Reductions of Hecke correspondences on Anderson modular objects
aa r X i v : . [ m a t h . N T ] F e b REDUCTIONS OF HECKE CORRESPONDENCESON ANDERSON MODULAR OBJECTS
A. Grishkov, D. Logachev Abstract.
We formulate some properties of a conjectural object X fun ( r, n )parametrizing Anderson t-motives of dimension n and rank r . Namely, we giveformulas for p -Hecke correspondences of X fun ( r, n ) and its reductions at p (where p is a prime of F q [ θ ]). Also, we describe their geometric interpretation. These resultsare analogs of the corresponding results of reductions of Shimura varieties. Finally,we give conjectural formulas for Hodge numbers (over the fields generated by Heckecorrespondences) of middle cohomology submotives of X fun ( r, n ).
0. Introduction.
Let X be a Shimura variety of PEL-type. Its points pa-rametrize abelian varieties with some fixed endomorphism rings, polarization andlevel. There is a problem to describe relations between the rings H p ( X ) of p -Heckecorrespondences of X ( p is a prime) and of ˜ X p - the reduction of X at p , particu-larly, to find the characteristic polynomial of the Frobenius correspondence of ˜ X p over H p ( X ). Also, we can ask what are geometric interpretations of reductions of p -Hecke correspondences.There is also a problem of extreme complexity — to prove Langlands theoremsfor Shimura varieties (relations between L -functions of submotives of X and ofautomorphic representations of the corresponding reductive group G ). It is solvedonly in a few cases of X of low dimension. Even exact statements of theoremsgiving these relations is a complicated problem.Analogs of abelian varieties for the function field case are Anderson t-motives (orAnderson modules — their categories are anti-equivalent). It is natural to consideran object (function field analog of a Shimura variety) whose points parametrize Mathematics Subject Classification . 11G09.
Key words and phrases.
Moduli objects of Anderson t-motives, Hecke correspondences,reduction.Thanks: The authors are grateful to FAPESP, S˜ao Paulo, Brazil for a financial support (processNo. 2017/19777-6). The first author is grateful to SNPq, Brazil, to RFBR, Russia, grant 16-01-00577a (Secs. 1-4), and to Russian Science Foundation, project 16-11-10002 (Secs. 5-8) for afinancial support. The second author is grateful to Laurent Lafforgue for invitation to IHESwhere this paper was started. Discussions with Laurent Fargues and Alain Genestier on thesubject of these notes were very important. Particularly, they indicated to the second author theanalogy between Anderson varieties and unitary Shimura varieties, and Alain Genestier attractedhis attention to the paper [F] where the notion of the dual Anderson module is defined, andindicated the statement of Theorem 3.4.1a. Finally, the second author is grateful to NicholasKatz for a detailed explanation of his paper [K]. E-mails: [email protected]; [email protected] (corresponding author)Typeset by
AMS -TEX n , rank r ,type of a nilpotent operator N , see Section 2.4, and of an analog of PEL-type). Wedenote this object by X fun = X fun ( r, n ).Unfortunately, at the moment for n > n = 1 we have a good theory of moduli spaces of Drinfeld modules).For example, if we restrict ourselves by pure uniformizable Anderson t-motivesand assume that there is 1 – 1 (or near 1 – 1) correspondence between these t-motives and their lattices (which is rather likely, see [GL17]), then the modulivariety of lattices would be the quotient set of Siegel matrices by an (almost)action of GL r ( F q [ θ ]). But this (almost) action does not have desired properties,see [GL17], Proposition 1.7.1.So, the whole contents of the present paper concerns conjectural objects X fun ( r, n ) that we shall call Anderson modular objects. Nevertheless, we can getsome information about them. Let us give more definitions. Naively, X fun ( r, n )parametrize pure abelian t-motives of rank r and dimension n whose nilpotent op-erator N is 0. The corresponding reductive group G is GL r and the dominantcoweight µ is (1 , . . . , , , . . . ,
0) ( n ones and r − n zeroes). For more informationfor any G on X fun and µ see [V].Let p be a prime ideal of F q [ θ ] ( q is a power of p and θ an abstract variable, seeSection 2). The contents of the present paper is the following.(1) We formulate (for some cases) in Section 2 the theorems concerning reduc-tions at p of Hecke correspondences T p ,i ( i = 0 , . . . , r ) on X fun , and their geomet-ric interpretation. These results are functional analogs of [FCh], Chapter 7, [BR],Chapter 6, and [W]. The case n = 1 (Drinfeld varieties) is treated with more detailsand explicit formulas.(2) Sketches of the proofs of these theorems are given in Section 3.We see that the function field case — Anderson varieties of rank r and dimension n , G fun = GL r ( F q ( θ )) is analogous to the number field case where G num = GU ( r − n, n ) corresponds to Shimura varieties (called unitary for brevity) of PEL-typeparametrizing abelian r -folds with multiplication by an imaginary quadratic field K , of signature ( r − n, n ). We indicate in Section 4 that really, properties ofunitary Shimura varieties are similar to the properties of X fun ( r, n ). By the way,this analogy is a source of more results, see for example [GL21].