Motivic Cohomology of Quaternionic Shimura varieties and level raising
Abstract.
We study the motivic cohomology of the special fiber of quaternionic Shimura varieties at a prime of good reduction. We exhibit classes in these motivic cohomology groups and use this to give an explicit geometric realization of level raising between Hilbert modular forms. The main ingredient for our construction is a form of Ihara’s Lemma for compact quaternionic Shimura surfaces which we prove by generalizing a method of Diamond–Taylor. Along the way we also verify the Hecke orbit conjecture for these quaternionic Shimura varieties which is a key input for our proof of Ihara’s Lemma.
Key words and phrases:
Shimura varieties, motivic cohomology, level raising, Ihara’s Lemma2010 Mathematics Subject Classification:
11F33, 11F41, 14F42,1. Introduction
1.1. Main Theorem
The aim of this paper is to study the motivic cohomology of the special fiber of certain quaternionic Shimura varieties. For a scheme of finite type over a field, its motivic cohomology groups are a generalization of the usual Chow groups, and the main new observation of this paper is that for certain Shimura varieties, these groups can encode very rich arithmetic information. More precisely, we will show that the cycle class map from motivic cohomology to étale cohomology gives a geometric realization of level raising between Hilbert modular forms.
We now state our main result. Let be a totally real field of even degree and a prime which is inert in . Let be a totally indefinite quaternion algebra over which is unramified at the unique prime above and the associated reductive group over . Let be a sufficiently small compact open subgroup of such that where is the standard hyperspecial maximal compact and . Then there is a Shimura variety defined over ; it extends to a smooth integral model over . We let denote its special fiber over and its base change to .
Fix an irreducible cuspidal automorphic representation of of parallel weight 2 defined over a number field . Let be a finite set of places of not containing and away from which is unramified and is hyperspecial. We also choose a prime of whose residue characteristic is coprime to and write . We write for the motivic cohomology group with coefficients defined in [SuVo]. By [Voe2], we may identify this with the higher Chow group defined in [Bloch]. When , this group is just the usual Chow group of codimension cycles modulo rational equivalence (with coefficients in ). The group is equipped with the following cycle class map to the absolute étale cohomology:
(1.1.1) 
We let denote the abstract Hecke algebra of away from ; it is the algebra generated by elements where runs over primes of away from . Then the Hecke eigenvalues of induce a map
We write a maximal ideal of and the preimage in .
The Hecke algebra acts on the étale cohomology and higher Chow groups of . Upon making a large image assumption on the Galois representation associated to (see Assumption LABEL:ass:_property_of_l) and localizing at the maximal ideal , there is an isomorphism
The cycle class map then induces the Abel–Jacobi map:
(1.1.2) 
In §LABEL:sec:_Motivic_Cohomology_and_Levelraising, we will define a subgroup of using the geometry of Goren–Oort cycles^{1}^{1}1In fact the cycles we consider arise from the supersingular locus. on as studied in [TX], [TX1] and [LT]. As the notation suggests, this subgroup is related to level raising. The main Theorem of the paper is the following; we refer to §LABEL:sec:_Motivic_Cohomology_and_Levelraising for the precise statement.
Theorem 1.1.1.
Suppose that is a level raising prime in the sense of Definition LABEL:def:_level_raising_prime and that Assumptions LABEL:ass:_property_of_l and LABEL:ass:_Dim_Jacquet_Langlands are satisfied; in particular and . Then the map
induced by (1.1.2) is surjective.
We note that as in [LT, Remark 4.2, 4.6], if there exist rational primes inert in , and is not dihedral and not isomorphic to a twist by a character of any of its internal conjugates, then for all but finitely many , the set of primes which are level raising primes has positive density.
In general it is difficult problem to produce nonzero classes in motivic cohomology. The key input to proving the surjectivity in Theorem 1.1.1 is a form of Ihara’s Lemma which we prove by generalizing a method of Diamond–Taylor [DT]; see the next subsection for more details.
