Arithmetic manifolds associated to orthogonal groups

Hodge type theorems for arithmetic manifolds associated to orthogonal groups

Nicolas Bergeron Institut de Mathématiques de Jussieu
Unité Mixte de Recherche 7586 du CNRS
Université Pierre et Marie Curie
4, place Jussieu 75252 Paris Cedex 05, France
bergeron@math.jussieu.fr http://people.math.jussieu.fr/ bergeron
John Millson Department of Mathematics
University of Maryland
College Park, Maryland 20742, USA
jjm@math.umd.edu http://www-users.math.umd.edu/ jjm/
 and  Colette Moeglin Institut de Mathématiques de Jussieu
Unité Mixte de Recherche 7586 du CNRS
4, place Jussieu 75252 Paris Cedex 05, France
moeglin@math.jussieu.fr http://www.math.jussieu.fr/ moeglin
Abstract.

We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree of compact congruence -dimensional hyperbolic manifolds “of simple type” as long as is strictly smaller than . We also prove that for connected Shimura varieties associated to the Hodge conjecture is true for classes of degree . The proof of our general theorem makes use of the recent endoscopic classification of automorphic representations of orthogonal groups by [5]. As such our results are conditional on the hypothesis made in this book, whose proofs have only appear on preprint form so far; see the second paragraph of subsection 1.19 below.

N.B. is a member of the Institut Universitaire de France.
J.M. was partially supported by NSF grant DMS-0907446

1. Introduction

1.1.

Let be the -dimensional hyperbolic space and let be a compact hyperbolic manifold.

Thirty-five years ago one of us (J.M.) proved, see [59], that if was a compact hyperbolic manifold of simple arithmetic type (see subsection 1.3 below) then there was a congruence covering such that contained a nonseparating embedded totally geodesic hypersurface . Hence the associated homology class was nonzero and the first Betti number of was nonzero. Somewhat later the first author refined this result to

1.2 Proposition.

Assume is arithmetic and contains an (immersed) totally geodesic codimension one submanifold

(1.2.1)

Then, there exists a finite index subgroup such that the map (1.2.1) lifts to an embedding and the dual class is non zero.

This result was the first of a series of results on non-vanishing of cohomology classes in hyperbolic manifolds, see [10] for the best known results in that direction. In this paper we investigate to which extent classes dual to totally geodesic submanifolds generate the whole cohomology. We work with congruence hyperbolic manifolds.

1.3.

First we recall the general definition of congruence hyperbolic manifolds of simple type. Let be a totally real field and the ring of adeles of . Let be a nondegenerate quadratic space over with . We assume that is compact at all but one infinite place. We denote by the infinite place where is non compact and assume that .

Consider the image in of the intersection , where is a compact-open subgroup of the group of finite adèlic points of . According to a classical theorem of Borel and Harish-Chandra, it is a lattice in . It is a cocompact lattice if and only if is anisotropic over . If is sufficiently deep, i.e. is a sufficiently small compact-open subgroup of , then is moreover torsion-free.

The special orthogonal group is a maximal compact subgroup of , and the quotient — the associated symmetric space — is isometric to the -dimensional hyperbolic space .

A compact congruence hyperbolic manifolds of simple type is a quotient with a torsion-free congruence subgroup obtained as above. is an -dimensional congruence hyperbolic manifold. In general, a hyperbolic manifold is arithmetic if it shares a common finite cover with a congruence hyperbolic manifold.

1.4.

Compact congruence hyperbolic manifolds of simple type contains many (immersed) totally geodesic codimension one submanifolds to which Proposition 1.2 applies. In fact: to any totally positive definite sub-quadratic space of dimension we associate a totally geodesic (immersed) submanifold of codimension in . Set so that . There is a natural morphism . Recall that we can realize as the set of negative lines in . We then let be the subset of consisting of those lines which lie in . Let be the image of in . The cycle is the image of the natural map

It defines a cohomology class .

The following theorem can thus be thought as a converse to Proposition 1.2.

1.5 Theorem.

Suppose . Let be a -dimensional compact congruence hyperbolic manifold of simple type. Then is spanned by the Poincaré duals of classes of totally geodesic (immersed) submanifolds of codimension .

Remark. Note that this result is an analogue in constant negative curvature of the results that totally geodesic flat subtori span the homology groups of flat tori (zero curvature) and totally-geodesic subprojective spaces span the homology with -coefficients for the real projective spaces (constant positive curvature).

