ARITHMETIC QUOTIENTS OF THE COMPLEX BALL AND A CONJECTURE OF LANG MLADEN DIMITROV AND DINAKAR RAMAKRISHNAN Introduction n be the n-dimensional complex hyperbolic space, repreFor any integer n > 1, let HC sented by the unit ball in Cn 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. n. Given a lattice Γ in U(n, 1), we consider the quotient YΓ := Γ\HC Let M be a CM quadratic extension of a totally real number ﬁeld F of degree d and ring of integers OF , and let G be a reductive group over F deﬁned by a hermitian form on M 3 of signature (n, 1) at one inﬁnite place ι and (n + 1, 0) at the others. A subgroup Γ ⊂ G(F ) is arithmetic, if it is commensurable with G(OF ), and we will use the same notation for its image ι(Γ) ⊂ G(Fι ) = U(n, 1), which is by deﬁnition an arithmetic lattice. By the Baily-Borel theorem the arithmetic quotient YΓ has a structure of a normal, quasiprojective variety, in fact deﬁned over a number ﬁeld. It is projective if, and only if, G is anisotropic. We will prove below three results for arithmetic YΓ , the ﬁrst one concerning the compact case, the second dealing with the general non-compact case, and the third giving precise and explicit conclusions in the case of (non-compact) surfaces. A quasi-projective variety X deﬁned over a number ﬁeld k is said to be Mordellic if, and only if, for any ﬁnite extension k ′ of k, the set X(k ′ ) of k ′ -rational points of X is ﬁnite. Lang conjectured in [L, Conjecture VIII.1.2] (see also [T, p.xviii]) that any hyperbolic, projective X is Mordellic; this is consistent with the general philosophy of Vojta. Theorem 0.1. If Γ is an arithmetic cocompact subgroup of U(2, 1) such that all its torsion elements are scalar, then the projective surface YΓ is Mordellic. Note that even though this 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). Our results do not apply, however, to the analogous case of (a cocompact discrete subgroup of) a unitary group deﬁned by a division algebra of dimension 9 over an imaginary quadratic ﬁeld M with an involution of the second kind. In that case it is known that the 1 2 MLADEN DIMITROV AND DINAKAR RAMAKRISHNAN Albanese of YΓ is zero for any congruence subgroup Γ; this was proved by Rapoport and Zink [RZ] under a ramiﬁcation hypothesis, and later by Rogawski [R1] using a diﬀerent method, without the hypothesis. After an earlier version of this paper was written, we came to know of Ullmo’s work [U] on Shimura varieties of abelian type which contains Theorem 0.1 for suitable covers since Picard modular surfaces of congruence type are in this class. We hope that our (quite diﬀerent) approach is nevertheless of interest since we do not make use of the Shafarevich conjecture. Instead we use certain key theorems of Rogawski [R1, R2] and of Faltings [F1, F2] (see Propositions 2.4 and 3.2). We have also come to know recently of some related work of Yeung [Ye] on rational points on ball quotients. Consider now the more complicated case when G is isotropic, which necessarily implies that F = Q and M is imaginary quadratic. Its toroidal compactiﬁcation XΓ is not hyperbolic even if Γ is torsion free (for example, if n = 2 then XΓ is a union of YΓ with a ﬁnite number of elliptic curves indexed by the cusps). However, by a result of Mumford [M, Proposition 4.2], for Γ neat, XΓ is always of log general type. This implies that for Γ suﬃciently small, XΓ is of general type. The Bombieri-Lang conjecture predicts that the points of XΓ over a given number ﬁeld are not Zariski dense. This we establish in Proposition 3.2 using a deep Theorem due to Faltings, when XΓ is smooth and does not admit a dominant map to its Albanese variety, in particular when its irregularity is > n. This allows us to solve an alternative of Ullmo and Yafaev [UY] regarding the Lang locus. Theorem 0.2. For all Γ ⊂ U(n, 1) arithmetic and suﬃciently small, YΓ is Mordellic. Keeping the assumption that M is imaginary quadratic, say of fundamental discriminant −D, and supposing in addition that n = 2, we restrict our attention to the corresponding locally symmetric spaces YΓ , called Picard modular surfaces. For every ideal N ⊂ OM we consider the congruence subgroups Γ(N) and Γ1 (N) (see Deﬁnition 1.5). We state now our main theorem. Theorem 0.3. Let D | D be the conductor of a simplest Hecke character over M (see (12)). Γ1 (D) , if D ∈ / {3, 4, 7, 8, 11, 15, 19, 20, 23, 24, 31, 39, 43, 47, 67, 71, 163}, (i) Let Γ = Γ(D) , if D ∈ {8, 15, 20, 23, 24, 31, 39, 47, 71}, Γ(D2 ) , if D ∈ {3, 4, 7, 11, 19, 43, 67, 163}. Then YΓ is Mordellic, while XΓ 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 YΓ(N )∩Γ1 (D) is Mordellic, while XΓ(N )∩Γ1 (D) is a minimal surface of general type. At the heart of our proof are some arithmetic computations yielding, for