(3) Finally, in Section 5 we state conjectural formulas for Hodge numbers (overfields generated by Hecke correspondences) of submotives of middle cohomology of X fun . They are analogs of the corresponding formulas for Shimura varieties ([BR],SEction 4.3, p. 548). In Section 6 we consider the action of Hecke correspondenceson some non-ordinary Drinfeld modules. These results will be useful for a futureproof of analog of Kolyvagin’s theorem (finiteness of Tate-Shafarevich group) forthe case of Drinfeld varieties.In order to show the analogy between the number field and the function fielscases, we give in Section 1 some well-known results on Hecke correspondences andtheir reductions for the case of Siegel varieties.2 . Definitions and results for the number field case (Siegel varieties).1.1. Reductions of correspondences. For a comparison with the number field case, here we formulate well-knownresults for reductions of Siegel modular varieties. References for the results of thissection are: [FCh], Section 7, and [W].Let X be a Siegel variety of genus g (of any fixed level), i.e. a quotient of theSiegel upper half plane by a congruence subgroup of GSp g ( Z ), or, equivalently,a set of principally polarized abelian varieties of dimension g together with somelevel structure. Let the congruence subgroup be such that X is defined over Q . Wehave G = G X = GSp g ( Q ) is the corresponding reductive group. Let p be a fixedprime which does not divide the level, and ˜ X the reduction of X at p .Let Corr( X ) = Corr p ( X ) (resp. Corr( ˜ X ) = Corr p ( ˜ X )) be the algebra of p -Hecke correspondences on X (resp. ˜ X ). We have the Frobenius map on ˜ X ;considering it as a correspondence we get an element f r X ∈ Corr( ˜ X ).There is a map γ : Corr( X ) → Corr( ˜ X ) — the reduction of a correspondenceat p . There is a problem of description of γ and of finding of the characteristicpolynomial of f r over im ( γ ). The answer is the following. Let M be the followingblock diagonal subgroup of G : M = (cid:26)(cid:18) A λ · ( A t ) − (cid:19)(cid:27) ⊂ G, (1 . . g ), and let T be the subgroup of M consisting of diagonal matrices.The abstract p -Hecke algebras H ( G ) = H ( G )( Q p ) (resp. H ( M ) = H ( M )( Q p ), H ( T ) = H ( T )( Q p )) consist of double cosets KαK , where K = K G = G ( Z p ) (resp. K = K M = M ( Z p ), K = K T = T ( Z p )) and α ∈ G ( Q p ), resp. α ∈ M ( Q p ), α ∈ T ( Q p ). There are the Satake inclusions S GM : H ( G ) → H ( M ), S MT : H ( M ) → H ( T ).The Hecke algebra H ( T ) is the subalgebra of Z [ U ± , ..., U ± g , V ± , ..., V ± g ] (here U i , V i are abstract variables) generated by ( U V − ) ± , ..., ( U g V − g ) ± and ( U · ... · U g ) ± . The Weyl group W G of G is the semidirect product of the permutationgroup S g and of ( Z / Z ) g = ( ± g , where S g permutes coordinates in ( ± g . Thereis a section S g ֒ → W G , we denote its image by W G,M . Further, W G acts on H ( T )in the obvious manner: S g permutes indices and ( ± g interchanges U ∗ , V ∗ .We have: S MT ( H ( M )) = H ( T ) W G,M , S GT ( H ( G )) = H ( T ) W G . (1 . . β : H ( G ) → Corr( X ) , (1 . . β : H ( M ) → Corr( ˜ X ) (1 . . KpK = id , where p = pI g is the scalarmatrix in both G and M , and id is the trivial correspondence on both X , ˜ X .3 heorem 1.1.5. There exists a commutative diagram: S GM : H ( G ) → H ( M ) β ↓ β ↓ γ : Corr( X ) → Corr( ˜ X ) (1 . . τ p the matrix (cid:18) p (cid:19) , where entries are scalar g × g -blocks, andwe denote the corresponding elements K G τ p K G (resp. K M τ p K M ) of H ( G ) (resp. H ( M )) by T p (resp. f r M ). Theorem 1.1.7. β ( f r M ) = f r X . Remark.
Formulas 1.1.2 and theorems 1.1.5, 1.1.7 permit us to find theHecke polynomial of X (the characteristic polynomial of f r X over Corr( X ) ).Really, 1.1.2 implies that H ( M ) is a free module over S GM ( H ( G )) of dimension W G ) / W G,M ) = 2 g . An explicit description of f r M ∈ H ( M ) (see below) per-mits us to find easily its characteristic polynomial over H ( G ). Theorem 1.1.5 impliesthat it is also the characteristic polynomial of the Frobenius correspondence on ˜ X over the algebra Corr X . For i = 0 , . . . , g we consider diagonal matrices in a block form ϕ i = I i pI g − i pI i
00 0 0 I g − i ∈ M (sizes of diagonal blocks are i , g − i , i , g − i ), and we denote the correspondingelements K M ϕ i K M ∈ H ( M ) by Φ i . Particularly, Φ g = f r M .Let I be a subset of { , ..., g } . We denote U I := Q i ∈ I U i Q i I V i ∈ H ( T ). Wehave S MT (Φ i ) = X I )= i U I and S GM ( T p ) = Φ + Φ + · · · + Φ g ∈ H ( M ) . (1 . . β ( T p ) ∈ Corr( X ), β (Φ i ) ∈ Corr( ˜ X ) again by T p , Φ i respectively, so(1.2.1) and Theorem 1.1.5 give us the following equality on Corr( ˜ X ): γ ( T p ) = Φ + Φ + · · · + Φ g . (1 . . Z there exists an involution on Corr( Z ) (symmetrywith respect the coordinates). Also, there exist involutions on H ( M ), H ( G ) com-muting with involutions on Corr( X ), Corr( ˜ X ) with respect to (1.1.6). We denotethese involutions by hat; we have ˆ T p = T p , ˆΦ i = Φ g − i .4he geometric interpretation of (1.2.2) is the following. Let t ∈ X ( ¯ Q ) be ageneric point, A t the corresponding principally polarized abelian g -fold with a fixedpolarization form and ( A t ) p the F p -space of its p -torsion points. The polarizationon A t defines a skew form on ( A t ) p . T p ( t ) is a finite set of points; we have: t ′ ∈ T p ( t )iff there exists an isogeny α t,t ′ : A t → A t ′ of type (1 , ..., , p, ..., p ). The kernel of α t,t ′ is an isotropic g -dimensional subspace of ( A t ) p . So, we have a Theorem 1.2.3.
The set T p ( t ) is in 1–1 correspondence with the set of isotropic g -dimensional subspaces of ( A t ) p .Now let t ∈ X ( Q ) be a generic point such that A t has a good ordinary reductionat p . Let ( ˜ A t ) p,points be the set of closed points of ˜ A t of order p , and red : ( A t ) p → ( ˜ A t ) p,points the reduction map. We denote by D = D Siegel the kernel of red , it isan isotropic g -dimensional subspace of ( A t ) p .Let t ′ ∈ T p ( t ) and ˜ t ′ ∈ ˜ X its reduction. (1.2.2) shows that ˜ t ′ belongs to one ofΦ i (˜ t ). Theorem 1.2.4.
Number i is defined as follows: i = dim (Ker ( α t,t ′ ) ∩ D Siegel ) . (1 . . i = g ⇐⇒ ˜ t ′ = f r (˜ t ) ⇐⇒ Ker ( α t,t ′ ) = D Siegel .Further, we have the following
Theorem 1.2.6.
