{2}))| det(h_{2})|_{v}^{3/2}ν_{v}(h_{1})

Documenta Mathematica 20 (2015) 1185–1205

of the maximal parabolic of type (2, 1) in GL(3, F_{v}) (see [R2, §1]). For almost all v, π_{n}(λ_{v}, ν_{v}) is necessarily unramified. We set

Π′(λ, ν) = {⊗_{v}π_{v}|π_{v} ∈ Π′(λ_{v}, ν_{v}) for all v ,
and π_{v} ≃ π_{n}(λ_{v}, ν_{v}) for almost all v} .

Recall that a CM type Φ on M is the choice, for each Archimedean place v of F, of an isomorphism M ⊗_{F,v} ℝ ≃ ℂ.

Definition 3.1. Let Ξ denote the set of pairs (λ, ν) where λ is a unitary Hecke character of M whose restriction to F is ω, and ν is a unitary character of 𝔸_{M}^{1}/M^{1}, such that

(13)  λ_{∞}(z) = ∏_{v∈Φ} \frac{z̄_{v}}{|z_{v}|} , for all z ∈ M_{∞} and

ν_{∞}(z) = ∏_{v∈Φ} z_{v} , for all z ∈ M_{∞}^{1},

for some CM type Φ on M.

Theorem 3.2 (Rogawski [R1, R2]). (i) For every (λ, ν) ∈ Ξ, Π′(λ, ν) is a global Arthur packet for G′ such that for all infinite v, π_{n}(λ_{v}, ν_{v}) = π^{+} or π^{−}.

(ii) Π′(λ, ν) can be transferred to an Arthur packet Π(λ, ν) on G such that Π(λ_{v}, ν_{v}) = {𝟙} at all the Archimedean places v ≠ ι, and Π(λ_{v}, ν_{v}) = Π′(λ_{v}, ν_{v}) at the remaining places.

(iii) Denote by W (λν_{M}) ∈ {±1} the root number of the Hecke character λν_{M}, where ν_{M}(z) = ν(z̄/z) for z ∈ 𝔸_{M}^{×}, and by s(π) the number of finite places v such that π_{v} ≃ π_{s}(λ_{v}, ν_{v}). Then

π ∈ Π(λ, ν) is automorphic if and only if W (λν_{M}) = (−1)^{d−1+s(π)}.

Moreover, in this case the global multiplicity m(π) is 1. (iv) Any automorphic representation π of G(𝔸_{F}) such that π_{ι} = π^{+} or π^{−},

and π_{v} = 𝟙 at all the Archimedean places v ≠ ι, belongs to Π(λ, ν) for some (λ, ν) ∈ Ξ.

Proof. Let H = U(2)×U(1) be the unique elliptic endoscopic group, shared by G′ and all its inner forms over F. The embedding of L-groups^{L}H ↪ ^{L}G = ^{L}G′ depends on the choice of a Hecke character μ of M, whose restriction to F is ω, and allows one to transfer discrete L-packets on H to automorphic L-packets on G (see [R2, §13.3]). The character μ being fixed, any pair of characters (λ, ν) ∈ Ξ uniquely determines a (one-dimensional) character of H, whose endoscopic transfer is Π′(λ, ν) (see [R2, §1]). Denote by W_{F} (resp. W_{M}) the global Weil group of F (resp. M). By loc.cit., the restriction to W_{M} of the global Arthur parameter

W_{F} × SL(2, ℂ) → ^{L}G = GL(3, ℂ) ⋊ Gal(M/F)

of Π′(λ, ν) is given by the 3-dimensional representation (λ ⊗ St) ⊕ (ν_{M} ⊗ 𝟙), where St (resp. 𝟙) is the standard 2-dimensional (resp. trivial) representation of SL(2, ℂ). By [La, p.62] the restrictions to ℂ^{×} of the Langlands parameters of

