aa r X i v : . [ m a t h . N T ] F e b DUALITY OF ANDERSON t -MOTIVES A. Grishkov, D. Logachev Abstract.
Let M be a t-motive. We introduce the notion of duality for M . Mainresults of the paper (we consider uniformizable M over F q [ T ] of rank r , dimension n ,whose nilpotent operator N is 0):1. Algebraic duality implies analytic duality (Theorem 5). Explicitly, this meansthat the lattice of the dual of M is the dual of the lattice of M , i.e. the transposedof a Siegel matrix of M is a Siegel matrix of the dual of M .2. Let n = r −
1. There is a 1 – 1 correspondence between pure t-motives (allthey are uniformizable), and lattices of rank r in C n ∞ having dual (Corollary 8.4).3. Pure t-motives have duals which are pure t-motives as well (Theorem 10.3).4. Some explicit results are proved for M having complete multiplication. TheCM-type of the dual of M is the complement of the CM-type of M . Moreover, for M having multiplication by a division algebra there exists a simple formula for theCM-type of the dual of M (Section 12).5. We construct a class of non-pure t-motives (t-motives having the completelynon-pure row echelon form) for which duals are explicitly calculated (Theorem 11.5).This is the first step of the problem of description of all t-motives having duals.6. If M has good ordinary reduction then the kernels of reduction maps on groupsof torsion points for M and its dual are complementary with respect to a naturalpairing (proof is given for a particular case, Conjecture 13.4.1).
0. Introduction. t-motives are the function field analogs of abelian varieties (more exactly, ofabelian varieties with multiplication by an imaginary quadratic field, see [L09]).
Mathematics Subject Classification . Primary 11G09; Secondary 11G15, 14K22.
Key words and phrases. t-motives; duality; symmetric polarization form; Hodge conjecture;t-motives of complete multiplication; complementary CM-type.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 Gilles Lachaud for invitation to IML and toLaurent Lafforgue for invitation to IHES where this paper was started, and to Vladimir Drinfeldfor invitation to the University of Chicago where this paper was continued. Discussions withGreg Anderson, Vladimir Drinfeld, Laurent Fargues, Alain Genestier, David Goss, Richard Pink,Yuichiro Taguchi, Dinesh Thakur on the subject of this paper were very important. Particularly,Alain Genestier informed me about the paper of Taguchi where the notion of the dual of a Drinfeldmodule is defined. Further, Richard Pink indicated me an important reference (see Section 6 fordetails); proof of the main theorem of the present paper grew from it. Finally, Vladimir Drinfeldindicated me the proof of the Theorem 12.6 and Jorge Morales gave me a reference on classificationof quadratic forms over F q [ T ] (Remark 7.8). E-mails: [email protected]; [email protected] (corresponding author)Typeset by
AMS -TEX M (this is not the duality in a Tannakian category!),and we prove some properties of this notion, see the abstract. Particularly, if M is uniformizable and has dual then the lattice of the dual of M is the dual of thelattice of M (Theorem 5) . An immediate corollary of the above theorem and theresult of Drinfeld on 1 – 1 correspondence between Drinfeld modules and latticesin C ∞ (here C ∞ is the function field analog of C ) is Corollary 8.4: there is a 1 – 1correspondence between pure t-motives of dimension r − r , and latticesof rank r in C r − ∞ having dual (not all such lattices have dual).Let us give more details on the contents of the paper. For simplicity, mostresults are proved for t-motives over the ring A = F q [ T ], and we consider, withfew exceptions, only the case N = 0. The main definition of duality of t-motives(definition 1.8 — case A = F q [ T ] and definition 1.13 — general case) is given inSection 1. Lemma 1.10 gives the explicit matrix form of the definition of dualityof t-motives. Since Taguchi in [T] gave a definition of dual to a Drinfeld module,we prove in Proposition 1.12.3 that the definition of the present paper is equivalentto the original definition of Taguchi. Section 1.14 contains a definition of dualityfor abelian τ -sheaves ([BH], Definition 2.1), but we do not develop this subject.Section 2 contains the definition of the dual lattice. Section 3 contains explicitformulas for the dual lattice. Section 5 contains the statement and the proof ofthe main theorem 5 — coincidence of algebraic and analytic duality for the case A = F q [ T ] (section 4 contains the statement of the corresponding conjecture forthe case of general A ). Section 6 contains the theorem 6 describing the lattice ofthe tensor product of two t-motives (case N = 0; the proof for the general casewas obtained, but not published, by Anderson). Section 7 contains the notionof self-dual t-motives and polarization form on them. Some examples are given.We discuss in Section 8 the problem of correspondence between uniformizable t-motives and lattices. Section 9 gives the statement of the main result for the case N = 0 without proof and a reformulation of the theorem 5 in terms of Hodge-Pinkstructures of constant weight.Further on, we prove in Section 10 that pure t-motives have duals which are puret-motives as well, and some related results (a proof that the dual of an abelian τ -sheaf is also an abelian τ -sheaf can be obtained using ideas of Section 10). In Section11 we consider t-motives having the completely non-pure row echelon form, and wegive an explicit formula for their duals. In Section 12 we consider t-motives withcomplete multiplication, and we give for them a very simple version of the proof ofthe first part of the main theorem. Section 13 contains some explicit formulas fort-motives of complete multiplication. In 13.1 we describe the dual lattice, in 13.2we show that the results of Section 12 are compatible with (the first form of) themain theorem of complete multiplication. Section 13.3 contains an explicit proofof the main theorem for t-motives with complete multiplication by two types ofsimplest fields. Section 13.4 gives us an application of the notion of duality to thereduction of t-motives (subject in development, see [L]). Here this result is proved for M having the associated nilpotent operator N (see (1.9.2)) equalto 0. The same result for M having N = 0 is proved in [GL18]. A version of the definition of duality is obtained independently in [Tae], 2.2. otations. q is a power of a prime p ; Case of M over F q [ T ]: Z ∞ := F q [ θ ], R ∞ := F q ((1 /θ )), C ∞ is the completion of its algebraic closure( Z ∞ , R ∞ , C ∞ are the function field analogs of Z , R , C respectively); A := F q [ T ], K := F q ((1 /T )); ι : A → C ∞ ( ι ( T ) = θ ) is the standard map of generic characteristic (with oneexception (1.16), we shall not consider the case of finite characteristic); we extend ι to K , and we have Z ∞ = ι ( A ) ⊂ C ∞ , R ∞ = ι ( K ) ⊂ C ∞ . C (resp. C ) is the Carlitz module over A = F q [ T ] (resp. over F q [ T ]). Case of M over an extension of F q [ T ]: Q ∞ is a finite separable extension of F q ( θ ); ∞ is a fixed valuation of Q ∞ over the infinity of F q ( θ ); Z ∞ ⊂ Q ∞ is the subring of elements which are regular outside ∞ ; R ∞ is the completion of Q ∞ at infinity, and C ∞ — the completion of its algebraicclosure — is the same as of the case of M over F q [ T ]. A ⊃ F q [ T ], K ⊃ F q ((1 /T )) are defined by the condition that ι : A → Z ∞ , ι : K → R ∞ are isomorphisms. A C := A ⊗ F q C ∞ (i.e. A C = C ∞ [ T ] for the case of M over F q [ T ]). C is a Drinfeld module of rank 1 over A .If P = P a i T i P b i T i ∈ C ∞ ( T ) then P ( k ) := P a qki T i P b qki T i . For x ∈ A C , x = a ⊗ z , a ∈ A , z ∈ C ∞ we let x ( k ) := a ⊗ z q k . M r is the set of r × r matrices. If C = { c ij } is a matrix with entries c ij ∈ C ∞ ( T )then C ( k ) := { c ( k ) ij } , C t is the transposed of C , C ( k ) − = ( C ( k ) ) − , C t − = ( C t ) − .If M is an A C -module, we define M (1) as the tensor product M ⊗ A C , ∗ (1) A C with respect to the map ∗ (1) : A C → A C (this notation is concordant in the obvioussense with the above notation C (1) ).For a t-motive M we denote by E = E ( M ) the corresponding t-module (see [G],Theorem 5.4.11; Goss uses the inverse functor E M = M ( E )).Lie( M ) is Lie( E ( M )) ([G], 5.4). I k is the unit matrix of size k .Throughout the whole paper the word ”canonical” will mean ”canonical up tomultiplication by elements of F ∗ q ”.