1.6.

There are results for local coefficients analogous to Theorem 1.5 that are important for the deformation theory of locally homogeneous structures on hyperbolic manifolds. Let be the inclusion. Suppose is resp. . In each case we have a natural inclusion . For the case the image of is the subgroup leaving the first basis vector of fixed, in the second the inclusion is the “identity”. The representation is no longer rigid (note that has infinite covolume in ). Though there was some earlier work this was firmly established in the early 1980’s by Thurston, who discovered the “Thurston bending deformations”, which are nontrivial deformations associated to embedded totally geodesic hypersurfaces where , see [38] §5, for an algebraic description of these deformations. It is known that the Zariski tangent space to the real algebraic variety of representations at the point is the space of one cocycles in the first case and in the second case. Here denotes the space of harmonic (for the Minkowski metric) degree two polynomials on . Also trivial deformations correspond to -coboundaries. Then Theorem 5.1 of [38] proves that the tangent vector to the curve at is cohomologous to the Poincaré dual of the embedded hypersurface equipped with the coefficient in the first case and the harmonic projection of in the second where is a suitable vector in determining the parametrization of the curve .

We then have

1.7 Theorem.

Suppose . Let be a cocompact congruence lattice of simple type in . Then resp. are spanned by the Poincaré duals of (possibly non-embedded) totally-geodesic hypersurfaces with coefficients in resp. .

Remark. First, we remind the reader that the above deformation spaces of representations are locally homeomorphic to deformation spaces of locally homogeneous structures. In the first case, a hyperbolic structure on a compact manifold is a fortiori a flat conformal structure and a neighborhood of in the first space of representations (into ) is homeomorphic to a neighborhood of in the space of (marked) flat conformal structures. In the second case (representations into ) a neighborhood of is homeomorphic to a neighborhood of the hyperbolic manifold in the space of (marked) flat real projective structures. Thus it is of interest to describe a neighborhood of in these two cases. By the above theorem we know the infinitesimal deformations of are spanned modulo coboundaries by the Poincaré duals of totally-geodesic hypersurfaces with coefficients. The first obstruction (to obtaining a curve of structures or equivalently a curve of representations) can be nonzero, see [38] who showed that the first obstruction is obtained by intersecting the representing totally geodesic hypersurfaces with coefficients. By Theorem 1.21 we can compute the first obstruction as the restriction of the wedge of holomorphic vector-valued one-forms on . This suggests the higher obstructions will be zero and in fact the deformation spaces will be cut out from the above first cohomology groups by the vector-valued quadratic equations given by the first obstruction ( see [29]).

1.8.

Theorems 1.5 and 1.7 bear a strong ressemblance to the famous Hodge conjecture for complex projective manifolds: Let be a projective complex manifold. Then every rational cohomology class of type on is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of .

Hyperbolic manifolds are not complex (projective) manifolds, so that Theorem 1.5 is not obviously related to the Hodge conjecture. We may nevertheless consider the congruence locally symmetric varieties associated to orthogonal groups . These are connected Shimura varieties and as such are projective complex manifolds. As in the case of real hyperbolic manifolds, one may associate special algebraic cycles to orthogonal subgroups of .

The proof of the following theorem now follows the same lines as the proof of Theorem 1.5.

1.9 Theorem.

Let be a connected compact Shimura variety associated to the orthogonal group . Let be an integer . Then every rational cohomology class of type on is a linear combination with rational coefficients of the cohomology classes Poincaré dual to complex subvarieties of .

Note that the complex dimension of is . Hodge theory provides with a pure Hodge structure of weight and we more precisely prove that is defined over and that every rational cohomology class of type on is a linear combination with rational coefficients of the cup product with some power of the Lefschetz class of cohomology classes associated to the special algebraic cycles corresponding to orthogonal subgroups.111We should note that is of pure -type when but this is no longer the case in general when .

It is very important to extend Theorem 1.9 to the noncompact case since, for the case (and the correct isotropic rational quadratic forms of signature depending on a parameter , the genus), the resulting noncompact locally symmetric spaces are now the moduli spaces of quasi-polarized surfaces of genus . In Theorem 1.14 of this paper we state a theorem for general orthogonal groups, that as a special case extends Theorem 1.9 to the noncompact case by proving that the cuspidal projections, see subsection 1.11, onto the cuspidal Hodge summand of type of the Poincaré-Lefschetz duals of special cycles span the cuspidal cohomology of Hodge type for .