each imaginary quadratic ﬁeld M , explicit congruence subgroups Γ such that XΓ does not admit a dominant ARITHMETIC QUOTIENTS OF THE COMPLEX BALL AND A CONJECTURE OF LANG 3 map to its Albanese variety. A geometric ingredient of the proof is a result of Holzapfel et al that XΓ is of general type, implying by a theorem of Nadel [N] that YΓ does not contain curves of genus ≤ 1. In a related paper [DR], we will investigate further the structure of the Albanese variety of Picard modular surfaces XΓ for speciﬁc families of congruence subgroups Γ, and exhibit the abelian varieties B(p) of Gross as Albanese quotients (up to isogeny) and also explain their relationship to the elliptic curves over the cusps. In addition, we will exploit the Albanese quotients coming from residual automorphic forms, and present an alternate method to deal with the possible curves on genus ≤ 1 by means of a formal immersion at special points on the curves at inﬁnity by analyzing the Fourier-Jacobi coeﬃcients. This leads to another approach for proving ﬁniteness of rational points on the open surface. In a third paper we will analyze the situation over Spec(OM ) with a view towards showing the paucity, even lack, of rational points on the open surface for a suitable inﬁnite class of congruence subgroups. The ultimate aim of our program is to establish a weaker analogue of Mazur’s theorem on modular curves, with a consequence for the boundedness of torsion for principally polarized abelian 3-folds with multiplication by OM . Acknowledgments. We would like to thank Don Blasius, Laurent Clozel, Jean-Fran¸cois Dat, Najmuddin Fakhruddin, Dick Gross, Haruzo Hida, Barry Mazur, David Rohrlich, Matthew Stover and Shing-Tung Yau for helpful conversations. In fact it was Fakhruddin who suggested our use of Lang’s conjecture for abelian varieties. Needless to say, this Note owes much to the deep results of Faltings. 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 the following sources: the Agence Nationale de la Recherche grants ANR-10-BLAN-0114 and ANR-11-LABX-0007-01 for the ﬁrst author (M.D.), and from the NSF grant DMS-1001916 for the second author (D.R.). 1. Basics: Lattices, general type and neatness ¯ = Γ/Γ ∩ U(1) denote its Definition 1.1. Given a discrete subgroup Γ ⊂ U(n, 1) we let Γ image in the adjoint group PU(n, 1) = U(n, 1)/U(1), where U(1) is centrally embedded in n U(n, 1). We put YΓ¯ := YΓ = Γ\HC ¯ ⊂ PU(n, 1) = PSU(n, 1) is the image of a discrete Conversely any discrete subgroup Γ ¯ ∩ SU(n, 1). subgroup of U(n, 1), namely U(1)Γ Lemma 1.2. Let Γ be a lattice in U(n, 1). (i) The analytic variety YΓ is an orbifold and one has the following implications: ¯ torsion free ⇒ YΓ is a hyperbolic manifold. Γ neat ⇒ Γ torsion free ⇒ Γ 4 MLADEN DIMITROV AND DINAKAR RAMAKRISHNAN ¯ is torsion free. Then the natural projection Hn → YΓ is an ´etale (ii) Assume that Γ C ¯ Moreover for every ﬁnite index normal covering with deck transformation group Γ. ¯ Γ ¯ ′. subgroup Γ′ of Γ the natural morphism YΓ′ → YΓ is an ´etale covering of group Γ/ n is a compact group, hence its intersection Proof. The stabilizer in U(n, 1) of any point of HC with the discrete subgroup Γ is ﬁnite, showing that YΓ is an orbifold. If Γ is neat, then no element of it has a non trivial root of unity as an eigenvalue, in particular Γ is torsion free. ¯ is torsion free too. Under the latter assumption, Since Γ ∩ U(1) is ﬁnite, this implies that Γ n , and Γ ¯ acts freely and properly discontinuously on it, hence Γ ∩ U(1) acts trivially on HC n is simply connected, it is a universal covering space of Y with YΓ is a manifold. Since HC Γ ¯ group Γ. In particular, YΓ is hyperbolic. The last claim follows from the exact sequence: ¯′ → Γ ¯ → Γ/ ¯ Γ ¯ ′ → 1. 1→Γ ¯ is cocompact and torsion free. Then the projective Proposition 1.3. Assume that Γ variety YΓ is of general type and can be deﬁned over a number ﬁeld. n implies by the Kodaira embedProof. The existence of the positive Bergman metric on HC ding theorem that YΓ has ample canonical bundle, which results in YΓ being of general type; it even implies that any subvariety is of general type. For surfaces one may alternately use the hyperbolicity of YΓ to rule out all the cases in the Enriques-Kodaira classiﬁcation where the Kodaira dimension is less that 2, thus showing that YΓ is of general type. Calabi and Vesentini [CV] have proved that YΓ is locally rigid, hence by Shimura [Sh1] it can be deﬁned over a number ﬁeld. For the convenient of the reader, we will provide a second, more direct proof when n = 2 and Γ is arithmetic, based on Yau’s algebra-geometric characterization of compact K¨ahler 2 . Since Y has an ample canonical bundle it can be embedded in surfaces covered by HC Γ 2, some projective space, hence