1. Definitions.
If otherwise is not explicitly stated, throughout the whole paper we consider thecase of t-motives M over the ring A = F q [ T ] such that N = N ( M ) = 0. Exceptions:case of arbitrary A is treated in Sections 1.13, 1.14, 2, 4, 5.2. Case of arbitrary N istreated in Sections 1, 10 and in statements of some results of Anderson in Sections5, 6.In the present section we consider M such that N ( M ) is arbitary.3et C ∞ [ T, τ ] be the Anderson ring, i.e. the ring of non-commutative polynomialssatisfying the following relations (here a ∈ C ∞ ): T a = aT, T τ = τ T, τ a = a q τ (1 . C ∞ [ T, τ ] — the ring C ∞ ( T )[ τ ] which is the ring ofnon-commutative polynomials in τ over the field of rational functions C ∞ ( T ) withthe same relations (1.1). For a left C ∞ [ T, τ ]-module M we denote by M C ∞ [ T ] thesame M treated as a C ∞ [ T ]-module with respect to the natural inclusion C ∞ [ T ] ֒ → C ∞ [ T, τ ]. Analogously, we define M C ∞ [ τ ] ; we shall use similar notations also forthe left C ∞ ( T )[ τ ]-modules.Obviously we have: (1.2) For C ∈ M r ( C ∞ ( T )) operations C t , C − and C ( i ) commute. Definition 1.3. ([G], 5.4.2, 5.4.12, 5.4.10). A t-motive M is a left C ∞ [ T, τ ]-module which is free and finitely generated as both C ∞ [ T ]-, C ∞ [ τ ]-module andsuch that ∃ m = m ( M ) such that ( T − θ ) m M/τ M = 0 (1 . . Remark.
The above object is called ”abelian t-motive” (resp. ”t-motive”) in[G] (resp. [A]), while the name ”t-motive” is used in [G] for a more general object([G], Definition 5.4.2). Since we shall not use objects defined in [G], 5.4.2, I preferto use a shorter name for the above M .t-motives are main objects of the present paper. If we affirm that an objectexists this means that it exists as a t-motive if otherwise is not stated. We denotedimension of M over C ∞ [ τ ] (resp. C ∞ [ T ]) by n (resp. r ), these numbers are calleddimension and rank of M . Morphisms of abelian t-motives are morphisms of left C ∞ [ T, τ ]-modules.To define a left C ∞ [ T, τ ]-module M is the same as to define a left C ∞ [ T ]-module M C ∞ [ T ] endowed by an action of τ satisfying τ ( P m ) = P (1) τ ( m ), P ∈ C ∞ [ T ].In this situation we can also treate τ as a C ∞ [ T ]-linear map M (1) → M . Thisinterpretation is necessary if we consider the general case A ⊃ F q [ T ].We need two categories which are larger than the category of abelian t-motives. Definition 1.4.
A pr´e-t-motive is a left C ∞ [ T, τ ]-module which is free andfinitely generated as C ∞ [ T ]-module, and satisfies (1.3.1). Definition 1.5.
A rational pr´e-t-motive is a left C ∞ ( T )[ τ ]-module which is freeand finitely generated as C ∞ ( T )-module. Remark 1.6.
An analog of (1.3.1) does not exist for them.There is an obvious functor from the category of t-motives to the category ofpr´e-t-motives which is fully faithful, and an obvious functor from the category ofpr´e-t-motives to the category of rational pr´e-t-motives. We denote these functorsby i , i respectively. It is easy to see (Remark 10.2.3) that if M is a pr´e-t-motivethen the action of τ on i ( M ) is invertible.Let M , M be rational pr´e-t-motives such that the action of τ on ( M ) C ∞ ( T ) isinvertible. Definition 1.7. (1)
Hom( M , M ) is a rational pr´e-t-motive such that4om( M , M ) C ∞ ( T ) = Hom C ∞ ( T ) (( M ) C ∞ ( T ) , ( M ) C ∞ ( T ) )and the action of τ is defined by the usual manner: for ϕ : M → M , m ∈ M ( τ ϕ )( m ) = τ ( ϕ ( τ − ( m ))) (2) Let M , M be t-motives. Their tensor product is defined by M ⊗ C ∞ [ T ] M where the action of τ is given by τ ( m ⊗ m ) = τ ( m ) ⊗ τ ( m ). It is known(Anderson; see also [G]) that M ⊗ M is really a t-motive of rank r r , of dimension n r + n r . M ⊗ M has N = 0 even if M , M have N = 0.The Carlitz module C is the Anderson t-motive with r = n = 1, it is unique over C ∞ (see, for example, [G], 3.3). The µ -th tensor power of C is denoted by C ⊗ µ . Itsrank r is 1 and its dimension is µ . Definition 1.8.
Let M be a t-motive and µ a positive number. A t-motive M ′ = M ′ µ is called the µ -dual of M (dual if µ = 1) if M ′ = Hom( M, C ⊗ µ ) as arational pr´e-t-motive, i.e. i ◦ i ( M ′ ) = Hom( i ◦ i ( M ) , C ⊗ µ ) (1 . . Remark.
This definition generalizes the original one of Taguchi ([T], Section5), see 1.12 below. A similar definition is in [F].
We shall need the explicit matrix description of the above objects. Let e ∗ = ( e , ..., e n ) t be the vector column of elements of a basis of M over C ∞ [ τ ].There exists a matrix A ∈ M n ( C ∞ [ τ ]) such that T e ∗ = A e ∗ , A = l X i =0 A i τ i where A i ∈ M n ( C ∞ ) (1 . . A = θI n + N (1 . . N is a nilpotent matrix, and the condition m ( M ) = 1 is equivalent to thecondition N = 0.Let f ∗ = ( f , ..., f r ) t be the vector column of elements of a basis of M over C ∞ [ T ]. There exists a matrix Q = Q ( f ∗ ) ∈ M r ( C ∞ [ T ]) such that τ f ∗ = Qf ∗ (1 . . Lemma 1.10.
Let M be as above. A t-motive M ′ is the µ -dual of M iff thereexists a basis f ′∗ = ( f ′ , ..., f ′ r ) t of M ′ over C ∞ [ T ] such that its matrix Q ′ = Q ( f ′∗ )satisfies Q ′ = ( T − θ ) µ Q t − (1 . . Proof.
The matrix Q of C ⊗ µ is ( T − θ ) µ . This implies the formula. (cid:3) For further applications we shall need the following lemma. The above f ∗ , f ′∗ are the dual bases (i.e. if we consider f ′ i as elements of Hom( M, C ) then5 ′ i ( f j ) = δ ij f , where f is canonically defined by the condition that it generates C C ∞ [ T ] and satisfies τ f = ( T − θ ) f ). Let γ be an endomorphism of M and D itsmatrix in the basis f ∗ (i.e. γ ( f ∗ ) = Df ∗ ). Let γ ′ be the dual endomorphism. Lemma 1.10.3.
The matrix of γ ′ in the basis f ′∗ is D t . (cid:3) Remark 1.11.1.
For any M having dual there exists a canonical homomorphism δ : C → M ⊗ M ′ . This is a well-known theorem of linear algebra. Really, in theabove notations we have f P i f i × f ′ i . It is obvious that δ is well-defined, canonicaland compatible with the action of τ . Remark 1.11.2.
The µ -dual of M — if it exists — is unique, i.e. does notdepend on base change. This follows immediately from Definition 1.8, but can bededuced easily from 1.10.1. Really, let g ∗ = ( g , ..., g r ) t be another basis of M over C ∞ [ T ] and C ∈ GL r ( C ∞ [ T ]) the matrix of base change (i.e. g ∗ = Cf ∗ ). Then Q ( g ∗ ) = C (1) QC − . Let g ′∗ = ( g ′ , ..., g ′ r ) t be a basis of M ′ over C ∞ [ T ] satisfying g ′∗ = C t − f ′∗ . Elementary calculation shows that matrices Q ( g ∗ ), Q ( g ′∗ ) satisfy(1.10.1). Remark 1.11.3.