In [6] (with out new collaborator Zhiyuan Li), we extend this last result from the cuspidal cohomology to the reduced -cohomology, see [6], Theorem 0.3.1. and hence, by a result of Zucker, to the entire cohomology groups . In [6], we apply this extended result to the special case and prove that the Noether-Lefschetz divisors (the special cycles with ) generate the Picard variety of the moduli spaces thereby giving an affirmative solution of the Noether-Lefschetz conjecture formulated by Maulik and Pandharipande in [56]. We note that using work of Weissauer [84] the result that special cycles span the second homology of the noncompact Shimura varieties associated to for the special case , was proved earlier by Hoffman and He [32]. In this case the Shimura varieties are Siegel modular threefolds and the special algebraic cycles are Humbert surfaces and the main theorem of [32] states that the Humbert surfaces rationally generate the Picard groups of Siegel modular threefolds.

Finally, we also point out that in [7] we prove the Hodge conjecture — as well as its generalization in the version first formulated (incorrectly) by Hodge — away from the middle dimensions for Shimura varieties uniformized by complex balls. The main ideas of the proof are the same, although the extension to unitary groups is quite substantial; moreover, the extension is a more subtle. In the complex case sub-Shimura varieties do not provide enough cycles, see [7] for more details.

1.10. A general theorem

As we explain in Sections 11 and 12, Theorems 1.5, 1.7 and 1.9 follow from Theorem 10.10 (see also Theorem 1.14 of this Introduction) which is the main result of our paper. It is concerned with general (i.e. not necessarily compact) arithmetic congruence manifolds associated to as above but such that with . It is related to the Hodge conjecture through a refined Hodge decomposition of the cuspidal cohomology that we now describe. In the noncompact quotient case it is necesary to first project the Poincaré-Lefschetz dual form of the special cycle onto the cusp forms. We now construct this projection - for another construction not using Franke’s Theorem see Section 9.

1.11. The cuspidal projection of the class of a special cycle with coefficients

In case is not compact the special cycles (with coefficients) are also not necessarily compact. However, they are properly embedded and hence we may consider them as Borel-Moore cycles or as cycles relative to the Borel-Serre boundary of . The Borel-Serre boundaries of the special cycles with coefficients have been computed in [25]. The smooth differential forms on that are Poincaré-Lefschetz dual to the special cycles are not necessarily but by [21] they are cohomologous to forms with automorphic form coefficients (uniquely up to coboundaries of such forms) which can then be projected onto cusp forms since any automorphic form may be decomposed into an Eisenstein component and a cuspidal component. We will abuse notation and call the class of the resulting cusp form the cuspidal projection of the class of the special cycle. Since this cuspidal projection is it has a harmonic projection which can then be used to defined the refined Hodge type(s) of the cuspidal projection and hence of the original class according to the next subsection.

1.12. The refined Hodge decomposition

As first suggested by Chern [14] the decomposition of exterior powers of the cotangent bundle of under the action of the holonomy group, i.e. the maximal compact subgroup of , yields a natural notion of refined Hodge decomposition of the cohomology groups of the associated locally symmetric spaces. Recall that

and let be the corresponding (complexified) Cartan decomposition. As a representation of the space is isomorphic to where (resp. ) is the standard representation of (resp. ). The refined Hodge types therefore correspond to irreducible summands in the decomposition of as a -module. In the case of the group (then is the complex hyperbolic space) it is an exercise to check that one recovers the usual Hodge-Lefschetz decomposition. But in general the decomposition is much finer and in our orthogonal case it is hard to write down the full decomposition of into irreducible modules. Note that, as a -module, the decomposition is already quite complicated. We have (see [22, Equation (19), p. 121]):

(1.12.1)

Here we sum over all partition of (equivalently Young diagram of size ) and is the conjugate partition (or transposed Young diagram).

It nevertheless follows from work of Vogan-Zuckerman [79] that very few of the irreducible submodules of can occur as refined Hodge types of non-trivial coholomogy classes. The ones which can occur (and do occur non-trivially for some ) are understood in terms of cohomological representations of . We review cohomological representations of in section 5. We recall in particular how to associate to each cohomological representation of a strongly primitive refined Hodge type. This refined Hodge type correspond to an irreducible representation of which is uniquely determined by some special kind of partition as in (1.12.1), see [8] where these special partitions are called orthogonal. The first degree where these refined Hodge types can occur is . We will use the notation for the space of the cohomology in degree corresponding to this special Hodge type.