is algebraic over C by Chow. Since YΓ¯ is uniformized by HC the Chern numbers c1 , c2 of its complex tangent bundle satisfy the relation c21 = 3c2 . Since everything can be deﬁned algebraically, for any automorphism σ of C, the variety YΓ¯σ also σ has ample canonical bundle and cσ2 1 = 3c2 . By a famous result of Yau [Y, Theorem 4], this ¯ σ \H2 for some cocompact discrete is equivalent to the fact that YΓ¯σ may be realized as Γ C ¯σ. irreducible torsion free subgroup Γ ¯ is arithmetic, it has inﬁnite index in its commensurator in PU(2, 1), denoted Since Γ ¯ For every element g ∈ Comm(Γ) ¯ there is a Hecke correspondence Comm(Γ). (1) ∼ YΓ¯ ← YΓ∩g ¯ −1 Γg ¯ −→ Yg Γg ¯ ¯ −1 ∩Γ ¯ → YΓ g· ∼ and the correspondences for g and g ′ diﬀer by an isomorphism YgΓg ¯ −1 ∩Γ ¯ −→ Yg ′ Γg ¯ ′−1 ∩Γ ¯ ′ ¯ over YΓ¯ if, and only if, g ∈ Γg. By Chow (1) is deﬁned algebraically, hence yields a ARITHMETIC QUOTIENTS OF THE COMPLEX BALL AND A CONJECTURE OF LANG 5 correspondence on YΓ¯σ = YΓ¯ σ : ∼ YΓ¯ σ ← YΓ¯ 1 −→ YΓ¯ 2 → YΓ¯ σ , ¯ 1 and Γ ¯ 2 of Γ ¯ σ . By the universal property of the covering for some ﬁnite index subgroups Γ 2 , the middle isomorphism is given by an element of g ∈ PU(2, 1) ≃ Aut(H2 ). space HC σ C 2 ¯ 1 gσ−1 , and by applying ¯ ¯ Since Aut(HC /YΓ¯ i ) = Γi (i = 1, 2), it easily follows that Γ2 = gσ Γ ¯ σ gσ . It follows that gσ ∈ CommG (Γσ ) and one can check ¯1 = Γ ¯ σ ∩ g −1 Γ σ −1 one sees that Γ σ ¯ σ gσ if, and only if, g ′ ∈ Γg. ¯ Therefore Comm(Γ ¯ σ )/Γ ¯ σ ≃ Comm(Γ)/ ¯ Γ ¯ is inﬁnite that gσ′ ∈ Γ too, which by a major theorem of Margulis implies that Γσ is arithmetic, providing an alternative proof of a result of Kazhdan. Consider now the action of Aut(C) on the set of equivalence classes of cocompact arithmetic subgroups Γ (modulo their center and up to conjugation by an element of U(2, 1)). The group U(2, 1) has only countably many Q-forms, classiﬁed by central simple algebras of dimension 9 over a CM ﬁeld, endowed with an involution of a second kind and verifying some conditions at inﬁnity (see [PR, pp. 87-88]). Finally, 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. It follows that Γ is ﬁxed by an open subgroup of Aut(C), allowing one to conclude that YΓ is deﬁned over a number ﬁeld. It is a well known fact that any compact orbifold admits a ﬁnite cover which is a manifold. We will now provides such a cover explicitly for arithmetic quotients. Recall that an arithmetic subgroup Γ ⊂ G(F ) is a congruence subgroup if there exists an integer N such that Γ contains the principal congruence subgroup of level N , deﬁned as: Γ(N ) = ker (G(OF ) → G(OF /N OF )) . 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. Since G is deﬁned by a hermitian form on M n+1 , we have an embedding G(OF ) ,→ GL(n + 1, OM ), through which we may view elements of G(F ) as (n + 1) × (n + 1) matrices. This allows us to deﬁne the following ﬁner congruence subgroups. Definition 1.5. For every ideal N ⊂ OM we deﬁne the congruence subgroup Γ(N) (resp. Γ1 (N)) as the kernel (resp. the inverse image of upper unipotent matrices) of the composed homomorphism: G(OF ) ,→ GL(n + 1, OM ) → GL(n + 1, OM /N). Lemma 1.6. Suppose that n = 2 and that M is an imaginary quadratic ﬁeld of fundamental √ discriminant −D ∈ / {−3, −4, −7, −8, −24}. Then Γ1 ( −D) is neat. 6 MLADEN DIMITROV AND DINAKAR RAMAKRISHNAN √ Proof. Suppose that the subgroup of C× generated by the eigenvalues of some γ ∈ Γ1 ( −D) contains a non-trivial root of unity ζ. If γ is elliptic then it is necessarily of ﬁnite order. 2 ⊂ P2 (C) hence is conjugated in GL(3, C) to a Otherwise γ ﬁxes a boundary point of HC ( α¯ ∗ ∗ ) ∗ , where β is necessarily a root of unity. Hence, it both cases, matrix of the form 0 β −1 0 0 α one may assume that ζ is an eigenvalue of γ. By the Cayley-Hamilton theorem we have [M (ζ) : M ] ≤ 3 and since D ̸= 7 we may assume (after possibly lifting 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 only divisible by the primes 2 or 3. Hence D ∈ {3, 4, 8, 24} leading to a contradiction. 2. Irregularity of arithmetic surfaces Non-vanishing (and unboundedness) of irregularity for suﬃciently small arithmetic subgroups has been known since the works of Kazhdan [K] and Shimura [Sh2, Theorem 8.1]. 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 contributing to H1 (YΓ , C) allowed one to prove such result for explicit congruence subgroups Γ which can be chosen in various ways. Throughout this section we assume that n = 2. 