Chow groups and L-derivatives of automorphic motives for unitary groups, II
CCHOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVESFOR UNITARY GROUPS, II. CHAO LI AND YIFENG LIU
Abstract.
In this article, we improve our main results from [LL] in two direction: First, weallow ramified places in the CM extension
E/F at which we consider representations that arespherical with respect to a certain special maximal compact subgroup, by formulating andproving an analogue of the Kudla–Rapoport conjecture for exotic smooth Rapoport–Zinkspaces. Second, we lift the restriction on the components at split places of the automorphicrepresentation, by proving a more general vanishing result on certain cohomology of integralmodels of unitary Shimura varieties with Drinfeld level structures.
Contents
1. Introduction 21.1. Main results 21.2. Two new ingredients 51.3. Notation and conventions 62. Intersection of special cycles at ramified places 72.1. A Kudla–Rapoport type formula 82.2. Fourier transform of analytic side 102.3. Bruhat–Tits stratification 202.4. Linear invariance of intersection numbers 222.5. Proof of Theorem 2.9 when r = 1 262.6. Fourier transform of geometric side 282.7. Proof of Theorem 2.9 332.8. Comparison with absolute Rapoport–Zink spaces 363. Local theta lifting at ramified places 423.1. Weil representation and spherical module 433.2. Doubling zeta integral and doubling L-factor 454. Arithmetic inner product formula 474.1. Recollection on doubling method 474.2. Recollection on arithmetic theta lifting 524.3. Local indices at split places 564.4. Local indices at inert places 624.5. Local indices at ramified places 634.6. Local indices at archimedean places 664.7. Proof of main results 66References 67 Date : January 26, 2021.2010
Mathematics Subject Classification. a r X i v : . [ m a t h . N T ] J a n CHAO LI AND YIFENG LIU Introduction
In [LL], we proved that for certain cuspidal automorphic representations π on unitarygroups of even ranks, if the central derivative L (1 / , π ) is nonvanishing, then the π -nearlyisotypic localization of the Chow group of a certain unitary Shimura variety over its reflexfield does not vanish. This proved part of the Beilinson–Bloch conjecture for Chow groupsand L -functions. Moreover, assuming the modularity of Kudla’s generating functions ofspecial cycles, we further proved the arithmetic inner product formula relating L (1 / , π )and the height of arithmetic theta liftings. In this article, we improve the main results from[LL] in two directions: First, we allow ramified places in the CM extension E/F at which weconsider representations that are spherical with respect to a certain special maximal compactsubgroup, by formulating and proving an analogue of the Kudla–Rapoport conjecture forexotic smooth Rapoport–Zink spaces. Second, we lift the restriction on the components atsplit places of the automorphic representation, by proving a more general vanishing resulton certain cohomology of integral models of unitary Shimura varieties with Drinfeld levelstructures. However, for technical reasons, we will still assume F = Q (see Remark 4.31).Readers may read the introduction of [LL] for more background of those results.1.1. Main results.
Let
E/F be a CM extension of number fields with the complex con-jugation c . Denote by V ( ∞ ) F and V fin F the set of archimedean and non-archimedean places of F , respectively; and V spl F , V int F , and V ram F the subsets of V fin F of those that are split, inert, andramified in E , respectively. For every v ∈ V fin F , we denote by q v the residue cardinality of F v . Definition 1.1.
We define the subset V ♥ F of V spl F ∪ V int F consisting of v satisfying that for every v ∈ V ( p ) F ∩ V ram F , where p is the underlying rational prime of v , the subfield of F v generatedby F v and the Galois closure of E v is unramified over F v . Remark . The purpose of this technical definition is that for certain places v in V spl F ∪ V int F ,we need to have a CM type of E such that its reflex field does not contain more ramificationover p than F v does – this is possible for v ∈ V ♥ F . Note that • the complement ( V spl F ∪ V int F ) \ V ♥ F is finite; • when E is Galois, or contains an imaginary quadratic field, or satisfies V ram F = ∅ , we have V ♥ F = V spl F ∪ V int F .Take an even positive integer n = 2 r . We equip W r := E n with the skew-hermitian form(with respect to the involution c ) given by the matrix (cid:16) r − r (cid:17) . Put G r := U( W r ), theunitary group of W r , which is a quasi-split reductive group over F . For every v ∈ V fin F , wedenote by K r,v ⊆ G r ( F v ) the stabilizer of the lattice O nE v , which is a special maximal compactsubgroup. Setup 1.3.
Suppose that F = Q , that V spl F contains all 2-adic places, and that every prime in V ram F is unramified over Q . We consider a cuspidal automorphic representation π of G r ( A F )realized on a space V π of cusp forms, satisfying:(1) For every v ∈ V ( ∞ ) F , π v is the holomorphic discrete series representation of Harish-Chandra parameter { n − , n − , . . . , − n , − n } .(2) For every v ∈ V ram F , π v is spherical with respect to K r,v , that is, π K r,v v = { } .(3) For every v ∈ V int F , π v is either unramified or almost unramified (see Remark 1.4 below)with respect to K r,v ; moreover, if π v is almost unramified, then v is unramified over Q . HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 3 (4) For every v ∈ V fin F , π v is tempered. (5) We have R π ∪ S π ⊆ V ♥ F (Definition 1.1), where • R π ⊆ V spl F denotes the (finite) subset for which π v is ramified, • S π ⊆ V int F denotes the (finite) subset for which π v is almost unramified.Comparing Setup 1.3 with [LL, Setup 1.3], we have lifted the restriction that V ram F = ∅ (by allowing π v to be a certain type of representations for v ∈ V ram F ), and also the restrictionon π v for v ∈ V spl F . Note that (5) is not really a new restriction since when V ram F = ∅ , it isautomatic by Remark 1.2. Remark . The notion of almost unramified representations of G r ( F v ) at v ∈ V int F is definedin [Liub, Definition 5.3]. Roughly speaking, an irreducible admissible representation π v of G r ( F v ) is almost unramified (with respect to K r,v ) if π I r,v v contains a particular characteras a module over C [ I r,v \ K r,v /I r,v ], where I r,v is an Iwahori subgroup contained in K r,v ,and that the Satake parameter of π v contains the pair { q v , q − v } ; it is not unramified. By[Liub, Theorem 1.2], when q v is odd, almost unramified representations are exactly thoserepresentations whose local theta lifting to the non-quasi-split unitary group of the samerank 2 r has nonzero invariants under the stabilizer of an almost self-dual lattice.Suppose that we are in Setup 1.3. Denote by L ( s, π ) the doubling L -function. Thenwe have ε ( π ) = ( − r [ F : Q ]+ | S π | for the global (doubling) root number, so that the vanishingorder of L ( s, π ) at the center s = has the same parity as r [ F : Q ] + | S π | . The cuspidalautomorphic representation π determines a hermitian space V π over A E of rank n via localtheta dichotomy (so that the local theta lifting of π v to U( V π )( F v ) is nontrivial for everyplace v of F ), unique up to isomorphism, which is totally positive definite and satisfies thatfor every v ∈ V fin F , the local Hasse invariant (cid:15) ( V π ⊗ A F F v ) = 1 if and only if v S π .Now suppose that r [ F : Q ] + | S π | is odd hence ε ( π ) = −
1, which is equivalent to that V π is incoherent. In what follows, we take V = V π in the context of [LL, Conjecture 1.1], hence H = U( V π ). Let R be a finite subset of V fin F . We fix a special maximal subgroup L R of H ( A ∞ , R F )that is the stabilizer of a lattice Λ R in V ⊗ A F A ∞ , R F (see Setup 4.2(H6) for more details). Fora field L , we denote by T R L the (abstract) Hecke algebra L [ L R \ H ( A ∞ , R F ) /L R ], which is acommutative L -algebra. When R contains R π , the cuspidal automorphic representation π gives rise to a character χ R π : T R Q ac → Q ac , where Q ac denotes the subfield of C of algebraic numbers; and we put m R π := ker χ R π , whichis a maximal ideal of T R Q ac . The following is the first main theorem of this article. Theorem 1.5.
Let ( π, V π ) be as in Setup 1.3 with r [ F : Q ] + | S π | odd, for which we assume [LL, Hypothesis 5.6] . If L ( , π ) = 0 , that is, ord s = L ( s, π ) = 1 , then as long as R satisfies R π ⊆ R and | R ∩ V spl F ∩ V ♥ F | (cid:62) , the nonvanishing lim −→ L R (cid:16) CH r ( X L R L R ) Q ac (cid:17) m R π = { } holds, where the colimit is taken over all open compact subgroups L R of H ( F R ) . Our remaining results rely on Hypothesis 4.11 on the modularity of Kudla’s generatingfunctions of special cycles, hence are conditional at this moment. In fact, (4) is implied by (1). See [LL, Remark 1.3(2)].
CHAO LI AND YIFENG LIU
Theorem 1.6.
Let ( π, V π ) be as in Setup 1.3 with r [ F : Q ] + | S π | odd, for which we assume [LL, Hypothesis 5.6] . Assume Hypothesis 4.11 on the modularity of generating functions ofcodimension r .(1) For every test vectors • ϕ = ⊗ v ϕ v ∈ V π and ϕ = ⊗ v ϕ v ∈ V π such that for every v ∈ V ( ∞ ) F , ϕ v and ϕ v have the lowest weight and satisfy h ϕ c v , ϕ v i π v = 1 , • φ ∞ = ⊗ v φ ∞ v ∈ S ( V r ⊗ A F A ∞ F ) and φ ∞ = ⊗ v φ ∞ v ∈ S ( V r ⊗ A F A ∞ F ) ,the identity h Θ φ ∞ ( ϕ ) , Θ φ ∞ ( ϕ ) i \X,E = L ( , π ) b r (0) · C [ F : Q ] r · Y v ∈ V fin F Z \π v ,V v ( ϕ c v , ϕ v , φ ∞ v ⊗ ( φ ∞ v ) c ) holds. Here, • Θ φ ∞ i ( ϕ i ) ∈ lim −→ L CH r ( X L ) C is the arithmetic theta lifting (Definition 4.12), whichis only well-defined under Hypothesis 4.11; • h Θ φ ∞ ( ϕ ) , Θ φ ∞ ( ϕ ) i \X,E is the normalized height pairing (Definition 4.17), whichis constructed based on Beilinson’s notion of height pairing; • b r (0) is defined in Setup 4.1(F4), which equals L ( M ∨ r (1)) where M r is the motiveassociated to G r by Gross [Gro97] , and is in particular a positive real number; • C r = ( − r r ( r − π r Γ(1) ··· Γ( r )Γ( r +1) ··· Γ(2 r ) , which is the exact value of a certain archimedeandoubling zeta integral; and • Z \π v ,V v ( ϕ c v , ϕ v , φ ∞ v ⊗ ( φ ∞ v ) c ) is the normalized local doubling zeta integral [LL, Sec-tion 3] , which equals for all but finitely many v .(2) In the context of [LL, Conjecture 1.1] , take ( V = V π and) ˜ π ∞ to be the theta lifting of π ∞ to H ( A ∞ F ) . If L ( , π ) = 0 , that is, ord s = L ( s, π ) = 1 , then Hom H ( A ∞ F ) ˜ π ∞ , lim −→ L CH r ( X L ) C ! = { } holds.Remark . We have the following remarks concerning Theorem 1.6.(1) Part (1) verifies the so-called arithmetic inner product formula , a conjecture proposedby one of us [Liu11a, Conjecture 3.11].(2) The arithmetic inner product formula in part (1) is perfectly parallel to the classicalRallis inner product formula. In fact, suppose that V is totally positive definite but coherent . We have the classical theta lifting θ φ ∞ ( ϕ ) where we use standard Gaussianfunctions at archimedean places. Then the Rallis inner product formula in this casereads as h θ φ ∞ ( ϕ ) , θ φ ∞ ( ϕ ) i H = L ( , π ) b r (0) · C [ F : Q ] r · Y v ∈ V fin F Z \π v ,V v ( ϕ c v , ϕ v , φ ∞ v ⊗ ( φ ∞ v ) c ) , in which h , i H denotes the Petersson inner product with respect to the Tamagawameasure on H ( A F ).In the case where R π = ∅ , we have a very explicit height formula for test vectors that arenew everywhere. HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 5 Corollary 1.8.
Let ( π, V π ) be as in Setup 1.3 with r [ F : Q ] + | S π | odd, for which we assume [LL, Hypothesis 5.6] . Assume Hypothesis 4.11 on the modularity of generating functions ofcodimension r . In the situation of Theorem 1.6(1), suppose further that • R π = ∅ ; • ϕ = ϕ = ϕ ∈ V [ r ] ∅ π (see Setup 4.3(G8) for the precise definition of the one-dimensionalspace V [ r ] ∅ π of holomorphic new forms) such that for every v ∈ V F , h ϕ c v , ϕ v i π v = 1 ; and • φ ∞ = φ ∞ = φ ∞ such that for every v ∈ V fin F , φ ∞ v = (Λ ∅ v ) r .Then the identity h Θ φ ∞ ( ϕ ) , Θ φ ∞ ( ϕ ) i \X,E = ( − r · L ( , π ) b r (0) · | C r | [ F : Q ] · Y v ∈ S π q r − v ( q v + 1)( q r − v + 1)( q rv − holds.Remark . Assuming the conjecture on the injectivity of the étale Abel–Jacobi map, onecan show that the cycle Θ φ ∞ ( ϕ ) is a primitive cycle of codimension r . By [Be˘ı87, Conjec-ture 5.5], we expect that ( − r h Θ φ ∞ ( ϕ ) , Θ φ ∞ ( ϕ ) i \X,E (cid:62) L ( , π ) (cid:62) Example . Suppose that
E/F satisfies the conditions in Setup 1.3 and that r (cid:62)
2. Con-sider an elliptic curve A over F without complex multiplication, satisfying that Sym r − A hence Sym r − A E are modular. Let Π be the cuspidal automorphic representation ofGL n ( A E ) corresponding to Sym r − A E , which satisfies Π ∨ ’ Π ◦ c . Then there exists acuspidal automorphic representation π of G r ( A F ) as in Setup 1.3 with Π its base change ifand only if A has good reduction at every v ∈ V fin F \ V spl F . Moreover, if this is the case, thenwe have S π = ∅ hence ε ( π ) = ( − r [ F : Q ] ; in particular, above results apply when both r and[ F : Q ] are odd.1.2. Two new ingredients.
The proofs of our main theorems follow the same line in [LL],with two new (main) ingredients, responsible for the two improvements we have mentionedat the beginning.The first new ingredient is formulating and proving an analogue of the Kudla–Rapoportconjecture in the case where
E/F is ramified and the level structure is the one that givesthe exotic smooth model. Here, F is a p -adic field with p odd. Let L be an O E -latticeof a nonsplit (nondegenerate) hermitian space V over E of (even) rank n . Then one canassociate an intersection number Int( L ) of special divisors on a formally smooth relativeRapoport–Zink space classifying quasi-isogenies of certain unitary O F -divisible groups, andalso the derivative of the representation density function ∂ Den( L ) given by L . We show inTheorem 2.9 the formula Int( L ) = ∂ Den( L ) . This is parallel to the Kudla–Rapoport conjecture proved in [LZ], originally stated for thecase where
E/F is unramified. The proof follows from the same strategy as in [LZ], namely,we write L = L [ + h x i for a sublattice L [ of L such that V L [ := L [ ⊗ O F F is nonde-generate, and regard x as a variable. Thus, it motivates us to define a function Int L [ on V \ V L [ by the formula Int L [ ( x ) = Int( L [ + h x i ) and similarly for ∂ Den L [ . For Int L [ , thereis a natural decomposition Int L [ = Int h L [ + Int v L [ according to the horizontal and verticalparts of the special cycle defined by L [ . In a parallel manner, we have the decomposition ∂ Den L [ = ∂ Den h L [ + ∂ Den v L [ by simply matching ∂ Den h L [ with Int h L [ . Thus, it suffices to CHAO LI AND YIFENG LIU show that Int v L [ = ∂ Den v L [ . By some sophisticated induction argument on L [ , it suffices toshow the following remarkable property for both Int v L [ and ∂ Den v L [ : they extend (uniquely)to compactly supported locally constant functions on V , whose Fourier transforms are sup-ported in the set { x ∈ V | ( x, x ) V ∈ O F } . However, there are some new difficulties in ourcase: • The isomorphism class of an O E -lattice is not determined by its fundamental invariants,and there is a parity constraint for the valuation of an O E -lattice. This will make theinduction argument on L [ much more complicated than the one in [LZ] (see Subsection2.7). • The comparison of our relative Rapoport–Zink space to an (absolute) Rapoport–Zinkspace is not known. This is needed even to show that our relative Rapoport–Zink spaceis representable, and also in the p -adic uniformization of Shimura varieties. We solvethis problem when F/ Q p is unramified, which is the reason for us to assume that everyprime in V ram F is unramified over Q in Setup 1.3. See Subsection 2.8. • Due to the parity constraint, the computation of Int v L [ can only be reduced to the casewhere n = 4 (rather than n = 3 in [LZ]). After that, we have to compute certainintersection multiplicity, for which we use a new argument based on the linear invarianceof the K-theoretic intersection of special divisors. See Lemma 2.56.Here comes three more remarks: • First, we need to extend the result of [CY20] on a counting formula for ∂ Den( L ) tohermitian spaces over a ramified extension E/F (Lemma 2.20). • Second, we have found a simpler argument for the properties of ∂ Den v L [ (Proposition2.23), which does not use any functional equation or induction formula. This argumentis applicable to [LZ] to give a new proof of the main result on the analytic side there.Also note that we prove the vanishing property in Proposition 2.23 directly, while in [LZ]it is only deduced after proving Int v L [ = ∂ Den v L [ . • Finally, unlike the case in [LZ], the parity of the dimension of the hermitian space playsa crucial role in the exotic smooth case. In particular, we will not study the case where V has odd dimension.The second new ingredient is a vanishing result on certain cohomology of integral modelsof unitary Shimura varieties with Drinfeld level structures. For v ∈ V spl F ∩ V ♥ F with p theunderlying rational prime, we have a tower of integral models {X m } m (cid:62) defined by Drinfeldlevel structures (at v ), with an action by T R ∪ V ( p ) F Q ac via Hecke correspondences. We show inTheorem 4.21 that H r ( X m , Q ‘ ( r )) m = 0with ‘ = p and m := m R π ∩ S R ∪ V ( p ) F Q ac , where S R ∪ V ( p ) F Q ac is the subalgebra of T R ∪ V ( p ) F Q ac consisting of thosesupported at split places. We reduce this vanishing property to some other vanishing prop-erties for cohomology of Newton strata of X m , by using a key result of Mantovan [Man08]saying that the closure of every refined Newton stratum is smooth. For the vanishing proper-ties for Newton strata, we generalize an argument of [TY07, Proposition 4.4]. However, sincein our case, the representation π v has arbitrary level and our group has nontrivial endoscopy,we need a more sophisticated trace formula, which was provided in [CS17].1.3. Notation and conventions.
HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 7 • When we have a function f on a product set A × · · · × A m , we will write f ( a , . . . , a m )instead of f (( a , . . . , a m )) for its value at an element ( a , . . . , a m ) ∈ A × · · · × A m . • For a set S , we denote by S the characteristic function of S . • All rings (but not algebras) are commutative and unital; and ring homomorphisms pre-serve units. • For a (formal) subscheme Z of a (formal) scheme X , we denote by I Z the ideal sheaf of Z , which is a subsheaf of the structure sheaf O X of X . • For a ring R , we denote by Sch /R the category of schemes over R , by Sch /R the subcat-egory of locally Noetherian schemes over R , and when R is discretely valued, by Sch v /R the subcategory of schemes on which uniformizers of R are locally nilpotent. • If a base ring is not specified in the tensor operation ⊗ , then it is Z . • For an abelian group A and a ring R , we put A R := A ⊗ R . • For an integer m (cid:62)
0, we denote by 0 m and 1 m the null and identity matrices of rank m , respectively. We also denote by w m the matrix (cid:16) m − m (cid:17) . • We denote by c : C → C the complex conjugation. For an element x in a complex spacewith a default underlying real structure, we denote by x c its complex conjugation. • For a field K , we denote by K the abstract algebraic closure of K . However, for aestheticreason, we will write Q p instead of Q p and will denote by F p its residue field. On theother hand, we denote by Q ac the algebraic closure of Q inside C . • For a number field K , we denote by ψ K : K \ A K → C × the standard additive character,namely, ψ K := ψ Q ◦ Tr K/ Q in which ψ Q : Q \ A → C × is the unique character such that ψ Q , ∞ ( x ) = e πix . • Throughout the entire article, all parabolic inductions are unitarily normalized.
Acknowledgements.
We thank Xuhua He and Yichao Tian for helpful discussion. The re-search of C. L. is partially supported by the NSF grant DMS–1802269. The research of Y. L.is partially supported by the NSF grant DMS–2000533.2.
Intersection of special cycles at ramified places
Throughout this section, we fix a ramified quadratic extension
E/F of p -adic fields with p odd, with c ∈ Gal(
E/F ) the Galois involution. We fix a uniformizer u ∈ E satisfying u c = − u . Let k be the residue field of F and denote by q the cardinality of k . Let n = 2 r be an even positive integer.In Subsection 2.1, we introduce our relative Rapoport–Zink space and state the maintheorem (Theorem 2.9) on the relation between intersection numbers and derivatives of rep-resentation densities. In Subsection 2.2, we study derivatives of representation densities. InSubsection 2.3, we recall the Bruhat–Tits stratification on the relative Rapoport–Zink spacefrom [Wu] and deduce some consequences. In Subsection 2.4, we prove the linear invarianceon the K-theoretic intersection of special divisors, following [How19]. In Subsection 2.5, weprove Theorem 2.9 when r = 1, which is needed for the proof when r >
1. In Subsection2.6, we study intersection numbers. In Subsection 2.7, we prove Theorem 2.9 for general r . In Subsection 2.8, we compare our relative Rapoport–Zink space to certain (absolute)Rapoport–Zink space assuming F/ Q p is unramified.Here are two preliminary definitions for this section: • A hermitian O E -module is a finitely generated free O E -module L together with an O F -bilinear pairing ( , ) L : L × L → E such that the induced E -valued pairing on L ⊗ O F F CHAO LI AND YIFENG LIU is a nondegenerate hermitian pairing (with respect to c ). When we say that a hermitian O E -module L is contained in a hermitian O E -module or a hermitian E -space M , werequire that the restriction of the pairing ( , ) M to L coincides with ( , ) L . • Let X be an object of an additive category with a notion of dual. – We say that a morphism σ X : X → X ∨ is a symmetrization if σ X is an isomorphismand the composite morphism X → X ∨∨ σ ∨ X −→ X ∨ coincides with σ X . – Given an action ι X : O E → End( X ), we say that a morphism λ X : X → X ∨ is ι X -compatible if λ X ◦ ι X ( α ) = ι X ( α c ) ∨ ◦ λ X holds for every α ∈ O E .2.1. A Kudla–Rapoport type formula.
We fix an embedding ϕ : E → C p and let ˘ E bethe maximal complete unramified extension of ϕ ( E ) in C p . We regard E as a subfield of ˘ E via ϕ hence identify the residue field of ˘ E with an algebraic closure k of k . Definition 2.1.
Let S be an object of Sch /O ˘ E . We define a category Exo ( n − , ( S ) whoseobjects are triples ( X, ι X , λ X ) in which • X is an O F -divisible group over S of dimension n and (relative) height 2 n ; • ι X : O E → End( X ) is an action of O E on X satisfying: – (Kottwitz condition): the characteristic polynomial of ι X ( u ) on the O S -moduleLie( X ) is ( T − u ) n − ( T + u ) ∈ O S [ T ], – (Wedge condition): we have ^ ( ι X ( u ) − u | Lie( X )) = 0 , – (Spin condition): for every geometric point s of S , the action of ι X ( u ) on Lie( X s ) isnonzero; • λ X : X → X ∨ is a ι X -compatible polarization such that ker( λ X ) = X [ ι X ( u )].A morphism (resp. quasi-morphism) from ( X, ι X , λ X ) to ( Y, ι Y , λ Y ) is an O E -linear isomor-phism (resp. quasi-isogeny) ρ : X → Y of height zero such that ρ ∗ λ Y = λ X .When S belongs to Sch v /O ˘ E , we denote by Exo b( n − , ( S ) the subcategory of Exo ( n − , ( S )consisting of ( X, ι X , λ X ) in which X is supersingular. Remark . Giving a ι X -compatible polarization λ X of X satisfying ker( λ X ) = X [ ι X ( u )]is equivalent to giving a ι X -compatible symmetrization σ X of X . In fact, since ker( λ X ) = X [ ι X ( u )], there is a unique morphism σ X : X → X ∨ satisfying λ X = σ X ◦ ι X ( u ), which is infact an isomorphism, satisfying σ ∨ X = ι X ( u − ) ∨ ◦ λ ∨ X = − ι X ( u − ) ∨ ◦ λ X = − λ X ◦ ι X ( u − , c ) = λ X ◦ ι X ( u − ) = σ X , and is clearly ι X -compatible. Conversely, given a ι X -compatible symmetrization σ X of X ,we may recover λ X as σ X ◦ ι X ( u ). In what follows, we call σ X the symmetrization of λ X .To define our relative Rapoport–Zink space, we fix an object ( X , ι X , λ X ) ∈ Exo b( n − , ( k ). Definition 2.3.
We define a functor N := N ( X ,ι X ,λ X ) on Sch v /O ˘ E such that for every object S of Sch v /O ˘ E , N ( S ) consists of quadruples ( X, ι X , λ X ; ρ X ) in which • ( X, ι X , λ X ) is an object of Exo b( n − , ( S ); An O F -divisible group is also called a strict O F -module. HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 9 • ρ X is a quasi-morphism from ( X, ι X , λ X ) × S ( S ⊗ O ˘ E k ) to ( X , ι X , λ X ) ⊗ k ( S ⊗ O ˘ E k ) inthe category Exo b( n − , ( S ⊗ O ˘ E k ). Hypothesis 2.4.
The functor N is (pro-)represented by a separated formal scheme overSpf O ˘ E . Remark . When F is unramified over Q p , Hypothesis 2.4 is known. In fact, by Corollary2.66, N is isomorphic to an absolute Rapoport–Zink space N Φ which is known to be aseparated formal scheme over Spf O ˘ E by [RZ96].In what follows, we will assume Hypothesis 2.4. Lemma 2.6.
The functor N is a separated formal scheme formally smooth over Spf O ˘ E ofrelative dimension n − . Moreover, N has two connected components.Proof. The formal smoothness of N follow from the smoothness of its local model, which is[RSZ17, Proposition 3.10]; and the dimension also follows. For the last assertion, our modulifunctor N is the disjoint union of N (0 , and N (0 , from [Wu, Section 3.4], each of which isconnected by [Wu, Theorem 5.18(2)]. (cid:3) To study special cycles on N , we fix a triple ( X , ι X , λ X ) where • X is a supersingular O F -divisible group over Spec O ˘ E of dimension 1 and height 2; • ι X : O E → End( X ) is an O E -action on X such that the induced action on Lie( X ) isgiven by ϕ ; • λ X : X → X ∨ is a ι X -compatible principal polarization.Note that ι X induces an isomorphism ι X : O E ∼ −→ End O E ( X ). Put V := Hom O E ( X ⊗ O ˘ E k, X ) ⊗ Q , which is a vector space over E of dimension n . We have a pairing( , ) V : V × V → E (2.1)sending ( x, y ) ∈ V to the composition of quasi-homomorphisms X x −→ X λ X −−→ X ∨ y ∨ −→ X ∨ u − λ − X −−−−→ X as an element in End O E ( X ) ⊗ Q hence in E via ι − X . It is known that ( , ) V is a nondegenerateand nonsplit hermitian form on V [RSZ17, Lemma 3.5]. Definition 2.7.