Note that since the group acts on . In this paper we will be mainly concerned with elements of — that is elements that are trivial on the -side. Note that in general is strictly contained in . Recall that if is even there exists an invariant element

the Euler class/form (see subsection 5.12.1 for the definition). We define if is odd. We finally note that if is the partition ( times) then is the trivial representation of and the special Hodge type associated to occurs in ; in that case we use the notation .

1.13. The refined Hodge decomposition of special cycles with coefficients

We also consider general local systems of coefficients. Let be a dominant weight for with at most nonzero entries and let be the associated finite dimensional representation of .

The reader will verify that the subalgebra of is invariant under . Hence, we may form the associated subbundle

of the bundle

of exterior powers of the cotangent bundle of twisted by . The space of sections of is invariant under the Laplacian and hence under harmonic projection, compare [14, bottom of p. 105]. In case is trivial the space of sections of is a subalgebra of the algebra of differential forms.

We denote by the corresponding subspace (subalgebra if is trivial) of . Note that when we have and when we have

As above we may associate to -dimensional totally positive sub-quadratic spaces of special cycles of codimension in with coefficients in the finite dimensional representation . They yield classes in . In fact we shall show that these classes belong to the subspace and it follows from Proposition 5.15 that

(1.13.1)

(Compare with the usual Hodge-Lefschetz decomposition.) We call the primitive part of . We see then that if is odd the above special classes have pure refined Hodge type and if is even each such class is the sum of at most refined Hodge types. In what follows we will consider the primitive part of the special cycles i.e. their projections into the subspace associated to the refined Hodge type :

The notion of refined Hodge type is a local Riemannian geometric one. However since we are dealing with locally symmetric spaces there is an equivalent global definition in terms of automorphic representations. Let be the cohomological Vogan-Zuckerman -module where is a -stable parabolic subalgebra of whose associated Levi subgroup is isomorphic to . Let denote the space of cuspidal harmonic -forms such that the corresponding automorphic representations of the adelic orthogonal group have distinguished (corresponding to the noncompact factor) infinite component equal to the unitary representation corresponding to . Then we have

Now Theorem 10.10 reads as:

1.14 Theorem.

Suppose and . Then the space is spanned by the cuspidal projections of classes of special cycles.

Remark. See Subsection 1.11 for the definition of the cuspidal projection of the class of a special cycle. In case we make the slightly stronger assumption it is proved in [25], see Remark 1.2, that the form of Funke-Millson is square integrable. We can then immediately project it into the space of cusp forms and arrive at the cuspidal projection without first passing to an automorphic representative.

In degree one may deduce from the Vogan-Zuckerman classification of cohomological representations that is generated by cup-products of invariant forms with primitive subspaces or . Exchanging the role of and we may therefore apply Theorem 1.14 to prove the following:

1.15 Corollary.

Let be an integer . Then the full cohomology group is generated by cup-products of classes of totally geodesic cycles and invariant forms.

Beside proving Theorem 1.14 we also provide strong evidence for the following:

1.16 Conjecture.

If or the space is not spanned by projections of classes of special cycles.

In the special case , we give an example of a cuspidal class of degree one in the cohomology of Bianchi hyperbolic manifolds which does not belong to the subspace spanned by classes of special cycles, see Proposition 15.14.

1.17. Organisation of the paper

The proof of Theorem 10.10 is the combination of three main steps.

The first step is the work of Kudla-Millson [47] — as extended by Funke and Millson [24]. It relates the subspace of the cohomology of locally symmetric spaces associated to orthogonal groups generated by special cycles to certain cohomology classes associated to the “special theta lift” using vector-valued adelic Schwartz functions with a fixed vector-valued component at infinity. More precisely, the special theta lift restricts the general theta lift to Schwartz functions that have at the distinguished infinite place where the orthogonal group is noncompact the fixed Schwartz function taking values in the vector space , see §7.12, at infinity. The Schwartz functions at the other infinite places are Gaussians (scalar-valued) and at the finite places are scalar-valued and otherwise arbitrary. The main point is that is a relative Lie algebra cocycle for the orthogonal group allowing one to interpret the special theta lift cohomologically.