We now give an example of the construction of which makes clear the relationship with level raising. We assume so that .
We write for the totally definite quaternion algebra which agrees with at all finite places. We fix an isomorphism
which allows us to consider as a compact open subgroup of . We let and denote the discrete Shimura sets
where the compact open subgroup agrees with away from and is the standard Iwahori subgroup of at . We let
denote the natural degeneracy maps so that the diagram
is the usual Hecke correspondence for . For any finite set , we write for the abelian group of valued functions on .
We may think of as a moduli space of abelian varieties with multiplication by some maximal order in . We let be the locus where the underlying abelian variety is supersingular. Using the geometry of one can show that under the assumptions of Theorem 1.1.1, admits a map from
The construction uses an interpretation of classes in as cycles together with a rational function on the cycle; see §LABEL:sec:_Motivic_Cohomology_and_Levelraising for the details. Then is defined to be the image of . Theorem 1.1.1 in this case follows from the following stronger result:
Theorem 1.1.2.
Let . Suppose that is a level raising prime and that Assumption LABEL:ass:_property_of_l is satisfied. Then the map
(1.1.3) 
is surjective.
The relationship with levelraising should now be clear. Indeed under the Jacquet–Langlands correspondence, acts on left hand side of (1.1.3) via the quotient in the sense of [Ribet], whereas it is well known that it acts via the quotient on the right hand side. In this sense, the Abel–Jacobi map gives an explicit realization of the congruence between old and new forms.
It is known by the work of many authors that the motivic cohomology groups satisfy many of the formal properties of a cohomology theory. However there is much that is still not understood, we refer to [Geisser] for a brief survey. We may use Theorem 1.1.2 to show that in certain cases of Shimura surfaces, motivic cohomology coincides with étale cohomology, upon localizing at .
Theorem 1.1.3.
Let . Suppose that is a level raising prime and that Assumption LABEL:ass:_property_of_l is satisfied. Then the cycle class map induces an isomorphism
When , Voevodsky [Voe1] has shown that is isomorphic to , for proper smooth over any base field. For , not much seems to be known.
Remark 1.1.4.
When is odd, there is an Abel–Jacobi map
(1.1.4) 
In this case the supersingular locus is equidimensional of dimension and we may consider the subgroup generated by the irreducible components in . Then [LT, Theorem 1.3] have shown the surjectivity of (1.1.4) modulo restricted to this subgroup. Thus our Theorem 1.1.1 can be thought of as the even dimensional analogue of the Theorem of Liu–Tian. The main new observation of this work is that we are able to produce certain classes in motivic cohomology, or higher Chow groups, as opposed to ordinary Chow groups. Its conceptual importance lies in the fact that we are able to obtain a geometric interpretation of even dimensional Galois cohomology.
Remark 1.1.5.
In [LT], the geometric realization of level raising was a key ingredient in their proof of certain cases of the Bloch–Kato conjecture, see [LT, Theorem 5.7]. Our work should have applications to cases of this conjecture for noncentral values; we aim to carry this out in a future work.
1.2. Proof of main result and Ihara’s Lemma
We now explain the proof of Theorem 1.1.1. Our approach follows that of [LT], but there are many new difficulties in the even dimensional case.
Firstly, using the construction of the group and the intersection pairing between certain Goren–Oort strata proved in [TX1], we reduce to proving the surjectivity statement in the case of quaternionic Shimura surfaces, see Proposition LABEL:prop:_AJ_surjective_for_surface. The statement in this case follows from the following form of Ihara’s Lemma. These type of results first appeared in Ribet’s ICM article [Ribet1] for the case of modular curves and over the last thirty years they have seen many important arithmetic applications. Therefore our result in the case of surfaces should certainly be of independent interest.
For simplicity, we only state the result in the totally indefinite case; we refer to Theorem LABEL:thm:_Ihara's_Lemma for the more general statement. Thus we assume as in the example of the previous subsection.