For every nonzero element x ∈ V , we define the special divisor N ( x ) of N to be the maximal closed formal subscheme over which the quasi-homomorphism ρ − X ◦ x : ( X ⊗ O ˘ E k ) ⊗ k ( S ⊗ O ˘ E k ) → X × S ( S ⊗ O ˘ E k )lifts (uniquely) to a homomorphism X ⊗ O ˘ E S → X . The article [Wu] only studied the case F = Q p . In fact, except for Hypothesis 2.4, all arguments henceresults work for general F . This footnote applies to the proof of Proposition 2.29 as well. Readers may notice that we have an extra factor u − in the definition of the hermitian form. This isbecause we want to ensure that N ( x ) is nonempty if and only if ( x, x ) V ∈ O F . Definition 2.8.
For an O E -lattice L of V , the Serre intersection multiplicity χ (cid:18) O N ( x ) L ⊗ O N · · · L ⊗ O N O N ( x n ) (cid:19) does not depend on the choice of a basis { x , . . . , x n } of L by Corollary 2.36, which we defineto be Int( L ). Theorem 2.9.
Assume Hypothesis 2.4. For every O E -lattice L of V , we have Int( L ) = ∂ Den( L ) , where ∂ Den( L ) is defined in Definition 2.17. By Remark 2.5, this theorem is unconditional if F is unramified over Q p .The strategy of proving this theorem described in Subsection 1.2 motivates the followingdefinition, which will be frequently used in the rest of Section 2. Definition 2.10.
We define [ ( V ) to be the set of hermitian O E -modules contained in V ofrank n −
1. In what follows, for L [ ∈ [ ( V ), we put V L [ := L [ ⊗ O F F and write V ⊥ L [ for theorthogonal complement of V L [ in V . Remark . Let S be an object of Sch /O ˘ E . We have another category Exo ( n, ( S ) whoseobjects are triples ( X, ι X , λ X ) in which • X is an O F -divisible group over S of dimension n and (relative) height 2 n ; • ι X : O E → End( X ) is an action of O E on X such that ι X ( u ) − u annihilates Lie( X ); • λ X : X → X ∨ is a ι X -compatible polarization such that ker( λ X ) = X [ ι X ( u )].Morphisms are defined similarly as in Definition 2.1. The category Exo ( n, ( S ) is a connectedgroupoid.For later use, we fix a nontrivial additive character ψ F : F → C × of conductor O F . For alocally constant compactly supported function φ on a hermitian space V over E , its Fouriertransform b φ is defined by b φ ( x ) = Z V φ ( y ) ψ F (Tr E/F ( x, y ) V ) d y where d y is the self-dual Haar measure on V .2.2. Fourier transform of analytic side.
In this subsection, we study local densities ofhermitian lattices. We first introduce some notion about O E -lattices in hermitian spaces. Definition 2.12.
Let V be a hermitian space over E of dimension m , equipped with thehermitian form ( , ) V .(1) For a subset X of V , • we denote by X int the subset { x ∈ X | ( x, x ) V ∈ O F } ; • we denote by h X i the O E -submodule of V generated by X ; when X = { x, . . . } isexplicitly presented, we simply write h x, . . . i instead of h{ x, . . . }i .(2) For an O E -lattice L of V , we put L ∨ := { x ∈ V | Tr E/F ( x, y ) V ∈ O F for every y ∈ L } = { x ∈ V | ( x, y ) V ∈ u − O E for every y ∈ L } . We say that L is • integral if L ⊆ L ∨ ; HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 11 • vertex if it is integral such that L ∨ /L is annihilated by u ; and • self-dual if L = L ∨ .(3) For an integral O E -lattice L of V , we define • the fundamental invariants of L unique integers 0 (cid:54) a (cid:54) · · · (cid:54) a m such that L ∨ /L ’ O E / ( u a ) ⊕ · · · ⊕ O E / ( u a m ) as O E -modules; • the type t ( L ) of L to be the number of nonzero elements in its fundamental invari-ants; and • the valuation of L to be val( L ) := P mi =1 a i ; when L is generated by a single element x , we simply write val( x ) instead of val( h x i ).The above notation and definitions make sense without specifying V , namely, they apply tohermitian O E -modules. Remark . For an integral hermitian O E -module L of rank m with fundamental invariants( a , . . . , a m ), we have(1) L is vertex if and only if a m (cid:54) a m = 0;(2) t ( L ) and val( L ) must have the same parity with m . Remark . For a hermitian O E -module L , we say that a basis { e , . . . , e m } of L is a normalbasis if its moment matrix T = (( e i , e j ) L ) mi,j =1 is conjugate to (cid:16) β u b (cid:17) ⊕ · · · ⊕ (cid:16) β s u b s (cid:17) ⊕ u c − − u c − ! ⊕ · · · ⊕ u c t − − u c t − ! by a permutation matrix, for some β , . . . , β s ∈ O × F and b , . . . , b s , c , . . . , c t ∈ Z . We have(1) normal basis exists;(2) the invariants s, t and b , . . . , b s , c , . . . , c t depend only on L ;(3) when L is integral, the fundamental invariants of L are the unique nondecreasingrearrangement of (2 b + 1 , . . . , b s + 1 , c , c , . . . , c t , c t ). Definition 2.15.
Let M and L be two hermitian O E -modules. We denote by Herm L,M thescheme of hermitian O E -module homomorphisms from L to M , which is a scheme of finitetype over O F . We define the local density to beDen( M, L ) := lim N → + ∞ (cid:12)(cid:12)(cid:12) Herm
L,M ( O F / ( u N )) (cid:12)(cid:12)(cid:12) q N · d L,M where d L,M is the dimension of Herm
L,M ⊗ O F F .Denote by H the standard hyperbolic hermitian O E -module (of rank 2) given by thematrix (cid:16) u − − u − (cid:17) . For an integer s (cid:62)
0, put H s := H ⊕ s . Then H s is a self-dual hermitian O E -module of rank 2 s . The following lemma is a variant of a result of Cho–Yamauchi [CY20]when E/F is ramified.
Lemma 2.16.
Let L be a hermitian O E -module of rank m . Then we have Den( H s , L ) = X L ⊆ L ⊆ L | L /L | m − s Y s − m + t ( L )2
Put V := L ⊗ O F F . For an integral O E -lattice L of V , we equip the k -vector space L k := L ⊗ O E O E / ( u ) with a k -valued pairing h , i L k by the formula h x, y i L k := u · ( x ] , y ] ) V mod ( u )where x ] and y ] are arbitrary lifts of x and y , respectively. Then L k becomes a symplecticspace over k of dimension m whose radical has dimension t ( L ). Similarly, we have H s,k ,which is a nondegenerate symplectic space over k of dimension 2 s . We denote by Isom L k ,H s,k the k -scheme of isometries from L k to H s,k .By the same argument in [CY20, Section 3.3], we haveDen( H s , L ) = q − m (4 s − m +1) / · X L ⊆ L ⊆ L | L /L | m − s | Isom L k ,H s,k ( k ) | . Thus, it remains to show that | Isom L k ,H s,k ( k ) | = q m (4 s − m +1) / Y s − m + t ( L )2
For an O E -lattice L of V , define the (normalized) local Siegel series of L to be the polynomial Den( X, L ) ∈ Z [ X ], which exists by Lemma 2.20 below, such that forevery integer s (cid:62)
0, Den( q − s , L ) = Den( H r + s , L ) Q r + si = s +1 (1 − q − i ) , where Den is defined in Definition 2.15. We then put ∂ Den( L ) := − dd X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X =1 Den( X, L ) . Remark . Since V is nonsplit, we have Den(1 , L ) = Den( H r , L ) = 0. Remark . Let L be an O E -lattice of V . Let T ∈ GL n ( E ) be a representing matrix of L ,and consider the T -th Whittaker function W T ( s, r , H rr ) of the Schwartz function H rr atthe identity element 1 r . By [KR14, Proposition 10.1], we have W T ( s, r , H rr ) = Den( H r + s , L )for every integer s (cid:62)
0. Thus, we obtainlog q · ∂ Den( L ) = W T (0 , r , H rr ) Q ri =1 (1 − q − i )by Definition 2.17. Lemma 2.20.
For every O E -lattice L of V , we have Den( X, L ) = X L ⊆ L ⊆ L ∨ X OE ( L/ L ) t ( L )2 − Y i =0 (1 − q i X ) , (2.3) and ∂ Den( L ) = 2 X L ⊆ L ⊆ L ∨ t ( L )2 − Y i =1 (1 − q i ) , (2.4) where both sums are taken over integral O E -lattices of V containing L . Proof.
The identity (2.3) is a direct consequence of Lemma 2.16 and Definition 2.17. Theidentity (2.4) is a consequence of (2.3). (cid:3)
Definition 2.21.
Let L [ be an element of [ ( V ) (Definition 2.10). For x ∈ V \ V L [ , we put ∂ Den L [ ( x ) := ∂ Den( L [ + h x i ) ,∂ Den h L [ ( x ) := 2 X L [ ⊆ L ⊆ L ∨ t ( L ∩ V L[ )=1 L ( x ) ,∂ Den v L [ ( x ) := ∂ Den L [ ( x ) − ∂ Den h L [ ( x ) . Here in the second formula, L in the summation is an O E -lattice of V . Remark . We have In [KR14, Proposition 10.1] and its proof, the lattice L r,r should be replaced by H r . In (2.4), when t ( L ) = 2, we regard the empty product Q t ( L )2 − i =1 (1 − q i ) as 1. (1) The summation in ∂ Den h L [ ( x ) equals twice the number of integral O E -lattices L of V that contains L [ + h x i and such that t ( L ∩ V L [ ) = 1.(2) There exists a compact subset C L [ of V such that ∂ Den L [ , ∂ Den h L [ , and ∂ Den v L [ vanishoutside C L [ and are locally constant functions on C L [ \ V L [ .(3) For an integral O E -lattice L of V , if t ( L ∩ V L [ ) = 1, then t ( L ) = 2 by Lemma 2.24(1)below and the fact that V is nonsplit.(4) By (3) and Lemma 2.20, we have ∂ Den v L [ ( x ) = 2 X L [ ⊆ L ⊆ L ∨ t ( L ∩ V L[ ) > t ( L )2 − Y i =1 (1 − q i ) L ( x )for x ∈ V \ V L [ .The following is our main result of this subsection. Proposition 2.23.
Let L [ be an element of [ ( V ) . Then ∂ Den v L [ extends (uniquely) toa (compactly supported) locally constant function on V , which we still denote by ∂ Den v L [ .Moreover, the support of (cid:92) ∂ Den v L [ is contained in V int (Definition 2.12). We need some lemma for preparation.
Lemma 2.24.
Let L be an integral hermitian O E -module of with fundamental invariants ( a , . . . , a m ) .(1) If T = (( e i , e j ) L ) mi,j =1 is the moment matrix of an arbitrary basis { e , . . . , e m } of L ,then for every (cid:54) i (cid:54) m , a + · · · + a i − i equals the minimal E -valuation of thedeterminant of all i -by- i minors of T .(2) If L = L + h x i for some (integral) hermitian O E -module L contained in L of rank m − , then we have t ( L ) = ( t ( L ) + 1 , if x ∈ uL + L , t ( L ) − , otherwise,where x is the unique element in L such that ( x , y ) L = ( x, y ) L for every y ∈ L .Proof. Part (1) is simply the well-known method of computing the Smith normal form of uT (over O E ) using ideals generated by determinants of minors. For (2), take a normal basis { x , . . . , x m − } of L (Remark 2.14) such that h x , . . . , x m − − t ( L ) i is self-dual. Applying (1)to the basis { x , . . . , x m − , x } of L , we know that t ( L ) = t ( L ) + 1 if ( x i , x ) L ∈ O E for every m − t ( L ) (cid:54) i (cid:54) m −
1; otherwise, we have t ( L ) = t ( L ) −
1. In particular, (2) follows. (cid:3)
In the rest of this subsection, in order to shorten formulae, we put µ ( t ) := t − Y i =1 (1 − q i )for every positive even integer t . Lemma 2.25.
Take L [ ∈ [ ( V ) that is integral. For every compact subset X of V notcontained in V L [ , we denote by δ X the maximal integer such that the image of X underthe projection map V → V ⊥ L [ induced by the orthogonal decomposition V = V L [ ⊕ V ⊥ L [ iscontained in u δ X ( V ⊥ L [ ) int . We denote by L the set of O E -lattices of V containing L [ , and by HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 15 E the set of triples ( L [ , δ, ε ) in which L [ is an O E -lattice of V L [ containing L [ , δ ∈ Z , and ε : u δ ( V ⊥ L [ ) int → L [ ⊗ O F F/O F is an O E -linear map.(1) The map L → E sending L to the triple ( L ∩ V L [ , δ L , ε L ) is a bijection, where ε L isthe is the extension map u δ X ( V ⊥ L [ ) int → ( L ∩ V L [ ) ⊗ O F F/O F induced by the short exactsequence → L ∩ V L [ → L → u δ X ( V ⊥ L [ ) int → . Moreover, L is integral if and only if the following hold: • L ∩ V L [ is integral; • the image of ε is contained in ( L ∩ V L [ ) ∨ / ( L ∩ V L [ ) ; • ε L ( x ) + x ⊆ V int for every x ∈ u δ X ( V ⊥ L [ ) int . (2) For L ∈ L that is integral and corresponds to ( L [ , δ, ε ) ∈ E , we have t ( L ) = t ( L [ ) + 1 , if the image of ε is contained in ( u ( L [ ) ∨ + L [ ) /L [ , t ( L [ ) − , otherwise.(3) For every fixed integral O E -lattice L [ of V L [ containing L [ , the sum X L ⊆ L ∨ L ∩ V L[ = L [ q − δ L | µ ( t ( L )) | is convergent; and if t ( L [ ) > , then we have X L ⊆ L ∨ L ∩ V L[ = L [ z ∈ L ∨ q − δ L µ ( t ( L )) = 0 for every z ∈ V \ V int .(4) For every fixed integral O E -lattice L [ of V L [ containing L [ with t ( L [ ) > , we have X L ⊆ L ∨ L ∩ V L[ = L [ δ L =0 µ ( t ( L )) = 0 . Proof.
For (1), the inverse map E → L is the one that sends ( L [ , δ, ε ) to the O E -lattice L generated by L [ and ε L ( x ) + x for every x ∈ u δ X ( V ⊥ L [ ) int . The rest of (1) is straightforward.Part (2) is simply Lemma 2.24(2).Part (4) follows by applying (3) to generators z of O E -modules u − ( V ⊥ L [ ) int and u − ( V ⊥ L [ ) int and then taking the difference.Now we prove (3), which is the most difficult one. For every x ∈ V , we denote by x ∈ V L [ the first component of x with respect to the orthogonal decomposition V = V L [ ⊕ V ⊥ L [ . PutΩ := { x ∈ V int | x ∈ ( L [ ) ∨ } , Ω ◦ := { x ∈ V int | x ∈ u ( L [ ) ∨ + L [ } . Note that both Ω and Ω ◦ are open compact subsets of V stable under the translation by L [ . For an element L ∈ L corresponding to ( L [ , δ, ε ) ∈ E from (1), L is integral if and only For ( L [ , δ, ε ) ∈ E , we regard ε ( x ) + x as an L [ -coset in V as long as we write ε ( x ) + x ⊆ Ω for a subsetΩ of V . ε ( x ) + x ⊆ Ω for every x ∈ u δ ( V ⊥ L [ ) int . By (2), for such L , t ( L ) = t ( L [ ) + 1 , if ε ( x ) + x ⊆ Ω ◦ for every x ∈ u δ ( V ⊥ L [ ) int \ u δ +1 ( V ⊥ L [ ) int , t ( L [ ) − , if ε ( x ) + x ⊆ Ω \ Ω ◦ for every x ∈ u δ ( V ⊥ L [ ) int \ u δ +1 ( V ⊥ L [ ) int .Thus, we may replace the term corresponding to L in the summation in (3) by an integrationover the region S x ∈ u δ ( V ⊥ L[ ) int \ u δ +1 ( V ⊥ L[ ) int ( ε ( x ) + x ) of Ω. It follows that X L ⊆ L ∨ L ∩ V L[ = L [ q − δ L | µ ( t ( L )) | = 1 C Z Ω ◦ \ V L[ | µ ( t ( L [ ) + 1) | d x + Z Ω \ (Ω ◦ ∪ V L[ ) | µ ( t ( L [ ) − | d x ! which is convergent, where C = vol( L [ ) · vol(( V ⊥ L [ ) int \ u ( V ⊥ L [ ) int ) . Now we take an element z ∈ V \ V int . We may assume z ∈ ( L [ ) ∨ since otherwise thesummation in (3) is empty. PutΩ z := { x ∈ Ω | ( x, z ) V ∈ u − O E } , Ω ◦ z := { x ∈ Ω ◦ | ( x, z ) V ∈ u − O E } , both stable under the translation by L [ since z ∈ ( L [ ) ∨ . Similarly, we have X L ⊆ L ∨ L ∩ V L[ = L [ z ∈ L ∨ q − δ L µ ( t ( L )) = 1 C Z Ω ◦ z \ V L[ µ ( t ( L [ ) + 1) d x + Z Ω z \ (Ω ◦ z ∪ V L[ ) µ ( t ( L [ ) −
1) d x ! = µ ( t ( L [ ) − C (cid:16) vol(Ω z \ Ω ◦ z ) + (cid:16) − q t ( L [ ) − (cid:17) vol(Ω ◦ z ) (cid:17) = µ ( t ( L [ ) − C (cid:16) vol(Ω z ) − q t ( L [ ) − vol(Ω ◦ z ) (cid:17) , where we have used t ( L [ ) > z ) = q t ( L [ ) − vol(Ω ◦ z ) . (2.5)We fix an orthogonal decomposition L [ = L ⊕ L in which L is self-dual and L is ofboth rank and type t ( L [ ). Since both Ω z and Ω ◦ z depend only on the coset z + L [ , we mayassume z ∈ L ∨ and anisotropic. Let V ⊆ V be the orthogonal complement of L + h z i . Weclaim( ∗ ) There exists an integral O E -lattice L of V of of type t ( L [ ) such that( u i L ∨ ) int = { x ∈ V int2 | x ∈ u i L ∨ } (2.6)for i = 0 , ∗ ), by construction, we have { x ∈ V | ( x, z ) V ∈ u − O E } = L ⊗ O F F ⊕ h z i ∨ ⊕ V . Now we use the condition z V int , which implies that h z i ∨ ⊆ u h z i ∩ V int . Combining with(2.6), we obtain Ω z = L × h z i ∨ × ( L ∨ ) int , Ω ◦ z = L × h z i ∨ × ( uL ∨ ) int . Thus, (2.5) follows from Lemma 2.26 below. Part (3) is proved.Now we show ( ∗ ). There are two cases. HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 17 First, we assume z = z , that is, z V L [ . Let L be the unique O E -lattice of V satisfying L ∨ = { x ∈ V | x ∈ L ∨ } . (2.7)Then (2.6) clearly holds. Thus, it remains to show that L is integral of type t ( L [ ). Put w := z − z ∈ V ⊥ L [ which is nonzero hence anisotropic. Then¯ z := z − ( z , z ) V ( w, w ) V w is the unique element in V such that ¯ z = z . To compute L , we write L ∨ = M + h y + αz i for some y ∈ V L [ ∩ V and α ∈ E \ uO E , where M := L ∨ ∩ V . Then M † := L ∩ V = { x ∈ M ∨ | ( x, y ) V ∈ u − O E } . Since M ∨ /M † is isomorphic to an O E -submodule of E/u − O E , we may take an element y † ∈ M ∨ that generates M ∨ /M † . Then we have L = M † + h y † + α † z i for some α † ∈ E × such that ( y † , y ) V + α † α c ( z , z ) V ∈ u − O E . Now by (2.7), we have L ∨ = M + h y + α ¯ z i . By the same argument, we have L = M † + h y † + α † ρ ¯ z i , where ρ := ( z , z ) V (¯ z, ¯ z ) V . By Lemma 2.24(2), we have t ( L ) = t ( L ) = t ( L [ ) as long as L is integral. Thus, it sufficesto show that y † + α † ρ ¯ z ∈ V int . We compute( y † + α † ρ ¯ z, y † + α † ρ ¯ z ) V − ( y † + α † z , y † + α † z ) V = ( α † ρ ¯ z, α † ρ ¯ z ) V − ( α † z , α † z ) V = Nm E/F ( α † ) ( z , z ) V (¯ z, ¯ z ) V − ( z , z ) V ! = Nm E/F ( α † )( z , z ) V ( z , z ) V ( z , z ) V + ( z ,z ) V ( w,w ) V − = Nm E/F ( α † )( z , z ) V ( w, w ) V ( z , z ) V + ( w, w ) V − ! = − ( α † ) c α † ( α † z , z ) V ( z, z ) V . As z ∈ L ∨ , we have ( α † z , z ) V ∈ u − O E . As z V int , we have ( z, z ) V u − O E . Together,we have ( α † z ,z ) V ( z,z ) V ∈ O F . Thus, y † + α † ρ ¯ z ∈ V int as y † + α † z ∈ V int , hence L meets therequirement in ( ∗ ).Second, we assume z = z , that is, z ∈ V L [ . Take L = ( L ∨ ∩ V ) ∨ ⊕ u δ ( V ⊥ L [ ) int for someinteger δ (cid:62) L ∨ ∩ V ) ∨ is an integral hermitian O E -moduleof type t ( L [ ) −
1. As in the previous case, we write L ∨ = M + h y + αz i for some y ∈ V L [ ∩ V and α ∈ E \ uO E , where M := L ∨ ∩ V . Then L = M † + h y † + α † z i so that M ∨ is generated by M † and y † . As L is of type t ( L [ ) which is its rank, we have L ⊆ uL ∨ , that is, M † + h y † + α † z i ⊆ uM + u h y + αz i hence M † ⊆ uM . As z ∈ L ∨ , we have ( αz , z ) V ∈ u − O E . As z = z V int , wehave ( z , z ) V u − O E hence α † ∈ uO E . Again as z ∈ L ∨ , we have α † z ∈ uL ∨ hence y † ∈ uL ∨ ∩ V = uM . Together, we obtain M ∨ ⊆ uM , that is, ( L ∨ ∩ V ) ∨ is an integralhermitian O E -module of type t ( L [ ) − L is an integral O E -lattice of V of type t ( L [ ). Since L ∨ = ( L ∨ ∩ V ) ⊕ u − δ − ( V ⊥ L [ ) int , it is clear that for δ sufficiently large, (2.6) holds for i = 0 ,
1. Thus, ( ∗ ) isproved.The lemma is all proved. (cid:3) Lemma 2.26.
Let L be an integral hermitian O E -module of rank m + 1 for some integer m (cid:62) with t ( L ) = 2 m + 1 . Then we have (cid:12)(cid:12)(cid:12) ( L ∨ ) int /L (cid:12)(cid:12)(cid:12) = q m · (cid:12)(cid:12)(cid:12) ( uL ∨ ) int /L (cid:12)(cid:12)(cid:12) . (2.8) Note that both ( L ∨ ) int and ( uL ∨ ) int are stable under the translation by L as t ( L ) = 2 m + 1 .Proof. Put V := L ⊗ O F F . We prove by induction on val( L ) for integral O E -lattices L of V with t ( L ) = 2 m + 1 that (2.8) holds.The initial case is such that val( L ) = 2 m + 1, that is, L ∨ = u − L . The pairing u ( , ) V induces a nondegenerate quadratic form on L ∨ /L . It is clear that ( L ∨ ) int /L is exactly theset of isotropic vectors in L ∨ /L under the previous form. In particular, we have (cid:12)(cid:12)(cid:12) ( L ∨ ) int /L (cid:12)(cid:12)(cid:12) = q m = q m · (cid:12)(cid:12)(cid:12) ( uL ∨ ) int /L (cid:12)(cid:12)(cid:12) . Now we consider L with val( L ) > m + 1, and suppose that (2.8) holds for such L with val( L ) < val( L ). Choose an orthogonal decomposition L = L ⊕ L in which L isan integral hermitian O E -module with fundamental invariants (1 , . . . ,
1) and such that allfundamental invariants of L are at least 2. In particular, L has positive rank. It is easy tosee that we may choose a hermitian O E -module L contained in u − L satisfying L (cid:32) L and t ( L ) = t ( L ). Put L := L ⊕ L . By the induction hypothesis, we have (cid:12)(cid:12)(cid:12) ( L ) int /L (cid:12)(cid:12)(cid:12) = q m · (cid:12)(cid:12)(cid:12) ( uL ) int /L (cid:12)(cid:12)(cid:12) . It remains to show that (cid:12)(cid:12)(cid:12) (( L ∨ ) int \ ( L ) int ) /L (cid:12)(cid:12)(cid:12) = q m · (cid:12)(cid:12)(cid:12) (( uL ∨ ) int \ ( uL ) int ) /L (cid:12)(cid:12)(cid:12) . (2.9)We claim that the map(( L ∨ ) int \ ( L ) int ) /L → (( uL ∨ ) int \ ( uL ) int ) /L given by the multiplication by u is q m -to-1. Take an element x ∈ ( uL ∨ ) int \ ( uL ) int . Itspreimage is bijective to the set of elements ( y , y ) ∈ L /uL ⊕ L /uL such that u − ( x +( y , y )) ∈ V int , which amounts to the equation( x, x ) V + Tr E/F ( x, y ) V + Tr E/F ( x, y ) V + ( y , y ) V ∈ u O F . Since x ∈ ( uL ∨ ) × (( uL ∨ ) int \ ( u L ∨ ) int ), there exists y ∈ L such that ( x, y ) V ∈ O × E . Inother words, for each y , the above relation defines a nontrivial linear equation on L /uL .Thus, the preimage of x has cardinality q m . We obtain (2.9) hence complete the inductionprocess. (cid:3) HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 19 Proof of Proposition 2.23.