π^{+} and π^{−} are given by z ↦ ( \begin{matrix}z̄ & 0 & 0 \\ 0 & z/z̄ & 0 \\ 0 & 0 & z^{−1} \end{matrix} ) and its complex conjugate, hence for

every Archimedean place v one has π_{n}(λ_{v}, ν_{v}) = π^{+} or π^{−}, depending on the choice of isomorphism M ⊗_{F,v}ℝ ≃ ℂ in the CM type Φ. It follows that for every Archimedean place v, Π′(λ_{v}, ν_{v}) is a packet containing a discrete series representation of G′_{v}, and thus by [R1, §14.4], there will be a corresponding Arthur packet Π(λ, ν) of representations of G(𝔸_{F}) such that at any Archimedean place v ≠ ι, Π(λ_{v}, ν_{v}) is a singleton consisting of a finite-dimensional representation of the compact real group G(F_{v}) = U(3). In the notation of [R2, p.397] the representations π^{+} and π^{−} have parameters (r, s) = (1, −1) and (r, s) = (0, 1), respectively, and hence, by the recipe on the same page, the highest weight of the associated finite-dimensional representation equals (1, 0, −1). Therefore at every Archimedean v ≠ ι we have Π(λ_{v}, ν_{v}) = {𝟙}. So far we have established (i) and (ii), while (iii) is the content of [R2, Theorem 1.1]. Conversely, any π as in (iv) is discrete, hence belongs to an Arthur packet Π on G, which can be transferred to an Arthur packet Π′ on G′ (see [R1, §14.4 and Proposition 14.6.2]). By definition, Π_{v} = Π′_{v} at v = ι and at all the finite places v (where, as noted earlier, G_{v} = G′_{v}). In particular π^{+} or π^{−} belongs to Π_{ι} = Π′_{ι}, hence Π′ arises by endoscopy from H, that is to say equals Π′(λ, ν) for some unitary Hecke character λ of M whose restriction to F is ω, and some unitary character of 𝔸_{M}^{1}/M^{1} (see [R1, Theorem 13.3.6]). Since Π_{v} = {𝟙} for all the Archimedean places v ≠ ι, by the above mentioned recipe Π(λ_{v}, ν_{v}) contains either π^{+} or π^{−}, implying that (λ, ν) ∈ Ξ (see Definition 3.1). □

3.2. Irregularity of the connected components. Let K be an open compact subgroup of G(𝔸_{F,f}). Using Theorem 3.2(iii) one can transform (10) into the formula:

(14) 4q(Y_{K}) = ∑_{(λ,ν)∈Ξ} ∑_{π∈Π(λ,ν)} dim(π_{f}^{K})(W (λν_{M}) + (−1)^{d−1+s(π)}).

We will now deduce a similar formula for the irregularity of the connected component of identity Y_{Γ} of Y_{K}, where Γ = G(F) ∩ KG_{∞}. Recall (see (7)) that π_{0}(Y_{K}) ≃ 𝔸_{M}^{1}/M^{1} det(K)M_{∞}^{1}, and denote by { \atop π_{0}(Y_{K})} its (finite, abelian) group of characters. Consider the free action of { \atop π_{0}(Y_{K})} on the set Ξ given by (χ, (λ, ν)) ↦ (λχ_{M}^{−1}, νχ), and denote by Ξ/{ \atop π_{0}(Y_{K})} the quotient set. Since for any π ∈ Π(λ, ν) and any χ ∈ { \atop π_{0}(Y_{K})} one has π ⊗ χ ∈ Π(λχ_{M}^{−1}, νχ), the group { \atop π_{0}(Y_{K})} acts freely on the set of automorphic representations contributing to q(Y_{K}). Moreover this action preserves λν_{M}, s(π) and the dimension of π_{f}^{K}. Hence, in the notations of (8), for any 1 ≤ i ≤ h, the image of the composite map

H^{1}(Lie(G_{∞}), K_{∞}; π_{∞}) ⊗ π_{f}^{K} → H^{1}(Y_{K}, ℂ) → H^{1}(Y_{Γ_{i}}, ℂ),

where the first map comes from (9) and the second from the inclusion Y_{Γ_{i}} ⊂ Y_{K}, does not change when replacing π by π ⊗ χ for any χ ∈ { \atop π_{0}(Y_{K})}. It follows that q(Y_{Γ_{i}}) ≤ \frac{1}{h}q(Y_{K}) for all 1 ≤ i ≤ h. Since ∑_{i=1}^{h} q(Y_{Γ_{i}}) = q(Y_{K}), we deduce that q(Y_{Γ_{i}}) = \frac{1}{h}q(Y_{K}) for all 1 ≤ i ≤ h. This establishes the formula:

(15) 4q(Y_{Γ}) = ∑_{(λ,ν)∈Ξ/{ \atop π_{0}(Y_{K})}} ∑_{π∈Π(λ,ν)} dim(π_{f}^{K})(W (λν_{M}) + (−1)^{d−1+s(π)}).