2.1. Automorphic forms contributing to the irregularity. Given a neat congruence subgroup Γ, we denote by q(YΓ ) the irregularity of YΓ , given by the dimension of H0 (YΓ , Ω1YΓ ). Fixing a maximal compact subgroup K∞ ≃ (U (2) × U (1)) × U (3)d−1 of the reductive Lie group G∞ = G(F ⊗Q R) ≃ U(2, 1) × U(3)d−1 , we obtain a decomposition: ⊕ (2) H1 (YΓ , C) ≃ H1 (Lie(G∞ ), K∞ ; π∞ )⊕m(π∞ ,Γ) , π∞ where π∞ runs over irreducible unitary representations of G∞ occurring with multiplicity m(π∞ , Γ) in the discrete spectrum of L2 (Γ\G∞ ). When Γ is cocompact, the entire L2 spectrum is discrete and this decomposition follows from [BW, Chap.XIII]. When Γ is a non-cocompact lattice, one gets such a decomposition a priori only for the L2 -cohomology of YΓ . However, it is known for Picard modular surfaces that H1 (YΓ , C) is isomorphic to the middle intersection cohomology (in degree 1) of the Baily-Borel compactiﬁcation of YΓ , which is in turn isomorphic to the L2 -cohomology H1(2) (YΓ , C) (see [MR, §1]). In addition, H1(2) (YΓ , C) is isomorphic to H1 (XΓ , C), where XΓ is a smooth toroidal compactiﬁcation of YΓ (see loc. cit.). In particular, H1 (YΓ , C) admits a pure Hodge structure and its dimension is given by 2q(YΓ ). ARITHMETIC QUOTIENTS OF THE COMPLEX BALL AND A CONJECTURE OF LANG 7 At the distinguished Archimedean place ι where G(Fι ) = U(2, 1) there are exactly two irreducible non-tempered unitary representations of G(Fι ), denoted πn+ and πn− , with nonzero relative Lie algebra cohomology in degree 1, while at the remaining inﬁnite places, the only irreducible unitary representation with non-zero relative Lie algebra cohomology in degree 0 is the trivial one. The restrictions to C× of the Langlands parameters of πn+ and πn− are given by: z¯ 0 0 (3) z 7→ 0 z/¯ z 0 ∈ L G0 = GL3 (C), 0 0 z −1 and its complex conjugate (see [La, p.62]). We will now introduce the adelic setting which is better suited for computing the irregularity. For K a neat open compact subgroup of G(AF,f ), where AF,f denotes the ring of ﬁnite adeles of F , we consider the adelic quotient (4) YK := G(F )\G(AF )/K∞ K. Since G1 := ker(det : G → M 1 ) is simply connected and G1∞ is non-compact, by strong approximation (see [PR, Theorem 7.12]) G1 (F ) is dense in G1 (AF,f ). It follows that YK is a ﬁnite disjoint union, indexed by the class group (5) 1 π0 (YK ) ≃ A1M /M 1 det(K)M∞ , of surfaces YΓ for some neat congruence subgroups Γ ⊂ G(F ) and (2) can be rewritten as: ⊕ H1 (YK , C) ≃ H1 (Lie(G∞ ), K∞ ; π∞ )⊕m(πf ,K) , π where π = π∞ ⊗ πf runs over all automorphic representation of G(AF ). By the above description of π∞ , and Rogawski’s multiplicity one theorem [R2] one deduces the formula: ∑ ∑ (6) q(YK ) = dim(πfK ) = dim(πfK ), π automorphic + π∞ =πι ⊗1⊗d−1 ,πι ≃πn π automorphic − π∞ =πι ⊗1⊗d−1 ,πι ≃πn where 1⊗d−1 denotes the trivial representation of U (3)d−1 . 2.2. Rogawski’s theory. Rogawski [R1, R2] gives an explicit description, in terms of global Arthur packets, of the automorphic representations π of G(A) such that πι ≃ πn± and πv = 1 at all Archimedean places v ̸= ι, which we will now present. Let G′ denote the quasi-split unitary group associated to M/F , so that G is an inner form of G′ (note that G ≃ G′ only for d = 1). Let λ be a unitary Hecke character of M whose restriction to F is the quadratic character associated to M/F , and let ν be a unitary character of A1M /M 1 . 8 MLADEN DIMITROV AND DINAKAR RAMAKRISHNAN At a place v of F which does not split in M , the local Arthur packet Π′ (λv , νv ) consists of a square-integrable representation πs (λv , νv ) and a non-tempered representation πn (λv , νv ) of G′ (Fv ). Those can be described (see [R1, §12.2]) as the unique sub-representation and the corresponding (Langlands) quotient of the induction of the character on the standard upper-triangular Borel subgroup B(Fv ) which is trivial on the unipotent subgroup and given on the diagonal torus T (Fv ) by: (7) (¯ α, β, α−1 ) 7→ λv (¯ α)|α|Mv νv (β), where α ∈ Mv× , β ∈ Mv1 . 3/2 If one considers unitary induction, then one has to divide the above character by the square root of the modular character of B(Fv ), that is to say by (¯ α, β, α−1 ) 7→ |α|Mv . At a place v of F which splits in M , G′ (Fv ) (resp. (M ⊗F Fv )1 ) can be identiﬁed with GL(3, Mw ) (resp. Mw× ) where w is a place of M dividing v. The local Arthur packet Π′ (λv , νv ) has a unique element πn (λv , νv ) which is induced from the character: ∗ h2 ∗ 7→ λw (det(h2 ))| det(h2 )|3/2 w νw (h1 ) 0 0 h1 of a maximal parabolic of G′ (Fv ) (see [R2, §1]). For almost all v, πn (λv , νv ) is necessarily unramiﬁed. We set { } Π′ (λ, ν) = ⊗v πv |πv ∈ Π′ (λv , νv ) for all v , and πv = πn (λv , νv ) for almost all v . Recall that a CM type of M is the choice, for each Archimedean places v of F , of an isomorphism M ⊗F,v R ≃ C. Suppose that λ (resp. ν) is algebraic of weight 1 (resp. −1) relatively to a CM type Φ of M , in the sense that: ( ) ∏ ∏ z¯v 1 (8) λ∞ (z) = , for all z ∈ M∞ resp. ν∞ (z) = zv , for all z ∈ M∞ . |zv | v∈Φ v∈Φ Denote by Ξ the set of such pairs (λ, ν). Theorem 2.1 (Rogaswski [R1, R2]). (i) For every (λ, ν) ∈ Ξ, Π′ (λ, ν) is a global Arthur packet for G′ such that for all inﬁnite v, πn (λv , νv ) = πn+ or πn− . (ii) Π′ (λ, ν) can be transferred to an Arthur packet Π(λ, ν) on G such that Π(λv , νv ) = {1} at all Archimedean places v ̸= ι, and Π(λv , νv ) = Π′ (λv , νv ) at the remaining places. (iii) Denote by W (λνM ) ∈ {±1} the root number of the weight 3 algebraic Hecke character λνM , where νM (z) = ν(¯ z /z) is the base change character, and by s(π) the number of ﬁnite places v such that πv ≃ πs (λv , νv ). Then π ∈ Π(λ, ν) is automorphic if, and only if, W (λνM ) = (−1)d−1+s(π) . ARITHMETIC QUOTIENTS OF THE COMPLEX BALL AND A CONJECTURE OF LANG 9 (iv) Any automorphic representation π of G(A) such that πι ≃ πn± and πv = 1 at all 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 the quadratic character associated to M/F , 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 couple of characters (λ, ν) ∈ Ξ uniquely determine a (one-dimensional) character of H, whose endoscopic transfer is Π′ (λ, ν) (see [R2, §1]). Denote by WF (resp. WM ) the global Weil group of F (resp. M ). By loc.cit., the restriction to WM of the global Arthur parameter WF × SL(2, C) → L G = GL(3, C) o Gal(M/F ) of Π(λ, ν) is given by the 3-dimensional representation (λ ⊗ St) ⊕ (νM ⊗ 1), where St (resp. 1) is the standard 2-dimensional (resp. trivial) representation of SL(2, C). Comparing this with the local parameter at inﬁnity (3) yields πn (λv , νv ) = πn± at each inﬁnite place v. It follows that for every archimedean place v, Π′ (λv , νv ) is a packet containing a discrete series representation of G′v , and hence by [R1, §14.4], there will be a corresponding Arthur packet Π(λ, ν) of representations of G(AF ) such that at any archimedean place v ̸= ι, Π(λv , νv ) is a singleton consisting of a ﬁnite-dimensional representation of the compact real group G(Fv ) = U(3). In the notation of [R2, p.397] the representations πn+ and πn− 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(λv , νv ) = {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 a Arthur packet Π on G, which can be transferred to an Arthur packet Π′ on G′ (see [R1, Proposition 14.6.2] and [R1, §14.4]). By deﬁnition Πv = Π′v at v = ι and at all the ﬁnite places v. In particular πn± ∈ Πι = 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 the quadratic character associated to M/F , and some unitary character ν of A1M /M 1 (see [R1, Theorem 13.3.6]). Since Πv = {1} for all Archimedean places v ̸= ι, by the above mentioned recipe Π(λv , νv ) contains either πn+ or πn− , implying that λ (resp. ν) is algebraic of weight 1 (resp. −1) relative to a unique choice of a CM type Φ of M . 2.3. Levels of induced representations. Let p be a prime of F divisible by a unique prime P of M and let Fq be the residue ﬁeld OF /p. In this section we exhibit open compact 10 MLADEN DIMITROV AND DINAKAR RAMAKRISHNAN subgroups K of G(Fp ) for which πn (λp , νp ) (resp. πs (λp , νp )) admit a non-zero K-invariant subspace, and compute in some cases the exact dimension of this space. For every integer n ≥ 1, we deﬁne the open compact subgroup K(Pn ) (resp. K1 (Pn )) of G(Fp ) as the kernel (resp. the inverse image of upper unipotent matrices) of the composed homomorphism: G(OF,p ) ,→ GL(n + 1, OM,P ) → GL(n + 1, OM /Pn ). Lemma 2.2. Let n ≥ 1 be an integer such that the character (7) is trivial on K1 (Pn ) ∩ T (Fp ). Then both πn (λp , νp ) and πs (λp , νp ) have non zero ﬁxed vectors under K1 (Pn ). Proof. Let J denote the Jacquet functor sending admissible G(Fp )-representations to admissible T (Fp )-representations. The Jacquet functor is exact and its basic properties imply: J(πs (λp , νp )) : (α ¯ , β, α−1 ) 7→ λp (¯ α)νp (β)|α|Mp = λp (¯ α)νp (β)|α|Mp · |α|Mp , 3/2 (9) 1/2 −1/2 J(πn (λp , νp )) : (α ¯ , β, α−1 ) 7→ λp (¯ α)νp (β)|α|Mp = λp (α−1 )νp (β)|α|Mp · |α|Mp . 1/2 One knows that the pro-p Iwahori subgroup K1 (Pn ) admits an Iwahori decomposition: ¯ (Fp )), K1 (Pn ) = (K1 (Pn ) ∩ N (Fp )) · (K1 (Pn ) ∩ T (Fp )) · (K1 (Pn ) ∩ N ¯ (Fp )) denotes the unipotent of the standard (resp. opposite) Borel where N (Fp ) (resp. N containing T (Fp ). This is proved for the principal congruence subgroup K(Pn ) in [Cs, Proposition 1.4.4] and the extension to K1 (Pn ) is an easy exercise. Now by [Cs, Proposition 3.3.6], given any admissible G(Fp )-representation V , one has a canonical surjection: n) V K1 (P J(V )K1 (P n )∩T (F p) . Since both characters in (9) are trivial on K1 (Pn ) ∩ T (Fp )), the claim follows. Lemma 2.3. Suppose that p is inert in M and that (λp , νp ) is unramiﬁed. Then the dimension of the K(p)-ﬁxed subspace of πs (λp , νp ) (resp. πn (λp , νp )) equals q 3 (resp. 1). Proof. Since (λp , νp ) is