Let t ′ , t ′′ be 2 points of T p ( t ). Then˜ t ′ = ˜ t ′′ ⇐⇒ Ker ( α t,t ′ ) ∩ D Siegel = Ker ( α t,t ′′ ) ∩ D Siegel . Recall that any correspondence C on X has the bidergee d ( C ) , d ( C ) — thedegrees of 2 projections π , π of its graph Γ( C ) to X . By definition, π i ( ˆ C ) = π ˆ i ( C )(here ˆ1 = 2 , ˆ2 = 1). Further, C has the separable (resp. non-separable) bidergee d s ( C ) , d s ( C ) (resp. d ns ( C ) , d ns ( C )) — the separable (resp. non-separable) degreesof π , π . We have d i ( C ) = d si ( C ) d nsi ( C ) and d ∗ i ( ˆ C ) = d ∗ ˆ i ( C ) , i = 1 , , ∗ = ∅ , s, ns. (1 . . a )We denote by g ( k, l ) the cardinality of the Grassmann variety Gr ( k, l )( F p ): g ( k, l ) = k Y i =1 p l − p i − p k − p i − . (1 . . Theorem 1.2.8. d s (Φ i ) = g ( i, g ) , d ns (Φ i ) = p ( g +1 − i )( g − i ) / , d s (Φ i ) = g ( i, g ) , d ns (Φ i ) = p ( i +1) i/ . (cid:3)
2. Definitions and statement of conjectures for the case of Andersonmodular objects.
We use standard notations for Anderson t-motives. Let q be a power of a prime p , F q the finite field of order q . The function field analog of Z is the ring of polynomials5 q [ θ ] where θ is an abstract variable. The analog of the archimedean valuation on Q is the valuation at infinity of the fraction field F q ( θ ) of F q [ θ ]; it is denoted by ord , it is uniquely determined by the property ord ( θ ) = −
1. The completion of analgebraic closure of the completion of F q ( θ ) with respect the valuation ”ord” is thefunction field analog of C . It is denoted by C ∞ .The definition of a t-motive M is given in [G], Definitions 5.4.2, 5.4.12 (Goss usesanother terminology: ”abelian t-motive” of [G] = ”t-motive” of the present paper).Particularly, M is a free C ∞ [ T ]-module of dimension r (this number r is called therank of M ) endowed by a C ∞ -skew-linear operator τ satisfying some properties. Itsdimension n is defined in [G], Remark 5.4.13.2 (Goss denotes the dimension by ρ ).A nilpotent operator N = N ( M ) associated to a t-motive is defined in [G], Remark5.4.3.2. Condition N = 0 implies n ≤ r . Except Section 2.4, we shall consider onlythe case N = 0.As it was written in the introduction, the main object of the present paper isconjectural. It is called an Anderson modular object, it is denoted by X fun = X fun ( r, n ), it is the function field analog of X . Naively, it parametrizes Andersont-motives of rank r and dimension n .An analog of p of Section 1 is a valuation (distinct of ord) of F q ( θ ) = a primeideal of F q [ θ ]. We denote by p both its generator and the prime ideal itself, and wedenote q = F q [ θ ] / p ). The corresponding algebraic group G fun — the functionfield analog of GSp g ( Q ) — is GL r ( F q ( θ )). Hence, the analogs of Q p , Z p for thefunctional case are F q ( θ ) p , F q [ θ ] p respectively, and the analog of K G of Section 1 is GL r ( F q [ θ ] p ) (it will be denoted by K G as well).In order to simplify the present version of the text, for the case n > r and dimension n over C ∞ is thequotient C ∞ /L , where L is a free r -dimensional F q [ θ ]-module. Since not all Ander-son t-motives are uniformizable, the exact statements of these conjectures must beslightly changed, see Remark 2.3.4a. Theorem 2.1.
The analog of M for this case is the group M r − n,n of blockdiagonal matrices (cid:18) ∗ ∗ (cid:19) ⊂ GL r ( F q ( θ )), sizes of blocks are r − n , n . We denote K M := M ( F q [ θ ] p ). The analogs of Hecke algebras and of algebras of correspon-dences are defined like in Section 1. The analogs of Theorems 1.1.5, 1.1.7 hold forthis case. Particularly, Corr( ˜ X fun ) is the quotient of H ( M ) by the trivial relation K M p K M = id . H ( G fun ) , H ( M ) and of the Satake inclusions. Let like in Section 1, T be the subgroup of G fun of diagonal matrices. Wehave H ( T ) = Z [ U ± , ..., U ± r ], and the Weyl group W G fun = S r , it acts on H ( T )permuting indices. An analog of W G,M is the subgroup W G fun ,M = S r − n × S n ֒ → S r = W G fun with the obvious inclusion. Formulas (1.1.2) hold for our case, explicitformulas are the following. H ( G fun ): For i = 0 , . . . , r we denote by τ p ,i the diagonal matrix (cid:18) I r − i p I i (cid:19) ∈ G fun , where sizes of blocks are r − i , i , and we denote the cor-responding elements K G τ p ,i K G ∈ H ( G fun ) by T p ,i . We have T p , = 1, T p ,r is the6rivial correspondence, and other T p ,i are free generators of H ( G fun ). We have S G fun T ( T p ,i ) = q − i ( i − / σ i ( U , ..., U r ) , (2 . . σ i is the i -th symmetric polynomial, andˆ T p ,i = T p ,r − i . (2 . . a ) H ( M ): (a) For i = 0 , . . . , r − n we denote by ϕ i the diagonal matrix I r − n − i p I i
00 0 I n ∈ M where sizes of blocks are r − n − i , i , n , and we denotethe corresponding elements K M ϕ i K M ∈ H ( M ) by Φ i .(b) For i = 0 , . . . , n we denote by ψ i the diagonal matrix I r − n p I i