This formula shows the importance of calculating dim(π_{f}^{K}) which, when K is of the form ∏_{v} K_{v} with v running over all the finite places of F, can be reduced to a local computation of dim(π_{v}^{K_{v}}). This will be taken up in the following section at places v where K_{v} is not the hyperspecial maximal compact subgroup K_{v}^{0}.

3.3. Levels of induced representations. Let 𝔭 be a prime of F divisible by a unique prime 𝔓 of M and let 𝔽_{q} be the residue field 𝔬/𝔭. In this section we exhibit open compact subgroups K_{𝔭} of G(F_{𝔭}) for which π_{n}(λ_{𝔭}, ν_{𝔭}) (resp. π_{s}(λ_{𝔭}, ν_{𝔭})) admit a non-zero K_{𝔭}-invariant subspace, and compute in some cases the exact dimension of this space. For every integer m ≥ 1, we define the open compact subgroup K(𝔓^{m}) (resp. K_{0}(𝔓^{m}), resp. K_{1}(𝔓^{m})) of G(F_{𝔭}) as the kernel (resp. the inverse image of upper triangular, resp. upper unipotent, matrices) of the composite homomorphism:

(16)  G(𝔬_{𝔭}) ↪ GL(3, 𝔒_{𝔓}) → GL(3, 𝔒/𝔓^{m}).

Lemma 3.3. Let m ∈ ℤ_{>0} be such that the character (12) is trivial on K_{1}(𝔓^{m})∩ T (F_{𝔭}). Then both π_{n}(λ_{𝔭}, ν_{𝔭}) and π_{s}(λ_{𝔭}, ν_{𝔭}) have non-zero fixed vectors under K_{1}(𝔓^{m}).

Proof. Let J denote the Jacquet functor sending admissible G(F_{𝔭})- representations to admissible T (F_{𝔭})-representations. The Jacquet functor is exact and its basic properties imply:

\begin{matrix}J(π_{s}(λ_{𝔭}, ν_{𝔭})) : (ᾱ, β, α^{−1}) & ↦ λ_{𝔭}(ᾱ)ν_{𝔭}(β)|α|_{M_{𝔭}}^{3/2} = λ_{𝔭}(ᾱ)ν_{𝔭}(β)|α|_{M_{𝔭}}^{1/2}|α|_{M_{𝔭}}, \\ J(π_{n}(λ_{𝔭}, ν_{𝔭})) : (ᾱ, β, α^{−1}) & ↦ λ_{𝔭}(ᾱ)ν_{p}(β)|α|_{M_{𝔭}}^{1/2} = λ_{𝔭}(ᾱ)ν_{𝔭}(β)|α|_{M_{𝔭}}^{−1/2}|α|_{M_{𝔭}}. \end{matrix} ; (17)

One knows that K_{1}(𝔓^{m}) admits an Iwahori decomposition:

K_{1}(𝔓^{m}) = (K_{1}(𝔓^{m}) ∩ N(F_{𝔭})) · (K_{1}(𝔓^{m}) ∩ T (F_{𝔭})) · (K_{1}(𝔓^{m}) ∩ N̄(F_{𝔭})),

where N(F_{𝔭}) (resp. N̄(F_{𝔭})) denotes the unipotent of the standard (resp. opposite) Borel containing T (F_{𝔭}). This is proved for the principal congruence subgroup K(𝔓^{m}) in [C, Proposition 1.4.4] and the extension to K_{1}(𝔓^{m}) is straightforward. Now by the proof of [C, Proposition 3.3.6], given any admissible G(F_{𝔭})-representation V, one has a canonical surjection:

V^{K_{1}(𝔓^{m})} ↠ J(V)^{K_{1}(𝔓^{m})∩T (F_{𝔭})}.

Since both characters in (17) are trivial on K_{1}(𝔓^{m}) ∩ T (F_{𝔭}), the claim follows.

□

Lemma 3.4. Suppose that 𝔭 is inert in M and that (λ_{𝔭}, ν_{𝔭}) is unramified. Then the dimension of the K_{0}(𝔭)-fixed subspace of both π_{s}(λ_{𝔭}, ν_{𝔭}) and π_{n}(λ_{𝔭}, ν_{𝔭}) is 1. Moreover the dimension of the K(𝔭)-fixed subspace of π_{s}(λ_{𝔭}, ν_{𝔭}) (resp. π_{n}(λ_{𝔭}, ν_{𝔭})) is q^{3} (resp. 1).