unramiﬁed, restriction to the standard hyperspecial maximal compact subgroup Kp of G(Fp ) yields, by Iwasawa decomposition, the following exact sequence: K p 0 → πs (λp , νp )|Kp → IndB(F (1) → πn (λp , νp )|Kp → 0. p )∩Kp K p The subspace of K(p)-invariant vectors in IndB(F (1) identiﬁes naturally with the p )∩Kp space of C-valued functions on the set: (B(Fp ) ∩ Kp ) \Kp /K(p) ≃ B(Fq )\G(Fq ), on which Kp /K(p) = G(Fq ) acts by right translation. By Iwahori decomposition, since G(F ) G(Fq ) has rank one, the representation IndB(Fqq ) (1) has exactly two irreducible constituents which are the trivial representation and the Steinberg representation. Since both πn (λp , νp )K(p) ARITHMETIC QUOTIENTS OF THE COMPLEX BALL AND A CONJECTURE OF LANG 11 and πs (λp , νp )K(p) are non-zero by Lemma 2.2, and since πs (λp , νp )Kp = 0, it follows that πn (λp , νp )K(p) (resp. πs (λp , νp )K(p) ) is isomorphic to the trivial (resp. Steinberg) representation of G(Fq ), hence its dimension equals 1 (resp. q 3 ). 2.4. Irregularity growth. The positivity of q(YΓ ) is an essential ingredient in the proof of our Diophantine results. Each step of the proof of Proposition 2.4 can be carried out explicitly providing a precise level Γ, depending on M , at which q(YΓ ) > r. After the completion of the work on this paper, we learned of Marshall’s interesting work [Ma] giving sharp asymptotic bounds for q(YΓ ) when Γ shrinks, also by using Rogawski’s theory. Proposition 2.4. For any arithmetic subgroup Γ of G(F ) and for any r > 0, there is a ﬁnite index torsion free subgroup Γ′ of Γ such that q(YΓ′ ) > r. Proof. Note that it suﬃces to ﬁnd a neat congruence subgroup Γ′ such that q(YΓ′ ) > r, since then, for any arithmetic Γ, the natural morphism YΓ∩Γ′ → YΓ′ is ﬁnite and surjective, hence q(YΓ∩Γ′ ) ≥ q(YΓ′ ) > r. Lemma 2.5. For any CM extension M/F and any CM type Φ, there exists an algebraic Hecke character λ of weight 1 and CM type Φ. ∏ z¯v × given by λ (z) = Proof. Consider the character on M∞ . Since M/F is a CM ∞ |zv | v∈Φ × × m extension, the index m of (OF× )2 in OM is ﬁnite, and λ∞ is trivial on (OM ) . By [C, × × m Th´eor`eme 1] there exists an open compact subgroup U of AM,f such that U ∩OM ⊂ (OM ) , × × × × × hence λ∞ can be extended (trivially) to M U M∞ . Finally, since AM /M U M∞ is a ﬁnite × abelian (class) group, there exists a character λ of A× M /M extending λ∞ . Let Π(λ, λ−1 ) be the global Arthur packet on G associated to a character λ as in the |M 1 lemma, and let v0 be a ﬁnite place of F which does not split in M . Choose a open compact ∏ K(λ)v K(λ) subgroup K(λ) = v K(λ)v of G(AF,f ) such that πs,v0 0 ̸= 0 and πn,v v ̸= 0 for all ﬁnite places v, with K(λ)v being the standard hyperspecial maximal compact for all v ̸= v0 relatively prime to the conductor of λ. Choose a ﬁnite place p of F inert in M relatively prime to v0 and to the conductor of λ, such that q = |OF /p| > r. Let K(λ, p) be the subgroup of K(λ) with the maximal compact K(λ)p replaced by K(p). Consider an element π = ⊗v πv ∈ Π(λ, λ−1 ) such that πι = πn± , πv = 1 for every inﬁnite |M 1 place v ̸= ι, πp = πs (λp , νp ), πv = πn,v for every ﬁnite v ̸= p, v0 , and ﬁnally: π , if W (λ3 ) = (−1)d , and n,v0 (10) πv0 = πs,v , if W (λ3 ) = (−1)d−1 . 0 12 MLADEN DIMITROV AND DINAKAR RAMAKRISHNAN K(λ,p) By Theorem 2.1(iii), π is automorphic, and by Lemma 2.3 we have dim(πf ) ≥ 3 q . By (5), for every character χ of the ﬁnite abelian group π0 (YK(λ,p) ), one still has dim(πf ⊗ χ)K(λ,p) ≥ q 3 . By (6) we have q(YK(λ,p) ) ≥ q 3 |π0 (YK(λ,p) )|, hence there must exist a connected component YΓ′ of YK(λ,p) such that q(YΓ′ ) ≥ q 3 > r. 2.5. Irregularity at low level. For the rest of this section M is imaginary quadratic. ˜ be an open compact subgroup of G(A ˜ Q,f ), where G ˜ ⊃ G is the group of unitary Let K ˜ ∩ G(AQ,f ). Then Shimura variety similitudes, and let K = K 2 ˜ ˜ ˜ SK˜ (C) = G(Q)\H C × G(AQ,f )/K has a canonical model SK˜ over its reﬂex ﬁeld M and the set of geometrically connected com¯ ) ≃ π0 (S ˜ (C)) is a principal homogeneous space under Gal(M ′ /M ) ponents π0 (SK˜ ×M M K for some abelian extension M ′ of M (see [Go, §4]). Hence they are all Galois conjugates, and in particular share the same irregularity. By [Go, Lemma 2.4] the identity component of SK˜ (C) can be identiﬁed with YΓ , where ˜ ˜ · G(R)). ˜ Γ = G(Q) ∩ (K ˜ Since for any g ∈ G(R) = GU(2, 1) we have ν(g)3 = | det(g)|2 > 0, implying ν(g) ∈ R× +, × × × b R = {1}. Hence Γ = G(Q) ∩ (K · G(R)) and the identity we deduce that ν(Γ) ⊂ Q ∩ Z + component of YK can also be identiﬁed with YΓ . Therefore the connected components of YK are a subset of those of SK˜ (C) and thus share the same irregularity. Suppose that Γ is neat. Using Theorem 2.1(iii) one can easily transform (6) into: ∑ ∑ (11) 2q(YΓ ) = dim(πfK )(1 + W (λνM )(−1)s(π) ), (λ,ν)∈Ξ/b π0 (YK ) π∈Π(λ,ν) + π∞ ≃πn where a (ﬁnite order) character χ of π0 (YK ) sends (λ, ν) ∈ Ξ to (λχ−1 M , νχ) ∈ Ξ. Note that this action preserves the root number W (λνM ). Proposition 2.6. Any Γ as in Theorem 0.3 is neat and q(YΓ ) > 2. ˜ of G(A ˜ Q,f ) such that Γ equals G(Q) ˜ Proof. There exists a compact open subgroup K ∩ ˜ ˜ ˜ (G(R) · K) as well as G(Q) ∩ (G(R) · K), where K = K ∩ G(AQ,f ). Let λ be a unitary simplest Hecke character as in [Ya, p.88] of conductor: √ −D , if D ̸= 3 is odd, √ 2 −D , if 8 divides D, (12) D= 3 , if D = 3, √ −2D , otherwise. In the ﬁrst two cases those are the canonical characters studied in Rohrlich [Roh]. ARITHMETIC QUOTIENTS OF THE COMPLEX BALL AND A CONJECTURE OF LANG 13 By deﬁnition, (λ, λ−1 ) ∈ Ξ and is trivial on K1 (D) ∩ T (AQ,f ). Lemma 2.2 implies that: |M 1 K1 (D) (13) πf ̸= 0, for all π ∈ Π(λ, λ−1 ). |M 1 In case (ii) we ﬁx a prime p dividing D and π = ⊗v πv ∈ Π(λ, λ−1 ) such that πv = |M 1 πn (λv , λ−1 ) for all v ̸= p, N , πN = πs (λN , λ−1 1 ) and |Mv1 |MN πn (λp , λ−1 1 ) , if W (λ3 ) = −1, |Mp πp = πs (λp , λ−1 ) , if W (λ3 ) = 1. |M 1 p Since Γ(N ) is neat by Lemma 1.4, we can apply (11) which combined with Lemma 2.3 yields K(N ) q(YΓ(N )∩Γ1 (D) ) ≥ dim(πN ) ≥ 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−1 ) ∈ Ξ giving h pairwise distinct elements in Ξ/b π0 (YK1 (D) ). |M 1 Fix a prime p | D and consider π = ⊗v πv ∈ Π(λξ, λ−1 ) such that πv = πn (λv ξv , λ−1 ) for |M 1 |Mv1 all v ̸= p and πn (λp ξp , λ−1 1 ) , if W (λ3 ) = 1, |Mp πp = −1 πs (λp ξp , λ ) , if W (λ3 ) = −1. |M 1 p Since Γ1 (D) is neat by Lemma 1.6, one can apply (11) which combined with (13) yields q(YΓ1 (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 simplest character λ on M , consider π ∈ Π(λ, λ−1 ) such that πv = πn (λv , λ−1 ) for all v ̸= p, q and |M 1 |Mv1 (πn (λp , λ−1 1 ), πn (λq , λ−1 1 )) or (πs (λp , λ−1 1 ), πs (λq , λ−1 1 )) , if W (λ3 ) = 1, |Mp |Mq |Mp |Mq (πp , πq ) = (πn (λp , λ−1 ), πs (λq , λ−1 )) or (πs (λp , λ−1 ), πn (λq , λ−1 )) , if W (λ3 ) = −1. |M 1 |M 1 |M 1 |M 1 p q p q If D ̸= 24 then Γ1 (D) is neat by Lemma 1.6 and (11) implies that q(YΓ1 (D) ) ≥ 2 · 2 = 4. If D = 24 then Γ(D) is neat by Lemma 1.4, since 4 divides D, and again q(YΓ(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 simplest character λ (the canonical one). √ Any character of (1 + −DOM /1 + DOM ) ≃ (Z/D) lifts to a ﬁnite order Hecke character ξ of M with trivial restriction to Q, hence (λξ, λ−1 ) ∈ Ξ. Let π = ⊗v πv ∈ Π(λξ, λ−1 ) be |M 1 |M 1 such that πv = πn (λv ξv , λ−1 ) for all v ̸= D and |Mv1 πn (λD ξD , λ−1 1 ) , if W (λ3 ) = 1, |MD πD = πs (λ ξ , λ−1 ) , if W (λ3 ) = −1. D D |M 1 D 14 MLADEN DIMITROV AND DINAKAR RAMAKRISHNAN K(D) Since Γ(D) is neat by Lemma 1.4, by (11) we get q(YΓ(D) ) ≥ D · dim(πD ) ≥ D. For D = 3 the same argument with D2 instead of D, implies that q(YΓ(9) ) ≥ 3. √ For D = 4 (resp. D = 8) the group Γ(8) (resp. Γ(2 −8)) is neat by Lemma 1.4 and it is an easy exercise in class ﬁeld theory to show that there are at least 3 weight one Hecke characters on M whose restriction to Q is the quadratic character attached to M , and √ whose conductor divides 8 (resp. 2 −8). It follows then from (11) and (13) that for D = 4 (resp. D = 8) one has q(YΓ(8) ) ≥ 3 (resp. q(YΓ(2√−8) ) ≥ 3). Remark 2.7. (i) The computation of the smallest level K for which there exists an automorphic representation π ∈ Π(λ, ν) such that πfK ̸= 0 is analyzed in detail in [DR]. In particular, if λ is a canonical character, we check that the level subgroup at any p | D is precisely the one conjectured by B. Gross, namely the index 2 subgroup of the maximal parahoric subgroup with reductive quotient PGL(2). (ii) One consequence of Rogawski’s theory is that the Albanese variety is of CM type for any congruence subgroup (see [MR]). If M is imaginary quadratic and D is prime, we will show in [DR] that the factor of the Albanese corresponding to the canonical character λ turns out (at an appropriate prime to D level) to be isogenous to the CM abelian variety B(D) deﬁned by B. Gross in [G]. (iii) When Γ is not a congruence subgroup, there are examples of C. Schoen where the Albanese is not of CM type (see [Sc]). 