Theorem 1.2.1 (Ihara’s Lemma).
Under the Assumption LABEL:ass:_surface, the map
is injective.
Here denotes the quaternionic Shimura surfaces with Iwahori level structure at and are the natural degeneracy maps. In fact the appropriate Abel–Jacobi map in this case can be related to the map in the statement of Theorem 1.2.1; they are essentially dual to one another. To show the existence of this duality requires a careful analysis of the global geometry of the mod fiber of the quaternionic Shimura surface with Iwahori level structure at . We note that in this case the Shimura surface has bad reduction at . The main result which is Corollary LABEL:cor:Iwahori_level_structure is proved in an appendix and is analogous to the results of [Stamm] in the case of Hilbert modular surfaces.
We now describe our approach to Theorem 1.2.1. The result in the case of Hilbert modular varieties has been proved by Dimitrov [Dim]. However his proof relied crucially on the existence of a expansion. Note that when , even if one is interested in Theorem 1.1.1 for Hilbert modular varieties, the reduction to the case of surfaces will necessitate that we consider compact Shimura surfaces where a expansion is not available. We therefore take another approach by generalizing a method of Diamond–Taylor who proved the result for Shimura curves [DT].
We first apply a crystalline comparison isomorphism to reduce the problem to proving injectivity of a certain map between global sections of line bundles over the reduction of the Shimura surface (Proposition LABEL:prop:_Ihara_coherent). The property that a nonzero section lies in the kernel implies that the divisor corresponding to this section is invariant under power Hecke operators. In the case of Shimura curves, it’s known that the image of an ordinary point under power Hecke operators is infinite; this constrains to be supported on the supersingular locus. Since power Hecke operators act transitively on supersingular points, the support contains the supersingular locus and this is enough to deduce a contradiction for degree reasons.
In the case of surfaces, we need a stronger result to constrain the support of the divisor . In section §LABEL:sec:_Hecke_orbit_conjecture, we prove a version of the Hecke orbit conjecture of Chai–Oort [ChOort] for the ordinary locus on quaternionic Shimura varieties. We assume is a prime where the compact open is hyperspecial and we write for the reduction of the integral model at a prime of the reflex field above . We write for the locus where the universal abelian variety is ordinary.
Theorem 1.2.2 (Hecke orbit Conjecture).
Let . Then the primeto Hecke orbit is Zariski dense in .
In fact we prove this result in a more general situation; we refer to §LABEL:sec:_3_statement for the statement. Using the strong approximation theorem, we deduce that the power Hecke orbit of is Zariski dense in the connected component of containing it. This allows one to show that is supported on the complement of . A computation involving intersection numbers of with certain cycles on then gives the desired contradiction.
Remark 1.2.3.
In [LT], the authors reduce their surjectivity result to a form of Ihara’s Lemma for Shimura curves, for which the method of [DT] is directly applicable. In our case, the most pertinent case is that of Shimura surfaces which, as explained above, is more delicate.
We note that many of the quaternionic Shimura varieties we consider do not admit good moduli interpretations. Thus in order to obtain the geometric results we need, we study the geometry of certain auxiliary unitary Shimura varieties which are of PELtype. Using [TX, §2], the results for unitary Shimura varieties transfer easily to the quaternionic side. The moduli interpretation for the unitary Shimura varieties allow us to adapt many proofs in the case of Hilbert modular varieties to the quaternionic case.