Proof. Since (λ_{𝔭}, ν_{𝔭}) is unramified, restriction to the standard hyperspecial maximal compact subgroup K_{𝔭}^{0} of G(F_{𝔭}) yields, by Iwasawa decomposition G(F_{𝔭}) = B(F_{𝔭}) · K_{𝔭}^{0}, the following exact sequence:

0 → π_{s}(λ_{𝔭}, ν_{𝔭})_{|K_{𝔭}^{0}} → Ind_{B(F_{𝔭})∩K_{𝔭}^{0}}^{K_{𝔭}^{0}}(𝟙) → π_{n}(λ_{𝔭}, ν_{𝔭})_{|K_{𝔭}^{0}} → 0.

The subspace of K(𝔭)-invariant vectors in Ind_{B(F_{𝔭})∩K_{𝔭}^{0}}^{K_{𝔭}^{0}}(𝟙) identifies naturally with the space of ℂ-valued functions on the set:

(B(F_{𝔭}) ∩ K_{𝔭}^{0}) \K_{𝔭}^{0}/K(𝔭) ≃ B(𝔽_{q})\G(𝔽_{q}),

on which K_{𝔭}^{0}/K(𝔭) = G(𝔽_{q}) acts by right translation. By the Iwahori decomposition, since G(𝔽_{q}) has rank 1, the representation Ind_{B(𝔽_{q})}^{G(𝔽_{q})}(𝟙) has exactly two irreducible constituents which are the trivial representation and the Steinberg representation, implying that both π_{n}(λ_{𝔭}, ν_{𝔭})^{K_{0}(𝔭)} and π_{s}(λ_{𝔭}, ν_{𝔭})^{K_{0}(𝔭)} are one-dimensional. Since π_{s}(λ_{𝔭}, ν_{𝔭})^{K_{𝔭}^{0}} = 0, it follows that π_{n}(λ_{𝔭}, ν_{𝔭})^{K(𝔭)} (resp. π_{s}(λ_{𝔭}, ν_{𝔭})^{K(𝔭)} ) is isomorphic to the trivial (resp. Steinberg) representation of G(𝔽_{q}), hence its dimension equals 1 (resp. q^{3}). □

3.4. Surfaces with positive irregularity. The existence of Hecke characters λ of M satisfying (13) goes back to Chevalley and Weil. We will show that there are still such characters if one further imposes their restriction to F to be ω.

Lemma 3.5. For any CM extension M/F and any CM type Φ on M, there exist Hecke characters λ of M whose restriction to F equals ω, such that λ_{∞}(z) =

∏_{v∈Φ} \frac{z̄_{v}}{|z_{v}|} for all z ∈ M_{∞}^{×}.

Proof. Since M is totally imaginary, λ_{∞} and ω agree on F_{∞}^{×}, hence there is a character λ_{0} of 𝔸_{F}^{×}M_{∞}^{×} extending both. We show now that λ_{0} can be extended to a Hecke character λ of M, which will obviously satisfy the assumptions of the lemma. Since M/F is totally imaginary, 𝔬^{×2} has finite index in 𝔒^{×}. By [Ch, Théorème 1] there exists an open compact subgroup U of 𝔸_{M,f}^{×} such that U ∩ 𝔒^{×} ⊂ 𝔬^{×2}. We may, and we will, assume that U is contained in the congruence subgroup whose level is the relative different of M/F and by replacing U by U ∩ Ū we can further assume that U = Ū. Since the Artin conductor of ω is the relative discriminant of M/F, it follows that ω is trivial on U ∩ 𝔸_{F,f}^{×}. Hence one can extend λ_{0} to a character of 𝔸_{F}^{×}UM_{∞}^{×} by letting it be trivial on U. Suppose we knew that

(18)  M^{×} ∩ 𝔸_{F}^{×}UM_{∞}^{×} = F^{×}.