00 0 I n − i ∈ M where sizes of blocks are r − n , i , n − i , and we denote the corresponding elements K M ψ i K M ∈ H ( M ) by Ψ i . We have Φ = Ψ = 1, Φ i = 0 (resp. Ψ i = 0) if i isout of the range 0 , . . . , r − n (resp. 0 , . . . , n ) and other Φ i , Ψ i are free generatorsof H ( M ). Obviously S MT (Φ i ) = q − i ( i − / σ i ( U , ..., U r − n ) , S MT (Ψ i ) = q − i ( i − / σ i ( U r − n +1 , ..., U r ) . (2 . . S G fun M ( T p ,j ) = j X i =0 q − i ( j − i ) Ψ i Φ j − i . (2 . . n Φ r − n = q n ( r − n ) , (2 . . a )ˆΦ i = q − n ( r − n − i ) Ψ n Φ r − n − i , (2 . . b )ˆΨ i = q − ( n − i )( r − n ) Ψ n − i Φ r − n . (2 . . c )particularly, ˆΦ r − n = Ψ n . Coefficients of (2.2.5b,c) can be found from the propertythat equations (2.2.2a) and (2.2.5) are concordant with respect to the duality. Remark. (2.2.5), (2.1), (1.1.5) imply that for the case n = 1 (Drinfeld modules)the explicit formulas are the following ( f r = Ψ ):˜ T p , = f r + Φ ;˜ T p , = 1 q f r Φ + Φ ; . . . (2 . . T p ,r − = 1 q r − f r Φ r − + Φ r − ;7 T p ,r = 1 q r − f r Φ r − . H ( M ) is a free module over S G fun M ( H ( G fun )) respectively the Satake inclusion.Its dimension is W G fun ) / W G fun ,M ) = (cid:18) rn (cid:19) .We denote Ψ n by f r M . Its characteristic polynomial over S G fun M ( H ( G fun )) (theHecke polynomial) can be easily found by elimination of Φ , ..., Φ r − n , Ψ , ..., Ψ n − in the system (2.2.5). We denote it by P r,n , it belongs to H p ( G fun )[ f r M ]. For n = 1 we have P r, = P ri =0 ( − i q i ( i − / T p ,i f r r − i = f r r − T p , f r r − + q T p , f r r − ± · · · + ( − r q r ( r − / T p ,r . (2 . . Theorems 1.2.4, 1.2.6 can be rewritten almost word-to-word as conjectures forthe functional case. Let us do it. It is more convenient to use Anderson modules([G], 5.4.5; Goss calls them Anderson T -modules) instead of Anderson t-motives.The categories of Anderson t-motives and modules are anti-isomorphic, so there isno essential difference which object to use.Let t ∈ X fun be such that the corresponding Anderson module E t is uniformiz-able: E t = C n ∞ /L (as earlier, n is the dimension and r is the rank). This condition is“closed under Hecke correspondences”: if t ′ ∈ T p ,j ( t ) then E t ′ is also uniformizable.Obviously, we have Theorem 2.3.1. ( E t ) p — the set of p -torsion points of E t — is p − L/L andhence is an r -dimensional F q [ θ ] / p -vector space. Theorem 2.3.2. t ′ ∈ T p ,j ( t ) iff there exists an isogeny α t,t ′ : E t → E t ′ of type(1 , ..., , p , ..., p ) ( r − j j p ’s). Theorem 2.3.3.
The set T p ,j ( t ) is in 1–1 correspondence with the set of j -dimensional subspaces of ( E t ) p .Now we can formulate the conjecture on reductions at p . Let ( ˜ E t ) p ,points and red : ( E t ) p → ( ˜ E t ) p ,points be analogs of the corresponding objects in Section 1.2.We denote by D fun the kernel of red . Firstly, we have a Theorem 2.3.4.
For a generic t ∈ X fun D fun is an n -dimensional subspaceof ( E t ) p . Remark 2.3.4a.
For an arbitrary t ∈ X fun (such that E t is not uniformizable)statements of the above and below theorems and conjectures require minor changes.For example, Theorem 2.3.1 becomes Theorem 2.3.4b. ( E t ) p is an r -dimensional A/ p -vector space.(we cannot claim that it is p − L/L because L does not exist). Geometric interpretation.
Here we formulate analogs of (1.2.4) – (1.2.8) for the function field case. Let t ′ ∈ T p ,j ( t ) and ˜ t ′ ∈ ˜ X fun its reduction. Conjecture 2.1 and (2.2.5) show that ˜ t ′ belongs to one of q − i ( j − i ) Ψ i Φ j − i (˜ t ). 8 heorem 2.3.5. Number i is defined as follows: i = dim (Ker ( α t,t ′ ) ∩ D fun ) . (2 . . j = i = n ⇐⇒ ˜ t ′ = f r X fun (˜ t ) ⇐⇒ Ker ( α t,t ′ ) = D fun .Now g ( k, l ) will mean the cardinality of Grassmann variety Gr ( k, l ) over F q [ θ ] / p ,i.e. p in (1.2.7) must be replaced by q . Obviously we have d ( T p ,i ) = d ( T p ,i ) = g ( i, r ) . For the reader’s convenience, we formulate the following conjecture separatelyfor the case n = 1. Theorem 2.3.7.
For n = 1 we have: d s (Ψ ) = d ns (Ψ ) = 1 , d s (Ψ ) = 1 , d ns (Ψ ) = q r − ; d s (Φ i ) = g ( i, r − , d ns (Φ i ) = q i , d s (Φ i ) = g ( i, r − , d ns (Φ i ) = 1 . Corollary 2.3.7.1.
For correspondences q i f r Φ i — summands in the righthand side of (2.2.6) — we have d s ( 1 q i f r Φ i ) = g ( i, r − , d ns ( 1 q i f r Φ i ) = 1 ,d s ( 1 q i f r Φ i ) = g ( i, r − , d ns ( 1 q i f r Φ i ) = q r − − i . An analog of the Theorem 1.2.6 is the following. Let t ′ , t ′′ be 2 points of T p ,i ( t ).Firstly we consider the case whenKer ( α t,t ′ ) ∩ D fun = Ker ( α t,t ′′ ) ∩ D fun = 0 , i.e. both ˜ t ′ , ˜ t ′′ ∈ Φ i (˜ t ) (the second summand in (2.2.6)). Theorem 2.3.8.
Under this condition we have (for any n ):˜ t ′ = ˜ t ′′ as closed points iff the linear spans coincide: < Ker ( α t,t ′ ) , D fun > = < Ker ( α t,t ′′ ) , D fun > . If (for n = 1) Ker ( α t,t ′ ) ⊃ D fun , Ker ( α t,t ′′ ) ⊃ D fun , i.e. both ˜ t ′ , ˜ t ′′ ∈ q i − Ψ Φ i − (˜ t ) = q i − f r Φ i − (˜ t ) (the first summand in (2.2.6))then (2.3.7.1) implies that ˜ t ′ , ˜ t ′′ are always different.Now let us consider the case of arbitrary n . Theorem 2.3.10. d s (Ψ i ) = g ( i, n ) , d ns (Ψ i ) = 1; d s (Φ i ) = g ( i, r − n ) , d ns (Φ i ) = q in . emark 2.3.10a. Numbers d ∗ (Φ i ), d ∗ (Ψ i ) ( ∗ = ∅ , s, ns ) can be found fromthe above formulas using (1.2.6a), (2.2.5b,c). Particularly, for the Frobenius Ψ n wehave d s (Ψ n ) = 1 , d ns (Ψ n ) = q n ( r − n ) . Corollary 2.3.11.