The second step, accomplished in Theorem 10.5 and depending essentially on Theorem 7.31, is to show that the intersections of the images of the general theta lift and the special theta lift just described with the subspace of the cuspidal automorphic forms that have infinite component the Vogan-Zuckerman representation coincide (of course the first intersection is potentially larger). In other words the special theta lift accounts for all the cohomology of type that may be obtained from theta lifting. This is the analogue of the main result of the paper of Hoffman and He [32] for the special case of and our arguments are very similar to theirs. Combining the first two steps, we show that, in low degree (small ), all cuspidal cohomology classes of degree and type that can by obtained from the general theta lift coincide with the span of the special chomology classes dual to the special cycles of Kudla-Millson and Funke-Millson.

The third step (and it is here that we use Arthur’s classification [5]) is to show that in low degree (small ) any cohomology class in can be obtained as a projection of the class of a theta series. In other words, we prove the low-degree cohomological surjectivity of the general theta lift (for cuspidal classes of the refined Hodge type ). In particular in the course of the proof we obtain the following (see Theorem 8.11):

1.18 Theorem.

Assume that is anisotropic and let be an integer such that and . Then the global theta correspondence induces an isomorphism between the space of cuspidal holomorphic Siegel modular forms, of weight at and weight at all the others infinite places, on the connected Shimura variety associated to the symplectic group and the space

Combining the two steps we find that in low degree the space

is spanned by images of duals of special cycles. From this we deduce (again for small ) that is spanned by totally geodesic cycles.

The injectivity part of the previous theorem is not new. It follows from Rallis inner product formula [71]. In our case it is due to Li, see [53, Theorem 1.1]. The surjectivity is the subject of [65, 28] that we summarize in section 2. In brief a cohomology class (or more generally any automorphic form) is in the image of the theta lift if its partial -function has a pole far on the right. This condition may be thought of as asking that the automorphic form — or rather its lift to — is very non-tempered in all but a finite number of places. To apply this result we have to relate this global condition to the local condition that our automorphic form is of a certain cohomological type at infinity.

1.19.

This is where the deep theory of Arthur comes into play. We summarize Arthur’s theory in Section 3. Very briefly: Arthur classifies automorphic representations of classical groups into global packets. Two automorphic representations belong to the same packet if their partial -functions are the same i.e. if the local components of the two automorphic representations are isomorphic almost everywhere. Moreover in loose terms: Arthur shows that if an automorphic form is very non tempered at one place then it is very non tempered everywhere. To conclude we therefore have to study the cohomological representations at infinity and show that those we are interested in are very non-tempered, this is the main issue of section 6. Arthur’s work on the endoscopic classification of representations of classical groups relates the automorphic representations of the orthogonal groups to the automorphic representations of twisted by some outer automorphism . Note however that the relation is made through the stable trace formula for the orthogonal groups (twisted by an outer automorphism in the even case) and the stable trace formula for the twisted (non connected) group .

Thus, as pointed above, our work uses the hypothesis made in Arthur’s book. The twisted trace formula has now been stabilized (see [80] and [62]). As opposed to the case of unitary groups considered in [7], there is one more hypothesis to check. Indeed: in Arthur’s book there is also an hypothesis about the twisted transfer at the Archimedean places which, in the case of orthogonal groups, is only partially proved by Mezo. This is used by Arthur to find his precise multiplicity formula. We do not use this precise multiplicity formula but we use the fact that a discrete twisted automorphic representation of a twisted is the transfer from a stable discrete representation of a unique endoscopic group. So we still have to know that: at a real place, the transfer of the stable distribution which is the sum of the discrete series in one Langlands packet is the twisted trace of an elliptic representation of normalized using a Whittaker functional as in Arthur’s book.

Mezo [57] has proved this result up to a constant which could depend on the Langlands’ packet. Arthur’s [5, §6.2.2] suggests a local-global method to show that this constant is equal to . This is worked out by the third author in [63] which will eventually be part of a joint work with N. Arancibia and D. Renard.

1.20.

Part 4 is devoted to applications. Apart from those already mentioned, we deduce from our results and recent results of Cossutta [16] and Cossutta-Marshall [17] an estimate on the growth of the small degree Betti numbers in congruence covers of hyperbolic manifolds of simple type. We also deduce from our results an application to the non-vanishing of certain periods of automorphic forms.

