THE ARITHMETIC INTERSECTION CONJECTURE In
THE ARITHMETIC INTERSECTION CONJECTURE M. RAPOPORT
In my talk I reported on my joint work with B. Smithling and W. Zhang [10]. The theorem of Gross and Zagier [3] relates the N´eron–Tate heights of Heegner points on modular curves to special values of derivatives of certain L-functions. Ever since the appearance of the paper of Gross/Zagier, the problem of generalizing this fundamental result to higher dimension has attracted considerable attention. The generalization that is most relevant to my talk is the Arithmetic Gan–Gross–Prasad conjecture (AGGP conjecture). This conjectural generalization concerns Shimura varieties attached to orthogonal groups of signature (2, n − 2), and to unitary groups of signature (1, n − 1) (note that modular curves are closely related to Shimura varieties associated to orthogonal groups of signature (2, 1) and to unitary groups of signature (1, 1)). In [2], algebraic cycles of codimension one on such Shimura varieties are defined by exploiting embeddings of Shimura varieties attached to orthogonal groups of signature (2, n−3), resp. to unitary groups of signature (1, n−2). By taking the graphs of these embeddings, one obtains cycles in codimension just above half the (odd) dimension of the ambient variety. For any algebraic variety X smooth and proper of odd dimension over a number field, Beilinson and Bloch have defined a height pairing on the rational Chow group Ch(X)Q,0 of cohomologically trivial cycles of codimension just above half the dimension. Their definition makes use of some widely open unsolved conjectures on algebraic cycles and the existence of regular proper integral models of X. By suitably replacing in the case at hand the graph cycle by a cohomologically trivial avatar, one obtains a linear form on Ch(X)Q,0 , where now X is the product of the two Shimura varieties in question. The AGGP conjecture relates a special value of the derivative of an L-function to the non-triviality of the restriction of this linear form to a Hecke eigenspace in Ch(X)Q,0 . It is stated in a very succinct way in [2], for orthogonal groups and for unitary groups. In the present talk, I restrict myself to unitary groups, and my first aim is to give in this case a variant of this conjecture. Notation √ • F/F0 is a CM field, where F = F0 [ ∆], with ∆ ∈ F0 totally negative. √ √ • Φ = {ϕ : F → C | ϕ( ∆) ∈ R>0 −1}, the corresponding CM-type. • ϕ0 ∈ Φ a distinguished element. • W a F/F0 -hermitian vector space such that sgn(Wϕ0 ) = (1, n − 1); sgn(Wϕ ) = (0, n), ∀ϕ 6= ϕ0 . • G = ResF0 /Q (U(W )) and hG : C× → G(R) in coordinates hG (z)ϕ0 = diag(z/z, 1, . . . , 1),
hG (z)ϕ = 1 for ϕ 6= ϕ0 .
Let Sh(G, hG ) be the corresponding Shimura variety with its (weakly) canonical model over F , which is a subfield of C via ϕ0 . Let u ∈ W with (u, u)ϕ < 0, ∀ϕ (a totally negative vector ). Let W [ = u⊥ . Then W [ is like W , but with n replaced by n − 1. The corresponding Shimura variety is denoted by Sh(H, hH ). We obtain inclusions of Shimura varieties, Sh(H, hH ) ⊂ Sh(G, hG ),
Sh(H, hH ) ⊂ Sh(H, hH ) × Sh(G, hG ) = Sh(H × G, hH×G ).