For correspondences q i ( j − i ) Ψ i Φ j − i — summands in the righthand side of (2.2.5) — we have d s ( 1 q i ( j − i ) Ψ i Φ j − i ) = g ( i, n ) g ( j − i, r − n ) , d ns ( 1 q i ( j − i ) Ψ i Φ j − i ) = q ( j − i )( n − i ) . Let t ′ , t ′′ ∈ T p ,j ( t ) be as above such thatdim (Ker ( α t,t ′ ) ∩ D fun ) = dim (Ker ( α t,t ′′ ) ∩ D fun ) = i. According Theorem 2.3.5, both ˜ t ′ , ˜ t ′′ ∈ q i ( j − i ) Ψ i Φ j − i . Theorem 2.3.12. ˜ t ′ = ˜ t ′′ as closed points iff both intersections and linear spanscoincide: Ker ( α t,t ′ ) ∩ D fun = Ker ( α t,t ′′ ) ∩ D fun ; < Ker ( α t,t ′ ) , D fun > = < Ker ( α t,t ′′ ) , D fun > . Remark 2.3.13. (a) If we fix an i -dimensional subspace V i ⊂ D fun and a( n + j − i )-dimensional overspace V n + j − i ⊃ D fun then the quantity of j -dimensionalspaces α such that α ∩ D fun = V i , < α, D fun > = V n + j − i , is equal to q ( j − i )( n − i ) = d ns ( q i ( j − i ) Ψ i Φ j − i ), as it is natural to expect.(b) Thanks to existence of skew pairing in the number case, we have < Ker ( α t,t ′ ) , D fun > = (Ker ( α t,t ′ ) ∩ D fun ) dual , ( dual is with respect to the skew pairing), so in Theorem 1.2.6 it is sufficient toclaim only coincidence of intersections.(c) For n = 1 (Theorem 2.3.8) intersections always coincide, so it is sufficient toclaim only coincidence of linear spans. N = 0. This is a subject of further research. Here we do noteven give statements of results, we indicate only the discrete invariants of t-motiveshaving N = 0 and explain the methods how these statements can be obtained.Let M be a uniformizable Anderson t-motive such that its N is not (necessarily)0, of dimension n and of rank r . M is a C ∞ [ T ]-module with a skew map τ : M → M such that M /τ M is annihilated by a power of T − θ . We have Lie( M ) = C n ∞ , and N is a nilpotent operator acting on Lie( M ). Also, T acts on Lie( M ); we have T = θI n + N on End (Lie( M )). Particularly, we can consider Lie( M ) as a F q [ T ]-module. 10he lattice L ( M ) of M is a free F q [ T ]-submodule of Lie( M ) considered as a F q [ T ]-module. We have a natural inclusion F q [ T ] ֒ → C ∞ [[ N ]] ( T N + θ ). Hence,the tautological inclusion L ( M ) ֒ → Lie( M ) defines a surjection L ( M ) ⊗ F q [ T ] C ∞ [[ N ]] ։ Lie( M ) . Its kernel is denoted by q M ; the exact sequence0 → q M → L ( M ) ⊗ F q [ T ] C ∞ [[ N ]] → Lie( M ) → M are the discrete invariants of the pair of lattices( q M ; L ( M ) ⊗ F q [ T ] C ∞ [[ N ]]) over a discrete valuation ring C ∞ [[ N ]]. Let us give adescription of these invariants from [GL18], (3.3). Let ν be the minimal numbersuch that N ν = 0. These invariants are numbers k ≥ , . . . , k ν +1 ≥ l , . . . , l r of L ( M ) over F q [ T ] and its partition on ν + 1sets of lengths k , . . . , k ν +1 (if some k i = 0 then the i -th set is empty); the i -th setis denoted by l i, . . . , l i,k i , having the following properties: N ν − ( l ν +1 ,i ), i = 1 , . . . , k ν +1 , form a C ∞ -basis of N ν − Lie( M ) ([GL18], (3.6); N ν − ( l ν,i ), i = 1 , . . . , k ν , N ν − ( l ν +1 ,i ), i = 1 , . . . , k ν +1 , N ν − ( l ν +1 ,i ), i = 1 , . . . , k ν +1 , form a C ∞ -basis of N ν − Lie( M ) ([GL18], (3.8); N ν − ( l ν − ,i ), i = 1 , . . . , k ν − , N ν − ( l ν,i ), N ν − ( l ν,i ), i = 1 , . . . , k ν ,and N ν − ( l ν +1 ,i ), N ν − ( l ν +1 ,i ), N ν − ( l ν +1 ,i ), i = 1 , . . . , k ν +1 ,form a C ∞ -basis of N ν − Lie( M ) ([GL18], (3.8);etc., until l ,i , i = 1 , . . . , k , . . . , N ν − ( l ν +1 ,i ), i = 1 , . . . , k ν +1 ,form a C ∞ -basis of Lie( M ).See [GL18], (3.3) - (3.10) for more details.Particularly, we have: r = k + ... + k ν +1 ; n = k + 2 k + 3 k + ... + νk ν +1 . ([GL18], (3.10) and (3.5)). If ν = 1, i.e. N = 0 — this is the case considered above,then k = r − n, k = n . Therefore, numbers k , . . . , k ν +1 are N = 0-generalizationsof numbers r − n, n . Conjecture 2.4.1.
The analog of the subgroup M of G fun = GL r for the setof Anderson t-motives having N = 0 and invariants k , . . . , k ν +1 is the subgroup of GL r of block diagonal matrices with block sizes k , . . . , k ν +1 .11 emark 2.4.2. Some of k i can be 0. In this case we cannot distinguish between M for different sets of k , . . . , k ν +1 . Hence, maybe it is necessary to modify thestatement of Conjecture 2.4.1.As it was written, finding of analogs of statements of Sections 2.2, 2.3 for thesets of Anderson t-motives having invariants k , . . . , k ν +1 is a subject of furtherresearch. Remark 2.4.3.