1.3. Outline of paper
In §2 we begin with some basics on Shimura varieties and define the quaternionic Shimura varieties of interest. We recall the construction of the auxiliary unitary Shimura varieties of PELtype as in [TX, §3], and recall the description of Goren–Oort cycles obtained in [TX1]. In §3 we prove the Hecke orbit conjecture for quaternionic Shimura varieties. We deduce our results from the corresponding statement for the auxiliary unitary Shimura varieties. Using the moduli interpretation, the proof in the unitary case follows the strategy of [Chai] who proved the result for Hilbert modular varieties. A key input here is Moonen’s generalization of Serre–Tate theory for ordinary abelian varieties [Mo]. In §4 we study the intersection pairing of cycles on the reduction of Shimura surfaces and use this to prove Theorem 1.2.1. Finally in §5, we recall the definition of motivic cohomology groups and higher Chow groups, paying extra attention in the most pertinent case of surfaces, and we construct the level raising subgroup . We then prove Theorem 1.1.1 using the strategy outlined above. In the Appendix we describe the bad reduction of quaternionic Shimura surfaces with Iwahori level structure.
Acknowledgments: The author would like to thank Tony Feng, Bao Le Hung, Chao Li, Ananth Shankar, Richard Taylor, Liang Xiao and Xinwen Zhu for useful discussions and comments about this work. Above all the author would like to thank Akshay Venkatesh for his suggestion that there could be interesting arithmetic information contained in the motivic cohomology of Shimura varieties and for many hours of enlightening and enjoyable discussions. The author was partially supported by NSF grant No. DMS1638352 through membership at the Institute for Advanced Study.
1.4. Notations

If is a number field we write for its ring of integers. If is a place of , we write for the completion of at and if is finite, we write for its residue field at .

If is a local field we write for its ring of integers.

For any field , we write for a fixed algebraic closure of .

We write for the ring of adeles and the ring of finite adeles. If is a prime, denotes the finite adeles with trivial component.

If is a map of algebras and is an scheme, we write for the base change of to .

If is a scheme, we write for its structure sheaf. We write for the étale cohomology of . For any closed subscheme , we write for the étale cohomology supported on .
2. Geometry of quaternionic Shimura varieties and Goren–Oort strata
In this section we recall the results concerning the geometry of quaternionic Shimura varieties and Goren–Oort cycles following [TX] and [TX1] that we will need.
2.1. Basics on quaternionic Shimura varieties
Let be a totally real field with such that is unramified in . We are mainly interested in the case when is inert in ; this is the case considered in [TX] and [LT]. However, we will sometimes need to consider the reduction mod of these Shimura varieties, so we will keep the more general assumption for now. We write (resp. ) for the set of adic places (resp. infinite places). We fix once and for all an isomorphism , which we will use to identify with the set of adic embeddings of . For we let and the set of adic embeddings which induce . As is unramified in , the Frobenius induces an action on .
We fix a totally indefinite quaternion algebra over which is split at all the places above . Let be a subset of even cardinality. We set and for each we set . We will make the assumption that only if .
We write for the quaternion algebra over whose ramification set is precisely the union of and the places in over which ramifies. For each a place of away from the ramification set for , we fix an isomorphism We define to be the reductive group over such that for any algebra we have
When , we simply write for the above group. For we have an isomorphism . Hence we may fix an isomorphism
Let and . We consider the following homomorphism:
Then and the conjugacy class of forms a (weak) Shimura datum in the sense of [TX, §2.2]. We let denote the reflex field which is the subfield of the Galois closure of in fixed by the subgroup of stabilizing and . We let be the adic place of induced by the embedding . We define the compact open subgroup , where
if .
the unique maximal compact of if .
For a sufficiently small compact open subgroup , we write and let denote the Shimura variety associated to the above data. We use the notation of [TX] so that determines the Deligne homomorphism. It is an algebraic variety over whose complex points are given by
where is the union of the complex upper and lower half planes. We note that the algebraic variety is independent of . However, different choices of will give rise to different varieties, see for example [TX1, p9]. We also point out the abuse of notation here, where the compact open subgroup implicitly depends on the choice of . When , is a discrete set and the action of can be described explicitly as in [LT, §2.1].
We also set
and we write for the neutral connected component of . This is the component containing the image of the point
2.2. Unitary Shimura varieties
In this section we define an auxiliary unitary Shimura variety which is of PELtype in order to define integral models for .