Using Hecke correspondences one can construct an element in the cohomologically trivial rational Chow group, z ∈ Chn−1 := Chn−1 Sh(H × G, hH×G ) . 0 0 The element z defines by the Bloch-Beilinson pairing Chn−1 × Chn−1 → R a linear form 0 0 ` : Chn−1 −→ R. 0 Date: May 4, 2018. 1
2
M. RAPOPORT
Our version in [10] of the AGGP conjecture is now as follows. Conjecture 0.1. Let π be a cohomological generic automorphic representation of (H × G)(A). The following statements are equivalent. (1) `|Chn−1 [πf ] 6= 0. 0
(2) HomH(Af ) (πf , C) is one-dimensional and L0 ( 12 , π, R) 6= 0. Here Chn−1 [πf ] denotes the πf -isotopic subspace in Chn−1 . Also, the Langlands L-function 0 0 appearing here can be interpreted as the Rankin-Selberg L-function of the base change of π to (GLn−1 × GLn )(AF ). As indicated above, the AGGP conjecture is based on conjectures of Beilinson and Bloch which seem out of reach at present. As a consequence, the conjecture in [2] has not been proved in a single case of higher dimension. A variant of the AGGP conjecture, inspired by the relative trace formula of Jacquet-Rallis, has been proposed by W. Zhang [11]. More precisely, this variant relates the height pairing with distributions that appear in the relative trace formula. This variant leads to local conjectures (on intersection numbers on Rapoport–Zink spaces), namely the Arithmetic Fundamental Lemma conjecture and the Arithmetic Transfer conjecture, cf. [11, 9]—and these have been proved in various cases [11, 8, 9]. In [10], we formulate a global conjecture whose proof in various cases is a realistic goal. My second aim is to formulate this conjecture. To formulate this conjecture, we define variants of the Shimura varieties Sh(G, hG ) which are of PEL type, i.e. are related to moduli problems of abelian varieties with polarizations, endomorphisms, and level structures. In fact, we even define integral models of these Shimura varieties, in a global version and a semi-global version. Once these models are defined, we replace the Bloch–Beilinson pairing on the cohomologically trivial Chow group by the Gillet– Soul´e pairing on the arithmetic Chow group of the (global or semi-global) integral model. More notation • E ⊂ Q, the composite of the reflex field of (sgnϕ )ϕ∈Φ and the reflex field of Φ. Then E contains F via ϕ0 . • Z Q the torus Z Q := z ∈ ResF/Q (Gm ) NmF/F0 (z) ∈ Gm , with hZ Q , depending on Φ. e := Z Q × G and h ˜ = hZ Q × hG , and the corresponding Shimura variety Sh(G, ˜ h ˜ ). • G G
G
e of abelian varieties with additional We are able to formulate a PEL moduli problem MKGe (G) structure (endomorphisms and polarization) which defines a model over E of the Shimura variety e = MK (G) e ⊗E C. ShK (G) (0.1) e G
e G
(In fact, we demand that KGe = KZ Q × KG , where KZ Q is the unique maximal compact subgroup ˜ differs from the of Z Q (Af ) and where KG is an open compact subgroup of G(Af ).) The group G group of unitary similitudes GU(W ) by a central isogeny. The Shimura variety corresponding to the latter group is considered by Kottwitz [5], and he formulates a PEL moduli problem over the corresponding reflex field. When n is even, the corresponding moduli space defines a model for it; when n is odd, this is almost true but not quite, because of the possible failure of the Hasse principle for GU(W ). This Shimura variety is also considered by Harris–Taylor [4]. In the setup of [4], we have E = F and both their Shimura variety and ours are defined over F .1 e over Spec OE (at least when F/F0 is not We also define global integral models of MKGe (G) everywhere unramified) and semi-global integral models over Spec OE,(ν) , where ν is a fixed nonarchimedean place of E, of residue characteristic p. The definition of these integral models is based on a sign invariant invrv (A0 , A) ∈ {±1} for every non-archimedean place v of F0 which is non-split in F . Here (A0 , ι0 , λ0 ) is a polarized abelian variety of dimension d = [F0 : Q] with complex multiplication of CM type Φ of F and (A, ι, λ) is a polarized abelian variety of dimension nd with complex multiplication of generalized CM type r of F . This sign invariant is similar to the one in [6, 7], but much simpler. This simplicity is another reflection of the advantage of our Shimura varieties over those considered by Kottwitz [5]. These integral models generalize those in [1] when F0 = Q and when KG is the stabilizer of a self-dual lattice in W . Here we allow 1Kottwitz [5] does not need any assumptions on the signature of W ; neither do we.