Hartl and Juschka use some other invariants of M , see [HJ],Section 2. First, they consider slightly more general objects, namely, their q = q M isa subset not of L ( M ) ⊗ F q [ T ] C ∞ [[ N ]] but of L ( M ) ⊗ F q [ T ] C ∞ (( N )) (also, they considera weight filtration on L ( M ) ). Further, their Hodge-Pink weights ω , . . . , ω r arerelated with k , . . . , k ν +1 as follows: for all i = 1 , . . . , ν + 1 the number − i + 1occurs k i times among ω , . . . , ω r (i.e. among ω , . . . , ω r there are k zeroes, k minus ones etc.).
3. Proofs.
We follow [FCh], Ch. 7, Section 4 using the same notations ifpossible, and indicating results that are not completely analogous to the numberfield case.Recall that p is a prime ideal of F q [ θ ]. We denote by F q [ θ ] p , F q ( θ ) p the com-pletions at p of F q [ θ ], F q ( θ ) respectively, and by F q [ θ ] nr p the ring of integers of themaximal unramified extension of F q ( θ ) p . As usual, bar means an algebraic closure.There are maps F q [ θ ] nr p ֒ → F q ( θ ) p , F q [ θ ] nr p ։ F q [ θ ] / p . The corresponding mapsof schemes Spec F q ( θ ) p → Spec F q [ θ ] nr p , Spec F q [ θ ] / p → Spec F q [ θ ] nr p are denotedby ξ k , ξ p respectively. The inverse image ξ ∗ p of an object (i.e. the reduction of thisobject) is denoted by tilde.We fix i , and let Γ be the graph of T p ,i over Spec F q [ θ ] nr p .It is known that it exists. For t ∈ Γ (resp. t ∈ ˜Γ) let φ t : E t → E ′ t be thecorresponding map of Anderson modules over Spec F q [ θ ] nr p (resp. Spec F q [ θ ] / p ).We consider the ordinary locus Γ of Γ: t ∈ Γ ⇐⇒ ξ k ( E t ) , ξ k ( E ′ t )are ordinary. Lemma 3.1. ˜Γ is dense in ˜Γ. (cid:3) Now let τ p ∈ G fun be any diagonal matrix, T G, p the element of Heckealgebra H ( G fun ) corresponding to the double coset K G τ p K G , Γ the graph of T G, p over Spec F q [ θ ] nr p , and ˜Γ for this Γ is defined as earlier.Let c be the highest power of p that appears in the diagonal entries of τ p (forexample, if τ p = τ p ,i then c = 1). Let s ∈ ˜Γ and E s , E ′ s the corresponding Andersont-motives over Spec F q [ θ ] / p . This means that we have a direct sum decompositionof the finite F q [ θ ]-module scheme ( E s )[ p c ] over Spec F q [ θ ] / p on its multiplicativeand etale part: ( E s )[ p c ] = ( E s )[ p c ] mult ⊕ ( E s )[ p c ] et , (3 . . E s )[ p c ] mult = ( µ p c ) n , (3 . . E s )[ p c ] et = (Spec F q [ θ ] / p c ) r − n . (3 . . φ s to ( E s )[ p c ] getting a map( φ s )[ p c ] : ( E s )[ p c ] → ( E ′ s )[ p c ] . (3 . . φ s )[ p c ] mult : ( E s )[ p c ] mult → ( E ′ s )[ p c ] mult (3 . . φ s )[ p c ] et : ( E s )[ p c ] et → ( E ′ s )[ p c ] et . (3 . . φ s defines elements in H ( GL n ) (resp. H ( GL r − n )). In concordance of notations of [FCh], we denote themby a (resp. d ). This pair ( a , d ) defines us an element of H ( M ). It is called the typeof s . Remark.
Unlike in the number case, here the elements a , d are independent.In order to formulate the below proposition 3.4, we need the following notations: Let δ : E → E be a map of Anderson modules over Spec F q [ θ ] / p of typeΨ i Φ j .This means that c of (3.1.1) is 1, and kernels of the map (3.1.5) (resp. (3.1.6))is isomorphic to ( µ p ) i (resp. (Spec F q [ θ ] / p ) j ). We denote them by ˜ K m , ˜ K e respec-tively.Further, let E be a Anderson module over Spec F q [ θ ] p such that ˜ E = E . Lemma 3.3.
We can identify ˜ K m (resp. ˜ K e ) with some i (resp. j )-dimensionalsubspaces in D fun ( E ) (resp. ( E ) p /D fun ( E ) ).We denote these subspaces by K m , K e respectively.Now let us consider the set of pairs ( φ, E ) where φ : E → E is a map ofAnderson modules over Spec F q [ θ ] p , such that ˜ φ = δ (and hence ˜ E = E ). Proposition 3.4.
The set of the above ( φ, E ) is isomorphic to the set ofsubspaces W ⊂ ( E ) p such that W ∩ D fun ( E ) = K m , W + D fun ( E ) /D fun ( E ) = K e . (3 . . Proof.
We need the function field analog of [K], Th. 2.1. Let R be an Artinianlocal ring with residue field F q [ θ ] / p and the maximal ideal m . We consider only thecase R = R η = F q [ θ ] nr p / p η for some η . Let E be an ordinary Anderson module over F q [ θ ] / p . Let us consider (3.1.1) for E , and let T p ( E ) be the Tate module of the etalepart: T p ( E ) = invlim c →∞ E [ p c ] et . E t is defined in [L], [F], it is of rank r and dimension n . Let E R be an Anderson module over R such that its reduction to F q [ θ ] / p is E (alift of E on R ).The function field analog of [K], Th. 2.1, (1) is the following: Theorem 3.4.1a.
The set of E R is in 1 – 1 correspondence with the set of mapsHom ( T p ( E ) ⊗ T p ( E t ) , m ) , (3 . . F q [ T ] / p -modules. Notation.
For a fixed E R we denote this map by q E R . Idea of the proof.