The operation M M ′ µ is obviously contravariant functorial.It is an exercise to the reader to give an exact definition of the correspondingcategory such that the functor of duality is defined on it, and is involutive (recallthat not all t-motives have duals, and the dual of a map of t-motives is a priori amap of rational pr´e-t-motives). The original definition of duality ([T], Definition 4.1; Theorem 5.1) fromthe first sight seems to be more restrictive than the definition 1.8 of the presentpaper, but really they are equivalent. We recall some notations and definitions of [T]in a slightly less general setting (rough statements; see [T] for the exact statements).Let G be a finite affine group scheme over C ∞ , i.e. G = Spec R where R is a finite-dimensional C ∞ -algebra. Let µ : R → R ⊗ R be the comultiplication of R . Suchgroup G is called a finite v -module ([T], Definition 3.1) if there is a homomorphism ψ : A → End gr. sch. ( G ) satisfying some natural conditions (for example, an analogof 1.3.1). Further, let E G be a C ∞ -subspace of R defined as follows: E G = { x ∈ R | µ ( x ) = x ⊗ ⊗ x } The map x x q is a C ∞ -linear map fr : E (1) G → E G . Further, the map ψ ( T ) : G → G can be defined on E G . Let v : E G → E (1) G be a map satisfying fr ◦ v = ψ ( T ) − θ .We consider two finite v -modules G , H , the above objects fr, v etc. will carrythe respective subscript. Let * be the dual in the meaning of linear algebra. Definition 1.12.1 ([T], 4.1). Two finite v -modules G , H are called dual if thereexists an isomorphism α : E ∗ H → E G such that if we denote by v : E G → E (1) G a mapwhich enters in the commutative diagram E ∗ H fr ∗ H −→ E ∗ (1) H α ↓ α (1) ↓E G v −→ E (1) G G ◦ v = ψ G ( T ) − θ (1 . . v = v G .Let M be a t-motive having m ( M ) = 1, E = E ( M ) the corresponding t-moduleand a ∈ A . We denote E a — the set of a -torsion elements of E — by M a . It is afinite v -module. Proposition 1.12.3.
Let M , M ′ be t-motives which are dual in the meaningof Definition 1.8. Then ∀ a ∈ A , a = 0 we have: M a , M ′ a are dual in the meaningof 1.12.1 = [T], Definition 4.1. Proof.
Condition a ∈ F q [ T ] implies that multiplication by τ is well-defined on M/aM . Lemma 1.12.3.1.
We have canonical isomorphisms i : M/aM → E M a , i (1) : M/aM → E (1) M a such that the following diagrams are commutative: M/aM τ −→ M/aM M/aM T −→ M/aMi (1) ↓ i ↓ i ↓ i ↓E (1) M a fr −→ E M a E M a ψ T −→ E M a Proof.
Let R be a ring such that Spec R = M a . The pairing between M and E shows that there exists a map M → R which is obviously factorized via an inclusion M/aM → R . It is easy to see that the image of this inclusion is contained in E M a ,i.e. we get i . Since dim C ∞ ( M/aM ) = deg a · r ( M ) and dim C ∞ ( R ) = q deg a · r ( M ) we get from [T], Definition 1.3 that i is an isomorphism. Other statements of thelemma are obvious. (cid:3) This lemma means that we can rewrite Definition 1.12.1 for the case G = M a , H = N a by the following way: Two finite v -modules M a , N a are dual if there exists an isomorphism α : ( N/aN ) ∗ → M/aM such that after identification via α of τ ∗ : ( N/aN ) ∗ → ( N/aN ) ∗ with a map v : M/aM → M/aM we have on
M/aM : τ ◦ v = t − θ (1 . . . Lemma 1.12.3.4.
For i = 1 , N i be a free C ∞ [ T ]-module of dimension r with a base f i ∗ = ( f i , ..., f ir ), let ϕ i : N i → N i be C ∞ [ T ]-linear maps havingmatrices Q i in f i ∗ such that Q = Q t , and let a be as above. Let, further, ϕ i,a : N i /aN i → N i /aN i be the natural quotient of ϕ i . Then there exist C ∞ -bases ˜ f i ∗ of N i /aN i such that the matrix of ϕ ,a in the base ˜ f ∗ is transposed to the matrix of ϕ ,a in the base ˜ f ∗ . Here and below a t-motive N should not be confused with N of 1.9.2. roof. We can identify elements of N with C ∞ [ T ]-linear forms on N (notation:for x ∈ N the corresponding form is denoted by χ x ) such that χ ϕ ( x ) = χ x ◦ ϕ .Any C ∞ [ T ]-linear form χ on N i defines a C ∞ [ T ] /a C ∞ [ T ]-linear form on N i /aN i which is denoted by χ a . Let now x ∈ N /aN , ¯ x its lift on N , then χ x,a = ( χ ¯ x ) a is a well-defined C ∞ [ T ] /a C ∞ [ T ]-linear form on N /aN . For x ∈ N /aN we have χ ϕ ,a ( x ) ,a = χ x,a ◦ ϕ ,a Further, let λ : C ∞ [ T ] → C ∞ be a C ∞ -linear map such that Its kernel does not contain any non-zero ideal of C ∞ [ T ] /a C ∞ [ T ].(such λ obviously exist.) For x ∈ N /aN we denote λ ◦ χ x,a by ψ x , it is a C ∞ -linear form on C ∞ -vector space N /aN . Obviously condition (1.12.3.5) impliesthat the map x ψ x is an isomorphism from N /aN to the space of C ∞ -linearforms on C ∞ -vector space N /aN , and we have ψ ϕ ,a ( x ) = ψ x ◦ ϕ ,a which is equivalent to the statement of the lemma. (cid:3) Finally, the proposition follows immediately from this lemma multiplied by T − θ ,formula 1.10.1 and 1.12.3.2. (cid:3) Remark.
Let a = P ki =0 g i T i , g i ∈ F q , g k = 1. Taguchi ([T], proof of 5.1 (iv))uses the following λ : λ ( T j ) = 0 for j < k − λ ( T k − ) = 1. It is easy to check thatfor x = ( T i + T i − g k − + T i − g k − + ... + g k − i ) f j for this λ we have: ψ x ( T i f j ) = 1, ψ x ( T i ′ f j ′ ) = 0 for other i ′ , j ′ . We consider in Sections 1.13, 1.14 the case of arbitrary A ⊃ F q [ T ].A t-motive over A is defined for example in [BH], p.1. Let us reproduce thisdefinition for the case of characteristic 0. Let J be an ideal of A C generated by theelements a ⊗ − ⊗ ι ( a ) for all a ∈ A . The ring A C [ τ ] is defined by the formula τ · ( a ⊗ z ) = ( a ⊗ z q ) · τ , a ∈ A , z ∈ C ∞ . Definition 1.13.1.
A t-motive M over A is a pair ( M, τ ) where M is a locallyfree A C -module and τ is an A C -linear map M (1) → M satisfying the followinganalog of 1.3.1, 1.9.2: ∃ m such that J m ( M/τ ( M (1) )) = 0 (1 . . Remark 1.13.3.