We fix an element L [ ∈ [ ( V ). If L [ is not integral, then ∂ Den v L [ ≡ L [ integral and will freely adoptnotation from Lemma 2.25.To show that ∂ Den v L [ extends to a compactly supported locally constant function on V ,it suffices to show that for every y ∈ V L [ /L [ , there exists an integer δ ( y ) > ∂ Den v L [ ( y + x ) is constant for x ∈ u δ ( y ) ( V ⊥ L [ ) int \ { } . If L [ + h y i is not integral, then thereexists δ ( y ) > L [ + h y + x i is not integral for x ∈ u δ ( y ) ( V ⊥ L [ ) int \ { } , which implies ∂ Den v L [ ( y + x ) = 0.Now we fix an element y ∈ V L [ /L [ such that L [ + h y i is integral. We claim that we maytake δ ( y ) = a n +1 , which is the maximal element in the fundamental invariants of L [ . Itamounts to showing that for every fixed pair ( f , f ) of generators of the O E -module ( V ⊥ L [ ) int ,we have ∂ Den v L [ ( y + u δ f ) − ∂ Den v L [ ( y + u δ − f ) = 0(2.10)for δ > a n − . For every δ ∈ Z , we define two sets L δ := { L ∈ L | L ⊆ L ∨ , δ L = δ , y + u δ f ∈ L } , L δ := { L ∈ L | L ⊆ L ∨ , δ L = δ , y + u δ − f ∈ L } . By Remark 2.22(4), we have ∂ Den v L [ ( y + u δ f ) = 2 X δ (cid:54) δ X L ∈ L δ t ( L ∩ V L[ ) > µ ( t ( L )) = 2 X L [ ⊆ L [ ⊆ ( L [ ) ∨ t ( L [ ) > X δ (cid:54) δ X L ∈ L δ L ∩ V L[ = L [ µ ( t ( L )) ,∂ Den v L [ ( y + u δ − f ) = 2 X δ (cid:54) δ − X L ∈ L δ t ( L ∩ V L[ ) > µ ( t ( L )) = 2 X L [ ⊆ L [ ⊆ ( L [ ) ∨ t ( L [ ) > X δ (cid:54) δ − X L ∈ L δ L ∩ V L[ = L [ µ ( t ( L )) . Now we claim that X δ (cid:54) δ X L ∈ L δ L ∩ V L[ = L [ µ ( t ( L )) − X δ (cid:54) δ − X L ∈ L δ L ∩ V L[ = L [ µ ( t ( L )) = 0(2.11)for every L [ in the summation. Since δ > a n +1 , it follows that for δ <
0, we have L δ = L δ = { L ∈ L | L ⊆ L ∨ , δ L = δ , y ∈ L } . Thus, the left-hand side of (2.11) equals δ X δ =0 X L ∈ L δ L ∩ V L[ = L [ µ ( t ( L )) − δ − X δ =0 X L ∈ L δ L ∩ V L[ = L [ µ ( t ( L )) . (2.12)However, we also have L = { L ∈ L | L ⊆ L ∨ , δ L = δ , y ∈ L } , which implies X L ∈ L L ∩ V L[ = L [ µ ( t ( L )) = L [ ( y ) X L ⊆ L ∨ L ∩ V L[ = L [ δ L =0 µ ( t ( L )) , which vanishes by Lemma 2.25(4). Thus, we obtain(2.12) = δ X δ =1 X L ∈ L δ L ∩ V L[ = L [ µ ( t ( L )) − δ − X δ =0 X L ∈ L δ L ∩ V L[ = L [ µ ( t ( L )) . (2.13)Finally, the automorphism of E sending ( L [ , δ , ε ) to ( L [ , δ − , ε ◦ ( uα · )), where α ∈ O × E isthe element satisfying f = αf , induces a bijection from L δ to L δ − preserving both L ∩ V L [ and t ( L ). Thus, (2.13) vanishes hence (2.11) and (2.10) hold.Now we show that the support of (cid:92) ∂ Den v L [ is contained in V int . Take an element z ∈ V \ V int . Using Remark 2.22(4), we have (cid:92) ∂ Den v L [ ( z ) = Z V (cid:92) ∂ Den v L [ ( x ) ψ (Tr E/F ( x, z ) V ) d z = 2 X L [ ⊆ L ⊆ L ∨ t ( L ∩ V L[ ) > µ ( t ( L )) vol( L ) L ∨ ( z )= 2 X L [ ⊆ L [ ⊆ ( L [ ) ∨ t ( L [ ) > X L ⊆ L ∨ L ∩ V L[ = L [ z ∈ L ∨ µ ( t ( L )) vol( L )= 2 X L [ ⊆ L [ ⊆ ( L [ ) ∨ t ( L [ ) > vol( L [ ) vol(( V ⊥ L [ ) int ) X L ⊆ L ∨ L ∩ V L[ = L [ z ∈ L ∨ q − δ L µ ( t ( L )) , which is valid and vanishes by Lemma 2.25(3).Proposition 2.23 is proved. (cid:3) Bruhat–Tits stratification.
Let the setup be as in Subsection 2.1. We assume Hy-pothesis 2.4.We first generalize Definition 2.7 to a more general context. For every subset X of V suchthat h X i is finitely generated, we put N ( X ) := \ x ∈ X N ( x ) , which is always a finite intersection, and depends only on h X i . Clearly, we have N ( X ) ⊆N ( X ) if h X i ⊆ h X i . When X = { x, . . . } is explicitly presented, we simply write N ( x, . . . )instead of N ( { x, . . . } ). Remark . When h X i is an O E -lattice of V , the formal subscheme N ( X ) is a properclosed subscheme of N . This can be proved by the same argument for [LZ, Lemma 2.10.1]. Definition 2.28.
Let Λ be a vertex O E -lattice of L (Definition 2.12).(1) We equip the k -vector space Λ ∨ / Λ with a k -valued pairing ( , ) Λ ∨ / Λ by the formula( x, y ) Λ ∨ / Λ := u Tr E/F ( x ] , y ] ) V mod ( u )where x ] and y ] are arbitrary lifts of x and y , respectively. Then Λ ∨ / Λ becomes anonsplit (nondegenerate) quadratic space over k of (even positive) dimension t (Λ). HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 21 (2) Let V Λ be the reduced subscheme of N (Λ), and put V ◦ Λ := V Λ − [ Λ (cid:36) Λ V Λ . Proposition 2.29 (Bruhat–Tits stratification, [Wu]) . The reduced subscheme N red is a dis-joint union of V ◦ Λ for all vertex O E -lattices Λ of V in the sense of stratification, such that V Λ ∩ V Λ coincides with V Λ+Λ (resp. is empty) if Λ + Λ is (resp. is not) a vertex O E -lattice.Moreover, for every vertex O E -lattice Λ ,(1) V Λ is canonically isomorphic to the generalized Deligne–Lusztig variety of O(Λ ∨ / Λ) over k classifying maximal isotropic subspaces U of (Λ ∨ / Λ) ⊗ k k satisfying dim( U ∩ δ ( U )) = t (Λ)2 − , where δ ∈ Gal( k/k ) denotes the Frobenius element;(2) the intersection of V Λ with each connected component of N red is connected, nonempty,and smooth projective over k of dimension t (Λ)2 − .Proof. This follows from [Wu, Proposition 5.13 & Theorem 5.18]. Note that we use latticesin V , which is different from the hermitian space C used in [Wu], to parameterize strata.By the obvious analogue of [KR11, Lemma 3.9], we may naturally identify V with C , afterwhich the stratum S Λ in [Wu] coincides with our stratum V u Λ ∨ . (cid:3) Remark . In the above proposition, when t (Λ) = 4, V Λ is isomorphic to two copies of P k , though we do not need this explicit description in the following. Corollary 2.31.
For every nonzero element x ∈ V , we have N ( x ) red = [ x ∈ Λ V ◦ Λ where the union is taken over all vertex O E -lattices of V containing x .Proof. Since N ( x ) red is a reduced closed subscheme of N red , it suffices to check that N ( x )( k ) = [ x ∈ Λ V ◦ Λ ( k ) . By Definition 2.28(2), we have N ( x )( k ) ⊇ [ x ∈ Λ V ◦ Λ ( k ) . For the other direction, by Proposition 2.29, we have to show that if Λ does not contain x ,then N ( x )( k ) ∩ V ◦ Λ ( k ) = ∅ . Suppose that we have s ∈ N ( x )( k ) ∩ V ◦ Λ ( k ), then s should belongto V Λ ( k ) where Λ is the O E -lattice generated by Λ and x . In particular, Λ is vertex andstrictly contains Λ. But this contradicts with the definition of V ◦ Λ . The corollary follows. (cid:3) Corollary 2.32.
Suppose that r (cid:62) . For every x ∈ V , the intersection of N ( x ) with eachconnected component of N red is strictly a closed subscheme of the latter.Proof. By Corollary 2.31 and Proposition 2.29(2), it suffices to show that the intersection ofall vertex O E -lattices of V is { } .Take a nonsplit hermitian subspace V of V of dimension 2 and an O E -lattice L of V offundamental invariants (1 , V ⊥ of V in V admits a self-dual O E -lattice L . Choose a normal basis (Remark 2.14) { e , . . . , e r − } of L under which the moment matrix is given by (cid:16) u − − u − (cid:17) ⊕ r − . For every tuple a = ( a , . . . , a r − ) ∈ Z r − satisfying a i − + a i = 0 for 1 (cid:54) i (cid:54) r −
1, the O E -latticeΛ a := L ⊕ h u a e , . . . , u a r − e r − i is integral with fundamental invariants (0 , . . . , , , a is L . Since r (cid:62)
2, the intersection of all 2-dimensional nonsplithermitian subspaces of V is { } . Thus, the intersection of all vertex O E -lattices of V is { } . (cid:3) Lemma 2.33.
Let Λ be a vertex O E -lattice of V . For each connected component V +Λ of V Λ and integer d (cid:62) , the group of d -cycles of V +Λ , up to ‘ -adic homological equivalence for everyrational prime ‘ = p , is generated by V Λ ∩ V +Λ for all vertex O E -lattices Λ containing Λ with t (Λ ) = 2 d + 2 .Proof. Let k be the quadratic extension of k in k . Note that V +Λ has a canonical structureover k , so that V ◦ +Λ := V ◦ Λ ∩ V +Λ (over k ) is the classical Deligne–Lusztig variety of SO(Λ ∨ / Λ)of Coxeter type.Recall that δ is the Frobenius element of Gal( k/k ). Fix a rational prime ‘ different from p . For every finite dimensional Q ‘ -vector space V with an action by δ , we denote by V † the subspace consisting of elements on which δ acts by roots of unity. Then for the lemma,it suffices to show that for every d (cid:62)
0, H d ( V +Λ , Q ‘ ( − d )) † is generated by (the cycle classof) V Λ ∩ V +Λ for all vertex O E -lattices Λ containing Λ with t (Λ ) = 2 d + 2. By the sameargument for [LZ, Theorem 5.3.2], it reduces to the following claim:( ∗ ) The action of δ on V := L j (cid:62) H j ( V ◦ +Λ , Q ‘ ( j )) is semisimple, and V † = H ( V ◦ +Λ , Q ‘ ).There are three cases.When t (Λ) = 2, V ◦ +Λ is isomorphic to Spec k hence ( ∗ ) is trivial.When t (Λ) = 4, V ◦ +Λ is an affine curve hence ( ∗ ) is again trivial.When t (Λ) (cid:62)
6, by Case D n (with n = t (Λ)2 (cid:62)
3) in [Lus76, Section 7.3], the action of δ on L j (cid:62) H jc ( V ◦ +Λ , Q ‘ ) has eigenvalues { , q , q , . . . , q t (Λ) − } and that the eigenvalue q j appearsin H j + t (Λ)2 − c ( V ◦ +Λ , Q ‘ ). Moreover by [Lus76, Theorem 6.1], the action of δ is semisimple.Thus, ( ∗ ) follows from the Poincaré duality.The lemma is proved. (cid:3) Linear invariance of intersection numbers.
Let the setup be as in Subsection 2.1.We assume Hypothesis 2.4.For every nonzero element x ∈ V , we define a chain complex of locally free O N -modules C ( x ) := (cid:16) · · · → → I N ( x ) → O N → (cid:17) supported in degrees 1 and 0 with the map I N ( x ) → O N being the natural inclusion. Weextend the definition to x = 0 by setting C (0) := (cid:16) · · · → → ω −→ O N → (cid:17) (2.14)supported in degrees 1 and 0, where ω is the line bundle from Definition 2.39.The following is our main result of this subsection. HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 23 Proposition 2.34.
Let (cid:54) m (cid:54) n be an integer. Suppose that x , . . . , x m ∈ V and y , . . . , y m ∈ V generate the same O E -submodule. Then we have an isomorphism H i ( C ( x ) ⊗ O N · · · ⊗ O N C ( x m )) ’ H i ( C ( y ) ⊗ O N · · · ⊗ O N C ( y m )) of O N -modules for every i . Proposition 2.34 has the following two immediate corollaries.
Corollary 2.35.
Let (cid:54) m (cid:54) n be an integer. Suppose that x , . . . , x m ∈ V and y , . . . , y m ∈ V generate the same O E -submodule. Then we have [ C ( x ) ⊗ O N · · · ⊗ O N C ( x m )] = [ C ( y ) ⊗ O N · · · ⊗ O N C ( y m )] in K ( N ) , where K ( N ) denotes the K-group of N [LL, Section B] . Corollary 2.36.
Suppose that x , . . . , x n ∈ V generate an O E -lattice of V . The Serreintersection multiplicity χ (cid:18) O N ( x ) L ⊗ O N · · · L ⊗ O N O N ( x n ) (cid:19) := X i,j (cid:62) ( − i + j length O ˘ E H j (cid:18) N , H i (cid:18) O N ( x ) L ⊗ O N · · · L ⊗ O N O N ( x n ) (cid:19)(cid:19) depends only on the O E -lattice of V generated by x , . . . , x n . Note that by Remark 2.27, theabove number is finite. Now we start to prove Proposition 2.34, following [How19]. Let (
X, ι X , λ X ) be the universalobject over N . We have a short exact sequence0 → Fil( X ) → D( X ) → Lie( X ) → O N -modules, where D( X ) denotes the covariant crystal of X restricted tothe Zariski site of N . Then ι X induces actions of O E on all terms so that the short exactsequence is O E -linear.We define an O N -submodule F X of Lie( X ) as the kernel of ι X ( u ) − u on Lie( X ), which isstable under the O E -action. Lemma 2.37.
The O N -submodule F X is locally free of rank n − and is locally a directsummand of Lie( X ) .Proof. Let s ∈ N ( k ) be a closed point. By the Wedge condition and the Spin condition inDefinition 2.1, we know that the map ι X ( u ) − u : Lie( X ) ⊗ O N O N ,s → Lie( X ) ⊗ O N O N ,s has rank 1 on both generic and special fibers. Thus, F X ⊗ O N O N ,s is a direct summand ofLie( X ) ⊗ O N O N ,s of rank n −
1. The lemma follows. (cid:3)
The symmetrization σ X of the polarization λ X (Remark 2.2) induces a perfect symmetric O N -bilinear pairing ( , ) : D( X ) × D( X ) → O N satisfying ( ι X ( α ) x, y ) = ( x, ι X ( α c ) y ) for every α ∈ O E and x, y ∈ D( X ). As Fil( X ) is amaximal isotropic O N -submodule of D( X ) with respect to ( , ), we have an induced perfect O N -bilinear pairing ( , ) : Fil( X ) × Lie( X ) → O N , under which we denote by F ⊥ X ⊆ Fil( X ) the annihilator of F X . Then the O N -submodule F ⊥ X is locally free of rank 1 and is locally a direct summand of Fil( X ).Following [How19, Section 3], we put (cid:15) := u ⊗ ⊗ u ∈ O E ⊗ O F O N ,(cid:15) c := u ⊗ − ⊗ u ∈ O E ⊗ O F O N . Lemma 2.38.
There are inclusions of O U -modules F ⊥ X ⊆ (cid:15) D( X ) ⊆ D( X ) which are locallydirect summands. The map (cid:15) : D( X ) → (cid:15) D( X ) descends to a surjective map Lie( X ) (cid:15) −→ (cid:15) D( X ) /F ⊥ X , whose kernel L X is locally a direct summand O U -submodule of Lie( X ) of rank . Moreover,the O E -action stabilizes L X , and acts on Lie( X ) /L X and L X via ϕ and ϕ c , respectively.Proof. This follows from the same proof for [How19, Proposition 3.3]. (cid:3)
Definition 2.39.
We define the line bundle of modular forms ω to be L − X , where L X is theline bundle on N from Lemma 2.38.For every closed formal subscheme Z of N , we denote by e Z the closed formal subschemedefined by the sheaf I Z . Take a nonzero element x ∈ V . By the definition of N ( x ), we havea distinguished morphism X | N ( x ) x −→ X | N ( x ) of O F -divisible groups, which induces an O E -linear mapD( X ) | N ( x ) x −→ D( X ) | N ( x ) of vector bundles. By the Grothendieck–Messing theory, the above map admits a canonicalextension D( X ) | (cid:94) N ( x ) ˜ x −→ D( X ) | (cid:94) N ( x ) , which further restricts to a mapFil( X ) | (cid:94) N ( x ) ˜ x −→ Lie( X ) | (cid:94) N ( x ) . (2.15)From now on, we fix a generator γ of the rank 1 free O ˘ E -module Fil( X ). Lemma 2.40.
The image ˜ x ( γ ) is a section of L X over (cid:93) N ( x ) , whose vanishing locus coincideswith N ( x ) , where ˜ x is the map (2.15) .Proof. This follows from the same proof for [How19, Proposition 4.1]. (cid:3)
The following lemma is parallel to [KR11, Proposition 3.5].
Lemma 2.41.
For every nonzero element x ∈ V , the closed formal subscheme N ( x ) of N is either empty or a relative Cartier divisor.Proof. The case r = 1 has been proved in [RSZ17, Proposition 6.6]. Thus, we now assume r (cid:62) N ( x ) is nonempty. By the same argument in the proof of [How19,Proposition 4.3], N ( x ) is locally defined by one equation. It remains to show that suchequation is not divisible by u . Since r (cid:62)
2, this follows from [KR11, Lemma 3.6], Lemma2.6, and Corollary 2.32. (cid:3)
HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 25 Proof of Proposition 2.34.
The proof of [How19, Theorem 5.1] can be applied in the sameway to Proposition 2.34, using Lemma 2.40 and Lemma 2.41. (cid:3)
To end this subsection, we prove some results that will be used later.
Lemma 2.42.
The O N -submodule L X from Lemma 2.38 coincides with the image of themap ι X ( u ) − u : Lie( X ) → Lie( X ) .Proof. Denote by L X the image of the map ι X ( u ) − u : Lie( X ) → Lie( X ). As we have L X ’ Lie( X ) /F X , L X is a locally free O N -submodule of Lie( X ) of rank 1 by Lemma 2.37.By the Spin condition in Definition 2.1, for every closed point s ∈ N ( k ), the induced map L X ⊗ O N k → Lie( X ) ⊗ O N k over the residue field at s is injective. Thus, the quotient O N -module Lie( X ) /L X is locally free. It remains to show that L X ⊆ L X .By definition, every section of L X can be locally written as the image of ( ι X ( u ) − u ) x forsome section x of D( X ). We need to show that(1) (cid:15) ( ι X ( u ) − u ) x is a section of Fil( X );(2) ( (cid:15) ( ι X ( u ) − u ) x, y ) = 0 for every section y of F X .For (1), we have (cid:15) ( ι X ( u ) − u ) x = ( ι X ( u ) + u )( ι X ( u ) − u ) x = ( ι X ( u ) − u ) x . Since ι X ( u ) − u acts by zero on Lie( X ), (1) follows. For (2), we have( (cid:15) ( ι X ( u ) − u ) x, y ) = (( ι X ( u ) − u ) x, ( − ι X ( u ) + u ) y ) = 0as y is a section of ker( ι X ( u ) − u ). Thus, (2) follows.The lemma is proved. (cid:3) Lemma 2.43.
Let Λ be a vertex O E -lattice of V with t (Λ) = 4 . Then ω has degree q − on each connected component of (the smooth projective curve) V Λ (Definition 2.28).Proof. Let δ be the Frobenius element of Gal( k/k ).Let s ∈ N ( k ) be a closed point represented by the quadruple ( X, ι X , λ X ; ρ X ). Let M bethe covariant O F -Dieudonné module of X equipped with the O E -action ι X , which becomesa free O ˘ E -module. We have Lie( X ) = M / VM . By Definition 2.39 and Lemma 2.42, thefiber ω − | s is canonically identified with (( u ⊗ M + VM ) / VM , which is further canonicallyisomorphic to (( u ⊗ V − M + M ) / M . By the identification between V Λ and the generalizedDeligne–Lusztig variety of O(Λ ∨ / Λ) in Proposition 2.29 given in [Wu, Proposition 4.29], weknow that ω − | V Λ coincides with ( δ ( U ) + U ) /U where U is the tautological subbundle of(Λ ∨ / Λ) ⊗ k O V Λ .To compute the degree of ( δ ( U )+ U ) /U , let V +Λ and V − Λ be the two connected components of V Λ . Let L Λ be the scheme over k classifying lines in Λ ∨ / Λ with the tautological bundle L . Wemay identify V +Λ and V − Λ as two closed subschemes of L Λ via the assignment U δ ( U ) ∩ U (see[HP14, Section 3.2] for more details). Then, V +Λ and V − Λ are the two irreducible componentsof the locus where L and δ ( L ) generate an isotropic subspace, and the assignment L δ ( L )switches V +Λ and V − Λ . Let I Λ be the locus where L is isotropic and L = δ ( L ). Then I Λ is adisjoint union of q + 1 copies of Spec k since there are exactly q + 1 isotropic lines in Λ ∨ / Λ,and is contained in V +Λ ∩ V − Λ . Note that the map δ ( U ) / ( δ ( U ) ∩ U ) → ( δ ( U ) + U ) /U is anisomorphism, and there is a short exact sequence0 → δ ( δ ( U ) ∩ U ) → δ ( U ) / ( δ ( U ) ∩ U ) → O I Λ → of O V ± Λ -modules. Since δ ( U ) ∩ U is the restriction of the tautological bundle L on L Λ , wehave deg (cid:16) ω − | V ± Λ (cid:17) = deg (cid:16) ( δ ( U ) + U ) /U | V ± Λ (cid:17) = deg (cid:16) δ ( δ ( U ) ∩ U ) | V ± Λ (cid:17) + ( q + 1)= deg (cid:16) L ⊗ q | V ± Λ (cid:17) + ( q + 1) = − q deg( V ± Λ ) + ( q + 1) , where deg( V ± Λ ) denotes the degree of the curve V ± Λ in the projective space L Λ . Thus, itremains to show that deg( V ± Λ ) = q + 1.To compute the degree, take a 3-dimensional quadratic subspace H of Λ ∨ / Λ. Let L H Λ bethe hyperplane of L Λ that consists of lines contained in H . Then L H Λ ∩ V Λ is the subschemeof lines L ⊆ H that is isotropic and fixed by δ , which is a disjoint union of q + 1 copies ofSpec k since there are exactly q + 1 isotropic lines in H . As L H Λ ∩ V Λ is contained in I Λ , itis contained in V +Λ ∩ V − Λ . Therefore, we have deg( V ± Λ ) = q + 1.The lemma is proved. (cid:3) Proof of Theorem 2.9 when r = 1 . Let the setup be as in Subsection 2.1. Weassume Hypothesis 2.4. In this subsection, we assume r = 1. Note that since V is nonsplit,the fundamental invariants of an integral O E -lattice of V must consist of two positive oddintegers. Lemma 2.44.
Let L be an integral O E -lattice of V with fundamental invariants (2 b +1 , b + 1) . Then ∂ Den( L ) = 2 b X j =0 (cid:16) q + · · · + q j + ( b − j ) q j (cid:17) . Proof.
We denote by L the set of integral O E -lattices of V containing L . We now count L .Fix an orthogonal basis { e , e } of V with ( e , e ) V ∈ O × F and ( e , e ) V ∈ O × F and suchthat L = h u b e i + h u b e i . For every L ∈ L , we let j ( L ) be the unique integer such that L ∩ h e i ⊗ O F F = h u j ( L ) e i and let k ( L ) be the unique integer such that image of L under thenatural projection map V → h e i ⊗ O F F is h u k ( L ) e i . Then by Lemma 2.24(1), L is uniquelydetermined by j ( L ), k ( L ), and the extension map ε L : h u k ( L ) e i → h u j ( L ) e i ⊗ O F F/O F . Thecondition that L contains L is equivalent to that j ( L ) (cid:54) b , k ( L ) (cid:54) b , and that ε L vanisheson h u b e i . Since L is nonsplit, the condition that L is integral is equivalent to that j ( L ) (cid:62) k ( L ) (cid:62)
0, and that the image of ε L is contained h e i / h u j ( L ) e i . Thus, the number of L ∈ L with j ( L ) = j for some fixed 0 (cid:54) j (cid:54) b equals 1 + q + · · · + q j + ( b − j ) q j . Summing overall 0 (cid:54) j (cid:54) b , we obtain | L | = b X j =0 (cid:16) q + · · · + q j + ( b − j ) q j (cid:17) . The lemma then follows from (2.4) as t ( L ) = 2. (cid:3) Proposition 2.45.
Theorem 2.9 holds when r = 1 . More explicitly, for an integral O E -lattice L of V with fundamental invariants (2 b + 1 , b + 1) , we have Int( L ) = ∂ Den( L ) = 2 b X j =0 (cid:16) q + · · · + q j + ( b − j ) q j (cid:17) . HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 27 Proof. If L is not integral, then Int( L ) = ∂ Den( L ) = 0. If L is integral with fundamentalinvariants (2 b + 1 , b + 1). We may take an orthogonal basis { x , x } of L such thatval( x ) = 2 b + 1 and val( x ) = 2 b + 1.Put D := End O F ( X ) ⊗ Q , which is a division quaternion algebra over F with the F -linearembedding ι X : E → D . By the Serre construction, we may naturally identify D with V ,and we have an identity N ( x ) = b X j =0 W E x ,j (2.16)of divisors, decomposing the special divisor as a sum of quasi-canonical lifting divisors (see[RSZ17, Section 6 & Proposition 7.1]).We claim that for every 0 (cid:54) j (cid:54) b , the identitylength O ˘ E W E x ,j ∩ N ( x ) = 2 (cid:16) q + · · · + q j + ( b − l ) q j (cid:17) (2.17)holds. In fact, this can be proved in the same way as for [KR11, Proposition 8.4] usingKeating’s formula [Vol07, Theorem 2.1]. Notice that in [KR11, Proposition 8.4] we replace e s by 2 q j since E/F is ramified, and that the factor 2 comes from the fact that Z l has twoconnected components. By (2.16) and (2.17), we haveInt( L ) = length O ˘ E N ( x ) ∩ N ( x ) = b X j =0 (cid:16) q + · · · + q j + ( b − l ) q j (cid:17) . The proposition follows by Lemma 2.44. (cid:3)
Definition 2.46.
For L [ ∈ [ ( V ), we put N ( L [ ) ◦ := N ( L [ ) − N ( u − L [ )as an effective divisor by (the r = 1 case of) Lemma 2.41. Corollary 2.47.
Take an element L [ ∈ [ ( V ) . For every x ∈ V \ V L [ , we have length O ˘ E N ( L [ ) ◦ ∩ N ( x ) = 2 X L ⊆ L ∨ L ∩ V L[ = L [ L ( x ) . Proof.
By Proposition 2.45, we havelength O ˘ E N ( L [ ) ∩ N ( x ) = Int( L [ + h x i ) = ∂ Den( L [ + h x i ) = 2 X L ⊆ L ∨ L [ ⊆ L ∩ V L[ L ( x ) , in which the last identity is due to (2.4). Similarly, we havelength O ˘ E N ( u − L [ ) ∩ N ( x ) = 2 X L ⊆ L ∨ u − L [ ⊆ L ∩ V L[ L ( x ) . Taking the difference, we obtain the corollary. (cid:3)
Fourier transform of geometric side.
Let the setup be as in Subsection 2.1. Weassume Hypothesis 2.4. We will freely use notation concerning K-groups of formal schemesfrom [LL, Section B] and [Zha, Appendix B], based on the work [GS87].
Definition 2.48.
Let X be a formal scheme over Spf O ˘ E .(1) We denote by X h the closed formal subscheme of X defined by the ideal sheaf O X [ p ∞ ].(2) We say that an element in K ( X ) has proper support if it belongs to the subgroupK Z ( X ) for a proper closed subscheme Z of X . Definition 2.49.
Let X be a subset of V such that h X i is finitely generated of rank m .(1) We denote by N K ( X ) ∈ K ( N ) the element [ C ( x ) ⊗ O N · · ·⊗ O N C ( x m )] from Subsection2.4 for a basis { x , . . . , x m } of the O E -module generated by X , which is independentof the choice of the basis by Corollary 2.35.(2) We denote by N K ( X ) h ∈ K ( N ) the class of N ( X ) h .(3) We put N K ( X ) v := N K ( X ) − N K ( X ) h ∈ K ( N ). Lemma 2.50.