First, we define the analog of the map ϕ A /R , [K], p. 151for the present situation. Here it is ϕ E /R : T p ( E ) → m ⊕ n .Recall that η satisfies m η = 0. We choose k such that q k ≥ η , and we considerformulas of multiplication by p k for E : p k ( X ) = η X i = k C i X q i (3 . . a )where X ∈ C ⊕ n ∞ is a column vector and C i ∈ M n × n ( C ∞ ). Condition X ∈ E [ p k ] et means that P ηi = k C i X q i = 0.Let ˜ X ∈ R ⊕ n be a lift of X ∈ E [ p k ] et . Since for the first term C k X q k of 3.4.2awe have q k ≥ N , we get that ϕ E /R ( X ) := P ηi = k C i ˜ X q i ∈ m ⊕ n does not depend onthe choice of ˜ X . (cid:3) Now we need the function field analog of [K], Th. 2.1, (4). Let E , E be ordinaryAnderson modules over F q [ θ ] / p , α : E → E a map and E ,R , E ,R lifts of E , E on R . We denote by T p ( α ) : T p ( E ) → T p ( E ) (3 . . T p ( α t ) : T p ( E t ) → T p ( E t ) (3 . . Lemma 3.4.5.
A map α R : E ,R → E ,R such that its reduction is α exists ifffor any x ∈ T p ( E ), y ∈ T p ( E t ) we have q E ,R ( T p ( α )( x ) , y ) = q E ,R ( x, T p ( α t )( y )) . (3 . . α R is unique. (cid:3) Lemma 3.4.6a. (Conjectural statement). To define E over F q [ θ ] p is the sameas to define a concordant system of ( E ) η over R η (the concordance condition isclear). Remark.
Obviously E defines a concordant system of ( E ) η . But is the inversereally true? Maybe non-trivial automorphisms of ( E ) η give obstacles?14ow we return to the proof of Proposition 3.4. We fix η , we take E = E , E = E . According (3.2), there exist bases e t , . . . , e tn , e n +1 , . . . , e r , f t , . . . , f tn , f n +1 , . . . , f r of T p ( E t ), T p ( E ), T p ( E t ), T p ( E ) respectively such that the maps T p ( δ ), T p ( δ t ) inthese bases are the following: T p ( δ )( e n +1 ) = p f n +1 ... (maps of type 1) T p ( δ )( e n + j ) = p f n + j , T p ( δ )( e n + j +1 ) = f n + j +1 ... (maps of type 2) T p ( δ )( e r ) = f r , T p ( δ t )( f t ) = p e t ... (maps of type 3) T p ( δ t )( f ti ) = p e ti , T p ( δ t )( f ti +1 ) = e ti +1 ... (maps of type 4) T p ( δ t )( f tn ) = e tn .Now we apply formula (3.4.6) to these formulas. We consider 4 types of x , y : Type 13. x of type 1, y of type 3 ( λ ∈ [ n + 1 , · · · , n + j ], µ ∈ [1 , · · · , i ]):We get: p · q E ,R ( f λ , f tµ ) = p · q E ,R ( e λ , e tµ ) . (3 . . m had no p -torsion then we can divide the above equality by p and to get q E ,R ( f λ , f tµ ) = q E ,R ( e λ , e tµ ) , (3 . . q E ,R on these f λ , f tµ is defined uniquely.We think that in order to prove that we can really divide by p , we must considernot one fixed η , but all the values of them. The similar problem exists for the nexttype: Type 14. x of type 1, y of type 4 ( λ ∈ [ n + 1 , · · · , n + j ], µ ∈ [ i + 1 , · · · , n ]):We get: p · q E ,R ( f λ , f tµ ) = q E ,R ( e λ , e tµ ) . (3 . . m were p -divisible and had the p -torsion isomorphic to F q [ θ ] / p then OK: wehave q ( n − i ) j possibilities for ( E ) R as it should be.For other types of x , y there is no such problem. Really: Type 23. x of type 2, y of type 3 ( λ ∈ [ n + j + 1 , · · · , r ], µ ∈ [1 , · · · , i ]):We get: q E ,R ( f λ , f tµ ) = p · q E ,R ( e λ , e tµ ) . (3 . . q E ,R on these f λ , f tµ is defined uniquely; Type 24. x of type 2, y of type 4 ( λ ∈ [ n + j + 1 , · · · , r ], µ ∈ [ i + 1 , · · · , n ]):We get: q E ,R ( f λ , f tµ ) = q E ,R ( e λ , e tµ ) . (3 . . q E ,R on these f λ , f tµ is defined uniquely; We get that we have q ( n − i ) j modules E , this number is equal to thequantity of W satisfying (3.4.1). Now we need to prove that these W really satisfy (3.4.1). (cid:3) Now we can define the map β : H ( M ) → Corr( ˜ X ) from (1.1.4). Idea ofthe definition: let τ p have the form (cid:18) p A p B (cid:19) where A = ( a , . . . , a r − n ), B =( b , . . . , b n ), p A = diag ( p a , . . . , p a r − n ), p b = diag ( p b , . . . , p b n ). We denote by T M, p = T M, p ( A, B ) the element of Hecke algebra H ( M ) corresponding to the doublecoset K M τ p K M . Explicit formula for S G fun M shows that S G fun M ( T G, p ) = q − m A,B T M, p + other terms, (3 . T M, p ( A ′ , B ′ ) for pairs ( A ′ , B ′ )distinct from ( A, B ). Coefficient m A,B ≥ T M, p ( A, B ) = Ψ i Φ j we have m A,B = ij .Now we consider the reduction of the correspondence β ( T G, p ). Let Γ irr bean irreducible component of its graph, φ : E → E a map of Anderson modularobjects over A/ p corresponding to a point of Γ irr , and t ∈ H ( M ) its type. t dependsonly on Γ irr but not on φ : E → E because it is a discrete invariant, so we cancall it the type of Γ irr .First, we denote by C ( A, B ) the correspondence on ˜ X whose graph is the sumof all the irreducible components of the graph of reduction of the correspondence β ( T G, p ) whose type is T M, p ( A, B ) (really, for each (
A, B ) there exists only onesuch component). By abuse of notations we denote by C (Ψ i Φ j ) the C ( A, B ) where A , B are from 2.2.3 a,b. Finally, we define β ( T M, p ( A, B )) = q m A,B C ( A, B ) , (3 . β (Ψ i Φ j ) = q ij C (Ψ i Φ j ) . (3 . Corollary 3.8. d ns ( C (Ψ i Φ j )) = q ( n − i ) j . Proof.