We can consider M as an A C [ τ ]-module using the followingformula for the product τ · m : τ · m = τ ( m ⊗ m ∈ M , m ⊗ ∈ M (1) .The rank of M as a locally free A C -module is called the rank of the correspondingt-motive ( M, τ ). If A = F q [ T ] then M (1) is isomorphic to M , we can consider M asa C ∞ [ T, τ ]-module, and it is possible to show that in this case 1.13.2 implies that M C ∞ [ τ ] is a free C ∞ [ τ ]-module. In the general case, the dimension n of ( M, τ ) isdefined as dim C ∞ ( M/τ ( M (1) )). 8et us fix C = ( C , τ C ) — a t-motive of rank 1 over A . For a t-motive M =( M, τ M ) a t-motive M ′ C — the C -dual of M — is defined as follows. We put M ′ C = Hom A C ( M, C ). Since for any locally free A C -modules M , M we haveHom A C ( M , M ) (1) = Hom A C ( M (1)1 , M (1)2 )we can define τ ( M ′ C ) by the following formula:For ϕ ∈ Hom A C ( M, C ) (1) we have τ ( M ′ C )( ϕ ) = τ C ◦ ϕ ◦ τ − M τ -sheaves. We use notations of [BH], Definition 2.1if they do not differ from the notations of the present paper; otherwise we continueto use notations of the present paper (for example, d (resp. σ ∗ ( X ) for any object X ) of [BH] is n (resp. X (1) ) of the present paper). For any abelian τ -sheaf F wedenote its Π i , τ i by Π i ( F ), τ i ( F ) respectively. If M , N are invertible sheaves on X and ρ : M → N a rational map then we denote by ρ inv : N → M the rationalmap which is inverse to ρ with respect to the composition. We define τ r ,i − ( F )(the rational τ i ) as the composition map τ i − ( F ) ◦ Π (1) i − inv ( F ), it is a rational mapfrom F (1) i to F i .Let O be a fixed abelian τ -sheaf having r = n = 1. The O -dual abelian τ -sheaf F ′ = F ′O is defined by the formulas F ′ = Hom X ( F , O )where Hom is the sheaf’s one, and the map τ r , − ( F ′ ) : F ′ → F ′ is defined asfollows. We have F ′ = Hom X ( F , O ). Let γ ∈ Hom X ( F , O )( U )where U is a sufficiently small affine subset of X C ∞ , such that γ : F ( U ) →O ( U ). We define: [[ τ r , − ( F ′ )]( U )]( γ ) is the following composition map: F ( U ) [ τ inv r , − ( F )]( U ) −→ F (1)0 ( U ) γ → O (1)0 ( U ) [ τ r , − ( O )]( U ) −→ O ( U ) ∈ Hom X ( F , O )( U )Clearly that this definition and the definitions 1.8, 1.13 are compatible withthe forgetting functor M ( F ) from abelian τ -sheaves to pure Anderson t-motives of[BH], Section 3, page 8. Let L ⊃ F q ( θ ) be a field extension of F q ( θ ), and M a t-motive over L (i.e. a pair ( M , an L -structure on M )). Obviously we have Proposition 1.15.1.
The notion of duality for M over L is well-defined. (cid:3) Similarly, we have a proposition for Galois action:
Proposition 1.15.2.
Let M be defined over F q ( θ ) and γ ∈ Gal ( F q ( θ )). Then( γ ( M )) ′ = γ ( M ′ ). (cid:3) Let ι : A → ¯ F q be a map of finitecharacteristic, we denote Ker ι by P . The definition of t-motive for this case issimilar to 1.3, see [G] for the details. The definition of duality also is similar to9he one of the case of generic characteristic. Duality commutes with reduction.Namely, let M be from 1.15, P a prime of L not over the infinity of F q ( θ ), P ⊂ A is ι − ( P ∩ F q [ θ ]) — the finite characteristic. We consider the case of good reductionof M at P , we denote it by ˜ M . It is a t-motive in characteristic P . Let M havedual M ′ . Proposition 1.16.1. ˜ M has dual iff M ′ has good reduction at P ; in this casethey coincide. (cid:3) Remark 1.16.2.
Apparently if M has good reduction and dual, then M ′ alsohas good reduction (in this case 1.16.1 means that M ′ exists implies ( ˜ M ) ′ exists).For standard-3 t-motives (this is a simple tipe of t-motives, see 11.8.1) apparentlythis can be shown by explicit calculations. Remark 1.16.3.
Clearly 1.16.1 is true for the case of bad reductions. I do notgive exact definitions for this case.
Let M be of finite characteristic. By analogy with thenumber field case, M is called ordinary if its Newton polygon consists of 2 segments.If N = 0 then the Newton polygon of M ′ is the dual of the one of M (the notion ofduality of polygons is clear; apparently the condition N = 0 can be omitted). So,we have Proposition 1.16.5. M is ordinary ⇐⇒ M ′ is ordinary. (cid:3) See 13.4.1 for a more exact result.
2. Analytic duality.
We consider in the present section the case of arbitrary A ⊃ F q [ T ] (and N = 0as usually).Condition N = 0 implies that an element a ∈ A acts on Lie( M ) by multiplicationby ι ( a ). Hence, we have a Definition 2.1.
Let V be the space C n ∞ . A locally free r -dimensional Z ∞ -submodule L of V is called a lattice if(a) L generates V as a C ∞ -module and(b) The R ∞ -linear span of L has dimension r over R ∞ .Numbers n , r are called the dimension and the rank of L respectively. Attachedto ( L, V ) is the tautological inclusion ϕ = ϕ ( L, V ) : L → V . We shall consider thecategory of triples ( ϕ, L, V ); a map ψ : ( ϕ, L, V ) → ( ϕ , L , V ) is a pair ( ψ L , ψ V )where ψ L : L → L is a Z ∞ -linear map, ψ V : V → V is a C ∞ -linear map such that ϕ ◦ ψ L = ψ V ◦ ϕ .Inclusion ϕ can be extended to a map L ⊗ Z ∞ C ∞ → V (which is surjective by2.1a), we denote it by ϕ = ϕ ( L, V ) as well. We can also attach to (
L, V ) an exactsequence 0 → Ker ϕ → L ⊗ Z ∞ C ∞ ϕ → V → . I ∈
Cl( A ) be a class of ideals; we shall use the same notation I to denotea representative in the ι -image of this class. Let ( ϕ ′ , L ′ , V ′ ) be another lattice and10 a structure of a perfect I -pairing < ∗ , ∗ > D between L and L ′ . Let us fix anisomorphism α : I ⊗ Z ∞ C ∞ → C ∞ (2 . ′ ) D extends via α to a perfect C ∞ -pairing between L ⊗ Z ∞ C ∞ and L ′ ⊗ Z ∞ C ∞ , we denotethis pairing by D α, ∞ . Definition 2.3.
Two lattices ( ϕ, L, V ) and ( ϕ ′ , L ′ , V ′ ) are called ( α, I )-dual ifthere exists a perfect I -pairing D between L and L ′ such that Ker ϕ ⊂ L ⊗ Z ∞ C ∞ ,Ker ϕ ′ ⊂ L ′ ⊗ Z ∞ C ∞ are mutually orthogonal with respect to D α, ∞ .Let ( n, r ), ( n ′ , r ′ ) be the dimension and rank of ( ϕ, L, V ) and ( ϕ ′ , L ′ , V ′ ) re-spectively. If they are ( α, I )-dual then r ′ = r , n ′ = r − n . There exists thefollowing reformulation of the definition of duality. D α, ∞ induces an isomorphism γ α,D : ( L ⊗ Z ∞ C ∞ ) ∗ → L ′ ⊗ Z ∞ C ∞ (here and below for any object W we denote W ∗ = Hom C ∞ ( W, C ∞ ) ). Property 2.4. ( ϕ, L, V ) and ( ϕ ′ , L ′ , V ′ ) are ( α, I )-dual iff there exists an iso-morphism from (Ker ϕ ) ∗ to V ′ making the following diagram commutative:0 → V ∗ ϕ ∗ → ( L ⊗ Z ∞ C ∞ ) ∗ → (Ker ϕ ) ∗ → ↓ γ α,D ↓ ↓ → Ker ϕ ′ → L ′ ⊗ Z ∞ C ∞ ϕ ′ → V ′ → . dim V ′ = r − n ; The composition map ϕ ′ ◦ γ D ◦ ϕ ∗ : V ∗ → V ′ is 0.Both 2.4 and (2.6, 2.7) are obvious. Remark 2.8.
It is easy to see that the functor ( ϕ, L, V ) ( ϕ ′ , L ′ , V ′ ) is well-defined on a subcategory (not all lattices have duals, see below) of the category ofthe triples ( ϕ, L, V ), it is contravariant and involutive.