3. Mordellicity We will deduce our main theorems from a more general proposition which is a consequence of the following powerful result of Faltings on the rational points of subvarieties of abelian varieties. Theorem 3.1 (Faltings [F2], [V]). Suppose A is an abelian variety over a number ﬁeld k, Z ⊂ A a closed subvariety. Then there are ﬁnitely many translates Zi of k-rational abelian subvarieties of A, such that Zi ⊂ Z, and such that each k-rational point of Z lies on one of the Zi . Proposition 3.2. Let X be a smooth projective variety over a number ﬁeld k which is geometrically irreducible and does not admit a dominant map to its Albanese variety. Then X(k) is not Zariski dense in X. Proof of Proposition 3.2. If X(k) is empty, there is nothing to prove. Otherwise, use a k-rational point of X to deﬁne the Albanese map over k: j : X → Alb(X). ARITHMETIC QUOTIENTS OF THE COMPLEX BALL AND A CONJECTURE OF LANG 15 Then Z = j(X) is a closed, irreducible subvariety of Alb(X). Applying Theorem 3.1 with A = Alb(X), we get a ﬁnite number, say m ≥ 1, of k-rational translates Zi of abelian subvarieties of Alb(X) such that Z(k) ⊂ m ∪ Zi (k) and Zi ⊂ Z. i=1 Since the Albanese map is deﬁned over k, all the k-rational points of X are contained in those j −1 (Zi ). Finally each j −1 (Zi ) is a proper closed sub-scheme of X (possibly singular and reducible), since otherwise, the irreducibility of X would imply that Z = Zi = Alb(X), contradicting the assumption that X does not admit a dominant map to its Albanese variety. Remark 3.3. When Lang originally made his conjecture on Mordellicity, his deﬁnition of a variety X over k ⊂ C being hyperbolic required the Kobayashi semi-distance on X(C) to be in fact a metric. Later it was established by R. Brody [B] that in the compact case this was equivalent to requiring that there is no non-constant holomorphic map from C to X(C). It is expected that every smooth projective irreducible variety X of general type over C containing no curve of genus ≤ 1 is hyperbolic, and this is known if X is a surface not admitting a dominant map to its Albanese variety. Proof of Theorem 0.1. By Proposition 2.4 there exists a ﬁnite index subgroup Γ′ of Γ such that q(YΓ′ ) > 2, which can be assumed to be normal. It follows that YΓ′ cannot admit a dominant map to its Albanese variety. Moreover YΓ′ is a geometrically irreducible projective surface, hence by Proposition 3.2 YΓ′ (k) is not Zariski dense in YΓ′ . If YΓ′ (k) is inﬁnite, then YΓ′ contains 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 one by Faltings’ celebrated proof of Mordell’s conjecture [F1]. Taking a uniformization of C yields a non-constant holomorphic map from C to YΓ′ , which is impossible since by Lemma 1.2(i), YΓ′ (C) is a smooth compact hyperbolic manifold. Therefore YΓ′ is Mordellic. By Lemma 1.2(ii), the natural morphism f : YΓ′ → YΓ is ﬁnite, etale and deﬁned over a number ﬁeld k. Denote S the ﬁnite set of places of k where f ramiﬁes. Then, for any given number ﬁeld k ′ ⊃ k, ∪ f −1 (YΓ (k ′ )) ⊂ YΓ′ (k ′′ ) k′′ k ′′ k′ where runs over all extensions of of degree at most the degree of f which are unramiﬁed outside S. Since YΓ′ is Mordellic and there are only ﬁnitely many such extensions k ′′ (Hermite-Minkowski), it follows that YΓ (k ′ ) is ﬁnite, hence YΓ is Mordellic. Proof of Theorem 0.2. The Lang locus of a quasi-projective irreducible variety Z is deﬁned as the Zariski closure of the union, over all number ﬁelds k, of irreducible components of 16 MLADEN DIMITROV AND DINAKAR RAMAKRISHNAN positive dimension of the Zariski closure of Z(k). It is clear that Z is Mordellic if, and only if, its Lang locus is empty. The main theorem in [UY] states that, for Γ suﬃciently small, the Lang locus of a Baily-Borel compactiﬁcation of YΓ is either empty or full, which implies immediately that the same statement holds for YΓ itself. For Γ neat, YΓ admits a smooth toroidal compactiﬁcation XΓ deﬁned over a number ﬁeld and by [Sh2, Theorem 8.1] one can assume by further shrinking Γ that q(XΓ ) > n. By Proposition 3.2 the Lang locus of XΓ is not all, hence the Lang locus of YΓ is empty. Proof of Theorem 0.3. Let us ﬁrst show that XΓ is of general type, hence its canonical divisor KX is big (see [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 [Feu] the surface XΓ1 (D) is of general type for all D ∈ / {3, 4, 7, 8, 11, 15, 19, 20, 23, 24, 31, 39, 47, 71}. By the main theorem of Dˇzambi´c [Dˇz], the surface XΓ(D) is of general type for all D ∈ {11, 15, 19, 20, 23, 31, 39, 47, 71} and a careful inspection of his proof (using the prime above 3) shows that this is also true when D = 24. The remaining varieties (D ∈ {3, 4, 7, 8}) are of general type by [H1, Proposition 4.13]. ∑2 2 viewed as the unit ball If g = zj denotes the Bergman metric of HC j dzi d¯ i,j=1 gi¯ {z = (z1 , z2 ) ∈ C2 , |z| < 1}, normalized by requiring that Ric(g) = 2 ∑ i,j=1 − ∂ 2 log(g1¯1 g2¯2 − g2¯1 g1¯2 ) dzi d¯ zj = −g, ∂zi ∂ z¯j then the holomorphic sectional curvature h is constant and equals −4/3 (see [GKK, §3.3], 3((1−|z|2 )δij +¯ zi zj ) where gi¯j = ). (1−|z|2 )2 Since by Proposition 2.6 we have that Γ is neat and q(XΓ ) = q(YΓ ) > 2, Proposition 3.2 implies that XΓ (k) is not Zariski dense in XΓ . 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