Let L [ be an element of [ ( V ) (Definition 2.10). We have(1) N ( L [ ) h is either empty or finite flat over Spf O ˘ E ;(2) all of N K ( L [ ) , N K ( L [ ) h , and N K ( L [ ) v belong to F n − K ( N ) ;(3) N K ( L [ ) v has proper support.Proof. Part (1) follows from Lemma 2.55 and Lemma 2.54.Take a basis { x , . . . , x n − } of the O E -module L [ .For (2), it suffices to show N K ( L [ ) ∈ F n − K ( N ) by (1). By definition, N K ( L [ ) is thecup product of the classes in K ( N ) of N ( x ) , . . . , N ( x n − ), each being a divisor by Lemma2.41. Thus, N K ( L [ ) belongs to F n − K ( N ) by (the analogue for formal schemes of) [GS87,Proposition 5.5].For (3), by the same argument for [LZ, Lemma 5.1.1], we know that there exists a properclosed subscheme Z of N such that N ( L [ ) is contained in N ( L [ ) h S Z . By (1), the difference N K ( x ) h . · · · . N K ( x n − ) h − N K ( L [ ) h belongs to K Z ( N ). By definition, N K ( L [ ) v = N K ( x ) . · · · . N K ( x n − ) − N K ( L [ ) h = ( N K ( x ) v + N K ( x ) h ) . · · · . ( N K ( x n − ) v + N K ( x n − ) h ) − N K ( L [ ) h = C + (cid:16) N K ( x ) h . · · · . N K ( x n − ) h − N K ( L [ ) h (cid:17) , where C := n − X i =1 N K ( x i ) v . N K ( x ) . · · · . (cid:92) N K ( x i ) . · · · . N K ( x n − )belongs to K Z ( N ). Thus, N K ( L [ ) v belongs to K Z ( N ) hence has proper support. (cid:3) Definition 2.51.
Let L [ be an element of [ ( V ) (Definition 2.10). For x ∈ V \ V L [ , we putInt L [ ( x ) := N K ( L [ ) . N K ( x ) , Int h L [ ( x ) := N K ( L [ ) h . N K ( x ) , Int v L [ ( x ) := N K ( L [ ) v . N K ( x ) . HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 29 Here, the intersection numbers are well-defined since N ( L [ ) ∩ N ( x ) is a proper closed sub-scheme of N by Remark 2.27. Note that Int L [ ( x ) = Int( L [ + h x i ) (Definition 2.8).The following is our main result of this subsection. Proposition 2.52.
Let L [ be an element of [ ( V ) (Definition 2.10).(1) We have Int h L [ ( x ) = ∂ Den h L [ ( x ) for x ∈ V \ V L [ , where ∂ Den h L [ is from Definition 2.21.(2) The function Int v L [ extends (uniquely) to a (compactly supported) locally constant func-tion on V , which we still denote by Int v L [ . Moreover, we have (cid:91) Int v L [ = − Int v L [ . In particular, the support of (cid:91)
Int v L [ is contained in V int (Definition 2.12). The rest of this subsection is devoted to the proof of this proposition.
Remark . Let V be a hermitian subspace of V thatis nonsplit and of positive even dimension n . Let L be an integral hermitian O E -modulecontained in V such that L ∩ V is a self-dual O E -lattice of V . We may choose • an object ( X , ι X , λ X ) ∈ Exo b( n − , ( k ) (Definition 2.1), • an object ( Y, ι Y , λ Y ) ∈ Exo ( n − n , ( O ˘ E ) (Remark 2.11), • a quasi-morphism % from ( Y, ι Y , λ Y ) ⊗ O ˘ E k ⊕ ( X , ι X , λ X ) to ( X , ι X , λ X ) in the categoryExo b( n − , ( S ⊗ O ˘ E k ) satisfying – % identifies Hom O E ( X ⊗ O ˘ E k, X ) ⊗ Q with V as hermitian spaces; – % identifies Hom O E ( X ⊗ O ˘ E k, Y ⊗ O ˘ E k ) with L ∩ V as hermitian O E -modules.Let N := N ( X ,ι X ,λ X ) be the relative Rapoport–Zink space for the triple ( X , ι X , λ X )(Definition 2.3). We have a morphism N → N such that for every object S of Sch v /O ˘ E , N ( S ) it sends an object ( X , ι X , λ X ; ρ X ) ∈ N ( S ) to the object( Y ⊗ O ˘ E S ⊕ X , ι Y ⊗ O ˘ E S ⊕ ι X , λ Y ⊗ O ˘ E S ⊕ λ X ; % ◦ (id Y ⊗ O ˘ E S ⊕ ρ X )) ∈ N ( S ) . We have(1) The morphism N → N above identifies N with the closed formal subscheme N ( L ∩ V ) of N .(2) Suppose that L ∩ V = { } , then N ( L ) coincides with the image of N ( L ∩ V ) underthe morphism N → N above.(3) For a nonzero element x ∈ V written as x = y + x with respect to the orthogonaldecomposition V = V ⊕ V , we have N × N N ( x ) = ∅ , if y L ∩ V , N , if y ∈ L ∩ V and x = 0, N ( x ) , if y ∈ L ∩ V and x = 0.(4) If L is an O E -lattice of V , then we have Int( L ) = Int( L ∩ V ). Lemma 2.54.
Let L [ ∈ [ ( V ) be an element that is integral and satisfies t ( L [ ) = 1 .(1) The formal subscheme N ( L [ ) is finite flat over Spf O ˘ E . When n = n , we simply ignore ( Y, ι Y , λ Y ). (2) If we put N ( L [ ) ◦ := N ( L [ ) − N ( L [ ) as an element in F n − K ( N ) , then for every x ∈ V \ V L [ , N ( L [ ) ◦ . N K ( x ) = 2 X L ⊆ L ∨ L ∩ V L[ = L [ L ( x ) . Here, L [ is the unique element in [ ( V ) satisfying L [ ⊆ L [ ⊆ ( L [ ) ∨ with | L [ /L [ | = q (so that L [ is either not integral, or is integral with t ( L [ ) = 1 ).Proof. Since t ( L [ ) = 1, we may choose a 2-dimensional (nonsplit) hermitian subspace V of V such that L [ ∩ V is a self-dual O E -lattice of V . We adopt the construction in Remark2.53.For (1), we have N ( L [ ) = N ( L [ ∩ V ), which is finite flat over Spf O ˘ E by (the r = 1 caseof) Lemma 2.41.For (2), we write x = y + x with respect to the orthogonal decomposition V = V ⊕ V .Since x V L [ , we have x = 0. By Remark 2.53(2), N ( L [ ) ◦ coincides with (the class of) N ( L [ ∩ V ) ◦ in F K ( N ) under the map F K ( N ) → F n − K ( N ). There are two cases.If y L [ ∩ V , then N ( L [ ) ◦ . N K ( x ) = 0 by Remark 2.53(3), and there is no integral O E -lattice of V containing L [ + h x i . Thus, (2) follows.If y ∈ L [ ∩ V , then by Remark 2.53(3), we have N ( L [ ) ◦ . N K ( x ) = N ( L [ ∩ V ) ◦ . N K ( x ) = length O ˘ E N ( L [ ∩ V ) ◦ ∩ N ( x ) . By Corollary 2.47, we havelength O ˘ E N ( L [ ∩ V ) ◦ ∩ N ( x ) = 2 X L ⊆ L ( ⊆ V ) L ∩ ( V L[ ∩ V )= L [ ∩ V L ( x ) = 2 X L ⊆ L ∨ L ∩ V L[ = L [ L ( x ) . Thus, (2) follows. (cid:3)
Lemma 2.55.
Let L [ be an element of [ ( V ) (Definition 2.10). We have N ( L [ ) h = [ L [ ⊆ L [ ⊆ ( L [ ) ∨ t ( L [ )=1 N ( L [ ) ◦ as closed formal subschemes of N , and the identity N K ( L [ ) h = X L [ ⊆ L [ ⊆ ( L [ ) ∨ t ( L [ )=1 N ( L [ ) ◦ in F n − K ( N ) / F n K ( N ) , where N ( L [ ) ◦ is introduced in Lemma 2.54(2).Proof. This lemma can be proved by the same way as for [LZ, Theorem 4.2.1], as long as weestablish the following claim replacing [LZ, Lemma 4.5.1] in the case where
E/F is ramified. • Let L be a self-dual hermitian O E -module of rank n and L [ a hermitian O E -modulecontained in L . If L/L [ is free, then L [ is integral with t ( L [ ) = 1.However, this is just a special case of Lemma 2.24(2). (cid:3) Lemma 2.56.
Let Λ be a vertex O E -lattice of V with t (Λ) = 4 . Take an arbitrary connectedcomponent V +Λ of the smooth projective curve V Λ from Proposition 2.29, regarded as anelement in F n − K ( N ) . For every x ∈ V \ { } , put Int V +Λ ( x ) := V +Λ . N K ( x ) . Then Int V +Λ HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 31 extends (uniquely) to a compactly supported locally constant function on V , which we stilldenote by Int V +Λ . Moreover, we have (cid:91) Int V +Λ = − Int V +Λ . Proof.
Since t (Λ) = 4, we may choose a 4-dimensional (nonsplit) hermitian subspace V of V such that Λ ∩ V is a self-dual O E -lattice of V . We adopt the construction in Remark2.53. Write x = y + x with respect to the orthogonal decomposition V = V ⊕ V . PutΛ := Λ ∩ V . By Remark 2.53(2) and Definition 2.28(2), V Λ coincides with V Λ under thenatural morphism N → N . Denote by V +Λ the connected component of V Λ that correspondsto V +Λ . By Remark 2.53(3), we have V +Λ . N K ( x ) = , if y Λ ∩ V , V +Λ . N K ( x ) , if y ∈ Λ ∩ V .In other words, we have Int V +Λ = Λ ∩ V ⊗ Int V +Λ . Therefore, it suffices to consider the casewhere n = 4.We now give an explicit formula for Int V +Λ ( x ) when n = 4. Let N + be the connectedcomponent of N that contains V +Λ , and put Z + := Z ∩ N + for every formal subscheme Z of N . Put Λ( x ) := Λ + h x i . There are three cases.(1) Suppose that Λ( x ) is not integral. By Corollary 2.31, V Λ has empty intersection with N ( x ). Thus, we have Int V +Λ ( x ) = 0.(2) Suppose that Λ( x ) is integral but x Λ. Then Λ( x ) has fundamental invariants(0 , , , V +Λ ∩ N ( x ) red = V +Λ( x ) which is a k -point. Thus, wehave Int V +Λ ( x ) (cid:62)
1. Choose a normal basis (Remark 2.14) { x , x , x , x } of Λ andwrite x = λ x + λ x + λ x + λ x with λ i ∈ E . Without lost of generality, wemay assume λ O E . Since ux ∈ Λ, we have Λ( x ) = h x , x , x , x i . By Corollary2.31, N ( x ) ∩ N ( x ) ∩ N ( x ) contains V Λ as a closed subscheme. By Remark 2.53and Proposition 2.45 applied to V spanned by x and x , N (Λ( x )) is a 0-dimensionalscheme and Int(Λ( x )) = 2. It follows thatInt V +Λ ( x ) (cid:54) length O ˘ E ( N ( x ) ∩ N ( x ) ∩ N ( x )) ∩ N ( x ) + = Int + (Λ( x )) = 1by Lemma 2.57 below. Thus, we obtain Int + (Λ( x )) = 1 hence Int V +Λ ( x ) = 1.(3) Suppose that x ∈ Λ. Then V +Λ is a closed subscheme of N ( x ), which implies O V Λ+ L ⊗ O N O N ( x ) = (cid:18) O V Λ+ L ⊗ O N ( x ) O N ( x ) (cid:19) L ⊗ O N O N ( x ) = O V Λ+ L ⊗ O N ( x ) (cid:18) O N ( x ) L ⊗ O N O N ( x ) (cid:19) . However, by Corollary 2.35, we have O N ( x ) L ⊗ O N O N ( x ) = O N ( x ) ⊗ O N C (0) in K ( N ),where C (0) is the complex (2.14). Thus, we obtainInt V +Λ ( x ) = χ (cid:16) C (0) | V +Λ (cid:17) = deg (cid:16) O V +Λ (cid:17) − deg (cid:16) ω | V +Λ (cid:17) = − deg (cid:16) ω | V +Λ (cid:17) = 1 − q by Lemma 2.43.Since there are exactly q + 1 vertex O E -lattices of V properly containing Λ, combining(1–3), we obtain Int V +Λ = − q (1 + q ) Λ + X Λ (cid:32) Λ ⊆ Λ Λ . It follows that (cid:91)
Int V +Λ = − qq Λ ∨ + 1 q X Λ (cid:32) Λ ⊆ Λ Λ . (2.18) • If x ∈ Λ, then (cid:91)
Int V +Λ ( x ) = − qq + q +1 q = q − • If Λ( x ) is integral but x Λ, then the number of Λ in the summation of (2.18) such that x ∈ Λ is exactly 1 (namely, Λ( x ) itself). Thus, we have (cid:91) Int V +Λ ( x ) = − qq + q = − • If Λ( x ) is not integral but x ∈ Λ ∨ , then the set of Λ in the summation of (2.18) satisfying x ∈ Λ is bijective to the set of isotropic lines in Λ ∨ / Λ perpendicular to x . Now sinceΛ( x ) is not integral, x is anisotropic in Λ ∨ / Λ, which implies that the previous set hascardinality q + 1. Thus, we have (cid:91) Int V +Λ ( x ) = − qq + q +1 q = 0. • If x Λ ∨ , then (cid:91) Int V +Λ ( x ) = 0.Therefore, we have (cid:91) Int V +Λ = − Int V +Λ . The lemma is proved. (cid:3) Lemma 2.57.
Denote the two connected components of N by N + and N − , and Int ± ( L ) the intersection multiplicity in Definition 2.8 on N ± . Then Int + ( L ) = Int − ( L ) = Int( L ) . Proof.
Choose a normal basis (Remark 2.14) { x , . . . , x n } of L . Since V is nonsplit, thereexists an anisotropic element in the basis, say x n . Let θ the unique element in U( V )( F )satisfying θ ( x i ) = 1 for 1 (cid:54) i (cid:54) n − θ ( x n ) = − x n . Then θ induces an automorphismof N , preserving N ( x i ) for 1 (cid:54) i (cid:54) n , but switching N + and N − as det θ = −
1. Thus, wehave Int + ( L ) = Int − ( L ). Since Int( L ) = Int + ( L ) + Int − ( L ), the lemma follows. (cid:3) Proof of Proposition 2.52.
We first consider (1). By Lemma 2.55, we have for x ∈ V \ V L [ ,Int h L [ ( x ) = X L [ ⊆ L [ ⊆ ( L [ ) ∨ t ( L [ )=1 N ( L [ ) ◦ . N K ( x ) , which, by Lemma 2.54, equals2 X L [ ⊆ L [ ⊆ ( L [ ) ∨ t ( L [ )=1 X L ⊆ L ∨ L ∩ V L[ = L [ L ( x ) = 2 X L [ ⊆ L ⊆ L ∨ t ( L ∩ V L[ )=1 L ( x ) . Thus, Proposition 2.52(1) follows from Definition 2.21.We first consider (2). We may assume r (cid:62) v L [ ≡ N = N + ∪ N − for the two connected components. For every vertex O E -lattice Λof V , we put V ± Λ := V Λ ∩ N ± . By Lemma 2.50 and Proposition 2.29, there exists finitelyfinitely many vertex O E -lattices Λ , . . . , Λ m of V of type n such that N K ( L [ ) v ∈ m X i =1 F n − K V Λ i ( N ) ⊆ F n − K ( N ) . Since the natural map F t (Λ i )2 − K ( V Λ i ) → F n − K V Λ i ( N ) is an isomorphism for 1 (cid:54) i (cid:54) m , byLemma 2.33, there exist rational numbers c ± Λ for vertex O E -lattices Λ of V with t (Λ) = 4, HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 33 of which all but finitely many are zero, such that N K ( L [ ) v − X Λ c +Λ · V +Λ + c − Λ · V − Λ ! has zero intersection with F K ( N ). Thus, Proposition 2.52(2) follows from Lemma 2.56. (cid:3) Proof of Theorem 2.9.
Let the setup be as in Subsection 2.1. In this subsection, foran element L [ ∈ [ ( V ) (Definition 2.10), we set val( L [ ) = − L [ is not integral. Lemma 2.58.
Suppose that r (cid:62) and take an integral element L [ ∈ [ ( V ) whose fun-damental invariants ( a , . . . , a n − , a n − ) satisfy a n − < a n − (in particular, a n − is odd).Then the number of integral O E -lattices of V containing L [ with fundamental invariants ( a , . . . , a n − , a n − − , a n − − is either or . When the number is and those latticesare denoted by L [ + and L [ − , we have(1) L [ ± ∩ V L [ = L [ ;(2) a n − (cid:62) ;(3) there are orthogonal decompositions L [ = L [ ← ⊕ L [ → and L [ ± = L [ ← ⊕ L [ ±→ , in which L [ ← , L [ → , and L [ ±→ are integral hermitian O E -modules with fundamental invariants ( a , . . . , a n − ) , ( a n − ) , and ( a n − − , a n − − , respectively.Proof. Let L be an integral O E -lattice L of V containing L [ with fundamental invariants( a , . . . , a n − , a n − − , a n − − L ∩ V L [ ) (cid:62) a + · · · + a n − + a n − − L ∩ V L [ contains L [ and val( L ∩ V L [ ) is odd, we must have L ∩ V L [ = L [ .Choose a normal basis ( e , . . . , e n − ) of L [ (Remark 2.14), and rearrange them such thatfor every 1 (cid:54) i (cid:54) n −
1, exactly one of the following three happens:(a) ( e i , e i ) V = β i u a i − for some β i ∈ O × F ;(b) ( e i , e i +1 ) V = u a i − ;(c) ( e i , e i − ) V = − u a i − .By the claim on (1), we may write L = L [ + h x i in which x = λ e + · · · + λ n − e n − + x n for some λ i ∈ ( E \ O E ) ∪ { } and 0 = x n ∈ V ⊥ L [ . Let T be the moment matrix with respectto the basis { e , . . . , e n − , x } of L .We show by induction that for 1 (cid:54) i (cid:54) n − λ i = 0. Suppose we know λ = · · · λ i − = 0.For λ i (with 1 (cid:54) i (cid:54) n − • If e i is in the situation (a) above, then applying Lemma 2.24(1) to the i -by- i minor of T consisting of rows { , . . . , i } and columns { , . . . , i − , n } , we obtain val E ( λ i β i u a i − ) (cid:62) a i −
1, which implies λ i = 0. • If e i is in the situation (b) above, then applying Lemma 2.24(1) to the i -by- i minorof T consisting of rows { , . . . , i − , i + 1 } and columns { , . . . , i − , n } , we obtainval E ( − λ i u a i − ) (cid:62) a i −
1, which implies λ i = 0. • If e i is in the situation (c) above, then applying Lemma 2.24(1) to the i -by- i minor of T consisting of rows { , . . . , i } and columns { , . . . , i − , n } , we obtain val E ( λ i u a i − ) (cid:62) a i −
1, which implies λ i = 0.Note that e n − is in the situation (a). Applying Lemma 2.24(1) to the ( n − n −
1) minor of T consisting of rows { , . . . , n − } and columns { , . . . , n − , n } , we obtain val E ( λ n − β n − u a n − − ) (cid:62) a n − −
2, which implies λ n − ∈ u − O E . On the other hand, λ n − = 0since otherwise a n − will appear in the fundamental invariants of L , which is a contradiction.Thus, we have λ n − ∈ u − O E \ O E . After rescaling by an element in O × E , we may assume λ n − = u − . Applying Lemma 2.24(1) to the ( n − n −
1) minor of T consisting ofrows { , . . . , n − , n } and columns { , . . . , n − , n } , we obtainval E (cid:16) ( x n , x n ) V − u − β n − u a n − − (cid:17) (cid:62) a n − − . (2.19)We note the following facts. • The set of x n ∈ V ⊥ L [ satisfying (2.19) is stable under the multiplication by 1 + uO E . • The set of orbits of such x n under the multiplication by 1 + uO E is bijective to the setof L . • The number of orbits is either 0 or 2. • If the number is 2, then a n − (cid:62)
3, since V is nonsplit.Thus, the main part of the lemma is proved, with the properties (1) and (2) included. For(3), we simply take L [ ← = h e , . . . , e n − i with L [ → and L [ ±→ uniquely determined.The lemma is proved. (cid:3) In the rest of subsection, we say that L [ is special if L [ is like in Lemma 2.58 for whichthe number is 2. We now define an open compact subset S L [ of V for an integral element L [ ∈ [ ( V ) in the following way: S L [ := L [ + ∪ L [ − , if L [ is special, L [ + ( V ⊥ L [ ) int , if L [ is not special. Lemma 2.59.
Take an integral element L [ ∈ [ ( V ) . Then for every x ∈ V \ ( V L [ ∪ S L [ ) , wemay write L [ + h x i = L [ + h x i for some L [ ∈ [ ( V ) satisfying val( L [ ) < val( L [ ) .Proof. Take an element x ∈ V \ ( V L [ ∪ S L [ ). Put L := L [ + h x i . If L is not integral, thenby Remark 2.14, we may write L = L [ + h x i with L [ ∈ [ ( V ) that is not integral; hence thelemma follows.In what follows, we assume L integral and write its fundamental invariants as ( a , . . . , a n ).By Remark 2.14, it suffices to show that a + · · · + a n − (cid:54) a + · · · + a n − − e , . . . , e n − ) of L [ (Remark 2.14), and rearrange them such thatfor every 1 (cid:54) i (cid:54) n −
1, exactly one of the following three happens:(a) ( e i , e i ) V = β i u a i − for some β i ∈ O × F ;(b) ( e i , e i +1 ) V = u a i − ;(c) ( e i , e i − ) V = − u a i − .Write x = λ e + · · · + λ n − e n − + x n for some λ i ∈ ( E \ O E ) ∪ { } and 0 = x n ∈ V ⊥ L [ . Let T be the moment matrix with respect to the basis { e , . . . , e n − , x } of L .If λ = · · · = λ n − = 0, then since x S L [ , we have either h x i is not integral, or val( x ) (cid:54) a n − − L [ is special) which implies a + · · · + a n − (cid:54) a + · · · + a n − − λ i = 0 for some 1 (cid:54) i (cid:54) n − e i is in the situation (b) or (c), then applyingLemma 2.24(1) to the ( n − n −
1) minor of T deleting the i -th row and the i -th column,we obtain a + · · · + a n − (cid:54) a + · · · + a n − − HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 35 If λ i u − O E for some 1 (cid:54) i (cid:54) n − e i is in the situation (a), then applyingLemma 2.24(1) to the ( n − n −
1) minor of T deleting the i -th row and the n -thcolumn, we obtain a + · · · + a n − (cid:54) a + · · · + a n − − λ i = 0 and λ j = 0 for 1 (cid:54) i < j (cid:54) n − e i and e j are in the situation(a), then applying Lemma 2.24(1) to the ( n − n −
1) minor of T deleting the i -th rowand the j -th column, we obtain a + · · · + a n − (cid:54) a + · · · + a n − − λ i ∈ u − O E \ O E for a unique element 1 (cid:54) i (cid:54) n − e i is in the situation (a). Then L [ + h x i is the orthogonal sum of h e , . . . , b e i , . . . , e n − i and h e i , x i .In particular, if we write the fundamental invariants of h e i , x i as ( b , b ), then the fundamentalinvariants of L [ + h x i is the nondecreasing rearrangement of ( a , . . . , b a i , . . . , a n − , b , b ). Wehave two cases: • If ( x, x ) V ∈ u e i − O F , then ( b , b ) = ( a i − , a i − a + · · · + a n − (cid:54) a + · · · + a n − −
2, or i = n − a n − < a n − , and L [ + h x i has fundamental invariants( a , . . . , a n − , a n − − , a n − −
1) (hence L [ is special). The latter case is not possible as x S L [ . • If ( x, x ) V u e i − O F , then b (cid:54) a i −
2. Thus we have a + · · · + a n − (cid:54) a + · · · + a n − − i = n − (cid:3) Proof of Theorem 2.9.
For every element L [ ∈ [ ( V ), we define a functionΦ L [ := ∂ Den v L [ − Int v L [ , which is a compactly supported locally constant function on V by Proposition 2.23 andProposition 2.52(2). It enjoys the following properties:(1) For x ∈ V \ V L [ , we have Φ L [ ( x ) = ∂ Den L [ ( x ) − Int L [ ( x ) by Proposition 2.52(1).(2) Φ L [ is invariant under the translation by L [ , which follows from (1) and the similarproperties for ∂ Den L [ and Int L [ .(3) The support of d Φ L [ is contained in V int , by Proposition 2.23 and Proposition 2.52(2).We prove by induction on val( L [ ) that Φ L [ ≡ L [ ) = −
1, that is, L [ is not integral. Then we have ∂ Den L [ =Int L [ = 0 hence Φ L [ ≡ L [ that is integral, and assume Φ L [ ≡ L [ ∈ [ ( V ) satisfyingval( L [ ) < val( L [ ). For every x ∈ V \ ( V L [ ∪ S L [ ), by Lemma 2.59, we may write L [ + h x i = L [ + h x i with some L [ ∈ [ ( V ) satisfying val( L [ ) < val( L [ ); and we haveΦ L [ ( x ) = ∂ Den L [ ( x ) − Int L [ ( x )= ∂ Den( L [ + h x i ) − Int( L [ + h x i )= ∂ Den( L [ + h x i ) − Int( L [ + h x i )= Φ L [ ( x ) = 0by the induction hypothesis. Thus, the support of Φ L [ is contained in S L [ . There are twocases.Suppose that L [ is not special. By (2), we may write Φ L [ = L [ ⊗ φ for a locally constantfunction φ on V ⊥ L [ supported on ( V ⊥ L [ ) int . Then d Φ L [ = C · ( L [ ) ∨ ⊗ b φ for some C ∈ Q × . Nowsince b φ is invariant under the translation by u − ( V ⊥ L [ ) int , we must have b φ = 0 by (3), that is,Φ L [ ≡ Suppose that L [ is special. We fix the orthogonal decompositions L [ = L [ ← ⊕ L [ → and L [ ± = L [ ← ⊕ L [ ±→ from Lemma 2.58. Put V ← := L [ ← ⊗ O F F and denote by V → the orthogonalcomplement of V ← in V . Then both L [ + → and L [ −→ are integral O E -lattices of V → withfundamental invariants ( a n − − , a n − − S L [ = L [ ← × ( L [ + → ∪ L [ −→ ).Thus, by (2), we may write Φ L [ = L [ ← ⊗ φ for a locally constant function φ on V → supportedon L [ + → ∪ L [ −→ . Since a n − (cid:62) L [ + → ∪ L [ −→ ⊆ uV int → , which implies thatthe support of φ is contained in uV int → . On the other hand, by (3), the support of b φ is containedin V int → . Together, we must have φ = 0 by the Uncertainty Principle [LZ, Proposition 8.1.6],that is, Φ L [ ≡ ∂ Den L [ ( x ) = Int L [ ( x ) for every x ∈ V \ V L [ . In particular, Theorem 2.9follows as every O E -lattice L of V is of the form L [ + h x i for some L [ ∈ [ ( V ). (cid:3) Comparison with absolute Rapoport–Zink spaces.