3. Explicit formulas for analytic duality.
Here we consider the case A = F q [ T ]. In this case Cl( A ) = 0, and ( α, I )-dual iscalled simply dual. The coordinate description of the dual lattice is the following.Let e , ..., e r be a Z ∞ -basis of L such that ϕ ( e ) , ..., ϕ ( e n ) form a C ∞ -basis of V .Like in the theory of abelian varieties, we denote by Z = ( z ij ) the Siegel matrixwhose lines are coordinates of ϕ ( e n +1 ) , ..., ϕ ( e r ) in the basis ϕ ( e ) , ..., ϕ ( e n ), moreexactly, the size of Z is ( r − n ) × n and ∀ i = 1 , ..., r − n ϕ ( e n + i ) = n X j =1 z ij ϕ ( e j ) (3 . Z defines L , we denote L by L ( Z ). 11 roposition 3.2. [ L ( Z )] ′ = L ( − Z t ), i.e. a Siegel matrix of the dual lattice isthe minus transposed Siegel matrix. Proof.
Follows immediately from the definitions. Really, let f , ..., f r be a basisof L ′ , we define the pairing by the formula < e i , f j > = δ ji (3 . ϕ ′ by the formula ∀ i = 1 , ..., n ϕ ′ ( f i ) = r − n X j =1 − z ji ϕ ′ ( f n + j )(minus transposed Siegel matrix). Ker ϕ is generated by elements v i = e n + i − n X j =1 z ij e j , i = 1 , ..., r − n and Ker ϕ ′ is generated by elements w i = f i + r − n X j =1 z ji f n + j , i = 1 , ..., n (3 . ∀ i, j we have < v i , w j > = 0; this follows immediatelyfrom 3.3. (cid:3) Remark 3.5. L ′ exists not for all L . Trivial counterexample: case n = r = 1.To get another counterexamples, we use that for n = 1 (lattices of Drinfeld modules)a Siegel matrix is a column matrix Z = ( z ... z r − ) t and L ( Z ) is not a lattice ⇐⇒ , z , ..., z r − are linearly dependent over R ∞ (3 . n = r − Z = ( − z ... − z r − ) and L ( Z ) is not a lattice ⇐⇒ ∀ i z i ∈ R ∞ (3 . n = 1, r > n = r − r > Z are a Siegel matrice of a lattice”, i.e. if wechoose an (infinite) basis of C ∞ / R ∞ , then coordinates of the entries of Z in thisbasis must satisfy some polynomial relations in order that Z is not a Siegel matriceof a lattice. Remark 3.8.
The coordinate proof of the theorem that the notion of thedual lattice is well-defined, is the following. Two Siegel matrices Z , Z are calledequivalent iff there exists an isomorphism of their pairs ( L ( Z ) , V ), ( L ( Z ) , V ).Like in the classical theory of modular forms, Z , Z are equivalent iff there existsa matrix γ ∈ GL r ( Z ∞ ) = (cid:18) A BC D (cid:19) ( A, B, C, D are the ( n × n ), ( n × r − n ),12 r − n × n ), ( r − n × r − n )-blocks of γ respectively; we shall call this block structireby the ( n, r − n )-block structure) such that C + DZ = Z ( A + BZ ) (3 . . A , B , C , D be the ( n, r − n )-block structure of the matrix γ − . Theequality − C t + A t Z t = Z t ( D t − B t Z t ) (3 . . Z , Z are equivalent then − Z t , − Z t are equivalent. [Proof of (3.8.2):(3.8.1) implies Z = ( C + DZ )( A + BZ ) − ; substituting this value of Z to thetransposed (3.8.2), we get − C + ZA = ( D − ZB )( C + DZ )( A + BZ ) − , or( − C + ZA )( A + BZ ) = ( D − ZB )( C + DZ ). This formula follows immediatelyfrom (cid:18) A B C D (cid:19) (cid:18) A BC D (cid:19) = (cid:18) I n I r − n (cid:19) ].Further, let α : ( L ⊂ C n ∞ ) → ( L ⊂ C n ∞ ) be a map of lattices. If L ′ , L ′ exist, then the map α ′ : ( L ′ ⊂ C r − n ∞ ) → ( L ′ ⊂ C r − n ∞ ) is defined by the followingformulas. Let Z i be the Siegel matrices of L i in the bases e i , ...e ir of L i ( i = 1 , M = ( m ij ) ∈ M r ( Z ∞ ) of α in the bases e i , ..., e ir (i.e. α ( e i ) = P j m ij e j ). Let f i , ..., f ir be the dual base of L ′ i (see 3.3) and e ′ i , ...e ′ ir another base of L ′ i defined by e ′ ij = f i,j + n , j + n mod r (3 . . e ′ i , ..., e ′ ir , their Siegel matrices are − Z ti .Let M = (cid:18) M M M M (cid:19) be the ( n, r − n )-block structure of M . The matrix of α ′ in the bases f i , ..., f ir is M t , and using the matrix 3.8.3 of change of base, we get that M ′ — the matrix of α ′ in the bases e ′ i , ..., e ′ ir — has the following ( r − n, n )-block structure: M ′ = (cid:18) M t M t M t M t (cid:19) (3 . . M comes from a C ∞ -linear map C n ∞ → C n ∞ implies that M ′ comes from a C ∞ -linear map C r − n ∞ → C r − n ∞ . This follows immediately from thedefinition of dual lattice, or can be easily checked algebraically. Remark 3.9.
Taking γ = (cid:18) − (cid:19) we get that Z is equivalent to − Z , hence Z ′ is also a Siegel matrix of the dual lattice.
4. Main conjecture for arbitrary A .The main result of the paper is the following Theorem 5 on coincidence of alge-braic and analytic duality. We formulate it as a conjecture 4.1 for any A , but weprove it only for the case A = F q [ T ]. Let M be a uniformizable t-motive. Its lat-tice L ( M ) is really a lattice in the meaning of Definition 2.1, because [A], Corollary3.3.6 (resp. [G], Lemma 5.9.12) means that it satisfies 2.1a (resp. 2.1b); recall that13e consider the case N = 0, i.e. the action of T on Lie( M ) is simply multiplicationby θ . Let us fix (like in 1.13) C = ( C , τ C ) — a t-motive of rank 1 over A , and let L ( C ) be its lattice. It is a Z ∞ -module. Ω = Ω( A ) is an A -module, we consider a Z ∞ -module ι − (Ω). There exists the notion of the L ( C ) ⊗ ι − (Ω)-duality. Conjecture 4.1.
Let M be a uniformizable t-motive having N = 0 such thatits C -dual M ′ exists. Then M ′ is uniformizable, it has N ′ := N ( M ′ ) = 0, and( L ( M ) , Lie( M )) and ( L ( M ′ ) , Lie( M ′ )) are α, L ( C ) ⊗ ι − (Ω)-dual for some α from2 . ′ (it can be explicitly described).We give in Section 5 the first step of the proof of this conjecture.
5. Main theorem.
Recall that the word ”canonical” means ”canonical up to multiplication by ele-ments of F ∗ q ”. Theorem 5. Let M be a uniformizable t-motive over A = F q [ T ] having N = 0such that its dual M ′ exists and has N ′ := N ( M ′ ) = 0. Then M ′ is uniformizable,and ( L ( M ) , Lie( M )) and ( L ( M ′ ) , Lie( M ′ )) are dual. Remark 5A.
Condition N ′ = 0 holds for pure M (Theorem 10.3) and for alarge class of non-pure M (Theorem 11.5). Most likely, a modification of the endof the proof of the present theorem will permit us to prove that N ′ = 0 holds forall M having N = 0 and having dual. Remark 5B.
A reformulation of the theorem in terms of Hodge-Pink structuresis given in Section 9. Proof of the theorem for the case N = 0 is given in [GL18]. Corollary 5.1.1. If A = F q [ T ] then a Siegel matrix of M ′ is the minus trans-posed of a Siegel matrix of M .In the section 8 below we give a corollary of this theorem and some conjecturesrelated to the problem of 1 – 1 correspondence between t-motives and lattices. Recall that E = E ( M ) is isomorphic to C n ∞ . Thereis a structure of A -module on E ; multiplication by T is denoted by m T , and thisoperator m T is defined in coordinates by the formula m T ( x ) = l X i =0 A i x ( i ) where x ∈ E = C n ∞ is a vector column, A i are from 1.9.1. There is a map exp :Lie( M ) → E making the following diagram commutative:Lie( M ) Exp → Eθ ↓ m T ↓ Lie( M ) Exp → E (5 . . L ( M ) = Ker Exp. The proof of this theorem was inspired by a result of Anderson, see Section 6 for details.