We finally note that the symmetric space embeds as a totally geodesic and totally real submanifold in the Hermitian symmetric space associated to the unitary group . Also there exists a representation of whose restriction of contains the irreducible representation . Hence there is a homomorphism from to . As explained in §7.14 the form is best understood as the restriction of a holomorphic form on . Now any as in §1.10 embeds as a totally geodesic and totally real submanifold in a connected Shimura variety modelled on . And the proof of Theorem 1.14 implies:

1.21 Theorem.

Suppose and . Then the space is spanned by the restriction of holomorphic forms in .

As holomorphic forms are easier to deal with, we hope that this theorem may help to shed light on the cohomology of the non-Hermitian manifolds .

1.22. More comments

General arithmetic manifolds associated to are of two types: The simple type and the non-simple type. In this paper we only deal with the former, i.e. arithmetic manifolds associated to a quadratic space of signature at one infinite place and definite at all other infinite places. Indeed: the manifolds constructed that way contain totally geodesic submanifolds associated to subquadratic spaces. But when is even there are other constructions of arithmetic lattices in commensurable with the group of units of an appropriate skew-hermitian form over a quaternion field (see e.g. [54, Section 2]. Note that when there are further constructions that we won’t discuss here.

Arithmetic manifolds of non-simple type do not contain as many totally geodesic cycles as those of simple type and Theorem 1.14 cannot hold. For example the real hyperbolic manifolds constructed in this way in [54] do not contain codimension  totally geodesic submanifolds. We should nevertheless point out that there is a general method to produce nonzero cohomology classes for these manifolds: As first noticed by Raghunathan and Venkataramana, these manifolds can be embedded as totally geodesic and totally real submanifolds in unitary arithmetic manifolds of simple type, see [70]. On the latter a general construction due to Kazhdan and extended by Borel-Wallach [12, Chapter VIII] produces nonzero holomorphic cohomology classes as theta series. Their restrictions to the totally real submanifolds we started with can produce nonzero cohomology classes and Theorem 1.21 should still hold in that case. This would indeed follow from our proof modulo the natural extension of [28, Theorem 1.1 (1)] for unitary groups of skew-hermitian form over a quaternion field.

We would like to thank Jeffrey Adams for helpful conversations about this paper. The second author would like to thank Stephen Kudla and Jens Funke for their collaborations which formed a critical input to this paper.

Part I Automorphic forms

2. Theta liftings for orthogonal groups: some background

2.1. Notations

Let be a number field and the ring of adeles of . Let be a nondegenerate quadratic space over with .

2.2. The theta correspondence

Let be a symplectic -space with . We consider the tensor product . It is naturally a symplectic -space and we let be the corresponding symplectic -group. Then forms a reductive dual pair in , in the sense of Howe [33]. We denote by the metaplectic double cover of if is odd or simply is is even.

For a non-trivial additive character of , we may define the oscillator representation . It is an automorphic representation of the metaplectic double cover of , which is realized in the Schrödinger model. The maximal compact subgroup of is , the unitary group in variables. We denote by its preimage in . The associated space of smooth vectors of is the Bruhat-Schwartz space . The -module associated to is made explicit by the realization of known as the Fock model that we will brefly review in §7.3. Using it, one sees that the -finite vectors in is the subspace obtained by replacing, at each infinite place, the Schwartz space by the polynomial Fock space , i.e. the image of holomorphic polynomials on under the intertwining map from the Fock model of the oscillator representation to the Schrödinger model.

2.3.

We denote by , and the adelic points of respectively , and . The global metaplectic group acts in via and preserves the dense subspace . For each we form the theta function

(2.3.1)

on . There is a natural homomorphism

which is described with great details in [37]. We pull the oscillator representation back to . Then is a smooth, slowly increasing function on ; see [83, 33].

2.4. The global theta lifting

We denote by the set of irreducible cuspidal automorphic representations of , which occur as irreducible subspaces in the space of cuspidal automorphic functions in . For a , the integral

(2.4.1)

with (the space of ), defines an automorphic function on  : the integral (2.4.1) is well defined, and determines a slowly increasing function on . We denote by the space of the automorphic representation generated by all as and vary, and call the -theta lifting of to . Note that, since is dense in we may as well let vary in the subspace .