Let the setup be as in Sub-section 2.1. In this subsection, we compare N to certain (absolute) Rapoport–Zink spaceunder the assumption that F is unramified over Q p . Put f := [ F : Q p ] hence q = p f . Thissubsection is redundant if f = 1.To begin with, we fix a subset Φ of Hom( E, C p ) = Hom( E, ˘ E ) containing ϕ and satisfyingHom( E, ˘ E ) = Φ ‘ Φ c . Recall that we have regarded E as a subfield of ˘ E via ϕ . We introducemore notation. • For every ring R , we denote by W ( R ) the p -typical Witt ring of R , with F , V , [ ], and I ( R )its ( p -typical) Frobenius, the Verschiebung, the Teichmüller lift, and the augmentationideal, respectively. For an F i -linear map f : P → Q between W ( R )-modules with i (cid:62) f \ : W ( R ) ⊗ , W ( R ) F i P → Q its induced W ( R )-linear map. • For i ∈ Z /f Z , put ψ i := F i : O F → O F , define ˆ ψ i : O F → W ( O F ) to be the composition of ψ i with the Cartier homomorphism O F → W ( O F ), and denote by ϕ i the unique elementin Φ above ψ i . • For i ∈ Z /f Z , let (cid:15) i be the unique unit in W ( O F ) satisfying (cid:15) V i = [ ψ i ( u )] − ˆ ψ i ( u ), whichexists by [ACZ16, Lemma 2.24]. We then fix a unit µ u in W ( O ˘ F ), where ˘ F denotes thecomplete maximal unramified extension of F in ˘ E , such that µ F f u µ u = f − Y i =1 (cid:15) i F f − − i , (2.20)which is possible since the right-hand side is a unit in W ( O F ). • For a p -divisible group X over an object S of Sch v /O ˘ E with an action by O F , we have adecomposition Lie( X ) = f − M i =0 Lie ψ i ( X )of O S -modules according to the action of O F on Lie( X ). Definition 2.60.
Let S be an object of Sch /O ˘ E . We define a category Exo Φ( n − , ( S ) whoseobjects are triples ( X, ι X , λ X ) in which • X is a p -divisible group over S of dimension nf and height 2 nf ; • ι X : O E → End( X ) is an action of O E on X satisfying: – (Kottwitz condition): the characteristic polynomial of ι X ( u ) on the O S -moduleLie ψ ( X ) is ( T − u ) n − ( T + u ) ∈ O S [ T ], HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 37 – (Wedge condition): we have ^ ( ι X ( u ) − u | Lie ψ ( X )) = 0 , – (Spin condition): for every geometric point s of S , the action of ι X ( u ) on Lie ψ ( X s )is nonzero; – (Banal condition): for 1 (cid:54) i (cid:54) f − O E acts on Lie ψ i ( X ) via ϕ i ; • λ X : X → X ∨ is a ι X -compatible polarization such that ker( λ X ) = X [ ι X ( u )].A morphism (resp. quasi-morphism) from ( X, ι X , λ X ) to ( Y, ι Y , λ Y ) is an O E -linear isomor-phism (resp. quasi-isogeny) ρ : X → Y of height zero such that ρ ∗ λ Y = λ X .When S belongs to Sch v /O ˘ E , we denote by Exo Φ , b( n − , ( S ) the subcategory of Exo Φ( n − , ( S )consisting of ( X, ι X , λ X ) in which X is supersingular.Note that both Exo b( n − , and Exo Φ , b( n − , are prestacks (that is, presheaves valued ingroupoids) on Sch v /O ˘ E . Now we construct a morphism − rel : Exo Φ , b( n − , → Exo b( n − , (2.21)of prestacks on Sch v /O ˘ E . We will use the theory of displays [Zin02, Lau08] and O F -displays[ACZ16].Let S = Spec R be an affine scheme in Sch v /O ˘ E . Take an object ( X, ι X , λ X ) of Exo Φ , b( n − , ( S ).Write ( P , Q , F , ˙ F ) for the display of X (as a formal p -divisible group). The action of O F on P induces decompositions P = f − M i =0 P i , Q = f − M i =0 Q i , F = f − X i =0 F i , ˙ F = f − X i =0 ˙ F i , where P i is the W ( R )-submodule on which O F acts via ˆ ψ i , and Q i = Q ∩ P i . It is clear thatthe above decomposition is O E -linear, and P i is a projective O E ⊗ O F , ˆ ψ i W ( R )-module of rank n . Lemma 2.61.
For (cid:54) i (cid:54) f − , we have Q i = ( u ⊗ − ⊗ [ ϕ i ( u )]) P i + I ( R ) P i , and that the map F i := ˙ F i ◦ ( u ⊗ − ⊗ [ ϕ i ( u )]) · : P i → P i +1 is a Frobenius linear epimorphism hence isomorphism.Proof. The Banal condition in Definition 2.60 implies that for 1 (cid:54) i (cid:54) f − u ⊗ − ⊗ [ ϕ i ( u )]) P i + I ( R ) P i ⊆ Q i . To show the reverse inclusion, it suffices to show that the image of ( u ⊗ − ⊗ [ ϕ i ( u )]) P i in P i / I ( R ) P i = P i ⊗ W ( R ) R is projective of rank n . But the image is ( u ⊗ − ⊗ ϕ i ( u )) P i ⊗ W ( R ) R ,which has rank n since P i is projective over O E ⊗ O F , ˆ ψ i W ( R ) of rank n .Now we show that ( F i ) \ is surjective. It suffices to show that coker( F i ) \ ⊗ W ( R ) κ vanishes forevery homomorphism W ( R ) → κ with κ a perfect field of characteristic p . Since W ( R ) → κ necessarily vanishes on I ( R ), it lifts to a homomorphism W ( R ) → W ( κ ). Thus, we may justassume that R is a perfect field of characteristic p . Since( u ⊗ − ⊗ [ ϕ i ( u )])( − u ⊗ − ⊗ [ ϕ i ( u )]) = [ ψ i ( u )] − ˆ ψ i ( u ) = (cid:15) V i in which (cid:15) i is a unit in W ( O F ), the image of the map( u ⊗ − ⊗ [ ϕ i ( u )]) · : P i → P i (2.22)contains ( u ⊗ − ⊗ [ ϕ i ( u )]) P i + W ( R ) (cid:15) V i · P i . As R is a perfect field of characteristic p , wehave W ( R ) (cid:15) V i = I ( R ), hence (2.22) is surjective. Thus, F i is a Frobenius linear epimorphismas F i is.The lemma is proved. (cid:3) Now we put P rel := P , Q rel := Q , F rel := F f − ◦ · · · ◦ F ◦ F , ˙ F rel := F f − ◦ · · · ◦ F ◦ ˙ F . Then ( P rel , Q rel , F rel , ˙ F rel ) defines an f (- Z p )-display in the sense of [ACZ16, Definition 2.1] withan O E -action, for which the Kottwitz condition, the Wedge condition, and the Spin conditionare obviously inherited. It remains to construct the polarization λ X rel . By Remark 2.62below, we have the collection of perfect symmetric W ( R )-bilinear pairings { ( , ) i | i ∈ Z /f Z } coming from λ X . For x, y ∈ P , put x i := ( F i − ◦· · ·◦ F ◦ ˙ F )( x ) and y i := ( F i − ◦· · ·◦ F ◦ ˙ F )( y )for 1 (cid:54) i (cid:54) f , and we have( ˙ F rel x, ˙ F rel y ) = ( F f − x f − , F f − y f − ) = ( ˙ F f − (( u ⊗ − ⊗ [ ϕ f − ( u )]) x f − ) , ˙ F f − (( u ⊗ − ⊗ [ ϕ f − ( u )]) y f − )) = ( V − ( u ⊗ − ⊗ [ ϕ f − ( u )]) x f − , ( u ⊗ − ⊗ [ ϕ f − ( u )]) y f − ) f − = (cid:16) ([ ψ f − ( u )] − ˆ ψ f − ( u )) · ( x f − , y f − ) f − (cid:17) V − = (cid:16) (cid:15) V f − · ( x f − , y f − ) f − (cid:17) V − = (cid:15) f − · ( F x f − , y f − ) f − = · · · = f − Y i =1 (cid:15) i F f − − i · ( F f − x , y ) = f − Y i =1 (cid:15) i F f − − i · ( F f − V − x, y ) . Put ( , ) rel := µ u ( , ) , which satisfies ( ˙ F rel x, ˙ F rel y ) rel = ( F f − V − x, y ) rel by (2.20). Then the f (- Z p )-display ( P rel , Q rel , F rel , ˙ F rel ) with O E -action together with the pairing ( , ) rel define anobject ( X, ι X , λ X ) rel of Exo b( n − , ( S ), as explained in the proof of [Mih20, Proposition 3.4]and Remark 2.62 below. It is clear that the construction is functorial in S . Remark . For an object (
X, ι X , λ X ) of Exo Φ , b( n − , ( S ) with ( P , Q , F , ˙ F ) the display of X ,we have a similar claim as in Remark 2.2 concerning the polarization λ X . In particular,as discussed in [Mih20, Section 11.1], the polarization λ X , or rather its symmetrization, isequivalent to a collection of perfect symmetric W ( R )-bilinear pairings { ( , ) i : P i × P i → W ( R ) | i ∈ Z /f Z } , satisfying ( ι X ( α ) x, y ) i = ( x, ι X ( α c ) y ) i for every α ∈ O E and ( ˙ F i x, ˙ F i y ) i +1 = ( x, y ) i V − for i ∈ Z /f Z . HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 39 Similarly, for an object ( X , ι X , λ X ) of Exo b( n − , ( S ) with ( P , Q , F , ˙ F ) the f (- Z p )-displayof X , the polarization λ X is equivalent to a perfect symmetric W ( R )-bilinear pairing( , ) : P × P → W ( R ) , satisfying ( ι X ( α ) x, y ) = ( x, ι X ( α c ) y ) for every α ∈ O E and ( ˙ F x, ˙ F y ) = ( x, y ) F f − V − . Proposition 2.63.
The morphism (2.21) is an isomorphism.Proof.
It suffices to show that for every affine scheme S = Spec R in Sch v /O ˘ E , the functor − rel ( S ) is fully faithful and essentially surjective.We first show that − rel ( S ) is fully faithful. Take an object ( X, ι X , λ X ) of Exo Φ , b( n − , ( S ).It suffices to show that the natural map Aut(( X, ι X , λ X )) → Aut((
X, ι X , λ X ) rel ) is an iso-morphism, which follows from a stronger statement that the natural map End O E ( X ) → End O E ( X rel ) is an isomorphism, where X rel denotes the first entry of ( X, ι X , λ X ) rel which isan O F -divisible group. For the latter, it amounts to showing that the natural mapEnd O E (( P , Q , F , ˙ F )) → End O E (( P rel , Q rel , F rel , ˙ F rel ))(2.23)is an isomorphism. For the injectivity, let f be an element in the source, which decomposesas f = P f − i =0 f i for endomorphisms f i : P i → P i which preserve Q i and commute with F and ˙ F .Since for every i ∈ Z /f Z , ˙ F i is a Frobenius linear surjective map from Q i to P i +1 , the map f is determined by f . Thus, (2.23) is injective. For the surjectivity, let f rel be an element in thetarget. Put f := f rel : P → P . By Lemma 2.64(2) below, there is a unique endomorphism f of P rendering the following diagram W ( R ) ⊗ , W ( R ) F Q F \ (cid:47) (cid:47) ⊗ ( f | Q ) (cid:15) (cid:15) P f (cid:15) (cid:15) W ( R ) ⊗ , W ( R ) F Q F \ (cid:47) (cid:47) P commute. For 2 (cid:54) i (cid:54) f −
1, we define f i to be the unique endomorphism of P i satisfyingthat f i ◦ ( F i − ◦ · · · ◦ F ) \ = ( F i − ◦ · · · ◦ F ) \ ◦ (1 ⊗ f i ) . Then f := P f − i =0 f i is an O E -linear endomorphism of P which commutes with ˙ F hence F . Itremains to check that f ( Q ) ⊆ Q , which follows from Lemma 2.61.We then show that − rel ( S ) is essentially surjective. Take an object ( X , ι X , λ X ) ofExo b( n − , ( S ) in which X is given by an f (- Z p )-display ( P , Q , F , ˙ F ). For 0 (cid:54) i (cid:54) f − P i := W ( R ) ⊗ , W ( R ) F i P . Denote by u : P → P the endomorphism given by the actionof u ∈ O E on P . Put Q = Q and for 1 (cid:54) i (cid:54) f −
1, put Q i := ((1 ⊗ u ) ⊗ − (1 ⊗ ⊗ [ ϕ i ( u )]) P i + I ( R ) P i . Fix a normal decomposition P = L ⊕ T for Q and let ¨ F := ˙ F | L + F | T : P → P be thecorresponding F f -linear isomorphism. For 0 (cid:54) i < f −
1, let ¨ F i : P i → P i +1 be the Frobeniuslinear isomorphism induced by the identity map on P ; and finally let ¨ F f − : P f − → P bethe Frobenius linear isomorphism induced by ¨ F . Let ˙ F : Q → P be the map defined bythe formula ˙ F ( l + w V · t ) = ¨ F ( l ) + w ¨ F ( t ) for l ∈ L , t ∈ T , and w ∈ W ( R ), which is a Frobenius linear epimorphism. By Lemma 2.64(2) below, there is a unique endomorphism u of P rendering the following diagram W ( R ) ⊗ , W ( R ) F Q F \ (cid:47) (cid:47) ⊗ ( u | Q ) (cid:15) (cid:15) P u (cid:15) (cid:15) W ( R ) ⊗ , W ( R ) F Q F \ (cid:47) (cid:47) P commute. For 2 (cid:54) i (cid:54) f −
1, we define u i to be the unique endomorphism of P i satisfyingthat u i ◦ (¨ F i − ◦ · · · ◦ ¨ F ) \ = (¨ F i − ◦ · · · ◦ ¨ F ) \ ◦ (1 ⊗ u ) , and define a map ˙ F i : Q i → P i +1 by the following (compatible) formulae ˙ F i (( u i ⊗ − ⊗ [ ϕ i ( u )]) x ) = ¨ F i ( x ) , ˙ F i ( w V · x ) = w(cid:15) i · ( u i +1 ⊗ ⊗ [ F ϕ i ( u )])¨ F i ( x ) , for x ∈ P i and w ∈ W ( R ), which is a Frobenius linear epimorphism. Put P := f − M i =0 P i , Q := f − M i =0 Q i , ˙ F := f − X i =0 ˙ F i , u := f − X i =0 u i . Then it is straightforward to check that ( P , Q , F , ˙ F ) is a display with an action by O E for which u acts by u , where F is determined by ˙ F in the usual way. Now we construct a collectionof perfect symmetric W ( R )-bilinear pairings { ( , ) i | i ∈ Z /f Z } as in Remark 2.62. Put( , ) := µ − u ( , ) , where ( , ) is the pairing induced by λ X . Define inductively for 1 (cid:54) i (cid:54) f − W ( R )-bilinear) pairing ( , ) i satisfying ( ˙ F i − x, ˙ F i − y ) i =( x, y ) i − V − . It is clear that we also have ( ˙ F f − x, ˙ F f − y ) = ( x, y ) f − V − . Then the display( P , Q , F , ˙ F ) with the O E -action together with the collection of pairings { ( , ) i | i ∈ Z /f Z } define an object ( X, ι X , λ X ) ∈ Exo Φ , b( n − , ( S ), which satisfies ( X, ι X , λ X ) rel ’ ( X , ι X , λ X ) byconstruction.The proposition is proved. (cid:3) Lemma 2.64.
Let R be a ring on which p is nilpotent. For a pair ( P , Q ) in which P is a projective W ( R ) -module of finite rank and Q is a submodule of P containing I ( R ) P such that P / Q is a projective R -module, we define Q ? to be the image of J ( R ) P under themap W ( R ) ⊗ , W ( R ) F I ( R ) P → W ( R ) ⊗ , W ( R ) F Q that is the base change of the inclusion map I ( R ) P → Q , where J ( R ) denotes the kernel of ( V − ) \ : W ( R ) ⊗ , W ( R ) F I ( R ) → W ( R ) . Thenfor every Frobenius linear epimorphism ˙ F : Q → P with P a projective W ( R ) -module of thesame rank as P , we have(1) the kernel of ˙ F \ coincides with Q ? ; We warn the readers that the endomorphism u might be different from 1 ⊗ u as u does not necessarilypreserve the normal decomposition. However, the image of u − ⊗ u is contained in I ( R ) P . HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 41 (2) for every endomorphism f : P → P that preserves Q , there exists a unique endomor-phism f : P → P rendering the following diagram W ( R ) ⊗ , W ( R ) F Q ˙ F \ (cid:47) (cid:47) ⊗ ( f | Q ) (cid:15) (cid:15) P f (cid:15) (cid:15) W ( R ) ⊗ , W ( R ) F Q ˙ F \ (cid:47) (cid:47) P commute.Proof. We first claim that J ( R ) is contained in the kernel of the map W ( R ) ⊗ , W ( R ) F I ( R ) → W ( R ) ⊗ , W ( R ) F W ( R ) = W ( R )(2.24)that is the base change of the inclusion map I ( R ) → W ( R ). Take an element x = P a i ⊗ b V i in W ( R ) ⊗ , W ( R ) F I ( R ). If x ∈ J ( R ), then P a i b i = 0. But the image of x under (2.24) is P a i b FV i , which equals p P a i b i . Thus, J ( R ) is contained in the kernel of (2.24).For (1), choose a normal decomposition P = L ⊕ T of W ( R )-modules such that Q = L ⊕ I ( R ) T . By (the proof of) [Lau10, Lemma 2.5], there exists a Frobenius linear automorphismΨ of P such that ˙ F ( l + at ) = Ψ( l ) + a V − · Ψ( t ) for l ∈ L , t ∈ T , and a ∈ I ( R ). Thus ker ˙ F \ equals the submodule J ( R ) T of W ( R ) ⊗ , W ( R ) F Q . However, by the claim above, the imageof J ( R ) L under the map W ( R ) ⊗ , W ( R ) F I ( R ) P → W ( R ) ⊗ , W ( R ) F Q vanishes. Thus, we have J ( R ) T = Q ? .For (2), the uniqueness follows since ˙ F \ is surjective; and the existence follows since themap 1 ⊗ ( f | Q ) preserves Q ? , which is a consequence of the definition of Q ? . (cid:3) To define our (absolute) Rapoport–Zink space, we fix an object ( X , ι X , λ X ) ∈ Exo Φ , b( n − , ( k ). Definition 2.65.
We define a functor N Φ := N Φ( X ,ι X ,λ X ) on Sch v /O ˘ E such that for every object S of Sch v /O ˘ E , N ( S ) consists of quadruples ( X, ι X , λ X ; ρ X ) in which • ( X, ι X , λ X ) is an object of Exo Φ , b( n − , ( S ); • ρ X is a quasi-morphism from ( X, ι X , λ X ) × S ( S ⊗ O ˘ E k ) to ( X , ι X , λ X ) ⊗ k ( S ⊗ O ˘ E k ) inthe category Exo Φ , b( n − , ( S ⊗ O ˘ E k ). Corollary 2.66.
The morphism N Φ = N Φ( X ,ι X ,λ X ) → N := N ( X ,ι X ,λ X ) rel sending ( X, ι X , λ X ; ρ X ) to (( X, ι X , λ X ) rel ; ρ rel X ) is an isomorphism.Proof. This follows immediately from Proposition 2.63. (cid:3)
Now we study special divisors on N Φ and their relation with those on N . Fix a triple( X , ι X , λ X ) where • X is a supersingular p -divisible group over Spec O ˘ E of dimension f and height 2 f ; • ι X : O E → End( X ) is an O E -action on X such that for 0 (cid:54) i (cid:54) f −
1, the summandLie ψ i ( X ) has rank 1 on which O E acts via ϕ i ; • λ X : X → X ∨ is a ι X -compatible principal polarization. Note that ι X induces an isomorphism ι X : O E ∼ −→ End O E ( X ). Put V := Hom O E ( X ⊗ O ˘ E k, X ) ⊗ Q , which is a vector space over E of dimension n , equipped with a natural hermitian formsimilar to (2.1). By a construction similar to (2.21), we obtain a triple ( X , ι X , λ X ) rel as inthe definition of special divisors on N (Definition 2.7), and a canonical mapHom O E ( X ⊗ O ˘ E k, X ) → Hom O E ( X rel0 ⊗ O ˘ E k, X rel ) , which induces a map − rel : V → V rel := Hom O E ( X rel0 ⊗ O ˘ E k, X rel ) ⊗ Q . (2.25)For every nonzero element x ∈ V , we have similarly a closed formal subscheme N Φ ( x ) of N Φ defined similarly as in Definition 2.7. Corollary 2.67.
The map (2.25) is an isomorphism of hermitian spaces. Moreover, underthe isomorphism in Corollary 2.66, we have N Φ ( x ) = N ( x rel ) .Proof. By the definition of − rel , the map (2.25) is clearly an isometry. Since both V and V rel have dimension n , (2.25) is an isomorphism of hermitian spaces. The second assertionfollows from Corollary 2.66 and construction of − rel , parallel to [Mih20, Remark 4.4]. (cid:3) Remark . Let S be an object of Sch /O ˘ E . We have another category Exo Φ( n, ( S ) whoseobjects are triples ( X, ι X , λ X ) in which • X is a p -divisible group over S of dimension nf and height 2 nf ; • ι X : O E → End( X ) is an action of O E on X such that for 0 (cid:54) i (cid:54) f − O E acts onLie ψ i ( X ) via ϕ i ; • λ X : X → X ∨ is a ι X -compatible polarization such that ker( λ X ) = X [ ι X ( u )].Morphisms are defined similarly as in Definition 2.60. The category Exo Φ( n, ( S ) is a connectedgroupoid. Moreover, one can show that there is a canonical isomorphism Exo Φ( n, → Exo ( n, of prestacks after restriction to Sch v /O ˘ E similar to (2.21). Remark . It is desirable to extend the results in this subsection to a general finiteextension F/ Q p . We hope to address this problem in the future.3. Local theta lifting at ramified places
Throughout this section, we fix a ramified quadratic extension
E/F of p -adic fields with p odd, with c ∈ Gal(
E/F ) the Galois involution. We fix a uniformizer u ∈ E satisfying u c = − u , and denote by q the cardinality of O E / ( u ). Let n = 2 r be an even positive integer.We fix a nontrivial additive character ψ F : F → C × of conductor O F .The goal of this section is to compute the doubling L -function, the doubling epsilon factor,the spherical doubling zeta integral, and the local theta lifting for a tempered admissibleirreducible representation π of G r ( F ) that is spherical with respect to the standard specialmaximal compact subgroup. HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 43 Weil representation and spherical module.
We equip W r := E r with the skew-hermitian form given by the matrix (cid:16) r − r (cid:17) . We denote by { e , . . . , e r } the natural basisof W r . Denote by G r the unitary group of W r , which is a reductive group over F . We writeelements of W r in the row form, on which G r acts from the right. Let K r ⊆ G r ( F ) be thestabilizer of the lattice O rE ⊆ W r , which is a special maximal compact subgroup. We fix theHaar measure d g on G r ( F ) that gives K r volume 1. Let P r be the Borel subgroup of G r consisting of elements of the form a ba t c , − ! , in which a is a lower-triangular matrix in Res E/F GL r . Let P r be the maximal parabolicsubgroup of G r containing P r with the unipotent radical N r , such that the standard diagonalLevi factor M r of P r is isomorphic to Res E/F GL r .We fix a a split hermitian space ( V, ( , ) V ) over E of dimension n = 2 r , and a self-duallattice Λ V of V , namely, Λ V = Λ ∨ V := { x ∈ V | Tr E/F ( x, y ) V ∈ O F for every y ∈ Λ V } . Put H V := U( V ), and let L V be the stabilizer of Λ V in H V ( F ). We fix the Haar measure d h on H V ( F ) that gives L V volume 1. Remark . We have(1) There exists an isomorphism κ : W r → V of E -vector spaces satisfying ( κ ( e i ) , κ ( e j )) V =0, ( κ ( e r + i ) , κ ( e r + j )) V = 0, and ( κ ( e i ) , κ ( e r + j )) V = u − δ ij for 1 (cid:54) i, j (cid:54) r , and suchthat L V is generated by { κ ( e i ) | (cid:54) i (cid:54) r } as an O E -submodule.(2) The double coset K r \ G r ( F ) /K r has representatives u a . . . u a r ( − u ) − a . . . ( − u ) − a r where 0 (cid:54) a (cid:54) · · · (cid:54) a r are integers.We introduce two Hecke algebras: H W r := C [ K r \ G r ( F ) /K r ] , H V := C [ L V \ H V ( F ) /L V ] . Then by the remark above, both H W r and H V are commutative complex algebras, and arecanonically isomorphic to T r := C [ T ± , . . . , T ± r ] {± } r (cid:111) S r .Let ( ω W r ,V , V W r ,V ) be the Weil representation of G r ( F ) × H V ( F ) (with respect to theadditive character ψ F and the trivial splitting character). We recall the action under theSchrödinger model V W r ,V ’ C ∞ c ( V r ) as follows: • for a ∈ GL r ( E ) and φ ∈ C ∞ c ( V r ), we have ω W r ,V (cid:16)(cid:16) a a t c , − (cid:17)(cid:17) φ ( x ) = | det a | rE · φ ( xa ); • for b ∈ Herm r ( F ) and φ ∈ C ∞ c ( V r ), we have ω W r ,V (cid:16)(cid:16) r b r (cid:17)(cid:17) φ ( x ) = ψ F (tr bT ( x )) · φ ( x )where T ( x ) := (( x i , x j ) V ) (cid:54) i,j (cid:54) r is the moment matrix of x = ( x , . . . , x r ); • for φ ∈ C ∞ c ( V r ), we have ω W r ,V (cid:16)(cid:16) r − r (cid:17)(cid:17) φ ( x ) = b φ ( x ); • for h ∈ H V ( F ) and φ ∈ C ∞ c ( V r ), we have ω W r ,V ( h ) φ ( x ) = φ ( h − x ) . Here, we recall the Fourier transform C ∞ c ( V r ) → C ∞ c ( V r ) sending φ to b φ defined by theformula b φ ( x ) := Z V r φ ( y ) ψ F r X i =1 Tr E/F ( x i , y i ) V ! d y, where d y is the self-dual Haar measure on V r . Definition 3.2.
We define the spherical module S W r ,V to be the subspace of V W r ,V consistingof elements that are fixed by K r × L V , as a module over H W r ⊗ C H V via the representation ω W r ,V . We denote by Sph( V r ) the corresponding subspace of C ∞ c ( V r ) under the Schrödingermodel. Lemma 3.3.
The function Λ rV belongs to Sph( V r ) .Proof. It suffices to check that ω W r ,V (cid:16)(cid:16) r − r (cid:17)(cid:17) Λ rV = Λ rV , which follows from the fact that Λ ∨ V = Λ V . The lemma follows. (cid:3) Proposition 3.4.