14e need another space Lie T ( M ) together with an isomorphism a : Lie T ( M ) → Lie( M ) and a structure of A -module on Lie T ( M ) such that the multiplication by T on Lie T ( M ) is simply the multiplication by θ on Lie( M ), i.e. a ( T x ) = θ · ( a ( x )) (5 . . x ∈ Lie T ( M ). Commutativity of 5.1.3 means that Exp ◦ a : Lie T ( M ) → E isa map of A -modules. We shall work merely with L T ( M ) := Ker (Exp ◦ a ) ⊂ Lie T ( M ) ratherthan L ( M ). Clearly L T ( M ) is an A -module, a : L T ( M ) → L ( M ) is an isomorphismsatisfying 5.1.4 for x ∈ L T ( M ).The proof of Theorem 5 consists of two steps. We formulate and prove Step 1for the case of arbitrary A . Step 1.
For the above M , M ′ we have:(A) Uniformizability of M implies uniformizability of M ′ .(B) There exists a canonical A -linear L T ( C ) ⊗ Ω-valued perfect pairing < ∗ , ∗ > M between L T ( M ) and L T ( M ′ ) (by 5.1.5, this is the same as the Z ∞ -linear pairingbetween L ( M ) and L ( M ′ ), which, in its turn, is D of Definition 2.3). It is functorial. Remark 5.1.6.
Practically, (B) comes from [T], Theorem 4.3 (case A = F q [ T ]).Really, to define a pairing between L ( M ) and L ( M ′ ) it is sufficient to define (con-cordant) pairings between L ( M ) /aL ( M ) and L ( M ′ ) /aL ( M ′ ) for any a ∈ A . Since M a := E ( M ) a = L ( M ) /aL ( M ) and because of Proposition 1.12.3 which affirmsthat M a and M ′ a are Taguchi-dual, we see that [T], Theorem 4.3 gives exactly thedesired pairing.We give two versions of the proof of Step 1: the first one — for the general caseof arbitrary A and the second one — for the case A = F q [ T ] — is based on explicitcalculations, it is used for the proof of Step 2. Here we consider the general case of arbitrary A . Let Ω = Ω( A / F q ) be the module of differential forms; we can consider it asan element of Cl( A ). We use formulas and notations of [G], Section 5.9 modifyingthem to the case of arbitrary A . For example, A (resp. K ) of [G], 5.9.16 is A (resp. K ) of the present paper (recall that ¯ K (resp. ¯ K [ T, τ ]) of [G] is C ∞ (resp. A C [ τ ], see 1.13) of the present paper). Hence, we denote ¯ K { T } of [G], Definition5.9.10 by C ∞ { T } . For the general case it must be replaced by a ring Z defined bythe formula Z := A ⊗ F q [ T ] C ∞ { T } (5 . . Z is a A C [ τ ]-module, i.e. τ acts on Z , and Z τ = A . Z for the present case is defined by the same formula [G], 5.9.22. Explicitly, Z := Hom cont A ( K / A , C ∞ ) (5 . . a )It is a locally free Z -module of dimension 1 (the module structure is compatiblewith the action of τ ; see [G], p. 168, lines 3 - 4 for the case A = F q [ T ]). We have: Z τ is a Z τ -module ( = A -module) which is isomorphic to Ω( A ) (see the last lines15f the proof of [G], Corollary 5.9.35 for the case A = F q [ T ]), and Z is isomorphicto Z ⊗ A Ω( A ).We shall consider M as a A C [ τ ]-module, like in 1.13.3. We denote M { T } := M ⊗ A C Z ( = [G], Definition 5.9.11.1 for the case A = F q [ T ]) and H ( M ) := M { T } τ like in [G], Definition 5.9.11.2. Analogous to [G], Corollary 5.9.25 we get that forthe present case H ( M ) := Hom A C [ τ ] ( M, Z ) = L T ( M )( H ( M ) = H ( E ) of [G], 5.9). Particularly, for M = C we have L T ( C ) = Hom A C [ τ ] ( C , Z ) Lemma 5.2.2. H ( M ′ ) = H ( M ) ⊗ A L T ( C ). Proof.
By definition, Hom A C ( M ′ , Z ) = Hom A C (Hom A C ( M, C ) , Z ). Fur-ther, Hom A C (Hom A C ( M, C ) , Z ) = ( M ⊗ A C Z ) ⊗ Z (Hom A C ( C , Z )) (5 . . τ -invariantsubspaces, we need the following objects. Let I be an ideal of A , M = IZ . Itis clear that M τ = I . Further, let M be a locally free Z -module. We have aformula: ( M ⊗ Z M ) τ = M τ ⊗ A M τ (5 . . M ⊗ Z M = I M , and ( I M ) τ = I M τ (5 . . I M ) τ ⊃ I M τ . Let J be an ideal of A such that IJ is a principal ideal. We have ( IJ ( J − M )) τ = IJ ( J − M ) τ and ( IJ ( J − M )) τ ⊃ I ( J ( J − M )) τ ⊃ IJ ( J − M ) τ , hence allthese objects are equal and we get 5.2.5 and hence 5.2.4.The action of τ on both sides of 5.2.3 coincide. Considering τ -invariant elementsof both sides of 5.2.3 and taking into consideration 5.2.4 ( M = Hom A C ( C , Z )and M = M ⊗ A C Z ) we get the lemma. (cid:3) This lemma proves (A) of Step 1.
Lemma 5.2.6.
Let M i ( i = 0 ,
1) be two locally free Z -modules with τ -actionsatisfying τ ( cm ) = τ ( c ) τ ( m ) ( c ∈ Z , m ∈ M i ), and ψ : M ⊗ Z M → Z a perfectpairing of Z -modules with τ -action. Let, further, both M i satisfy M τi ⊗ A Z = M i . Then the restriction of ψ to M τ ⊗ A M τ → Ω is a perfect pairing as well.
Proof.
Let α : M τ → Ω be an A -linear map. We prolonge it to a map¯ α : M → Z by Z - τ -linearity. By perfectness of ψ , there exists m ∈ M suchthat ¯ α ( m ) = ψ ( m ⊗ m ). It is easy to see that m is τ -invariant (we use the factthat τ : Z → Z is surjective). (cid:3) emma 5.2.7. There is a natural perfect A -linear Ω-valued pairing between H ( M ) and H ( M ): H ( M ) ⊗ A H ( M ) → Ω. Proof.
For the case A = F q [ T ] this is [G], Corollary 5.9.35. General case: wehave a perfect Z -pairingHom A C ( M, Z ) ⊗ Z ( M ⊗ A C Z ) → Z Now we take M = Hom A C ( M, Z ), M = M ⊗ A C Z and we apply Lemma 5.2.6. (cid:3) Step 1 of the theorem follows from these lemmas.
Remark 5.2.8.
The pairing can be defined also as the composition of H ( M ) ⊗ A H ( M ′ ) = Hom A C [ τ ] ( M, Z ) ⊗ A Hom A C [ τ ] ( M ′ , Z ) → Hom A C [ τ ] ( M ⊗ A C M ′ , Z ⊗ Z Z ) → Hom A C [ τ ] ( C , Z ⊗ Z Z ) = L T ( C ) ⊗ A Ω(5 . . δ : C → M ⊗ A C M ′ of Remark1.11.1 (more exactly, of its analog for arbitrary A ). Remark 5.2.10.