Then there is a unique character of M^{×}𝔸_{F}^{×}UM_{∞}^{×} extending both λ_{0} and the trivial character of M^{×}U. Since 𝔸_{M}^{×}/M^{×}𝔸_{F}^{×}UM_{∞}^{×} is a finite abelian (idele class) group, the character above can be further extended to a character λ of 𝔸_{M}^{×}/M^{×}, and any such extension has the desired properties. It remains to prove (18). Let x ∈ M^{×} ∩ 𝔸_{F}^{×}UM_{∞}^{×}. Then

x̄/x ∈ M^{1} ∩ UM_{∞}^{1} = 𝔒^{×} ∩ UM_{∞}^{1} ⊂ 𝔬^{×2}.

Since F^{×2} ∩ M^{1} = {1}, we have x = x̄ ∈ F^{×}. □

Proposition 3.6. Fix any Hecke character λ of M satisfying (13) whose restriction to F is ω. Let 𝔭 be a prime of F which splits in M and is relatively prime to the conductor ℭ of λ. If W (λ^{3}) = (−1)^{d} we choose a prime 𝔮 of F which does not split in M; if not, we take 𝔮 = 𝔬. Then q(Y_{Γ_{1}(ℭ)∩Γ_{0}(𝔭𝔮)}) > 2.

Proof. Let K = K_{1}(ℭ) ∩ K_{0}(𝔭𝔮), so that Γ = G(F) ∩ KK_{∞}. Let Π(λ, λ_{|M^{1}}^{−1}) be the global Arthur packet on G associated to λ. Let π = ⊗_{v}π_{v} ∈ Π(λ, λ_{|M^{1}}^{−1}) be such that π_{ι} = π^{+}, π_{v} = 𝟙 for every infinite place v ≠ ι, π_{v} = π_{n,v} for every finite v ≠ 𝔮, and finally if W (λ^{3}) = (−1)^{d} then π_{𝔮} = π_{s,𝔮}. By Theorem 3.2(iii), π is automorphic, and by Lemma 3.3, we have π_{n,v}^{K_{v}} ≠ 0 for all finite places v ≠ 𝔭, 𝔮. Moreover, if 𝔮 ≠ 𝔬, then π_{s,𝔮}^{K_{𝔮}} ≠ 0 by Lemma 3.3 (resp. Lemma 3.4 ) if 𝔮 divides (resp. does not divide) ℭ. Finally (11) implies that dim(π_{𝔭}^{K_{0}(𝔭)}) ≥ 3, hence q(Y_{Γ}) ≥ 3 by (15) as claimed. □

Remark 3.7. Since restriction of λ to 𝔸_{F}^{×} equals ω, its conductor ℭ is divisible by the different of M/F. Hence, unless M/F is unramified everywhere, one might take as 𝔮 a place where M/F is ramified and Proposition 3.6 applies then to K = K_{1}(ℭ) ∩ K_{0}(𝔭). Given a totally real number field F, there exists a totally imaginary quadratic extension M/F unramified everywhere if and only if all the units in F have norm 1.

For the rest of this section we assume that F = ℚ, so that G is quasi-split.

Proposition 3.8. Any Γ as in Theorem 0.3 is neat and q(Y_{Γ}) > 2.

Proof. There exists an open compact subgroup K of G(𝔸_{ℚ,f}) such that Γ = G(ℚ) ∩ KG(ℝ). We claim that there exists a Hecke character λ of M of conductor 𝔇 satisfying (13), whose restriction to F equals ω. If D ≠ 3 is odd or if 8 divides D such characters, called canonical, are proved to exists by Rohrlich [Ro]. If D > 4 is even but not divisible by 8, then by Yang (see [Ya, p.88]) there are such characters, called the simplest. Finally for D = 3 (resp. D=4) the claim follows from the existence of a CM elliptic curve over ℚ of conductor 27 (resp. 32). By definition, (λ, λ_{|M^{1}}^{−1}) ∈ Ξ and is trivial on K_{1}(𝔇) ∩ T (𝔸_{ℚ,f}). Then Lemma 3.3 implies that:

(19)  π_{f}^{K_{1}(𝔇)} ≠ 0 , for all π ∈ Π(λ, λ_{|M^{1}}^{−1}).