We can similarly define and the -theta correspondence from to .

2.5.

It follows from [65] and from [37, Theorem 1.3] that if contains non-zero cuspidal automorphic functions on then the representation of in is irreducible (and cuspidal). We also denote by the corresponding element of In that case it moreover follows from [65] and [37, Theorem 1.1] that

We say that a representation is in the image of the cuspidal -theta correspondence from a smaller group if there exists a symplectic space with and a representation such that

2.6.

The main technical point of this paper is to prove that if is such that its local component at infinity is “sufficiently non-tempered” (this has to be made precise) then the global representation is in the image of the cuspidal -theta correspondence from a smaller group.

As usual we encode local components of into an -function. In fact we only consider its partial -function where is a sufficiently big finite set of places such that is unramified for each . For such a we define the local factor by considering the Langlands parameter of .

Remark. We will loosely identifies the partial -function of and that of its restriction to . However we should note that the restriction of to the special orthogonal group may be reducible: If we may associate to the Langlands parameter of representations from the principal series of the special orthogonal group. Each of these has a unique unramified subquotient and the restriction of to the special orthogonal group is then the sum of the non-isomorphic subquotients.222There are at most two such non-isomorphic subquotients. Anyway: the local -factor is the same for each summand of the restriction as it only depends on the Langlands parameter.

We may generalize these definitions to form the partial -functions for any automorphic character .

Now the following proposition is a first important step toward the proof that a “sufficiently non-tempered” automorphic representation is in the image of the cuspidal -theta correspondence from a smaller group. It is symmetric to [48, Theorem 7.2.5] and is the subject of [65] and [28, Theorem 1.1 (1)]; it is revisited and generalized in [26].

2.7 Proposition.

Let and let be a quadratic character of . Let be a nonnegative integer with . We assume that the partial -function is holomorphic in the half-plane and has a pole in . Let and be a symplectic -space with .

Then there exists an automorphic sign character of such that the -theta lifting of to does not vanish.

The big second step to achieve our first goal will rely on Arthur’s theory.

3. Arthur’s theory

3.1. Notations

Let be a number field, its ring of adeles and . Let be a nondegenerate quadratic space over with . We let be the special orthogonal group over . We set and .

The group is an inner form of a quasi-split form . As for now Arthur’s work only deals with quasi-split groups. We first describe the group according to the parity of and briefly recall the results of Arthur we shall need. We recall from the introduction that Arthur’s work rely on extensions to the twisted case of two results which have only been proved so far in the case of connected groups: The first is the stabilization of the twisted trace formula for the two groups and , see [5, Hypothesis 3.2.1]. The second is Shelstad’s strong spectral transfer of tempered archimedean characters. Taking these for granted we will explain how to deal with non-quasi-split groups in the next section.

3.2.

We first assume that is odd. Then the special orthogonal group is an inner form of the split form over associated to the symmetric bilinear form whose matrix is

The (complex) dual group of is and .

3.3.

We now assume that is even. We let be the split orthogonal group over associated to the symmetric bilinear form whose matrix is . The quasi-split forms of are parametrized by morphisms , which by class field theory correspond to characters on such that quadratic Artin characters. We denote by the outer twist of the split group determined by : the twisting is induced by the action of on the Dynkin diagram via the character .

When is even, there exists a quadratic Artin character such that is an inner form of the quasi-split group . The (complex) dual group of is then and , where acts on by an order automorphism — trivial on the kernel of — and fixes a splitting, see [9, p. 79] for an explicit description.

Remark. Let be an infinite real place of such that with even so that . Then is trivial if and only if is even. We are lead to the following dichotomy for real orthogonal groups: if is odd, is an inner form of and if is even, is an inner form of (split over ).

3.4. Global Arthur parameters

In order to extends the classification [61] of the discrete automorphic spectrum of to the classical groups, Arthur represents the discrete automorphic spectrum of by a set of formal tensor products

where is an irreducible, unitary, cuspidal automorphic representation of and is an irreducible representation of of dimension , for positive integers and such that . For any such , we form the induced representation

(normalized induction from the standard parabolic subgroup of type ). We then write for the unique irreducible quotient of this representation.

We may more generally associate an automorphic representation of to a formal sum of formal tensor products:

(3.4.1)

where each is an irreducible, unitary, cuspidal automorphic representation of , is an irreducible representation of of dimension and .