Recall that the explicit formula for functoriality is the follow-ing. Let α : M → M be a map of t-motives, α ′ : M ′ → M ′ the dual map and L T ( α ) : L T ( M ) → L T ( M ), L T ( α ′ ) : L T ( M ′ ) → L T ( M ′ ) the corresponding mapson lattices. For any l ′ ∈ L T ( M ′ ), l ∈ L T ( M ) we have: < L T ( α )( l ) , l ′ > M = < l , L T ( α ′ )( l ′ ) > M (5 . . Case A = F q [ T ]. We identify Z of [G], p.168,lines 3 – 4 with C ∞ { T } (see [G], Definition 5.9.10) and A with Ω. Like above, wehave an isomorphism of A -modules (recall that A is the center of C ∞ [ T, τ ]): L T ( M ) = Hom C ∞ [ T,τ ] ( M, Z ) (5 . . ϕ : M → Z , ϕ ′ : M ′ → Z be elements of L T ( M ), L T ( M ′ ) respectively, and let f ∗ , f ′∗ , Q , Q ′ be from 1.9.3, 1.10. We denote ϕ ( f ∗ ) = v ∗ (5 . . v ∗ ∈ ( Z ) r is a vector column (it is a column of the scattering matrix ([A],p. 486) of M , see 5.4.1 below). The same notation for the dual: ϕ ′ ( f ′∗ ) = v ′∗ .Condition that ϕ , ϕ ′ are τ -homomorphisms is equivalent to Qv ∗ = v (1) ∗ , Q ′ v ′∗ = v ′ (1) ∗ (5 . . P ∞ i =0 a i T i ∈ C ∞ { T } ⊂ C ∞ [[ T ]] of [G], p. 172, line 1; recall that it is the onlyelement (up to multiplication by F ∗ q ) satisfyingΞ = ( T − θ )Ξ (1) , lim i →∞ a i = 0 , | a | > | a i | ∀ i > . . a Q i ≥ (1 − T /θ q i ) where a satisfies a q − = − /θ ). Finally, we define < ϕ, ϕ ′ > = Ξ v t ∗ v ′∗ (5 . . < ϕ, ϕ ′ > does not depend on a choice of a basis f ∗ . Lemma 5.3.6. < ϕ, ϕ ′ > ∈ A . Proof.
Firstly, this element belongs to F q [[ T ]], becauseΞ v t ∗ v ′∗ − (Ξ v t ∗ v ′∗ ) (1) = Ξ( v t ∗ v ′∗ − ( T − θ ) − v (1) t ∗ v ′ (1) ∗ ) = Ξ v t ∗ ( I r − ( T − θ ) − Q t Q ′ ) v ′∗ because of (5.3.3). But we have (see (1.10.1) — the definition of Q ′ ) I r − ( T − θ ) − Q t Q ′ = 0Secondly, let < ϕ, ϕ ′ > = P ∞ i =0 c i T i . Since coefficients of all factors of (5.3.5): Ξ, v ∗ and v ′∗ — tend to 0, we get that c i also tend to 0. But c i ∈ F q , i.e. they arealmost all 0. (cid:3) Lemma 5.3.7.
The above pairing is perfect.
Proof.
We have an isomorphism (here M { T } = M ⊗ C ∞ [ T ] C ∞ { T } with thenatural action of τ (see [G], Definition 5.9.11)) α : Hom C ∞ [ T,τ ] ( M, Z ) → Hom A ( M { T } τ , A ) (5 . . C ∞ [ T,τ ] ( M, Z ) = Hom C ∞ [ T ] ( M, Z ) τ β ′ → Hom C ∞ { T } ( M { T } , C ∞ { T } ) τγ → Hom A ( M { T } τ , A )where β : Hom C ∞ [ T ] ( M, Z ) → Hom C ∞ { T } ( M { T } , C ∞ { T } ) is the natural map and β ′ is the restriction of β to τ -invariant elements. Using the Anderson’s criterion ofuniformizability of M (see, for example, [G], 5.9.14.3 and 5.9.13) we get immediatelythat both γ , β , and hence β ′ , and hence α are isomorphisms. Further, let us considera homomorphism i : Hom C ∞ [ T,τ ] ( M ′ , Z ) → M { T } τ (5 . . ϕ ′ , f ′∗ , v ′∗ be as above. We set i ( ϕ ′ ) = Ξ v ′ t ∗ f ∗ ∈ M ⊗ C ∞ [ T ] C ∞ [[ T ]]Since Ξ ∈ C ∞ { T } , we get that Ξ v ′ t ∗ f ∗ ∈ M { T } . A simple calculation (like in theLemma 5.3.6, but simpler) shows that i ( ϕ ′ ) is τ -invariant, hence i really definesa map from Hom C ∞ [ T,τ ] ( M ′ , Z ) to M { T } τ . Obviously it is an inclusion. Letus prove that i is surjective. Really, let c ∗ ∈ ( Z ) r be a column vector such that c t ∗ f ∗ ∈ M { T } τ . An analog of the above calculation shows that if we define ϕ ′ by theformula ϕ ′ ( f ′∗ ) = Ξ − c ∗ then ϕ ′ ∈ Hom C ∞ [ T,τ ] ( M ′ , Z ), and i ( ϕ ′ ) = c t ∗ f ∗ ∈ M { T } τ .18inally, the combination of isomorphisms (5.3.8) and (5.3.9) corresponds to thepairing (5.3.5). (cid:3) It is easy to see that theconverse of the Corollary 5.1.1 (taking into consideration Proposition 3.2) is alsotrue, i.e. in order to prove Theorem 5 it is sufficient to prove that a Siegel matrixof M ′ is − Z t where Z is a Siegel matrix of M . Let us consider a basis l , ..., l r of L T ( M ) and for each l i we consider the corresponding (under identification 5.3.1) ϕ i ∈ Hom C ∞ [ T,τ ] ( M, Z ). Let Ψ be the scattering matrix of M ([A], p. 486) withrespect to the bases l , ..., l r , f , ..., f r , and we denote ϕ i ( f ∗ ) by v i ∗ (notations of5.3.2). Lemma 5.4.1. v i ∗ is the i -th column of Ψ ( Z is identified with C ∞ { T } , seethe proof). Proof.
Follows from the definitions. Recall that K = F q ((1 /T )). The isomor-phism 5.3.1 is the composition of 2 isomorphisms i : L T ( M ) → Hom c A ( K / A , E )([G], 5.9.19) and i : Hom c A ( K / A , E ) → Hom C ∞ [ T,τ ] ( M, Hom c ( K / A , C ∞ ) ([G],5.9.24; recall that Z = Hom c ( K / A , C ∞ )). For l i ∈ L T ( M ) we have ( i ( l i ))( T − k ) =exp( θ − k l i ) ([G], line above the lemma 5.9.18) and(( i ◦ i ( l i ))( f j ))( T − k ) = < f j , exp( θ − k l i ) > ([G], two lines above the lemma 5.9.24). Using the identification of Z and C ∞ { T } ([G], p. 168, lines 3 - 4) and the definition of Ψ ([A], p. 486, first formula of 3.2)we get immediately the lemma. (cid:3) Let l ′ , ..., l ′ r be a basis of L T ( M ′ ) which is dual to a basis l , ..., l r of L T ( M ) withrespect to the pairing 5.3.5. Lemma 5.4.2.
The scattering matrix of M ′ with respect to the bases l ′ , ..., l ′ r , f ′ , ..., f ′ r (denoted by Ψ ′ ) is Ξ − Ψ t − . Proof.
Follows immediately from 5.4.1 applied to both M , M ′ , and formula5.3.5. (cid:3) Remark 5.4.3.