In case (ii) of Theorem 0.3 where Γ = Γ(N) ∩ Γ_{1}(𝔇) we fix a prime p dividing D and π = ⊗_{v}π_{v} ∈ Π(λ, λ_{|M^{1}}^{−1}) such that π_{v} = π_{n}(λ_{v}, λ_{|M_{v}^{1}}^{−1}) for all v ≠ p, N, π_{N} = π_{s}(λ_{N}, λ_{|M_{N}^{1}}^{−1} ) and

π_{p} = \begin{cases}π_{n}(λ_{p}, λ_{|M_{p}^{1}}^{−1}) , if W (λ^{3}) = −1, \\ π_{s}(λ_{p}, λ_{|M_{p}^{1}}^{−1}) , if W (λ^{3}) = 1. \end{cases}

Since Γ(N) is neat by Lemma 1.4, we can apply (15) which, when combined with Lemma 3.4, yields

q(Y_{Γ(N)∩Γ_{1}(𝔇)}) ≥ dim(π_{N}^{K(N)}) ≥ N^{3} ≥ 3.

We now turn to case (i) and suppose first that M has class number h ≥ 3. For any class character ξ one has (λξ, λ_{|M^{1}}^{−1}) ∈ Ξ giving h pairwise distinct

elements in Ξ/{ \atop π_{0}(Y_{K_{1}(𝔇)})}. Fix a prime p dividing D and consider π = ⊗_{v}π_{v} ∈ Π(λξ, λ_{|M^{1}}^{−1}) such that π_{v} = π_{n}(λ_{v}ξ_{v}, λ_{|M_{v}^{1}}^{−1}) for all v ≠ p and

π_{p} = \begin{cases}π_{n}(λ_{p}ξ_{p}, λ_{|M_{p}^{1}}^{−1}) , if W (λ^{3}) = 1, \\ π_{s}(λ_{p}ξ_{p}, λ_{|M_{p}^{1}}^{−1}) , if W (λ^{3}) = −1. \end{cases}

Since Γ_{1}(𝔇) is neat by Lemma 1.5, one can apply (15) which combined with (19) yields q(Y_{Γ_{1}(𝔇)}) ≥ h ≥ 3. If M is one of the 18 imaginary quadratic fields of class number 2, then its fundamental discriminant D has (exactly) two distinct prime divisors p < q. For each character λ on M as above, consider π ∈ Π(λ, λ_{|M^{1}}^{−1}) such that π_{v} = π_{n}(λ_{v}, λ_{|M_{v}^{1}}^{−1}) for all v ≠ p, q, and moreover if W (λ^{3}) = 1, then

(π_{p}, π_{q}) = (π_{n}(λ_{p}, λ_{|M_{p}^{1}}^{−1}), π_{n}(λ_{q}, λ_{|M_{q}^{1}}^{−1})) or (π_{s}(λ_{p}, λ_{|M_{p}^{1}}^{−1}), π_{s}(λ_{q}, λ_{|M_{q}^{1}}^{−1})),

whereas if W (λ^{3}) = −1, then

(π_{p}, π_{q}) = (π_{n}(λ_{p}, λ_{|M_{p}^{1}}^{−1}), π_{s}(λ_{q}, λ_{|M_{q}^{1}}^{−1})) or (π_{s}(λ_{p}, λ_{|M_{p}^{1}}^{−1}), π_{n}(λ_{q}, λ_{|M_{q}^{1}}^{−1})).

If D ≠ 24 then Γ_{1}(𝔇) is neat by Lemma 1.5 and (15) implies that q(Y_{Γ_{1}(𝔇)}) ≥ 2 · 2 = 4. If D = 24 then Γ(𝔇) is neat by Lemma 1.4, since 4 divides 𝔇, and again q(Y_{Γ(𝔇)}) ≥ 4. Finally, we consider the nine imaginary quadratic fields of class number 1. For D ∈ {7, 11, 19, 43, 67, 163} there is a unique character λ as in the beginning of the proof. Any character of (1 + \sqrt{−D}𝔒/1 + D𝔒) ≃ ℤ/Dℤ lifts to a finite order Hecke character ξ of M with trivial restriction to ℚ, hence (λξ, λ_{|M^{1}}^{−1}) ∈ Ξ. Let π = ⊗_{v}π_{v} ∈ Π(λξ, λ_{|M^{1}}^{−1}) be such that π_{v} = π_{n}(λ_{v}ξ_{v}, λ_{|M_{v}^{1}}^{−1}) for all v ≠ D and

π_{D} = \begin{cases}π_{n}(λ_{D}ξ_{D}, λ_{|M_{D}^{1}}^{−1}) , if W (λ^{3}) = 1, \\ π_{s}(λ_{D}ξ_{D}, λ_{|M_{D}^{1}}^{−1}) , if W (λ^{3}) = −1. \end{cases}

Since Γ(D) is neat by Lemma 1.4, by (15) we get q(Y_{Γ(D)}) ≥ D · dim(π_{D}^{K(D)}) ≥ D.