The annihilator of the H W r ⊗ C H V -module S W r ,V is I W r ,V , where I W r ,V denotes the diagonal ideal of H W r ⊗ C H V .Proof. The same proof of [Liub, Proposition 4.4] (with (cid:15) = + and d = r ) works in this caseas well, using Lemma 3.3. (cid:3) In what follows, we review the construction of unramified principal series of G r ( F ) and H V ( F ).We identify M r , the standard diagonal Levi factor of P r , with (Res E/F GL ) r , under whichwe write an element of M r ( F ) as a = ( a , . . . , a r ) with a i ∈ E × its eigenvalue on e i for1 (cid:54) i (cid:54) r . For every tuple σ = ( σ , . . . , σ r ) ∈ ( C / πi log q Z ) r , we define a character χ σr of M r ( F )hence P r ( F ) by the formula χ σr ( a ) = r Y i =1 | a i | σ i + i − / E . We then have the normalized principal seriesI σW r := { ϕ ∈ C ∞ ( G r ( F )) | ϕ ( ag ) = χ σr ( a ) ϕ ( g ) for a ∈ P r ( F ) and g ∈ G r ( F ) } , which is an admissible representation of G r ( F ) via the right translation. We denote by π σW r the unique irreducible constituent of I σW r that has nonzero K r -invariants.For V , we fix a basis { v r , . . . , v , v − , . . . , v − r } of the O E -lattice Λ V , satisfying ( v i , v j ) V = u − δ i, − j for every 1 (cid:54) i, j (cid:54) r . We have an increasing filtration { } = Z r +1 ⊆ Z r ⊆ · · · ⊆ Z (3.1)of isotropic E -subspaces of V where Z i be the E -subspaces of V spanned by { v r , . . . , v i } .Let Q V be the (minimal) parabolic subgroup of H V that stabilizes (3.1). Let M V be theLevi factor of Q V stabilizing the lines spanned by v i for every i . Then we have the canonical HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 45 isomorphism M V = (Res E/F GL ) r , under which we write an element of M V ( F ) as b =( b , . . . , b r ) with b i ∈ E × its eigenvalue on v i for 1 (cid:54) i (cid:54) r . For every tuple σ = ( σ , . . . , σ r ) ∈ ( C / πi log q Z ) r , we define a character χ σV of M V ( F ) hence Q V ( F ) by the formula χ σV ( b ) = r Y i =1 | b i | σ i + i − / E . We then have the normalized principal seriesI σV := { ϕ ∈ C ∞ ( H V ( F )) | ϕ ( bh ) = χ σV ( b ) ϕ ( h ) for b ∈ Q V ( F ) and h ∈ H V ( F ) } , which is an admissible representation of H V ( F ) via the right translation. We denote by π σV the unique irreducible constituent of I σV that has nonzero L V -invariants.3.2. Doubling zeta integral and doubling L-factor.
In this section, we compute certaindoubling zeta integrals and doubling L -factors for irreducible admissible representations π of G r ( F ) satisfying π K r = { } . We will freely use notation from [Liub, Section 5].We have the degenerate principal series I (cid:3) r ( s ) := Ind G (cid:3) r P (cid:3) r ( | | sE ◦ ∆) of G (cid:3) r ( F ). Let f ( s ) r be theunique section of I (cid:3) r ( s ) such that for every g ∈ pK r with p ∈ P (cid:3) r ( F ), f ( s ) r ( g ) = | ∆( p ) | s + rE . It is a holomorphic standard hence good section.
Remark . By definition, we have I (cid:3) r ( s ) ⊆ I σ (cid:3) s W r , where σ (cid:3) s := ( s + r − , s + r − , . . . , s − r + , s − r + ) ∈ ( C / πi log q Z ) r . Moreover, if we denote by ϕ σ (cid:3) s the unique section in I σ (cid:3) s W r that is fixed by K r and such that ϕ σ (cid:3) s (1 r ) = 1, then f ( s ) r = ϕ σ (cid:3) s .Let π be an irreducible admissible representation of G r ( F ). For every element ξ ∈ π ∨ (cid:2) π ,we denote by H ξ ∈ C ∞ ( G r ( F )) its associated matrix coefficient. Then for every meromorphicsection f ( s ) of I (cid:3) r ( s ), we have the (doubling) zeta integral: Z ( ξ, f ( s ) ) := Z G r ( F ) H ξ ( g ) f ( s ) ( w r ( g, r )) d g, which is absolutely convergent for Re s large enough and has a meromorphic continuation.We let L ( s, π ) and ε ( s, π, ψ F ) be the doubling L -factor and the doubling epsilon factor of π ,respectively, defined in [Yam14, Theorem 5.2].Take an element σ = ( σ , . . . , σ r ) ∈ ( C / πi log q Z ) r . We define an L -factor L σ ( s ) := r Y i =1 − q σ i − s )(1 − q − σ i − s ) . Since π σW r is self-dual, the space (( π σW r ) ∨ ) K r (cid:2) ( π σW r ) K r is one dimensional. Let ξ σ be agenerator of this one dimensional space; it satisfies H ξ σ (1 r ) = 0. We normalize ξ σ so that H ξ σ (1 r ) = 1, which makes it unique. Proposition 3.6.
For σ ∈ ( C / πi log q Z ) r , we have Z ( ξ σ , f ( s ) r ) = L σ ( s + ) b r ( s ) , where b r ( s ) := Q ri =1 11 − q − s − i .Proof. We have an isomorphism m : Res E/F GL r → M r sending a to (cid:16) a a t c , − (cid:17) . Let τ bethe unramified constituent of the normalized induction of (cid:2) ri =1 | | σ i E , as a representation ofGL r ( E ). We fix vectors v ∈ τ and v ∨ ∈ τ ∨ fixed by M r ( F ) ∩ K r = m (GL r ( O E )) such that h v ∨ , v i τ = 1.By a similar argument in [GPSR87, Section 6] or in the proof of [Liub, Proposition 5.6],we have Z ( ξ σ , f ( s ) r ) = C w r ( s ) Z GL r ( E ) ϕ w r σ (cid:3) s ( w r ( m ( a ) , r )) | det a | − r/ E h τ ∨ ( a ) v ∨ , v i τ d a, (3.2)where C w r ( s ) = r Y i =1 ζ E (2 s + 2 i ) ζ E (2 s + r + i ) r Y i =1 ζ F (2 s + 2 i − ζ F (2 s + 2 i ) = r Y i =1 ζ E (2 s + 2 i − ζ E (2 s + r + i ) . See the proof of [Liub, Proposition 5.6] for unexplained notation. By [GPSR87, Proposi-tion 6.1], we have Z GL r ( E ) ϕ w r σ (cid:3) s ( w r ( m ( a ) , r )) | det a | − r/ E h τ ∨ ( a ) v ∨ , v i τ d a = L ( s + , τ ) L ( s + , τ ∨ ) Q ri =1 ζ E (2 s + i ) . Combining with (3.2), we have Z ( ξ σ , f ( s ) r ) = r Y i =1 ζ E (2 s + 2 i − ζ E (2 s + r + i ) ! · L ( s + , τ ) L ( s + , τ ∨ ) Q ri =1 ζ E (2 s + i ) ! = L ( s + , τ ) L ( s + , τ ∨ ) Q ri =1 ζ E (2 s + 2 i ) = L σ ( s + ) b r ( s ) . The proposition is proved. (cid:3)
Proposition 3.7.
For σ ∈ ( C / πi log q Z ) r , we have L ( s, π σW r ) = L σ ( s ) and ε ( s, π σW r , ψ F ) = 1 .Proof. It follows from the same argument for [Yam14, Proposition 7.1], using Proposition3.6. (cid:3)
Remark . It is clear that the base change BC( π σW r ) is well-defined, which is an unramifiedirreducible admissible representation of GL n ( E ), and we have L ( s, π σW r ) = L ( s, BC( π σW r )) byProposition 3.7.For an irreducible admissible representation π of G r ( F ), let Θ( π, V ) be the π -isotypicquotient of V W r ,V , which is an admissible representation of H V ( F ), and θ ( π, V ) its maxi-mal semisimple quotient. By [Wal90], θ ( π, V ) is either zero, or an irreducible admissiblerepresentation of H V ( F ), known as the theta lifting of π to V (with respect to the additivecharacter ψ F and the trivial splitting character). Proposition 3.9.
For an irreducible admissible representation π of G r ( F ) of the form π σW r for an element σ = ( σ , . . . , σ r ) ∈ ( i R / πi log q Z ) r , we have θ ( π, V ) ’ π σV .Proof. By the same argument in the proof of [Liub, Theorem 6.2], we have Θ( π, V ) L V = { } .By our assumption on σ , π is tempered. By (the same argument for) [GI16, Theorem 4.1(v)],Θ( π, V ) is a semisimple representation of H V ( F ) hence Θ( π, V ) = θ ( π, V ). In particular, wehave θ ( π, V ) L V = { } . By Proposition 3.4, the diagonal ideal I W r ,V annihilates ( π σW r ) K r (cid:2) θ ( π, V ) L V , which implies that θ ( π, V ) ’ π σV . (cid:3) HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 47 Arithmetic inner product formula
In this section, we collect all local ingredients and deduce our main theorems, following thesame line as in [LL]. In Subsection 4.1 and 4.2, we recall the doubling method and the arith-metic theta lifting from [LL], respectively. In Subsection 4.3, we prove the vanishing of localindices at split places, by proving the second main ingredient of this article, namely, Theorem4.21. In Subsection 4.4, we recall the formula for local indices at inert places. In Subsection4.5, we compute local indices at ramified places, based on the Kudla–Rapoport type formulaTheorem 2.9. In Subsection 4.6, we recall the formula for local indices at archimedean places.The deduction of the main results of the article is explained in Subsection 4.7, which is astraightforward modification of [LL, Section 11].4.1.
Recollection on doubling method.
For readers’ convenience, we copy three groupsof setups from [LL, Section 2] to here. The only difference is item (H5), which reflects thefact that we study certain places in V ram F in the current article. Setup 4.1.
Let
E/F be a CM extension of number fields, so that c is a well-defined elementin Gal( E/F ).(F1) We denote by • V F and V fin F the set of all places and non-archimedean places of F , respectively; • V spl F , V int F , and V ram F the subsets of V fin F of those that are split, inert, and ramified in E , respectively; • V ( (cid:5) ) F the subset of V F of places above (cid:5) for every place (cid:5) of Q ; and • V ? E the places of E above V ? F .Moreover, • for every place u ∈ V E of E , we denote by u ∈ V F the underlying place of F ; • for every v ∈ V fin F , we denote by p v the maximal ideal of O F v , and put q v := | O F v / p v | ; • for every v ∈ V F , we put E v := E ⊗ F F v and denote by | | E v : E × v → C × thenormalized norm character.(F2) Let m (cid:62) • We denote by Herm m the subscheme of Res E/F
Mat m,m of m -by- m matrices b sat-isfying b t c = b . Put Herm ◦ m := Herm m ∩ Res
E/F GL m . • For every ordered partition m = m + · · · + m s with m i a positive integer, we denoteby ∂ m ,...,m s : Herm m → Herm m × · · · × Herm m s the homomorphism that extractsthe diagonal blocks with corresponding ranks. We simply write ∂ for ∂ , ,..., . • We denote by Herm m ( F ) + (resp. Herm ◦ m ( F ) + ) the subset of Herm m ( F ) of elementsthat are totally semi-positive definite (resp. totally positive definite). We simplywrite F + for Herm ◦ ( F ) + .(F3) For every w ∈ V ( ∞ ) F , we fix an embedding ι w : E , → C above w , and • put E w := ι w ( E ) as a subfield of C ; • identify E w with C via ι w ; and • put E w u := E w ⊗ E E u for every u ∈ V E .(F4) Let η := η E/F : A × F → C × be the quadratic character associated to E/F . For every v ∈ V F and every positive integer m , put b m,v ( s ) := m Y i =1 L (2 s + i, η m − iv ) . Put b m ( s ) := Q v ∈ V F b m,v ( s ). (F5) For every element T ∈ Herm m ( A F ), we have the character ψ T : Herm m ( A F ) → C × given by the formula ψ T ( b ) := ψ F (tr bT ).(F6) Let R be a commutative F -algebra. A (skew-)hermitian space over R ⊗ F E is a free R ⊗ F E -module V of finite rank, equipped with a (skew-)hermitian form ( , ) V withrespect to the involution c that is nondegenerate. Setup 4.2.
We fix an even positive integer n = 2 r . Let ( V, ( , ) V ) be a hermitian space over A E of rank n that is totally positive definite.(H1) For every commutative A F -algebra R and every integer m (cid:62)
0, we denote by T ( x ) := (( x i , x j ) V ) i,j ∈ Herm m ( R )the moment matrix of an element x = ( x , . . . , x m ) ∈ V m ⊗ A F R .(H2) For every v ∈ V F , we put V v := V ⊗ A F F v which is a hermitian space over E v , anddefine the local Hasse invariant of V v to be (cid:15) ( V v ) := η v (( − r det V v ) ∈ {± } . In whatfollows, we will abbreviate (cid:15) ( V v ) as (cid:15) v .(H3) Let v be a place of F and m (cid:62) • For T ∈ Herm m ( F v ), we put ( V mv ) T := { x ∈ V mv | T ( x ) = T } , and( V mv ) reg := [ T ∈ Herm ◦ m ( F v ) ( V mv ) T . • We denote by S ( V mv ) the space of (complex valued) Bruhat–Schwartz functionson V mv . When v ∈ V ( ∞ ) F , we have the Gaussian function φ v ∈ S ( V mv ) given by theformula φ v ( x ) = e − π tr T ( x ) . • We have a Fourier transform map b : S ( V mv ) → S ( V mv ) sending φ to b φ defined bythe formula b φ ( x ) := Z V mv φ ( y ) ψ E,v m X i =1 ( x i , y i ) V ! d y, where d y is the self-dual Haar measure on V mv with respect to ψ E,v . • In what follows, we will always use this self-dual Haar measure on V mv .(H4) Let m (cid:62) T ∈ Herm m ( F ), we putDiff( T, V ) := { v ∈ V F | ( V mv ) T = ∅} , which is a finite subset of V F \ V spl F .(H5) Take a nonempty finite subset R ⊆ V fin F that contains { v ∈ V ram F | either (cid:15) v = −
1, or 2 | v , or v is ramified over Q } . Let S be the subset of V fin F \ R consisting of v such that (cid:15) v = −
1, which is contained in V int F .(H6) We fix a Q v ∈ V fin F \ R O E v -lattice Λ R in V ⊗ A F A ∞ , R F such that for every v ∈ V fin F \ R , Λ R v is asubgroup of (Λ R v ) ∨ of index q − (cid:15) v v , where(Λ R v ) ∨ := { x ∈ V v | ψ E,v (( x, y ) V ) = 1 for every y ∈ Λ R v } is the ψ E,v -dual lattice of Λ R v .(H7) Put H := U( V ), which is a reductive group over A F .(H8) Denote by L R ⊆ H ( A ∞ , R F ) the stabilizer of Λ R , which is a special maximal subgroup.We have the (abstract) Hecke algebra away from R T R := Z [ L R \ H ( A ∞ , R F ) /L R ] , HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 49 which is a ring with the unit L R , and denote by S R the subringlim −→ T ⊆ V spl F \ R | T | < ∞ Z [( L R ) T \ H ( F T ) / ( L R ) T ] ⊗ ( L R ) T of T R .(H9) Suppose that V is incoherent , namely, Q v ∈ V F (cid:15) v = −
1. For every w ∈ V F \ V spl F , wedenote by V w the w -nearby space of V , which is a hermitian space over E . Moreprecisely, • for w ∈ V ( ∞ ) F , V w is the hermitian space over E , unique up to isomorphism, thathas signature ( n − ,
1) at w and satisfies V w ⊗ F A wF ’ V ⊗ A F A wF ; • for w ∈ V fin F \ V spl F , V w is the hermitian space over E , unique up to isomorphism,that satisfies V w ⊗ F A wF ’ V ⊗ A F A wF .We put H w := U( V w ), which is a reductive group over F . Setup 4.3.
Let m (cid:62) W m = E m and ¯ W m = E m the skew-hermitian forms given by the matrices w m and − w m , respectively.(G1) Let G m be the unitary group of both W m and ¯ W m . We write elements of W m and ¯ W m in the row form, on which G m acts from the right.(G2) We denote by { e , . . . , e m } and { ¯ e , . . . , ¯ e m } the natural bases of W m and ¯ W m , re-spectively.(G3) Let P m ⊆ G m be the parabolic subgroup stabilizing the subspace generated by { e r +1 , . . . , e m } , and N m ⊆ P m its unipotent radical.(G4) We have • a homomorphism m : Res E/F GL m → P m sending a to m ( a ) := a a t c , − ! , which identifies Res E/F GL m as a Levi factor of P m . • a homomorphism n : Herm m → N m sending b to n ( b ) := m b m ! , which is an isomorphism.(G5) We define a maximal compact subgroup K m = Q v ∈ V F K m,v of G m ( A F ) in the followingway: • for v ∈ V fin F , K m,v is the stabilizer of the lattice O mE v ; • for v ∈ V ( ∞ ) F , K m,v is the subgroup of the form[ k , k ] := 12 k + k − ik + ik ik − ik k + k ! , in which k i ∈ GL m ( C ) satisfying k i k t c i = 1 m for i = 1 ,
2. Here, we have identified G m ( F v ) as a subgroup of GL m ( C ) via the embedding ι v in Setup 4.1(F3).(G6) For every v ∈ V ( ∞ ) F , we have a character κ m,v : K m,v → C × that sends [ k , k ] todet k / det k . (G7) For every v ∈ V F , we define a Haar measure d g v on G m ( F v ) as follows: In fact, both K m,v and κ m,v do not depend on the choice of the embedding ι v for v ∈ V ( ∞ ) F . • for v ∈ V fin F , d g v is the Haar measure under which K m,v has volume 1; • for v ∈ V ( ∞ ) F , d g v is the product of the measure on K m,v of total volume 1 and thestandard hyperbolic measure on G m ( F v ) /K m,v .Put d g = Q v d g v , which is a Haar measure on G m ( A F ).(G8) We denote by A ( G m ( F ) \ G m ( A F )) the space of ( g m, ∞ , K m, ∞ )-finite automorphic formson G m ( A F ), where g m, ∞ is the Lie algebra of G m ⊗ F F ∞ . We denote by • A [ r ] ( G m ( F ) \ G m ( A F )) the maximal subspace of A ( G m ( F ) \ G m ( A F )) on which forevery v ∈ V ( ∞ ) F , K m,v acts by the character κ rm,v , • A [ r ] R ( G m ( F ) \ G m ( A F )) the maximal subspace of A [ r ] ( G m ( F ) \ G m ( A F )) on which – for every v ∈ V fin F \ ( R ∪ S ), K m,v acts trivially; and – for every v ∈ S , the standard Iwahori subgroup I m,v acts trivially and C [ I m,v \ K m,v /I m,v ] acts by the character κ − m,v ([Liub, Definition 2.1]), • A cusp ( G m ( F ) \ G m ( A F )) the subspace of A ( G m ( F ) \ G m ( A F )) of cusp forms, and by h , i G m the hermitian form on A cusp ( G m ( F ) \ G m ( A F )) given by the Petersson innerproduct with respect to the Haar measure d g .For a subspace V of A ( G m ( F ) \ G m ( A F )), we denote by • V [ r ] the intersection of V and A [ r ] ( G m ( F ) \ G m ( A F )), • V [ r ] R the intersection of V and A [ r ] R ( G m ( F ) \ G m ( A F )), • V c the subspace { ϕ c | ϕ ∈ V} . Setup 4.4.
In what follows, we will consider an irreducible automorphic subrepresentation( π, V π ) of A cusp ( G r ( F ) \ G r ( A F )) satisfying that(1) for every v ∈ V ( ∞ ) F , π v is the (unique up to isomorphism) discrete series representationwhose restriction to K r,v contains the character κ rr,v ;(2) for every v ∈ V fin F \ R , π v is unramified (resp. almost unramified) with respect to K r,v if (cid:15) v = 1 (resp. (cid:15) v = − v ∈ V fin F , π v is tempered.We realize the contragredient representation π ∨ on V c π via the Petersson inner product h , i G r (Setup 4.3(G8)). By (1) and (2), we have V [ r ] R π = { } , where V [ r ] R π is defined in Setup 4.3(G8). Remark . By [LL, Proposition 3.6(2)] and Proposition 3.9, we know that when R ⊆ V spl F , V coincides with the hermitian space over A E of rank n determined by π via local thetadichotomy. Definition 4.6.
We define the L -function for π as the Euler product L ( s, π ) := Q v L ( s, π v )over all places of F , in which(1) for v ∈ V fin F , L ( s, π v ) is the doubling L -function defined in [Yam14, Theorem 5.2];(2) for v ∈ V ( ∞ ) F , L ( s, π v ) is the L -function of the standard base change BC( π v ) of π v .By Setup 4.4(1), BC( π v ) is the principal series representation of GL n ( C ) that is thenormalized induction of arg n − (cid:2) arg n − (cid:2) · · · (cid:2) arg − n (cid:2) arg − n where arg : C × → C × is the argument character. Remark . Let v be a place of F .(1) For v ∈ V ( ∞ ) F , doubling L -function is only well-defined up to an entire function withoutzeros. However, one can show that L ( s, π v ) satisfies the requirement for the doubling L -function in [Yam14, Theorem 5.2]. HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 51 (2) For v ∈ V spl F , the standard base change BC( π v ) is well-defined and we have L ( s, π v ) = L ( s, BC( π v )) by [Yam14, Theorem 7.2].(3) For v ∈ V int F \ R , the standard base change BC( π v ) is well-defined and we have L ( s, π v ) = L ( s, BC( π v )) by [Liub, Remark 1.4].(4) For v ∈ V ram F \ R , the standard base change BC( π v ) is well-defined and we have L ( s, π v ) = L ( s, BC( π v )) by Remark 3.8.In particular, when R ⊆ V spl F , we have L ( s, π ) = Q v L ( s, BC( π v )).Recall that we have the normalized doubling integral Z \π v ,V v : π ∨ v ⊗ π v ⊗ S ( V rv ) → C from [LL, Section 3]. Proposition 4.8.
Let ( π, V π ) be as in Setup 4.4.(1) For every v ∈ V fin F , we have dim C Hom G r ( F v ) × G r ( F v ) (I (cid:3) r,v (0) , π v (cid:2) π ∨ v ) = 1 . (2) For every v ∈ ( V fin F \ R ) ∪ V spl F , V v is the unique hermitian space over E v of rank r , upto isomorphism, such that Z \π v ,V v = 0 .(3) For every v ∈ V fin F , Hom G r ( F v ) ( S ( V rv ) , π v ) is irreducible as a representation of H ( F v ) ,and is nonzero if v ∈ ( V fin F \ R ) ∪ V spl F .Proof. This is same as [LL, Proposition 3.6] except that in (2) we have to take care of thecase where v ∈ V ram F , which is a consequence of Proposition 3.9. (cid:3) Proposition 4.9.
Let ( π, V π ) be as in Setup 4.4 such that L ( , π ) = 0 . Take • ϕ = ⊗ v ϕ v ∈ V [ r ] R π and ϕ = ⊗ v ϕ v ∈ V [ r ] R π such that h ϕ c v , ϕ v i π v = 1 for v ∈ V F \ R , and • Φ = ⊗ v Φ v ∈ S ( V r ) such that Φ v is the Gaussian function (Setup 4.2(H3)) for v ∈ V ( ∞ ) F ,and Φ v = (Λ R v ) r for v ∈ V fin F \ R .Then we have Z G r ( F ) \ G r ( A F ) Z G r ( F ) \ G r ( A F ) ϕ ( g ) ϕ c ( g ) E (0 , ( g , g ) , Φ) d g d g = L ( , π ) b r (0) · C [ F : Q ] r · Y v ∈ V fin F Z \π v ,V v ( ϕ c v , ϕ v , Φ v )= L ( , π ) b r (0) · C [ F : Q ] r · Y v ∈ S ( − r q r − v ( q v + 1)( q r − v + 1)( q rv − · Y v ∈ R Z \π v ,V v ( ϕ c v , ϕ v , Φ v ) , where C r := ( − r r ( r − π r Γ(1) · · · Γ( r )Γ( r + 1) · · · Γ(2 r ) , and the measure on G r ( A F ) is the one defined in Setup 4.3(G7). Strictly speaking, what we fixed is a decomposition ϕ c = ⊗ v ( ϕ c ) v and have abused notation by writing ϕ c v instead of ( ϕ c ) v . Proof.
The proof is same as [LL, Proposition 3.7], with the additional input Z \π v ,V v ( ϕ c v , ϕ v , Φ v ) = 1for v ∈ V ram F \ R by Proposition 3.6. (cid:3) Suppose that V is incoherent. By [Liu11b, Section 2B], we have(1) Take an element w ∈ V F \ V spl F , and Φ w = ⊗ v Φ w v ∈ S ( V w r ⊗ F A F ), where we recallfrom Setup 4.2(H9) that V w is the w -nearby hermitian space, such that supp( Φ w v ) ⊆ ( V w rv ) reg (Setup 4.2(H3)) for v in a nonempty subset R ⊆ R . Then for every g ∈ P (cid:3) r ( F R ) G (cid:3) r ( A R F ), we have E (0 , g, Φ w ) = X T (cid:3) ∈ Herm ◦ r ( F ) Y v ∈ V F W T (cid:3) (0 , g v , Φ w v ) . (2) Take Φ = ⊗ v Φ v ∈ S ( V r ) such that supp(Φ v ) ⊆ ( V rv ) reg for v in a subset R ⊆ R ofcardinality at least 2. Then for every g ∈ P (cid:3) r ( F R ) G (cid:3) r ( A R F ), we have E (0 , g, Φ) = X w ∈ V F \ V spl F E ( g, Φ) w , where E ( g, Φ) w := X T (cid:3) ∈ Herm ◦ r ( F )Diff( T (cid:3) ,V )= { w } W T (cid:3) (0 , g w , Φ w ) Y v ∈ V F \{ w } W T (cid:3) (0 , g v , Φ v ) . Here, Diff( T (cid:3) , V ) is defined in Setup 4.2(H4). Definition 4.10.
Suppose that V is incoherent. Take an element w ∈ V F \ V spl F , and a pair( T , T ) of elements in Herm r ( F ).(1) For Φ w = ⊗ v Φ w v ∈ S ( V w r ⊗ F A F ), we put E T ,T ( g, Φ w ) := X T (cid:3) ∈ Herm ◦ r ( F ) ∂ r,r T (cid:3) =( T ,T ) Y v ∈ V F W T (cid:3) (0 , g v , Φ w v ) . (2) For Φ = ⊗ v Φ v ∈ S ( V r ), we put E T ,T ( g, Φ) w := X T (cid:3) ∈ Herm ◦ r ( F )Diff( T (cid:3) ,V )= { w } ∂ r,r T (cid:3) =( T ,T ) W T (cid:3) (0 , g w , Φ w ) Y v ∈ V F \{ w } W T (cid:3) (0 , g v , Φ v ) . Here, ∂ r,r : Herm r → Herm r × Herm r is defined in Setup 4.1(F2).4.2. Recollection on arithmetic theta lifting.
From this moment, we will assume F = Q .Take an element w ∈ V ( ∞ ) F , and fix an isomorphism − w : V ⊗ A F A ∞ F ∼ −→ V w ⊗ F A ∞ F of hermitian spaces over A ∞ E , which induces an isomorphism − w : H ( A ∞ F ) ∼ −→ H w ( A ∞ F ). Forevery open compact subgroup L ⊆ H ( A ∞ F ), we have the Shimura variety X w L associated toRes F/ Q H w of the level L w , which is a smooth projective scheme over E w of dimension n − u ∈ V E , we put X w L,u := X w L ⊗ E w E w u as a scheme over E w u . HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 53 For every φ ∞ ∈ S ( V m ⊗ A F A ∞ F ) L and T ∈ Herm m ( F ), we put Z w T ( φ ∞ ) L := X x ∈ L \ V m ⊗ A F A ∞ F T ( x )= T φ ∞ ( x ) Z w ( x ) L , where Z w ( x ) L is Kudla’s special cycle recalled in [LL, Construction 4.2]. As the abovesummation is finite, Z w T ( φ ∞ ) L is a well-defined element in CH m ( X w L ) C . For every g ∈ G m ( A F ), Kudla’s generating function is defined to be Z w φ ∞ ( g ) L := X T ∈ Herm m ( F ) + ω m, ∞ ( g ∞ ) φ ∞ ( T ) · Z w T ( ω ∞ m ( g ∞ ) φ ∞ ) L as a formal sum valued in CH m ( X w L ) C , where ω m, ∞ ( g ∞ ) φ ∞ ( T ) := Y v ∈ V ( ∞ ) F ω m,v ( g v ) φ v ( T ) . Here, we note that for v ∈ V ( ∞ ) F , the function ω m,v ( g v ) φ v factors through the moment map V mv → Herm m ( F v ) (see Setup 4.2(H1)). Hypothesis 4.11 (Modularity of generating functions of codimension m ) . For every opencompact subgroup L ⊆ H ( A ∞ F ), every φ ∞ ∈ S ( V m ⊗ A F A ∞ F ) L , and every complex linearmap l : CH m ( X w L ) C → C , the assignment g l ( Z w φ ∞ ( g ) L )is absolutely convergent, and gives an element in A [ r ] ( G m ( F ) \ G m ( A F )). In other words, thefunction Z w φ ∞ ( − ) L defines an element in Hom C (CH m ( X w L ) ∨ C , A [ r ] ( G m ( F ) \ G m ( A F ))). Definition 4.12.