THE ARITHMETIC INTERSECTION CONJECTURE
3
KG to be the stabilizer of certain vertex lattices. To achieve flatness, we sometimes have to impose conditions on the Lie algebras of the abelian varieties in play that are known in a similar context from our earlier local papers [8, 9] (the Pappas wedge condition, the spin condition and its refinement, the Eisenstein conditions). However, in contrast to Kottwitz, we do not need any unramifiedness conditions. e and its global or semi-global versions are defined, we can also create Once the model MKGe (G) a restriction situation in analogy with [2]. Namely, fixing a totally negative vector u ∈ W (satisfying additional integrality conditions for the global, resp. semi-global integral situation), we define W [ to be the orthogonal complement of u. Then W [ satisfies the same conditions as W , with n replaced by n − 1. We obtain a finite unramified morphism e −→ MK (G), e δ : MKH (H) e f G
(0.2)
e = Z Q × H, where H = U(W [ ), resp. their global, resp. semi-global integral versions. Here H considered as an algebraic group over Q. A graph construction defines a closed embedding e −→ MK (HG) g := MK (H) e × MK (G). e ∆ : MKH (H) e f g f G HG H
(0.3)
Now let us consider the global, or semi-global analogue. We then obtain, under certain hypotheses, elements in the rational Chow group, resp. rational arithmetic Chow group, of the g Let us consider the Gillet–Soul´e intersection (global, or semi-global) integral model MKHG (HG). g g We have a conjecture on product pairing on the rational arithmetic Chow group of MKHG (HG). g the value of the intersection product of ∆ and its image under a Hecke correspondence. Let us state the semi-global version, since this is the one for which we can produce concrete evidence. We also have a global version. Conjecture 0.2 (Semi-global conjecture). Fix a non-split place v0 of F0 over the place p ≤ ∞ of Q. Let f = ⊗` f` ∈ HKp g (= HKHG if p is archimedean) be a completely decomposed element g HG g and let f 0 = ⊗v fv0 ∈ H (G0 (AF )) be a Gaussian test function of the finite Hecke algebra of HG, 0 in the Hecke algebra of G0 = ResF/F0 (GLn−1 × GLn ) such that ⊗v
In my talk I reported on my joint work with B. Smithling and W. Zhang [10]. The theorem of Gross and Zagier [3] relates the N´eron–Tate heights of Heegner points on modular curves to special values of derivatives of certain L-functions. Ever since the appearance of the paper of Gross/Zagier, the problem of generalizing this fundamental result to higher dimension has attracted considerable attention. The generalization that is most relevant to my talk is the Arithmetic Gan–Gross–Prasad conjecture (AGGP conjecture). This conjectural generalization concerns Shimura varieties attached to orthogonal groups of signature (2, n − 2), and to unitary groups of signature (1, n − 1) (note that modular curves are closely related to Shimura varieties associated to orthogonal groups of signature (2, 1) and to unitary groups of signature (1, 1)). In [2], algebraic cycles of codimension one on such Shimura varieties are defined by exploiting embeddings of Shimura varieties attached to orthogonal groups of signature (2, n−3), resp. to unitary groups of signature (1, n−2). By taking the graphs of these embeddings, one obtains cycles in codimension just above half the (odd) dimension of the ambient variety. For any algebraic variety X smooth and proper of odd dimension over a number field, Beilinson and Bloch have defined a height pairing on the rational Chow group Ch(X)Q,0 of cohomologically trivial cycles of codimension just above half the dimension. Their definition makes use of some widely open unsolved conjectures on algebraic cycles and the existence of regular proper integral models of X. By suitably replacing in the case at hand the graph cycle by a cohomologically trivial avatar, one obtains a linear form on Ch(X)Q,0 , where now X is the product of the two Shimura varieties in question. The AGGP conjecture relates a special value of the derivative of an L-function to the non-triviality of the restriction of this linear form to a Hecke eigenspace in Ch(X)Q,0 . It is stated in a very succinct way in [2], for orthogonal groups and for unitary groups. In the present talk, I restrict myself to unitary groups, and my first aim is to give in this case a variant of this conjecture. Notation √ • F/F0 is a CM field, where F = F0 [ ∆], with ∆ ∈ F0 totally negative. √ √ • Φ = {ϕ : F → C | ϕ( ∆) ∈ R>0 −1}, the corresponding CM-type. • ϕ0 ∈ Φ a distinguished element. • W a F/F0 -hermitian vector space such that sgn(Wϕ0 ) = (1, n − 1); sgn(Wϕ ) = (0, n), ∀ϕ 6= ϕ0 . • G = ResF0 /Q (U(W )) and hG : C× → G(R) in coordinates hG (z)ϕ0 = diag(z/z, 1, . . . , 1),
hG (z)ϕ = 1 for ϕ 6= ϕ0 .