For D = 3 the same argument with D^{2} instead of D, implies that q(Y_{Γ(9𝔒)}) ≥ 3. For D = 4 (resp. D = 8) the group Γ(8𝔒) (resp. Γ(2\sqrt{−8}𝔒)) is neat by Lemma 1.4 and it is an exercise on idele class groups to show that there are at least three Hecke characters of M satisfying (13) whose restriction to ℚ is ω, and whose conductor divides 8 (resp. 2\sqrt{−8}). It follows then from (15) and (19) that for D = 4 (resp. D = 8) one has q(Y_{Γ(8𝔒)}) ≥ 3 (resp. q(Y_{Γ(2\sqrt{−8}𝔒)}) ≥ 3). □

Remark 3.9. The computation of the smallest level K for which there exists an automorphic representation π ∈ Π(λ, ν) such that π_{f}^{K} ≠ 0 is analyzed in detail in [DR]. In particular, if λ is a canonical character, we check that the level subgroup at any p dividing D is precisely the one conjectured by B. Gross, namely the index 2 subgroup of the maximal parahoric subgroup with reductive quotient PGL(2).

4. The Albanese map and Mordellicity

A major ingredient in the proof of our theorems is the Mordell-Lang conjecture for abelian varieties in characteristic zero, established by Faltings [F2] using some earlier work of himself [F1] and Vojta [V] (see Mazur’s detailed account [M]).

Theorem 4.1 (Mordell-Lang conjecture : theorem of Faltings). Suppose A is an abelian variety over ℂ and Z ⊂ A a closed subvariety. Then for any finitely generated field extension k of ℚ over which Z ⊂ A is defined, the set Z(k) is contained in a union of finitely many translates of abelian subvarieties of A, each of which is defined over k and contained in Z.

The following corollary was proved in Moriwaki [Mo, Theorem 1.1]. He stated it for number fields, but the proof is the same for finitely generated fields over ℚ.

Corollary 4.2. Let X be a connected smooth projective variety over ℂ which does not admit a dominant map to its Albanese variety. Then for any finitely generated field extension k of ℚ over which X is defined, the set X(k) is not Zariski dense in X.

Proof. The conclusion is obvious if X(k) is empty so we may choose a point of X(k) to define the Albanese map over k:

j : X → Alb(X).

Applying Theorem 4.1 to the closed subvariety Z = j(X) of Alb(X) we get a finite number, say m ≥ 1, of translates Z_{i} of abelian subvarieties of Alb(X) defined over k and such that

Z(k) ⊂ ⋃_{i=1}^{m} Z_{i}(k) and Z_{i} ⊂ Z.

Since j is defined over k, each k-rational point of X is contained in j^{−1}(Z_{i}) for some i. If j^{−1}(Z_{i}) were not a proper closed subvariety of X, the universal property of the Albanese map would imply that Z = Z_{i} = Alb(X), contradicting the assumption that j is not dominant. □

Proposition 4.3. For every arithmetic subgroup Γ ⊂ G(F) there exists a finite cover of Y_{Γ} whose points over any finitely generated field extension of ℚ are not Zariski dense, i.e., the Bombieri-Lang conjecture holds for that cover.

Proof. Applying Corollary 2.2 recursively yields a finite index subgroup Γ′ ⊂ Γ, which one can assume to be torsion free, such that q(Y_{Γ′}) > n = dim(Y_{Γ′}). It suffices then to apply Corollary 4.2 to Y_{Γ′} in the compact case and, in view of (4), to X_{Γ′} in the non-compact case. □

Proof of Theorem 0.1. By Proposition 3.6, we have q(Y_{Γ}) > 2, hence Y_{Γ} does not admit a dominant map to its Albanese variety. Moreover Y_{Γ} is a geometrically irreducible smooth projective surface, hence by Corollary 4.2 Y_{Γ}(k) is not Zariski dense in Y_{Γ} for any finitely generated field extension k of ℚ over which Y_{Γ} is defined. If Y_{Γ}(k) were infinite, then Y_{Γ} would contain an irreducible curve C defined over k and such that C(k) infinite. Since C(k) is Zariski dense in C, the curve C is geometrically irreducible and its geometric genus is at most 1 by Theorem 4.1 applied to the Albanese map of C. Taking a complex uniformization of C would provide a non-constant holomorphic map from ℂ to Y_{Γ}, which is impossible by Lemma 1.1(i) which we can apply as Γ is torsion free. Therefore Y_{Γ} is Mordellic. □