Follows immediately from 3.4.12. (cid:3) d ns ( β (Ψ i Φ j )) = q nj . (3 . Proposition 3.10. β is a ring homomorphism.16 dea of the proof. Let (
A, B ), ( A ′ , B ′ ) be 2 pairs of multiindices as aboveand let T M, p ( A, B ) · T M, p ( A ′ , B ′ ) = P i κ i T M, p ( A i , B i ) for some pairs ( A i , B i ) andcoefficients κ i . Lemma 3.10.1.
For all i we have q m A,B d ns ( C ( A, B )) · q m A ′ ,B ′ d ns ( C ( A ′ , B ′ )) = κ i q m Ai,Bi d ns ( C ( A i , B i )) . Proof.
Explicit calculation. For a particular case corresponding to (
A, B ) = Ψ i ,( A ′ , B ′ ) = Φ j , ( A i , B i ) = Ψ i Φ j this follows from the above results.We have: T G, p ( A, B ) · T G, p ( A ′ , B ′ ) = X i κ i T G, p ( A i , B i ) + other terms . (3 . . A, B ), ( A ′ , B ′ ), ( A i , B i );(b) (3.10.2) and Lemma 3.10.1;(c) Commutativity of the the function field analog of the diagram (1.1.6)imply that β ( T M, p ( A, B )) · β ( T M, p ( A ′ , B ′ )) + other terms == X i κ i β ( T M, p ( A i , B i )) + other terms. Now naive considerations show us that “other terms” in both sides ofthe above equality are equal. Really, let us denote by S Γ( A, B ) the support of thegraph Γ( β ( T M, p ( A, B ))) ⊂ ˜ X × ˜ X , and analogically for the pairs ( A ′ , B ′ ), ( A i , B i ).We have:( t , t ) ∈ S Γ( A, B ) ⇐⇒ there is a map E t → E t of type T M, p ( A, B ) . By definition of the product of correspondences,( t , t ) ∈ ∪ i S Γ( A i , B i ) ⇐⇒ there exists t such that( t , t ) ∈ S Γ( A, B ) , ( t , t ) ∈ S Γ( A ′ , B ′ ) . Since the type of the composition of maps of Anderson varieties is concordantwith the multiplication in H ( M ), we get 3.10.3. (cid:3) According Langlands, L-function L ( M , s ) of an irreducible submotive M of aShimura variety is related with L ( π, r , s ), where π is an automorphic representationof G ( A Q ) and r : L G → GL ( W ) a finite-dimensional representation of L G : L ( M , s ) ∼ L ( π, r , s ) . (3 a. r is given for example in [BR], Section 5.1, p. 550.17et us formulate an analog of this result for Anderson modular objects. For thiscase an analog of G ( A Q ) is G fun ( A F q ( θ ) ). Theorem 3a.1.
If an analog of (3a.0) is true for Anderson modular objects X fun ( r, n ) of any level then the restriction of r to ˆ G fun is the n -th skew powerrepresentation of GL r .This theorem follows from the below Theorem 3a.3.Let π = ⊗ π l be a representation of G fun ( A F q ( θ ) ) corresponding (according Lang-lands) to an irreducible submotive of an Anderson modular object, and θ p ∈ L G aLanglands element of π p (we consider the case of p such that π p is non-ramified).Let α i , i = 1 , ...r , be eigenvalues of θ p and a i the eigenvalues of T p ,i (analogs ofFourier coefficients of an automorphic form for the classical case). Standard formal-ism of Langlands elements for GL r in the non-ramified case together with (2.2.2)shows that a i = q − i ( i − / σ i ( α ∗ ) . (3 a. P ′ r,n the characteristic polynomial of r ( θ p ), it belongs to Z [ a , ..., a r ][ T ] where T is an abstract variable. The following theorem followsimmediately from (2.2.2), (3a.2) (like in the number case): Theorem 3a.3. P r,n = P ′ r,n (after identification of T and f r , a i and T p ,i ). (cid:3)
4. Unitary Shimura varieties.
We consider abelian varieties with multiplication by an imaginary quadraticfield (abbreviation: MIQF). Let K be such field, X num the corresponding Shimuravariety parametrizing abelian r -folds with multiplication by K , of signature ( r − n, n ). We shall call them unitary Shimura varieties. The corresponding reductivegroup over Q is G = G num = GU ( r − n, n ). We have dim X num = ( r − n ) n . Let p be a prime inert in K ; we shall consider p -Hecke correspondences and the reductionat p . Theorem 4.1. M for this case is the same as in Theorem 2.1. Corollary 4.2.
Satake maps for this case coincide with the ones for the func-tional case (formulas (2.2.4), (2.2.5)).Let A t be as in Subsection 1.2. ( A t ) p is an r -dimensional vector space over F p .Let D = D unitary be as in Subsection 1.2. Theorem 4.3. D unitary is a vector space over F p of dimension max ( r − n, n ). Remark 4.4.
There exists a symmetry between n and r − n . Nevertheless, herethe analogy between functional and unitary case apparently is not complete. Theorem 4.5.
Analog of the Theorem 2.3.5 (i.e. formula 2.3.6) holds for theunitary case (dimension is taken over F p ). Theorem 4.6. ([BR], Section 5.1, p. 550, example (b)). Restriction of r onˆ G ⊂ L G is the same as in Theorem 3a.1.We think that analogs of Theorems 2.3.7, 2.3.8, 2.3.10 also hold for this case.18 . Conjectural values of Hodge numbers. There are conjectural formulas for values of Hodge numbers h ij (over the fieldsof multiplications coming from Hecke correspondences) of irreducible submotivesof Shimura varieties (see, for example, [BR], Section 4.3, p. 548). For example, forthe case of Siegel modular varieties of genus g (their dimension is d g = g ( g + 1) / d g they are the following: Theorem 5.1. h i,d g − i = { the quantity of subsets ( j , . . . , j α ) of the set 1 , , ..., g such that j + ... + j α = i } , where α is arbitrary.For other types of submotives the formulas for h ij are similar but more long.For example, for the case of unitary Shimura variety of Section 4 and for thesame type of submotives the formula is the following: Theorem 5.2. h i, ( r − n ) n − i = { the quantity of subsets ( j , ...j n ) of the set(1 , , ..., r ) such that j + ... + j n − (1 + ... + n ) = i } .By analogy between functional and unitary case we can conjecture that the sameformula holds for the functional case.
6. Non-ordinary Drinfeld modules.