Let be a CMextension such that the following two conditions are satisfied:
(1) is inert at every place that is ramified
(2) For , is split at if and is even, and is inert if and is odd or if .
Let denote the set of complex embeddings of , which we identify with the set of adic embeddings via . For , we let denote its complex conjugate. For , let denote the subset of adic places of inducing . Similarly for a adic place of , we let denote the set of adic places of inducing .
We choose a subset satisfying the property that for each , the natural restriction map induces a bijection , where
For each , the choice of determines a collection of numbers given by:
(2.2.1) 
Let and . We let denote the compact open . We define the homomorphism
Here for , we identify with via the embedding , where is the unique lift of . We use the above data to define a Shimura datum for a unitary similitude group which will give rise to a moduli interpretation of the unitary Shimura variety.
Let , which is isomorphic to by our assumptions on . We let denote the involution on defined by the product of canonical involution on and complex conjugation on . Fix a totally imaginary element such that is a adic unit for every place above . Choose an element such that as in [TX, Lemma 3.8]. We define a new involution of by .
We consider as a right module of rank 1. It is equipped with the following pairing
(2.2.2) 
where is the reduced trace. It is easy to see this pairing is alternating and nondegenerate. Moreover it satisfies the following property
The unitary similitude group is defined to be
which arises as the points of a reductive group over . We may also describe this group in the following way. Since we have lies in if and only if
This identity is equivalent to , i.e. . Thus
where is the reduced norm. Here denotes the quotient of by the central embedding given by .
Let denote the torus . We define the group
Applying the above to points valued in a algebra, we see that is identified with the subgroup of corresponding to the preimage of under the map
We now let denote the morphism induced by ; it is independent of . The image of lies in and we let denote the induced map. Let the compact open subgroup given by the image of and . For sufficiently small compact open subgroups and , we obtain Shimura varieties where and . We also set
Let denote the common reflex field of these Shimura varieties, which is a subfield of . The isomorphism induces a adic place of . We let (resp. ) denote the neutral geometric connected component of (resp. ). Then both and can be descended to .
We have the following diagram of groups
compatible with Deligne homomorphisms and such that the induced maps on the derived and adjoint groups are isomorphisms. By [TX, Corollary 2.16] this induces isomorphisms
of neutral geometric connected components. Since Shimura varieties may be constructed from its neutral connected component and the action of Hecke and Galois, we may transfer integral models from one group to the other. See [TX1, §2.11] for the details.
2.3. Moduli interpretation for unitary Shimura varieties and integral models
The Shimura variety is a Shimura variety of PELtype and thus admits an integral model as a moduli space. Recall the module together with the nondegenerate alternating form introduced in the last subsection. We also fix some integral PEL data. Let be an order which is maximal at and an lattice such that and is selfdual. Let be a sufficiently small compact open subgroup of stabilizing . We consider the moduli functor that associates to each scheme the set of isomorphism classes of triples where:
is an abelian scheme over of dimension .
is an embedding such that the induced action of acting on satisfies
(2.3.1) 
is a polarization such that:
The Rosati involution on defined by induces the involution on .
If , induces an isomorphism of divisible groups .
If , is a finite flat group scheme contained in of rank and the cokernel of the map is a locally free module of rank 2 over . Here denotes the relative de Rham homology.
is a level structure, i.e. a orbit of isomorphisms
which respects the action of on both sides and preserves the pairings on both sides. Here denotes the primeto Tate module of .
By [TX, Theorem 3.14], the moduli problem is representable by a smooth quasiprojective scheme over , and its generic fiber is identified with . Moreover it is an integral canonical model for in the sense of [TX, 2.4], see also [Milne]. We write
Taking the closure of in and using Deligne’s recipe to transfer across to we obtain an integral canonical model for ; see [TX, §2.11]. We write (resp.