Now consider the outer automorphism:

There is an action on the set of representations . If is as in (3.4.1), set

Then .

Arthur’s main result [5, Theorem 1.5.2] (see also [4, Theorem 30.2]) then parametrizes the discrete automorphic spectrum of by formal sum of formal tensor products as in (3.4.1) such that:

  1. the in (3.4.1) are all distinct, and

  2. for each , .

3.5. Local Arthur parameters

Assume that is local and let be its Weil-Deligne group. We can similarly define packets over . We define a local packet over as a formal sum of formal tensor product (3.4.1) where each is now a tempered irreducible representation of that is square integrable modulo the center.333Because we do not know that the extension to of Ramanujan’s conjecture is valid, we do not know that the local components of automorphic representations of are indeed tempered. So that in principle the are not necessarily tempered: their central characters need not be unitary. This requires a minor generalization that Arthur addresses in [4, Remark 3 p. 247]. Anyway the approximation to Ramanujan’s conjecture proved by Luo, Rudnick and Sarnak [55] is enough for our purposes and it makes notations easier to assume that each is indeed tempered. The other components remain irreducible representations of . To each we associate the unique irreducible quotient of

(normalized induction from the standard parabolic subgroup of type ). We then define as the induced representation

(normalized induction from the standard parabolic subgroup of type ). It is irreducible and unitary. Finally, the local parameter consists of those representations such that . Then is theta-stable: .

3.6. Local Arthur packets

Assume that is local and that is quasi-split. We now recall how Arthur associates a finite packet of representations of to a local parameter .

We say that two functions in are stably equivalent if they have the same stable orbital integrals, see e.g. [51]. Thanks to the recent proofs by Ngo [69] of the fundamental lemma and Waldspurger’s work [81], we have a natural notion of transfer from a test fonction to a representative of a stable equivalence class of functions in such that and are associated i.e. they have matching stable orbital integrals 444Here the orbital integrals on the -side are twisted orbital integrals., see [42] for more details about twisted transfer. Over Archimedean places existence of transfer is due to Shelstad, see [76]; we note that in that case being -finite is preserved by transfer. When one must moreover ask that is invariant under an outer automorphism of ; we may assume that .

Let be a local parameter as above and be the space of . We fix an intertwining operator () intertwining and .

When we identify the irreducible representations of that are conjugated by . Then is well defined when is as explained above.

The following proposition follows from [5, Theorem 2.2.1] (see also [4, Theorem 30.1]).

3.7 Proposition.

There exists a finite family of representations of , and some multiplicities () such that, for associated and  :

(3.7.1)

where each is a sign .

We remark that (3.7.1) uniquely determines as a set of representations-with-multiplicities; it also uniquely determines the signs . In fact Arthur explicitely computes these signs for some particular choice of an intertwiner .

By the local Langlands correspondence, a local parameter for can be represented as a homomorphism

(3.7.2)

Arthur associates to such a parameter the -parameter given by

One key property of the local Arthur’s packet is that it contains all representations of Langlands’ -packet associated to . This is proved by Arthur, see also [67, Section 6].

Ignoring the minor generalization needed to cover the lack of Ramanujan’s conjecture, the global part of Arthur’s theory (see [5, Corollary 3.4.3] when is quasi-split and [5, Proposition 9.5.2] in general) now implies:

3.8 Proposition.

Let be an irreducible automorphic representation of which occurs (discretely) as an irreducible subspace of . Then there exists a global Arthur parameter and a finite set of places of containing all Archimedean ones such that for all , the group is quasi-split, the representation is unramified and the -parameter of is .

Remark. Proposition 3.8 in particular implies that the part of a local Arthur parameter has a global meaning. This puts serious limitations on the kind of non-tempered representations which can occur discretely: e.g. an automorphic representation of which occurs discretely in and which is non-tempered at one place is non-tempered at all places.

The above remark explains how Arthur’s theory will be used in our proof. This will be made effective through the use of -functions.

3.9. Application to -functions

Let be an irreducible automorphic representation of which occurs (discretely) as an irreducible subspace of and let

be its global Arthur parameter (Proposition 3.8). We factor each where runs over all places of . Let be a finite set of places of containing the set of Proposition 3.8, and all for which either one or is ramified. We can then define the formal Euler product

Note that is the partial -function of a very special automorphic representation of with