Let Sh(G, hG ) be the corresponding Shimura variety with its (weakly) canonical model over F , which is a subfield of C via ϕ0 . Let u ∈ W with (u, u)ϕ < 0, ∀ϕ (a totally negative vector ). Let W [ = u⊥ . Then W [ is like W , but with n replaced by n − 1. The corresponding Shimura variety is denoted by Sh(H, hH ). We obtain inclusions of Shimura varieties, Sh(H, hH ) ⊂ Sh(G, hG ),
Sh(H, hH ) ⊂ Sh(H, hH ) × Sh(G, hG ) = Sh(H × G, hH×G ).
Using Hecke correspondences one can construct an element in the cohomologically trivial rational Chow group, z ∈ Chn−1 := Chn−1 Sh(H × G, hH×G ) . 0 0 The element z defines by the Bloch-Beilinson pairing Chn−1 × Chn−1 → R a linear form 0 0 ` : Chn−1 −→ R. 0 Date: May 4, 2018. 1
2
M. RAPOPORT
Our version in [10] of the AGGP conjecture is now as follows. Conjecture 0.1. Let π be a cohomological generic automorphic representation of (H × G)(A). The following statements are equivalent. (1) `|Chn−1 [πf ] 6= 0. 0
(2) HomH(Af ) (πf , C) is one-dimensional and L0 ( 12 , π, R) 6= 0. Here Chn−1 [πf ] denotes the πf -isotopic subspace in Chn−1 . Also, the Langlands L-function 0 0 appearing here can be interpreted as the Rankin-Selberg L-function of the base change of π to (GLn−1 × GLn )(AF ). As indicated above, the AGGP conjecture is based on conjectures of Beilinson and Bloch which seem out of reach at present. As a consequence, the conjecture in [2] has not been proved in a single case of higher dimension. A variant of the AGGP conjecture, inspired by the relative trace formula of Jacquet-Rallis, has been proposed by W. Zhang [11]. More precisely, this variant relates the height pairing with distributions that appear in the relative trace formula. This variant leads to local conjectures (on intersection numbers on Rapoport–Zink spaces), namely the Arithmetic Fundamental Lemma conjecture and the Arithmetic Transfer conjecture, cf. [11, 9]—and these have been proved in various cases [11, 8, 9]. In [10], we formulate a global conjecture whose proof in various cases is a realistic goal. My second aim is to formulate this conjecture. To formulate this conjecture, we define variants of the Shimura varieties Sh(G, hG ) which are of PEL type, i.e. are related to moduli problems of abelian varieties with polarizations, endomorphisms, and level structures. In fact, we even define integral models of these Shimura varieties, in a global version and a semi-global version. Once these models are defined, we replace the Bloch–Beilinson pairing on the cohomologically trivial Chow group by the Gillet– Soul´e pairing on the arithmetic Chow group of the (global or semi-global) integral model. More notation • E ⊂ Q, the composite of the reflex field of (sgnϕ )ϕ∈Φ and the reflex field of Φ. Then E contains F via ϕ0 . • Z Q the torus Z Q := z ∈ ResF/Q (Gm ) NmF/F0 (z) ∈ Gm , with hZ Q , depending on Φ. e := Z Q × G and h ˜ = hZ Q × hG , and the corresponding Shimura variety Sh(G, ˜ h ˜ ). • G G
G
e of abelian varieties with additional We are able to formulate a PEL moduli problem MKGe (G) structure (endomorphisms and polarization) which defines a model over E of the Shimura variety e = MK (G) e ⊗E C. ShK (G) (0.1) e G
e G