Let ( π, V π ) be as in Setup 4.4. Assume Hypothesis 4.11 on the modularityof generating functions of codimension r . For every ϕ ∈ V [ r ] π , every open compact subgroup L ⊆ H ( A ∞ F ), and every φ ∞ ∈ S ( V r ⊗ A F A ∞ F ) L , we putΘ w φ ∞ ( ϕ ) L := Z G r ( F ) \ G r ( A F ) ϕ c ( g ) Z w φ ∞ ( g ) L d g, which is an element in CH r ( X w L ) C by [LL, Proposition 4.7]. It is clear that the image ofΘ w φ ∞ ( ϕ ) L in CH r ( X w ) C := lim −→ L CH r ( X w L ) C depends only on ϕ and φ ∞ , which we denote by Θ w φ ∞ ( ϕ ). Finally, we define the arithmetictheta lifting of ( π, V π ) to V w to be the complex subspace Θ( π, V w ) of CH r ( X w ) C spanned byΘ w φ ∞ ( ϕ ) for all ϕ ∈ V [ r ] π and φ ∞ ∈ S ( V r ⊗ A F A ∞ F ).We recall Beilinson’s height pairing for our particular use from [LL, Section 5]. We havea map h , i ‘ X w L , E w : CH r ( X w L ) h ‘ i C × CH r ( X w L ) h ‘ i C → C ⊗ Q Q ‘ that is complex linear in the first variable, and conjugate symmetric. Here, ‘ is a rationalprime so that X w L,u has smooth projective reduction for every u ∈ V ( ‘ ) E . For a pair ( c , c ) ofelements in Z r ( X w L ) h ‘ i C × Z r ( X w L ) h ‘ i C with disjoint supports, we have h c , c i ‘ X w L , E w = X u ∈ V ( ∞ ) E h c , c i X w L,u , E w u + X u ∈ V fin E log q u · h c , c i ‘ X w L,u , E w u , in which • q u is the residue cardinality of E u for u ∈ V fin E ; • h c , c i ‘ X w L,u , E w u ∈ C ⊗ Q Q ‘ is the non-archimedean local index recalled in [LL, Section B]for u ∈ V fin E , which equals zero for all but finitely many u ; • h c , c i X w L,u , E w u ∈ C is the archimedean local index for u ∈ V ( ∞ ) E , recalled in [LL, Sec-tion 10]. Definition 4.13.
We say that a rational prime ‘ is R -good if ‘ is unramified in E and satisfies V ( ‘ ) F ⊆ V fin F \ ( R ∪ S ). Definition 4.14.
For every open compact subgroup L R of H ( F R ) and every subfield L of C ,we define(1) ( S R L ) L R to be the subalgebra of S R L (Setup 4.2(H8)) of elements that annihilate M i =2 r − H i dR ( X w L R L R / E w ) ⊗ Q L , (2) for every rational prime ‘ , ( S R L ) h ‘ i L R to be the subalgebra of S R L of elements that annihilate M u ∈ V fin E \ V ( ‘ ) E H r ( X w L R L R ,u , Q ‘ ( r )) ⊗ Q L . Here, L R is defined in Setup 4.2(H8). Definition 4.15.
Consider a nonempty subset R ⊆ R , an R -good rational prime ‘ , and anopen compact subgroup L of H ( A ∞ F ) of the form L R L R where L R is defined in Setup 4.2(H8).An ( R , R , ‘, L ) -admissible sextuple is a sextuple ( φ ∞ , φ ∞ , s , s , g , g ) in which • for i = 1 , φ ∞ i = ⊗ v φ ∞ iv ∈ S ( V r ⊗ A F A ∞ F ) L in which φ ∞ iv = (Λ R v ) r for v ∈ V fin F \ R ,satisfying that supp( φ ∞ v ⊗ ( φ ∞ v ) c ) ⊆ ( V rv ) reg for v ∈ R ; • for i = 1 ,
2, s i is a product of two elements in ( S R Q ac ) h ‘ i L R ; • for i = 1 , g i is an element in G r ( A R F ).For an ( R , R , ‘, L )-admissible sextuple ( φ ∞ , φ ∞ , s , s , g , g ) and every pair ( T , T ) of ele-ments in Herm ◦ r ( F ) + , we define(1) the global index I w T ,T ( φ ∞ , φ ∞ , s , s , g , g ) ‘L to be h ω r, ∞ ( g ∞ ) φ ∞ ( T ) · s ∗ Z w T ( ω ∞ r ( g ∞ ) φ ∞ ) L , ω r, ∞ ( g ∞ ) φ ∞ ( T ) · s ∗ Z w T ( ω ∞ r ( g ∞ ) φ ∞ ) L i ‘ X w L , E w as an element in C ⊗ Q Q ‘ , where we note that for i = 1 ,
2, s ∗ i Z w T i ( ω ∞ r ( g ∞ i ) φ ∞ i ) L belongsto CH r ( X w L ) h ‘ i C by Definition 4.14(2);(2) for every u ∈ V fin E , the local index I w T ,T ( φ ∞ , φ ∞ , s , s , g , g ) ‘L,u to be h ω r, ∞ ( g ∞ ) φ ∞ ( T ) · s ∗ Z w T ( ω ∞ r ( g ∞ ) φ ∞ ) L , ω r, ∞ ( g ∞ ) φ ∞ ( T ) · s ∗ Z w T ( ω ∞ r ( g ∞ ) φ ∞ ) L i ‘ X w L,u , E w u as an element in C ⊗ Q Q ‘ ;(3) for every u ∈ V ( ∞ ) E , the local index I w T ,T ( φ ∞ , φ ∞ , s , s , g , g ) L,u to be h ω r, ∞ ( g ∞ ) φ ∞ ( T ) · s ∗ Z w T ( ω ∞ r ( g ∞ ) φ ∞ ) L , ω r, ∞ ( g ∞ ) φ ∞ ( T ) · s ∗ Z w T ( ω ∞ r ( g ∞ ) φ ∞ ) L i X w L,u , E w u as an element in C .Let ( π, V π ) be as in Setup 4.4, and assume Hypothesis 4.11 on the modularity of generatingfunctions of codimension r . HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 55 Remark . In the situation of Definition 4.12 (and suppose that F = Q ), suppose that L has the form L R L R where L R is defined in Setup 4.2(H8). We have, from [LL, Proposi-tion 5.10], that for every elements ϕ ∈ V [ r ] R π and φ ∞ ∈ S ( V r ⊗ A F A ∞ F ) L ,(1) s ∗ Θ w φ ∞ ( ϕ ) L = Θ w φ ∞ ( ϕ ) L for every s ∈ S R Q ac such that χ R π (s) = 1;(2) Θ w φ ∞ ( ϕ ) L ∈ CH r ( X w L ) C ;(3) under [LL, Hypothesis 5.6], Θ w φ ∞ ( ϕ ) L ∈ CH r ( X w L ) h ‘ i C for every R -good rational prime ‘ .We recall the normalized height pairing between the cycles Θ w φ ∞ ( ϕ ) in Definition 4.12,under [LL, Hypothesis 5.6]. Definition 4.17.
Under [LL, Hypothesis 5.6], for every elements ϕ , ϕ ∈ V [ r ] π and φ ∞ , φ ∞ ∈ S ( V r ⊗ A F A ∞ F ), we define the normalized height pairing h Θ w φ ∞ ( ϕ ) , Θ w φ ∞ ( ϕ ) i \ X w , E w ∈ C ⊗ Q Q ‘ to be the unique element such that for every L = L R L R as in Remark 4.16 (with R possiblyenlarged) satisfying ϕ , ϕ ∈ V [ r ] R π , φ ∞ , φ ∞ ∈ S ( V r ⊗ A F A ∞ F ) L , and that ‘ is R -good, we have h Θ w φ ∞ ( ϕ ) , Θ w φ ∞ ( ϕ ) i \ X w , E w = vol \ ( L ) · h Θ w φ ∞ ( ϕ ) L , Θ w φ ∞ ( ϕ ) L i ‘ X w L , E w , where vol \ ( L ) is introduced in [LL, Definition 3.8] and h Θ φ ∞ ( ϕ ) L , Θ w φ ∞ ( ϕ ) L i ‘ X w L , E w is well-defined by Remark 4.16(3). Note that by the projection formula, the right-hand side of theabove formula is independent of L .Finally, we review the auxiliary Shimura variety that will only be used in the computationof local indices I w T ,T ( φ ∞ , φ ∞ , s , s , g , g ) L,u . Construction 4.18.
We define a torus T over Q such that for every commutative Q -algebra R , we have T ( R ) = { a ∈ E ⊗ Q R | Nm E/F a ∈ R × } .We choose a CM type Φ of E containing ι w and denote by E w the subfield of C generatedby E w and reflex field of Φ. For a (sufficiently small) open compact subgroup L of T ( A ∞ ),we have the PEL type moduli scheme Y of CM abelian varieties with CM type Φ and level L , which is a smooth projective scheme over E w of dimension 0. In what follows, when weinvoke this construction, the data Φ and L will be fixed, hence will not be carried into thenotation E w and Y . For every open compact subgroup L ⊆ H ( A ∞ F ), we put X w L := X w L ⊗ E w Y as a scheme over E w .The following setup is parallel to [LL, Setup 6.6]. Setup 4.19.
In Subsections 4.3, 4.4, and 4.5, we will consider a place u ∈ V fin E . Let p be theunderlying rational prime of u . We will fix an isomorphism C ∼ −→ Q p under which ι w inducesthe place u . In particular, we may identify Φ as a subset of Hom( E, Q p ).We further require that Φ in Construction 4.18 is admissible in the following sense: ifΦ v ⊆ Φ denotes the subset inducing the place v for every v ∈ V ( p ) F , then it satisfies(1) when v ∈ V ( p ) F ∩ V spl F , Φ v induces the same place of E above v ;(2) when v ∈ V ( p ) F ∩ V int F , Φ v is the pullback of a CM type of the maximal subfield of E v unramified over Q p ; (3) void unless u ∈ V ♥ F (Definition 1.1): when v ∈ V ( p ) F ∩ V ram F , the subfield of Q p generatedby E w u and the reflex field of Φ v is unramified over E w u .To release the burden of notation, we denote by K the subfield of Q p generated by E w u and the reflex field of Φ, by k its residue field, and by ˘ K the completion of the maximalunramified extension of K in Q p with the residue field F p . It is clear that admissible CMtype always exists; and that when u ∈ V ♥ F , K is unramified over E w u .We also choose a (sufficiently small) open compact subgroup L of T ( A ∞ ) such that L ,p is maximal compact. We denote by Y the integral model of Y over O K such that for every S ∈ Sch /O K , Y ( S ) is the set of equivalence classes of quadruples ( A , ι A , λ A , η pA ) where • ( A , ι A , λ A ) is a unitary O E -abelian scheme over S of signature type Φ (see [LTXZZ,Definition 3.4.2 & Definition 3.4.3]) such that λ A is p -principal; • η pA is an L p -level structure (see [LTXZZ, Definition 4.1.2] for more details).By [How12, Proposition 3.1.2], Y is finite and étale over O K .4.3. Local indices at split places.
In this subsection, we compute local indices at almostall places in V spl E . Our goal is to prove the following proposition. Proposition 4.20.
Let R , R , ‘ , and L be as in Definition 4.15 such that the cardinality of R is at least . Let ( π, V π ) be as in Setup 4.4. For every u ∈ V spl E satisfying u R \ V ♥ F and V ( p ) F ∩ R ⊆ V spl F where p is the underlying rational prime of u , there exist elements s u , s u ∈ S R Q ac \ m R π such that I w T ,T ( φ ∞ , φ ∞ , s u s , s u s , g , g ) ‘L,u = 0 for every ( R , R , ‘, L ) -admissible sextuple ( φ ∞ , φ ∞ , s , s , g , g ) and every pair ( T , T ) in Herm ◦ r ( F ) + . Moreover, we may take s u = s u = 1 if u R .Proof. This is simply [LL, Proposition 7.1] but without the assumption that π u is a (tem-pered) principal series. The proof is same, after we slightly generalize the construction of theintegral model X m to take care of places in V ( p ) F ∩ V ram F , and use Theorem 4.21 below whichgeneralizes [LL, Lemma 7.3]. (cid:3) From now to the end of this section, we assume V ( p ) F ∩ R ⊆ V spl F . We also assume u ∈ V ♥ F when we need m (cid:62) C ∼ −→ Q p in Setup 4.19 identifies Hom( E, C ) with Hom( E, C p ). For every v ∈ V ( p ) F , let Φ v be the subset of Φ, regarded as a subset of Hom( E, C p ), of elements thatinduce the place v of F .For every integer m (cid:62)
0, we define a moduli functor X m over O K as follows: For every S ∈ Sch /O K , X m ( S ) is the set of equivalence classes of tuples( A , ι A , λ A , η pA ; A, ι A , λ A , η pA , { η A,v } v ∈ V ( p ) F ∩ V spl F \{ u } , η A,u,m )where • ( A , ι A , λ A , η pA ) is an element in Y ( S ); • ( A, ι A , λ A ) is a unitary O E -abelian scheme of signature type n Φ − ι w + ι c w over S , suchthat Here, our notation on objects is slightly different from [LTXZZ] or [LL] as we, in particular, retrieve the O E -action ι A . HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 57 – for every v ∈ V ( p ) F \ V ram F , λ A [ v ∞ ] is an isogeny whose kernel has order q − (cid:15) v v ; – Lie( A [ u c , ∞ ]) is of rank 1 on which the action of O E is given by the embedding ι c w ; – for every v ∈ V ( p ) F ∩ V ram F , the triple ( A [ v ∞ ] , ι A [ v ∞ ] , λ A [ v ∞ ]) ⊗ O K O ˘ K is an object ofExo Φ v ( n, ( S ⊗ O K O ˘ K ) (Remark 2.68, with E = E v , F = F v , and ˘ E = ˘ K ); • η pA is an L p -level structure; • for every v ∈ V ( p ) F ∩ V spl F \ { u } , η A,v is an L v -level structure; • η A,u,m is a Drinfeld level- m structure.See [LL, Section 7] for more details for the last three items. By [RSZ20, Theorem 4.5], forevery m (cid:62) X m is a regular scheme, flat (smooth, if m = 0) and projective over O K , andadmits a canonical isomorphism X m ⊗ O K K ’ X w L u,m L u ⊗ E w K of schemes over K . Note that for every integer m (cid:62) S R ∪ V ( p ) F naturally gives a ring of étalecorrespondences of X m . The following theorem confirms the conjecture proposed in [LL, Remark 7.4], and the restof this subsection will be devoted to its proof.
Theorem 4.21.
Let the situation be as in Proposition 4.20 and assume u ∈ V ♥ F . For everyinteger m (cid:62) , (H r ( X m , Q ‘ ( r )) ⊗ Q Q ac ) m = 0 holds, where m := m R π ∩ S R ∪ V ( p ) F Q ac . We temporarily allow n to be an arbitrary positive integer, not necessarily even. Put Y m := X m ⊗ O K k . For every point x ∈ Y m ( F p ), we know that A x [ u c , ∞ ] is a one-dimensional O F u -divisible group of (relative) height n , and we let 0 (cid:54) h ( x ) (cid:54) n − (cid:54) h (cid:54) n −
1, let Y [ h ] m be locus where h ( x ) (cid:54) h , which is Zariski closed hencewill be endowed with the reduced induced scheme structure, and put Y ( h ) m := Y [ h ] m − Y [ h − m ( Y [ − m = ∅ ). It is known that Y ( h ) m is smooth over k of pure dimension h .Now we suppose that m (cid:62)
1. Let S hm be the set of free O F u / p mu -submodules of ( p − mu /O F u ) n of rank n − h , and put S m := S n − h =0 S hm . For every M ∈ S hm , we denote by Y ( M ) m ⊆ Y ( h ) m the(open and closed) locus where the kernel of the Drinfeld level- m structure is M . Then wehave Y ( h ) m = a M ∈ S hm Y ( M ) m for every 0 (cid:54) h (cid:54) n −
1. Let Y [ M ] m be the scheme-theoretic closure of Y ( M ) m inside Y m . Thenwe have Y [ M ] m = [ M ∈ S m M ⊆ M Y ( M ) m (4.1)as a disjoint union of strata. Note that Hecke operators away from u (of level L u ) preserve Y ( M ) m hence Y [ M ] m for every M ∈ S m .We need some general notation. For a sequence ( g , . . . , g t ) of nonnegative integers with g = g + · · · + g t , we denote by P g ,...,g t the standard upper triangular parabolic subgroup of The sign condition is redundant in our case by [RSZ20, Remark 5.1(i)]. When m = 0, we do not need u ∈ V ♥ F as the same holds even when K is ramified over E w u . GL g of block sizes g , . . . , g t , and M g ,...,g t its standard diagonal Levi subgroup. Moreover,we denote by C g ,...,g t m the cardinality ofGL g ( O F u / p mu ) / P g ,...,g t ( O F u / p mu ) , which depends only on the partition g = g + · · · + g t . We also put L gu,m := ker (cid:16) GL g ( O F u ) → GL g ( O F u / p mu ) (cid:17) . For an irreducible admissible representation π of GL g ( F u ) and a positive integer s , we havethe representation Sp s ( π ) of GL sg ( F u ) defined in [HT01, Section I.3]. Lemma 4.22.
For ( g , . . . , g t ) with g = g + · · · + g t as above and another integer g (cid:62) g ,we have C g − g,gm C g ,...,g t m = C g − g + g ,g ,...,g t m . Proof.
It follows from the isomorphismP g − g,g ( O F u / p mu ) / P g − g + g ,g ,...,g t ( O F u / p mu ) ’ GL g ( O F u / p mu ) / P g ,...,g t ( O F u / p mu ) . (cid:3) Lemma 4.23.
Suppose that m (cid:62) . Take a sequence ( g , . . . , g t ) of nonnegative integerswith g = g + · · · + g t . Let π (cid:2) · · · (cid:2) π t be an admissible representation of M g ,...,g t ( F u ) . Thenwe have dim (cid:18) Ind GL g ( F u )P g ,...,gt ( F u ) π (cid:2) · · · (cid:2) π t (cid:19) L gu,m = C g ,...,g t m t Y i =1 dim π L giu,m i . Proof.
Pick a set X of representatives of the double cosetP g ,...,g t ( F u ) \ GL g ( F u ) /L gu,m contained in GL g ( O F u ), which is possible by the Iwasawa decomposition. Then an element f ∈ (cid:18) Ind GL g ( F u )P g ,...,gt ( F u ) π (cid:2) · · · (cid:2) π t (cid:19) L gu,m is determined by f | X . Since GL g ( O F u ) normalizes L gu,m , a function f on X is of the form f = f | X if and only if f takes values in N ti =1 π L giu,m i . As | X | = C g ,...,g t m , the lemma follows. (cid:3) Lemma 4.24.
Suppose that m (cid:62) . For every positive integer g and every unramifiedcharacter φ of F × u , we have g X h =0 ( − h C g − h,hm dim Sp h ( φ ) L hu,m = 0 . Proof.
We claim the identity g X h =0 ( − h (cid:20) Ind GL g ( F u )P h,g − h ( F u ) Sp h ( φ ) (cid:2) (cid:18) φ | | g + h − u ◦ det g − h (cid:19)(cid:21) = 0(4.2)in Groth(GL g ( F u )). Assuming it, we have g X h =0 ( − h dim (cid:18) Ind GL g ( F u )P h,g − h ( F u ) Sp h ( φ ) (cid:2) (cid:18) φ | | g + h − u ◦ det g − h (cid:19)(cid:19) L gu,m = 0 . By Lemma 4.23, the lemma follows.
HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 59 For the claim, put I( φ ) := Ind GL g ( F u )P ,..., ( F u ) φ (cid:2) φ | | u (cid:2) · · · (cid:2) φ | | g − u . By the transitivity of (normalized) parabolic induction, every irreducible constituent ofI( φ ) h,g − h := Ind GL g ( F u )P h,g − h ( F u ) Sp h ( φ ) (cid:2) (cid:18) φ | | g + h − u ◦ det g − h (cid:19) is a constituent of I( φ ). By [Zel80], there is a bijection between the set of irreducible sub-quotients of I( φ ) and the set of sequences of signs of length g −
1. For such a sequence σ ,we denote by I( φ ) σ the corresponding irreducible subquotient. For 0 (cid:54) h (cid:54) g −
1, we denoteby σ ( i ) the sequence starting from h negative signs followed by g − − h positive signs. Inparticular, I( φ ) σ ( g − = Sp g ( φ ) = I( φ ) g, , I( φ ) σ (0) = φ | | g − u ◦ det g = I( φ ) ,g . By [HT01, Lemma I.3.2], we have[I( φ ) h,g − h ] = [I( φ ) σ ( h ) ] + [I( φ ) σ ( h − ]in Groth(GL g ( F u )) for 0 < h < g . Thus, (4.2) follows. (cid:3) Proposition 4.25.
Fix an isomorphism Q ‘ ’ C . Suppose that m (cid:62) . For every (cid:54) h (cid:54) n − and M ∈ S hm , we have H j ( Y [ M ] m ⊗ k F p , Q ‘ ) m = 0 for every j = h . This is an extension of [TY07, Proposition 4.4]. However, we allow arbitrary principallevel at u and our case involves endoscopy. Proof.
In what follows, h will always denote an integer satisfying 0 (cid:54) h (cid:54) n −
1. Denote byD n − h the division algebra over F u of Hasse invariant n − h , with the maximal order O D n − h .For a T -scheme Y of finite type over k , and a (finite) character χ : T ( Q ) \ T ( A ∞ ) /L → Q × ‘ , we put [H ? ,χ ( Y, Q ‘ )] := X j ∈ Z ( − j H j ? ( Y ⊗ k F p , Q ‘ )[ χ ]as an element in Groth(Gal( F p /k )) for ? = { , c } .Let I hm be the Igusa variety (of the first kind) introduced in [HT01, Section IV.1] so that I hm is isomorphic to Y ( M ) m for every M ∈ S hm as schemes over k (but not as schemes over Y ( h )0 ). Combining with (4.1), we obtain the identity[H χ ( Y [ M ] m , Q ‘ )] = h X h =0 X M ∈ S h m M ⊆ M ( − h − h [H c,χ ( Y ( M ) m , Q ‘ )](4.3) = h X h =0 ( − h − h · (cid:12)(cid:12)(cid:12) { M ∈ S h m | M ⊆ M } (cid:12)(cid:12)(cid:12) · [H c,χ ( I h m , Q ‘ )]= h X h =0 ( − h − h C h − h ,h m · [H c,χ ( I h m , Q ‘ )]in Groth(Gal( F p /k )). Now to compute [H χ ( I h m , Q ‘ )], we use [CS17, Lemma 5.5.1] in which the corresponding J b ( Q p ) is D n − h × GL h ( F u ), and we take φ = φ u φ u where φ u is the characteristic function of L u and φ u is the characteristic function of O × D n − h × L h u,m . Then we have the identity[H c,χ ( I h m , Q ‘ )] = X n X Π n c ( n , Π n ) · Red h n ( π n u ) O × D n − h × L h u,m (4.4)in Groth(D × n − h /O × D n − h ), where • n runs through ordered pairs ( n , n ) of nonnegative integers such that n + n = n ,which gives an elliptic endoscopic group G n of U( V w ); • Π n runs through a finite set of certain isobaric irreducible cohomological (with respectto the trivial algebraic representation) automorphic representations of G n ( A F ), with π n u the descent of Π n u to G n ( F u ) ’ M n ,n ( F u ); • c ( n , Π n ) is a constant depending only on n and Π n but not on h ; • Red h n : Groth(M n ,n ( F u )) → Groth(D × n − h × GL h ( F u )) is the zero map if h < n , andotherwise is the composition of – Groth(M n ,n ( F u )) → Groth(M n − h ,h − n ,n ( F u )), which is the normalized Jacquetfunctor, – Groth(M n − h ,h − n ,n ( F u )) → Groth(M n − h ,h ( F u )), which is the normalized parabolicinduction, and – Groth(M n − h ,h ( F u )) → Groth(D × n − h × GL h ( F u )), which is the Langlands–Jacquetmap (on the first factor).The image of [H c,χ ( I h m , Q ‘ )] in Groth(Gal( F p /k )) is given by the mapGroth(D × n − h /O × D n − h ) → Groth(Gal( F p /k ))sending an (unramified) character φ ◦ Nm D × n − h to rec( φ − ) · ˘ χ , where ˘ χ is a finite characterof Gal( F p /k ) determined by χ . In what follows, we will regardRed h n ( π n u ) O × D n − h × L h u,m as an element of Groth(Gal( F p /k )) via the above map.Now let us compute for each n = ( n , n ), h X h =0 ( − h − h C h − h ,h m · Red h n ( π n u ) O × D n − h × L h u,m (4.5)in Groth(Gal( F p /k )), when π n u is tempered. Write π n u = π (cid:2) π where π α is an temperedirreducible admissible representation of GL n α ( F u ). In particular, π is a full induction of theform Ind GL n ( F u )P s g ,...,stgt ( F u ) Sp s ( π ) (cid:2) · · · (cid:2) Sp s t ( π t ) , where s , . . . , s t and g , . . . , g t are positive integers satisfying s g + · · · + s t g t = n ; andfor 1 (cid:54) i (cid:54) t , π i is an irreducible cuspidal representation of GL g i ( F u ) such that Sp s i ( π i )is unitary. Let I be the subset of { , . . . , t } such that π i is an unramified character (hence HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 61 g i = 1) and s i (cid:62) n − h . Then we have for h (cid:62) n ,Red h n ( π n u ) O × D n − h × L h u,m (4.6)= X i ∈ I s i (cid:62) n − h dim (cid:18) Ind GL h ( F u )P ? ( F u ) Sp s i + h − n ( π i ) (cid:2) (cid:16) (cid:2) j = i Sp s j ( π j ) (cid:17) (cid:2) π (cid:19) L h u,m [rec(( π i ) − | | − n u ) · ˘ χ ]in which the suppressed subscript in P ? is ( s i + h − n, s g , . . . , d s i g i , . . . , s t g t , n ).We claim that for each i ∈ I , h X h = n − s i ( − h − h C h − h ,h m dim (cid:18) Ind GL h ( F u )P ? ( F u ) Sp s i + h − n ( π i ) (cid:2) (cid:16) (cid:2) j = i Sp s j ( π j ) (cid:17) (cid:2) π (cid:19) L h u,m = 0(4.7)if s i > n − h . In fact, by Lemma 4.23, there is a nonnegative integer D independent of h such that the left-hand side of (4.7) equals h X h = n − s i ( − h − h C h − h ,h m · C s i + h − n,s g ,..., c s i g i ,...,s t g t ,n m · D · dim Sp s i + h − n ( π i ) L si + h nu,m = h X h = n − s i ( − h − h C h − h ,s i + h − n,s g ,..., c s i g i ,...,s t g t ,n m · D · dim Sp s i + h − n ( π i ) L si + h nu,m = h + s i − n X h =0 ( − h − h C h + s i − n − h ,h ,s g ,..., c s i g i ,...,s t g t ,n m · D · dim Sp h ( π i ) L h u,m = ( − h C h + s i − n,s g ,..., c s i g i ,...,s t g t ,n m · D h + s i − n X h =0 ( − h C h + s i − n − h ,h m dim Sp h ( π i ) L h u,m in which the last summation vanishes by applying Lemma 4.24 with g = h + s i − n > π i ) − | | − n u ) · ˘ χ ]with i ∈ I satisfying s i = n − h . Thus, (4.5) is strictly pure of weight h since Sp s i ( π i ) isunitary. By (4.3), (4.4), and the fact that localization at m annihilates all terms in (4.4)with π n u not tempered, we know that [H χ ( Y [ M ] m , Q ‘ )] m is strictly pure of weight h . Finally,by [Man08, Proposition 12], we know that Y [ M ] m is smooth over k of pure dimension h . Since Y [ M ] m is also proper, we have H j ( Y [ M ] m ⊗ k F p , Q ‘ )[ χ ] m = 0for every j = h and every character χ : T ( Q ) \ T ( A ∞ ) /L → Q × ‘ from the Weil conjecture.Then the proposition follows. (cid:3) Proof of Theorem 4.21.