Proof of Theorem 0.2. The Lang locus of a quasi-projective irreducible variety Z over a number field is defined as the Zariski closure of the union, over all number fields k, of irreducible components of positive dimension of the Zariski closure of Z(k). It is clear that Z is arithmetically Mordellic if and only if its Lang locus is empty. The main theorem in [UY] asserts that, for Γ neat and sufficiently small, the Lang locus of Y_{Γ}^{∗} is either empty or everything. By Corollary 2.2 one can assume by further shrinking Γ that q(Y_{Γ}) > n, and by (4) we also have q(X_{Γ}) > n for X_{Γ} a smooth toroidal compactification of Y_{Γ}. By Corollary 4.2 the Lang locus of X_{Γ} is not everything, which forces the Lang locus of Y_{Γ}^{∗} to be empty. □

Proof of Theorem 0.3. Let us first show that X_{Γ} is of general type, hence its canonical divisor K_{X} is big in the sense of [N, Definition 1.1]. Note that just like irregularity, the Kodaira dimension cannot decrease when going to a finite covering. By Holzapfel [H2, Theorem 5.4.15] and Feustel [Fe] the surface X_{Γ_{1}(𝔇)} is of general type for all

D ∉ {3, 4, 7, 8, 11, 15, 19, 20, 23, 24, 31, 39, 47, 71}.

Also by [H1, Proposition 4.13], X_{Γ(N)} is of general type for all integers N > 2, with the possible exceptions of N = 3 and N = 4 when D = 4, implying in particular that X_{Γ(𝔇^{2})} is of general type for D ∈ {3, 4, 7, 8, 11, 19}. Finally, the argument from loc. cit. transports in a straight forward way to the case when the level is an ideal of 𝔒, yielding that the remaining varieties X_{Γ(𝔇)}, D ∈ {15, 20, 23, 24, 31, 39, 47, 71}, are of general type as well. See [D] where this has been carried out (it should be noted that his argument works also when D = 24). If g = ∑_{i,j=1}^{2} g_{ij̄}dz_{i}dz̄_{j} denotes the Bergman metric of H_{ℂ}^{2} viewed as the unit ball {z = (z_{1}, z_{2}) ∈ ℂ^{2} , |z| < 1}, normalized by requiring that

Ric(g) = ∑_{i,j=1}^{2} −\frac{∂^{2} log(g_{1¯1}g_{2¯2} − g_{2¯1}g_{1¯2})}{∂z_{i}∂z̄_{j}}dz_{i}dz̄_{j} = −g,

then the holomorphic sectional curvature is constant and equals −4/3 (see

[GKK, §3.3]), where g_{ij̄} = \frac{3((1−|z|^{2})δ_{ij}+z̄_{i}z_{j})}{(1−|z|^{2})^{2}}. By (4) and Proposition 3.8 we have that Γ is neat and q(X_{Γ}) = q(Y_{Γ}) > 2. Corollary 4.2 then implies that X_{Γ}(k) is not Zariski dense in X_{Γ} for any finitely generated field extension k of ℚ over which X_{Γ} is defined. If X_{Γ}(k) is infinite, arguing as in the proof of Theorem 0.1 shows that X_{Γ} contains a geometrically irreducible curve C whose geometric genus is at most one. Now applying a result of Nadel [N, Theorem 2.1] with γ = 1 (so that −γ ≥ −4/3), we see that the bigness of K_{X} implies that C is contained in the compactifying divisor, which is a finite union of elliptic curves indexed by the cusps. It follows that Y_{Γ}^{∗}(k) is finite and that X_{Γ} does not contain any rational curves at all, let alone just those of self intersection −1, hence it is a minimal surface of general type. □

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Mladen Dimitrov Université Lille 1 UMR 8524, Mathématiques 59655 Villeneuve d’Ascq France mladen.dimitrov@gmail.com Dinakar Ramakrishnan Caltech Mathematics 253-37 Pasadena, CA 91125 USA dinakar@caltech.edu 1206

Documenta Mathematica 20 (2015)