(In fact, we demand that KGe = KZ Q × KG , where KZ Q is the unique maximal compact subgroup ˜ differs from the of Z Q (Af ) and where KG is an open compact subgroup of G(Af ).) The group G group of unitary similitudes GU(W ) by a central isogeny. The Shimura variety corresponding to the latter group is considered by Kottwitz [5], and he formulates a PEL moduli problem over the corresponding reflex field. When n is even, the corresponding moduli space defines a model for it; when n is odd, this is almost true but not quite, because of the possible failure of the Hasse principle for GU(W ). This Shimura variety is also considered by Harris–Taylor [4]. In the setup of [4], we have E = F and both their Shimura variety and ours are defined over F .1 e over Spec OE (at least when F/F0 is not We also define global integral models of MKGe (G) everywhere unramified) and semi-global integral models over Spec OE,(ν) , where ν is a fixed nonarchimedean place of E, of residue characteristic p. The definition of these integral models is based on a sign invariant invrv (A0 , A) ∈ {±1} for every non-archimedean place v of F0 which is non-split in F . Here (A0 , ι0 , λ0 ) is a polarized abelian variety of dimension d = [F0 : Q] with complex multiplication of CM type Φ of F and (A, ι, λ) is a polarized abelian variety of dimension nd with complex multiplication of generalized CM type r of F . This sign invariant is similar to the one in [6, 7], but much simpler. This simplicity is another reflection of the advantage of our Shimura varieties over those considered by Kottwitz [5]. These integral models generalize those in [1] when F0 = Q and when KG is the stabilizer of a self-dual lattice in W . Here we allow 1Kottwitz [5] does not need any assumptions on the signature of W ; neither do we.
THE ARITHMETIC INTERSECTION CONJECTURE
3
KG to be the stabilizer of certain vertex lattices. To achieve flatness, we sometimes have to impose conditions on the Lie algebras of the abelian varieties in play that are known in a similar context from our earlier local papers [8, 9] (the Pappas wedge condition, the spin condition and its refinement, the Eisenstein conditions). However, in contrast to Kottwitz, we do not need any unramifiedness conditions. e and its global or semi-global versions are defined, we can also create Once the model MKGe (G) a restriction situation in analogy with [2]. Namely, fixing a totally negative vector u ∈ W (satisfying additional integrality conditions for the global, resp. semi-global integral situation), we define W [ to be the orthogonal complement of u. Then W [ satisfies the same conditions as W , with n replaced by n − 1. We obtain a finite unramified morphism e −→ MK (G), e δ : MKH (H) e f G
(0.2)
e = Z Q × H, where H = U(W [ ), resp. their global, resp. semi-global integral versions. Here H considered as an algebraic group over Q. A graph construction defines a closed embedding e −→ MK (HG) g := MK (H) e × MK (G). e ∆ : MKH (H) e f g f G HG H
(0.3)
Now let us consider the global, or semi-global analogue. We then obtain, under certain hypotheses, elements in the rational Chow group, resp. rational arithmetic Chow group, of the g Let us consider the Gillet–Soul´e intersection (global, or semi-global) integral model MKHG (HG). g g We have a conjecture on product pairing on the rational arithmetic Chow group of MKHG (HG). g the value of the intersection product of ∆ and its image under a Hecke correspondence. Let us state the semi-global version, since this is the one for which we can produce concrete evidence. We also have a global version. Conjecture 0.2 (Semi-global conjecture). Fix a non-split place v0 of F0 over the place p ≤ ∞ of Q. Let f = ⊗` f` ∈ HKp g (= HKHG if p is archimedean) be a completely decomposed element g HG g and let f 0 = ⊗v fv0 ∈ H (G0 (AF )) be a Gaussian test function of the finite Hecke algebra of HG, 0 in the Hecke algebra of G0 = ResF/F0 (GLn−1 × GLn ) such that ⊗v