We may assume m (cid:62) X m → X is finite andflat. In what follows, h is always an integer satisfying 0 (cid:54) h (cid:54) n − r −
1. For a subsetΣ ⊂ S hm , we put Y (Σ) m := [ M ∈ Σ Y ( M ) m , Y [Σ] m := [ M ∈ Σ Y [ M ] m in which the first union is disjoint. If h (cid:62)
1, we also denote by Σ † the subset of S h − m consisting of M that contains an element in Σ.Fix an arbitrary isomorphism Q ‘ ’ C . We show by induction on h that for every Σ ⊂ S hm ,H jc ( Y (Σ) m ⊗ k F p , Q ‘ ) m = H j ( Y [Σ] m ⊗ k F p , Q ‘ ) m = 0(4.8)if j > h . To ease notation, we simply write H • ? ( − ) for H • ? ( − ⊗ k F p , Q ‘ ) m for ? ∈ { , c } .The case for h = 0 is trivial. Suppose that we know (4.8) for h − h (cid:62)
1. Forevery M ∈ S hm , we have the exact sequence · · · → H j − ( Y [ { M } † ] m ) → H jc ( Y ( M ) m ) → H j ( Y [ M ] m ) → · · · By Proposition 4.25 and the induction hypothesis, we have H jc ( Y ( M ) m ) = 0 for j > h . Nowtake a subset Σ of S hm . Then we have H jc ( Y (Σ) m ) = L M ∈ Σ H jc ( Y ( M ) m ) = 0 for j > h . By theexact sequence · · · → H jc ( Y (Σ) m ) → H j ( Y [Σ] m ) → H j ( Y [Σ † ] m ) → · · · and the induction hypothesis, we have H j ( Y [Σ] m ) = 0 for j > h . Thus, (4.8) holds for h .By (4.8) and the Poincaré duality, we haveH j ( Y ( h ) m ⊗ k F p , Q ‘ ) m = 0for j < h . Now we apply [LL, Corollary B.13(2)] with Y j = Y r − − jm for 0 (cid:54) j (cid:54) r − n j = j + 1, which is allowed since 2 r − n j < r − − j . Note that the vanishing ofH r ( X m ⊗ O K K, Q ‘ ( r )) m follows from the same argument for [LL, Lemma 7.3]. We obtainH r ( X m , Q ‘ ( r )) m = 0, hence the theorem follows. (cid:3) Local indices at inert places.
In this subsection, we compute local indices at placesin V int E not above R . Proposition 4.26.
Let R , R , ‘ , and L be as in Definition 4.15. Take an element u ∈ V int E such that its underlying rational prime p is odd and satisfies V ( p ) F ∩ R ⊆ V spl F .(1) Suppose that u S . Then we have log q u · vol \ ( L ) · I w T ,T ( φ ∞ , φ ∞ , s , s , g , g ) ‘L,u = E T ,T (( g , g ) , Φ ∞ ⊗ (s φ ∞ ⊗ (s φ ∞ ) c )) u for every ( R , R , ‘, L ) -admissible sextuple ( φ ∞ , φ ∞ , s , s , g , g ) and every pair ( T , T ) in Herm ◦ r ( F ) + .(2) Suppose that u ∈ S ∩ V ♥ F and is unramified over Q . Fix an isomorphism − u : V ⊗ A F A uF ∼ −→ V u ⊗ F A uF of hermitian spaces over A uE and a ψ E,u -self-dual lattice Λ ?u of V u u . Then there existelements s u , s u ∈ S R Q ac \ m R π such that log q u · vol \ ( L ) · I w T ,T ( φ ∞ , φ ∞ , s u s , s u s , g , g ) ‘L,u = E T ,T (( g , g ) , Φ ∞ ⊗ (s u s φ ∞ ⊗ (s u s φ ∞ ) c )) u − log q u q ru − E T ,T (( g , g ) , Φ u ∞ ⊗ ( u s u s φ ∞ ,u ⊗ (s u s φ ∞ ,u ) c ) ⊗ (Λ ?u ) r ) for every ( R , R , ‘, L ) -admissible sextuple ( φ ∞ , φ ∞ , s , s , g , g ) and every pair ( T , T ) in Herm ◦ r ( F ) + . HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 63 In both cases, the right-hand side is defined in Definition 4.10 with the Gaussian function Φ ∞ ∈ S ( V r ⊗ A F F ∞ ) (Setup 4.2(H3)), and vol \ ( L ) is defined in [LL, Definition 3.8] .Proof. Part (1) is proved in the same way as [LL, Proposition 8.1]. Part (2) is proved inthe same way as [LL, Proposition 9.1]. Note that we need to extend the definition of theintegral model due to the presence of places in V ( p ) F ∩ V ram F , as we do in the previous subsection.The requirement that u ∈ V ♥ F in (2) is to ensure that K is unramified over E w u (see Setup4.19). (cid:3) Local indices at ramified places.
In this subsection, we compute local indices atplaces in V ram E not above R . Proposition 4.27.
Let R , R , ‘ , and L be as in Definition 4.15. Take an element u ∈ V ram E such that its underlying rational prime p satisfies V ( p ) F ∩ R ⊆ V spl F . Then we have log q u · vol \ ( L ) · I w T ,T ( φ ∞ , φ ∞ , s , s , g , g ) ‘L,u = E T ,T (( g , g ) , Φ ∞ ⊗ (s φ ∞ ⊗ (s φ ∞ ) c )) u for every ( R , R , ‘, L ) -admissible sextuple ( φ ∞ , φ ∞ , s , s , g , g ) and every pair ( T , T ) in Herm ◦ r ( F ) + , where the right-hand side is defined in Definition 4.10 with the Gaussian func-tion Φ ∞ ∈ S ( V r ⊗ A F F ∞ ) (Setup 4.2(H3)), and vol \ ( L ) is defined in [LL, Definition 3.8] .Proof. The proof of the proposition follows the same line as in [LL, Proposition 8.1], as longas we accomplish the following three tasks. We invoke Construction 4.18 together with Setup4.19.(1) Construct a good integral model X ˜ L for X w ˜ L over O K for open compact subgroups˜ L ⊆ L satisfying ˜ L v = L v for v ∈ V ( p ) F \ V spl F , which is provided after the proof.(2) Establish the nonarchimedean uniformization of X ˜ L along the supersingular locus usingthe relative Rapoport–Zink space N from Definition 2.3, analogous to [LL, (8.1)], andcompare special divisors. This is done in Proposition 4.29 below.(3) Show that for x = ( x , . . . , x r ) ∈ V u r with T ( x ) ∈ Herm ◦ r ( F u ), we have χ (cid:18) O N ( x ) L ⊗ O N · · · L ⊗ O N O N ( x r ) (cid:19) = b r,u (0)log q u W T (cid:3) (0 , r , (Λ R u ) r )if T ( x ) = T (cid:3) . In fact, this follows from Theorem 2.9, Remark 2.19, and the identity b r,u (0) = r Y i =1 (1 − q − iu ) . The proposition is proved. (cid:3)
Let the situation be as in Proposition 4.27. The isomorphism C ∼ −→ Q p in Setup 4.19identifies Hom( E, C ) with Hom( E, C p ). For every v ∈ V ( p ) F , let Φ v be the subset of Φ,regarded as a subset of Hom( E, C p ), of elements that induce the place v of F .To ease notation, put U := { v ∈ V ( p ) F \ V spl F | v = u } . In particular, U ∩ R = ∅ .There is a projective system {X ˜ L } , for open compact subgroups ˜ L ⊆ L satisfying ˜ L v = L v for v ∈ V ( p ) F \ V spl F , of smooth projective schemes over O K (see [RSZ20, Theorem 4.7, AT type(2)]) with X ˜ L ⊗ O K K = X w ˜ L ⊗ E w K = (cid:16) X w ˜ L ⊗ E w Y (cid:17) ⊗ E w K, and finite étale transition morphisms, such that for every S ∈ Sch /O K , X ˜ L ( S ) is the set ofequivalence classes of tuples( A , ι A , λ A , η pA ; A, ι A , λ A , η pA , { η A,v } v ∈ V ( p ) F ∩ V spl F )where • ( A , ι A , λ A , η pA ) is an element in Y ( S ); • ( A, ι A , λ A ) is a unitary O E -abelian scheme of signature type n Φ − ι w + ι c w over S , suchthat – for every v ∈ V ( p ) F \ V ram F , λ A [ v ∞ ] is an isogeny whose kernel has order q − (cid:15) v v ; – for every v ∈ U ∩ V ram F , the triple ( A [ v ∞ ] , ι A [ v ∞ ] , λ A [ v ∞ ]) ⊗ O K O ˘ K is an object ofExo Φ v ( n, ( S ⊗ O K O ˘ K ) (Remark 2.68, with E = E v , F = F v , and ˘ E = ˘ K ); – for v = u , ( A [ v ∞ ] , ι A [ v ∞ ] , λ A [ v ∞ ]) ⊗ O K O ˘ K is an object of Exo Φ v ( n − , ( S ⊗ O K O ˘ K )(Definition 2.60, with E = E v , F = F v , and ˘ E = ˘ K ); • η pA is an ˜ L p -level structure; • for every v ∈ V ( p ) F ∩ V spl F , η A,v is an ˜ L v -level structure.In particular, S R is naturally a ring of étale correspondences of X L .Let φ ∞ ∈ S ( V ⊗ A F A ∞ F ) ˜ L be a p -basic element [LL, Definition 5.5]. For every element t ∈ F + , we have a cycle Z t ( φ ∞ ) ˜ L ∈ Z ( X ˜ L ) extending the restriction of Z w t ( φ ∞ ) to X w ˜ L ,defined similarly as in [LZ, Section 13.3].Now we study the nonarchimedean uniformization of X ˜ L along the supersingular locus.Fix a point P := ( A , ι A , λ A , η pA ) ∈ Y ( O ˘ K ). Put X := lim ←− ˜ L X ˜ L and denote by X the fiber of P along the natural projection X → Y . Let X ∧ be the comple-tion along the (closed) locus where A [ u ∞ ] is supersingular, as a formal scheme over Spf O K .Also fix a point P ∈ X ∧ ( F p ) represented by ( P ⊗ O ˘ K F p ; A , ι A , λ A , η p A , { η A ,v } v ∈ V ( p ) F ∩ V spl F ).Put V := Hom O E ( A ⊗ O ˘ E F p , A ) ⊗ Q . Fixing an element $ ∈ O F that has valuation 0(resp. 1) at places in U ∩ V int F (resp., U ∩ V ram F ), we have a pairing( , ) V : V × V → E sending ( x, y ) ∈ V to the composition of quasi-homomorphisms A x −→ X λ A −→ A ∨ y ∨ −→ A ∨ $ − λ − A −−−−−→ A as an element in End O E ( A ) ⊗ Q hence in E via ι − A . We have the following propertiesconcerning V : • V , ( , ) V is a totally positive definite hermitian space over E of rank n ; • for every v ∈ V fin F \ ( V ( p ) F \ V spl F ), we have a canonical isometry V ⊗ F F v ’ V ⊗ F F v ofhermitian spaces; • for every v ∈ U , the O E v -lattice Λ v := Hom O E ( A ⊗ O ˘ E F p , A ) ⊗ O F O F v is – self-dual if v ∈ U ∩ V int F and (cid:15) v = 1, – almost self-dual if v ∈ U ∩ V int F and (cid:15) v = − – self-dual if v ∈ U ∩ V ram F ; HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 65 • V ⊗ F F u is nonsplit, and we have a canonical isomorphism V ⊗ F F u ’ Hom O Eu ( A [ u ∞ ] ⊗ O ˘ K F p , A [ u ∞ ]) ⊗ Q of hermitian spaces over E u .We have a Rapoport–Zink space N (Definition 2.3, with E = E u , F = F u , ˘ E = ˘ K , and ϕ the natural embedding) with respect to the object( X , ι X , λ X ) := ( A [ u ∞ ] , ι A [ u ∞ ] , λ A [ u ∞ ]) rel ∈ Exo b( n − , ( F p ) , where − rel is the morphism (2.21). We now construct a morphismΥ rel : X ∧ → U( V )( F ) \ N × U( V )( A ∞ ,uF ) / Y v ∈ U L v ! (4.9)of formal schemes over Spf O ˘ K , where L v is the stabilizer of Λ v in U( V )( F v ), as follows.We have the Rapoport–Zink space N Φ u = N Φ u ( A [ u ∞ ] ,ι A [ u ∞ ] ,λ A [ u ∞ ]) from Definition 2.65. Wefirst define a morphismΥ : X ∧ → U( V )( F ) \ N Φ u × U( V )( A ∞ ,uF ) / Y v ∈ U L v ! , and then define Υ rel as the composition of Υ with the morphism in Corollary 2.66. Toconstruct Υ, we take a point P = ( P ⊗ O ˘ K S ; A, ι A , λ A , η pA , { η A,v } v ∈ V ( p ) F ∩ V spl F ) ∈ X ∧ ( S )for a connected scheme S in Sch /O ˘ K ∩ Sch v /O ˘ K with a geometric point s . In particular, A [ p ∞ ]is supersingular. By [RZ96, Proposition 6.29], we can choose an O E -linear quasi-isogeny ρ : A × S ( S ⊗ O ˘ K F p ) → A ⊗ F p ( S ⊗ O ˘ K F p )of height zero such that ρ ∗ λ A ⊗ F p ( S ⊗ O ˘ K F p ) = λ A × S ( S ⊗ O ˘ K F p ). We have • ( A [ u ∞ ] , ι A [ u ∞ ] , λ A [ u ∞ ]; ρ [ u ∞ ]) is an element in N Φ u ( S ); • the composite map V ⊗ Q A ∞ ,p ∼ −→ V ⊗ Q A ∞ ,p η pA −→ Hom E ⊗ Q A ∞ ,p (H ( A ,s , A ∞ ,p ) , H ( A s , A ∞ ,p )) ρ s ∗ ◦ −−→ Hom E ⊗ Q A ∞ ,p (H ( A ,s , A ∞ ,p ) , H ( A s , A ∞ ,p )) = V ⊗ Q A ∞ ,p is an isometry, which gives rise to an element h p ∈ U( V )( A ∞ ,pF ); • the same process as above will produce an element h spl p ∈ Q v ∈ V ( p ) F ∩ V spl F U( V )( F v ); • for every v ∈ U , the image of the map ρ s ∗ ◦ : Hom O Ev ( A ,s [ v ∞ ] , A s [ v ∞ ]) → Hom O Ev ( A ,s [ v ∞ ] , A s [ v ∞ ]) ⊗ Q = V ⊗ F F v is an O E v -lattice in the same U( V )( F v )-orbit of Λ v , which gives rise to an element h v ∈ U( V )( F v ) / L v .Together, we obtain an element (cid:16) ( A [ u ∞ ] , ι A [ u ∞ ] , λ A [ u ∞ ]; ρ [ u ∞ ]) , ( h p , h spl p , { h v } v ∈ U ) (cid:17) ∈ N Φ u ( S ) × U( V )( A ∞ ,uF ) / Y v ∈ U L v , and we define Υ( P ) to be its image in the quotient, which is independent of the choice of ρ . Remark . Both V and Υ rel depend on the choice of P , while the isometry class of V does not. Proposition 4.29.
The morphism Υ rel (4.9) is an isomorphism. Moreover, for every p -basicelement φ ∞ ∈ S ( V ⊗ A F A ∞ F ) ˜ L and every t ∈ F + , we have Υ rel (cid:16) Z t ( φ ∞ ) ˜ L | X ∧ (cid:17) = X x ∈ U( V )( F ) \ V ( x,x ) V = t X h ∈ U( V x )( F ) \ U( V )( A ∞ ,uF ) / Q v ∈ U L v φ ( h − x ) · ( N ( x rel ) , h ) , (4.10) where • V x denotes the orthogonal complement of x in V ; • φ is a Schwartz function on V ⊗ F A ∞ ,uF such that φ v = φ ∞ v for v ∈ V fin F \ ( V ( p ) F \ V spl F ) and φ v = Λ v for v ∈ U ; • x rel is defined in (2.25) ; and • ( N ( x rel ) , h ) denotes the corresponding double coset in (4.9) .Proof. By a similar argument for [RZ96, Theorem 6.30], the morphism Υ is an isomorphism.Thus, Υ rel is an isomorphism as well by Corollary 2.66.For (4.10), by a similar argument for [Liua, Theorem 5.22], the identity holds with N ( x rel )replaced by N Φ u ( x ). Then it follows by Corollary 2.67.The proposition is proved. (cid:3) Local indices at archimedean places.
In this subsection, we compute local indicesat places in V ( ∞ ) E . Proposition 4.30.
Let R , R , ‘ , and L be as in Definition 4.15. Let ( π, V π ) be as in Setup 4.4.Take an element u ∈ V ( ∞ ) E . Consider an ( R , R , ‘, L ) -admissible sextuple ( φ ∞ , φ ∞ , s , s , g , g ) and an element ϕ ∈ V [ r ] R π . Let K ⊆ G r ( A ∞ F ) be an open compact subgroup that fixes both φ ∞ and ϕ , and F ⊆ G r ( F ∞ ) a Siegel fundamental domain for the congruence subgroup G r ( F ) ∩ g ∞ K ( g ∞ ) − . Then for every T ∈ Herm ◦ r ( F ) + , we have vol \ ( L ) · Z F ϕ c ( τ g ) X T ∈ Herm ◦ r ( F ) + I w T ,T ( φ ∞ , φ ∞ , s , s , τ g , g ) L,u d τ = 12 Z F ϕ c ( τ g ) X T ∈ Herm ◦ r ( F ) + E T ,T (( τ g , g ) , Φ ∞ ⊗ (s φ ∞ ⊗ (s φ ∞ ) c )) u d τ , in which both sides are absolutely convergent. Here, the term E T ,T is defined in Definition4.10 with the Gaussian function Φ ∞ ∈ S ( V r ⊗ A F F ∞ ) (Setup 4.2(H3)), and vol \ ( L ) isdefined in [LL, Definition 3.8] .Proof. This is simply [LL, Proposition 10.1]. (cid:3)
Proof of main results.
The proofs of Theorem 1.5, Theorem 1.6, and Corollary 1.8follow from the same lines as for [LL, Theorem 1.5], [LL, Theorem 1.7], and [LL, Corol-lary 1.9], respectively, written in [LL, Section 11]. However, we need to take R to be a finitesubset of V spl F ∩ V ♥ F containing R π and of cardinality at least 2, and modify the referenceaccording to the table below. HOW GROUPS AND L -DERIVATIVES OF AUTOMORPHIC MOTIVES FOR UNITARY GROUPS, II. 67 This article [LL]Proposition 4.8 Proposition 3.6Proposition 4.9 Proposition 3.7Proposition 4.20 Proposition 7.1Proposition 4.26 Proposition 8.1 & Proposition 9.1Proposition 4.27 (not available)Proposition 4.30 Proposition 10.1
Remark . Finally, we explain the main difficulty on lifting the restriction F = Q (when r (cid:62) F = Q and r (cid:62)
2. Then the Shimura variety X w L from Subsection4.2 is never proper over the base field. Nevertheless, it is well-known that X w L admits acanonical toroidal compactification which is smooth. However, to run our argument, weneed suitable compactification of their integral models at every place finite place u of E aswell. As far as we can see, the main obstacle is the compactification of integral models usingDrinfeld level structures when u splits over F , together with a vanishing result like Theorem4.21. References [ACZ16] T. Ahsendorf, C. Cheng, and T. Zink, O -displays and π -divisible formal O -modules , J. Algebra (2016), 129–193, DOI 10.1016/j.jalgebra.2016.03.002. MR3490080 ↑
36, 37, 38[Be˘ı87] A. Be˘ılinson,
Height pairing between algebraic cycles , Current trends in arithmetical algebraicgeometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI,1987, pp. 1–24. MR902590 ↑ On the generic part of the cohomology of compact unitary Shimuravarieties , Ann. of Math. (2) (2017), no. 3, 649–766, DOI 10.4007/annals.2017.186.3.1.MR3702677 ↑
6, 60[CY20] S. Cho and T. Yamauchi,
A reformulation of the Siegel series and intersection numbers , Math.Ann. (2020), no. 3-4, 1757–1826, DOI 10.1007/s00208-020-01999-2. MR4126907 ↑
6, 11, 12[GI16] W. T. Gan and A. Ichino,
The Gross-Prasad conjecture and local theta correspondence , Invent.Math. (2016), no. 3, 705–799, DOI 10.1007/s00222-016-0662-8. MR3573972 ↑ Explicit constructions of automorphic L -functions ,Lecture Notes in Mathematics, vol. 1254, Springer-Verlag, Berlin, 1987. MR892097 ↑ Intersection theory using Adams operations , Invent. Math. (1987),no. 2, 243–277, DOI 10.1007/BF01388705. MR910201 ↑ On the motive of a reductive group , Invent. Math. (1997), no. 2, 287–313, DOI10.1007/s002220050186. MR1474159 ↑ The geometry and cohomology of some simple Shimura varieties , Annalsof Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With anappendix by Vladimir G. Berkovich. MR1876802 ↑
58, 59[How12] B. Howard,
Complex multiplication cycles and Kudla-Rapoport divisors , Ann. of Math. (2) (2012), no. 2, 1097–1171, DOI 10.4007/annals.2012.176.2.9. MR2950771 ↑ Linear invariance of intersections on unitary Rapoport-Zink spaces , Forum Math. (2019), no. 5, 1265–1281, DOI 10.1515/forum-2019-0023. MR4000587 ↑
7, 23, 24, 25[HP14] B. Howard and G. Pappas,
On the supersingular locus of the
GU(2 , Shimura variety , AlgebraNumber Theory (2014), no. 7, 1659–1699, DOI 10.2140/ant.2014.8.1659. MR3272278 ↑ Special cycles on unitary Shimura varieties I. Unramified local theory ,Invent. Math. (2011), no. 3, 629–682, DOI 10.1007/s00222-010-0298-z. MR2800697 ↑
21, 24,27[KR14] ,
Special cycles on unitary Shimura varieties II: Global theory , J. Reine Angew. Math. (2014), 91–157, DOI 10.1515/crelle-2012-0121. MR3281653 ↑ [Lau08] E. Lau, Displays and formal p -divisible groups , Invent. Math. (2008), no. 3, 617–628, DOI10.1007/s00222-007-0090-x. MR2372808 ↑ Frames and finite group schemes over complete regular local rings , Doc. Math. (2010),545–569, DOI 10.3846/1392-6292.2010.15.547-569. MR2679066 ↑ Chow groups and L -derivatives of automorphic motives for unitary groups .arXiv:2006.06139. ↑
1, 2, 3, 4, 5, 23, 28, 47, 50, 51, 52, 53, 54, 55, 56, 57, 62, 63, 64, 66, 67[LZ] C. Li and W. Zhang,
Kudla–Rapoport cycles and derivatives of local densities . arXiv:1908.01701. ↑
5, 6, 20, 22, 28, 30, 36, 64[Liu11a] Y. Liu,
Arithmetic theta lifting and L -derivatives for unitary groups, I , Algebra Number Theory (2011), no. 7, 849–921. MR2928563 ↑ Arithmetic theta lifting and L -derivatives for unitary groups, II , Algebra Number Theory (2011), no. 7, 923–1000. MR2928564 ↑ Fourier–Jacobi cycles and arithmetic relative trace formula (with an appendix by ChaoLi and Yihang Zhu) . https://users.math.yale.edu/~yl2269/FJcycle.pdf , preprint. ↑ Theta correspondence for almost unramified representations of unitary groups . https://gauss.math.yale.edu/~yl2269/theta_aur.pdf , preprint. ↑
3, 44, 45, 46, 50, 51[LTXZZ] Y. Liu, Y. Tian, L. Xiao, W. Zhang, and X. Zhu,
On the Beilinson–Bloch–Kato conjecture forRankin–Selberg motives . arXiv:1912.11942. ↑ Coxeter orbits and eigenspaces of Frobenius , Invent. Math. (1976/77), no. 2, 101–159, DOI 10.1007/BF01408569. MR453885 ↑ A compactification of Igusa varieties , Math. Ann. (2008), no. 2, 265–292, DOI10.1007/s00208-007-0149-4. MR2368980 ↑
6, 61[Mih20] A. Mihatsch,
Relative unitary RZ-spaces and the arithmetic fundamental lemma , J. Inst. Math.Jussieu (2020). online first. ↑
38, 42[RZ96] M. Rapoport and Th. Zink,
Period spaces for p -divisible groups , Annals of Mathematics Studies,vol. 141, Princeton University Press, Princeton, NJ, 1996. MR1393439 ↑
9, 65, 66[RSZ17] M. Rapoport, B. Smithling, and W. Zhang,
On the arithmetic transfer conjecture for exotic smoothformal moduli spaces , Duke Math. J. (2017), no. 12, 2183–2336, DOI 10.1215/00127094-2017-0003. MR3694568 ↑
9, 24, 27[RSZ20] ,
Arithmetic diagonal cycles on unitary Shimura varieties , Compos. Math. (2020),no. 9, 1745–1824, DOI 10.1112/s0010437x20007289. MR4167594 ↑
57, 63[TY07] R. Taylor and T. Yoshida,
Compatibility of local and global Langlands correspondences , J. Amer.Math. Soc. (2007), no. 2, 467–493, DOI 10.1090/S0894-0347-06-00542-X. MR2276777 ↑
6, 59[Vol07] I. Vollaard,
Endomorphisms of quasi-canonical lifts , Astérisque (2007), 105–112 (English,with English and French summaries). MR2340375 ↑ Démonstration d’une conjecture de dualité de Howe dans le cas p -adique, p = 2, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, PartI (Ramat Aviv, 1989), Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 267–324(French). MR1159105 ↑ The supersingular locus of unitary Shimura varieties with exotic good reduction .arXiv:1609.08775. ↑
7, 9, 21, 25[Yam14] S. Yamana,
L-functions and theta correspondence for classical groups , Invent. Math. (2014),no. 3, 651–732, DOI 10.1007/s00222-013-0476-x. MR3211043 ↑
45, 46, 50, 51[Zel80] A. V. Zelevinsky,
Induced representations of reductive p -adic groups. II. On irreducible represen-tations of GL( n ), Ann. Sci. École Norm. Sup. (4) (1980), no. 2, 165–210. MR584084 ↑ Weil representation and arithmetic fundamental lemma . arXiv:1909.02697. ↑ The display of a formal p -divisible group , Astérisque (2002), 127–248. Cohomologies p -adiques et applications arithmétiques, I. MR1922825 ↑ Department of Mathematics, Columbia University, New York NY 10027, United States
Email address : [email protected] Department of Mathematics, Yale University, New Haven CT 06511, United States
Email address ::