An alternative proof for the case of pure M (for some basisof L T ( M ′ )) is the following. We denote Ξ − Ψ t − by Ψ . It satisfies Ψ (1)1 = ( T − θ ) Q t − Ψ and other conditions of [A], 3.1. According [A], Theorem 5, p. 488, thereexists a pure uniformizable t-motive M with σ -structure such that its scatteringmatrix is Ψ . Since Ψ satisfies Ψ (1)1 = Q ′ Ψ we get that Q ( M ) = Q ′ , i.e. M = M ′ . (cid:3) Let us recall the statement of the crucial proposition 3.3.2 of [A]. Here we considerthe case of those M whose N is not necessarily 0. Let Ψ be a scattering matrix of M . We consider the ( T − θ )-Laurent series for Ψ (here k ( M ) < D − i ∈ M r ( C ∞ )): Ψ = ∞ X i = k ( M ) D − i ( T − θ ) i
19e consider its negative partΨ − := − X i = k ( M ) D − i ( T − θ ) i as an element of M r ( C ∞ )(( T − θ )) /M r ( C ∞ )[[ T − θ ]].We consider the space ( T − θ ) k ( M ) C ∞ [[ T − θ ]] / C ∞ [[ T − θ ]] as a C ∞ -vector spaceendowed by the action of A , and we denote by V its r -th direct sum written asvector columns of length r . Obviously k ( M ) = − ⇐⇒ the action of T on V coincides with multiplication by θ (5 . . a )We denote the i -th column of Ψ − by Ψ − i ∗ , it belongs to V . Following [A], we denoteby Prin( M ) (resp. by Prin ( M )) the C ∞ [ T ]-linear span (resp. the A -linear span)of all Ψ − i ∗ in V . Finally, we obviously extend the definition of Lie T ( M ), L T ( M ) tothe case N = 0; formula 5.1.4 becomes a ( T x ) = ( θ + N )( a ( x )) (5 . . b ) Proposition 3.3.2, [A] (see also Remark 5.5 below). There exists a C ∞ [ T ]-linear isomorphism ψ E : Lie T ( M ) → Prin( M ) such that its restriction to L T ( M ) ⊂ Lie T ( M ) defines an isomorphism L T ( M ) → Prin ( M ) (denoted by ψ E as well). (cid:3) Corollary 5.4.4. N = 0 ⇐⇒ k ( M ) = − N = 0 ⇐⇒ the action of T on both Lie T ( M ), V coincides with multiplication by θ , by 5.4.3a). (cid:3) We return to the case N = 0.Let us consider the ( T − θ )-Laurent series for Ψ ′ and Ξ − :Ψ ′ = ∞ X i = k ( M ′ ) D ′− i ( T − θ ) i , Ξ − = ∞ X i = k ( ξ ) a i ( T − θ ) i Since for both M , M ′ we have N = N ′ = 0, we get k ( M ) = k ( M ′ ) = −
1. Anelementary calculation shows that k ( ξ ) is also −
1. Hence, equality Ψ ′ Ψ t = Ξ − (Lemma 5.4.2) implies that D ′ D t = 0.Further, there exist n columns of D which are C ∞ -linerly independent (theyare ψ E -images of elements of L T ( M ) which form a C ∞ -basis of Lie T ( M )) and allother columns of D are their linear combinations. Interchanging columns of D ifnecessary we can assume that these columns are the last n columns. We denote by D (resp. D ) the r × n (resp. r × ( r − n ) ) matrix formed by the last n (resp.the first r − n ) columns of D . There exists a matrix S such that D = D S t .Again according Proposition 3.3.2, [A], we have: S is a Siegel matrix of L ( M ) (5 . . D ′ . We denote by D ′ (resp. D ′ ) the r × n -(resp. r × ( r − n ))-matrix formed by the last n (resp. the first r − n ) columns of20 ′ . Since D ′ D t = D ′ D t + D ′ D t we get that D ′ D t + D ′ SD t = 0. Since D t is a n × r -matrix of rank n , it is not a zero-divisor from the right, so D ′ = − D ′ S (5 . . D ′ is r − n and D ′ is a r × ( r − n ) matrix, (5.4.6) implies thatcolumns of D ′ are linearly independent, and by (5.4.6) and Proposition 3.3.2, [A]we get that − S is a Siegel matrix of M ′ . (cid:3) Remark 5.5.
Since the notations of [A] differ from the ones of the presentpaper, for the reader’s convenience we give here a sketch of the proof for the case N = 0 of two crucial facts: Corollary 5.4.4 and 5.4.5 ([A], Theorem 3.3.2).Let α : Lie( M ) → E ( M ) be a linear isomorphism which is the first term of theseries for exp : Lie( M ) → E ( M ), and let l ∈ Lie( M ), f ∈ M be arbitrary. Weconsider the ( T − θ )-Laurent series P ∞ i = k b i ( T − θ ) i of P ∞ j =0 < exp( θ j +1 l ) , f > T j . Lemma 5.6. If N = 0 then k = −
1, and b − = − < α ( l ) , f > (this is [A],3.3.4). Sketch of the proof.
For z ∈ Lie( M ) we denote exp( z ) − α ( z ) by ε ( z ), hence P ∞ j =0 < exp( θ j +1 l ) , f > T j = A + E , where A = ∞ X j =0 < α ( 1 θ j +1 l ) , f > T j ; E = ∞ X j =0 < ε ( 1 θ j +1 l ) , f > T j We consider their ( T − θ )-Laurent series: A = ∞ X i = k ( A ) a i ( T − θ ) i ; E = ∞ X i = k ( E ) e i ( T − θ ) i Since we have exp( z ) = P ∞ i =0 C i z ( i ) where C = I n we get that ε ( z ) = P ∞ i =1 C i z ( i ) .This means that for large j the element ε ( θ j +1 l ) is small, and hence k ( E ) = 0,because finitely many terms having small j do not contribute to the pole of the( T − θ )-Laurent series of E (the reader can prove easily the exact estimations himself,or to look [A], p. 491). Since α is C ∞ -linear, equality P ∞ j =0 1 θ j +1 T j = − ( T − θ ) − implies that k ( A ) = − a − = − < α ( l ) , f > (and other a i = 0), hence thelemma. (cid:3) This lemma obviously implies Corollary 5.4.4. Further, elements f , ..., f r gen-erate the C ∞ -space M/τ M , because multiplication by T on M/τ M coincides withmultiplication by θ , hence the fact that f , ..., f r C ∞ [ T ]-generate M/τ M impliesthat they C ∞ -generate M/τ M .Let l , ..., l n form a C ∞ -basis of Lie( M ) (here we identify Lie T ( M ) and Lie( M )via a ). Since the pairing < ∗ , ∗ > between E ( M ) and M/τ M is non-degenerateand α is an isomorphism, we get that columns < α ( l ) , f ∗ >, ..., < α ( l n ) , f ∗ > arelinearly independent. Again since α is an isomorphism and the pairing with f ∗ islinear, we get that( < α ( l n +1 ) , f ∗ > ... < α ( l r ) , f ∗ > ) = ( < α ( l ) , f ∗ > ... < α ( l n ) , f ∗ > ) Z t
6. Tensor products.
There exists an analog of the Theorem 5 for the case of tensor products of t-motives. It describes the lattice L ( M ⊗ M ) in terms of L ( M ), L ( M ). This is atheorem of Anderson; it is formulated in [P], end of page 3, but its proof was notpublished. We recall its statement for the case of arbitrary N = 0, and we giveits proof for the case N = 0 (case of arbitrary N can be obtained easily using thesame ideas).Let M be an uniformizable t-motive whose N is not necessarily 0. Since N isnilpotent, formula 5.4.3b shows that Lie T ( M ) is a C ∞ [[ T − θ ]]-module. There existsan epimorphism of C ∞ [[ T − θ ]]-modules L T ( M ) ⊗ A C ∞ [[ T − θ ]] → Lie T ( M )whose kernel q = q ( M ) carries information on the pair ( L ( M ) , Lie( M )). Theorem 6 (Anderson). Let M , ¯ M be any two uniformizable abelian t-motives.Then q ( M ⊗ ¯ M ) = q ( M ) ⊗ C ∞ [[ T − θ ]] q ( ¯ M ) (6 . Remark 6A. M ⊗ ¯ M is a uniformizable t-motive ([G], Corollary 5.9.38). Proof of Theorem 6 (case N = 0 ). We define notations for M , and allnotations for ¯ M will carry bar. Let e i and Z be from the beginning of Section 3.We denote a − ( e i ) ∈ Lie T ( M ) by e i (there is no possibility of confusion). So, { e i } is a C ∞ [[ T − θ ]]-basis of L T ( M ) ⊗ A C ∞ [[ T − θ ]]. Elements b i := ( T − θ ) e i , i = 1 , ..., n and b n + i := e n + i − P nj =1 z ij e j , i = 1 , ..., r − n form a C ∞ [[ T − θ ]]-basis of q . Weneed a Lemma 6.2. Ψ( M ⊗ ¯ M ) = Ψ( M ) ⊗ Ψ( ¯ M ) where Ψ( M ) (resp. Ψ( ¯ M ); Ψ( M ⊗ ¯ M ))is taken with respect to bases e ∗ of L T ( M ), f ∗ of M C ∞ [ T ] (resp. ¯ e ∗ of L T ( ¯ M ), ¯ f ∗ of ¯ M C ∞ [ T ] ; e ∗ ⊗ ¯ e ∗ of L T ( M ⊗ ¯ M ), f ∗ ⊗ ¯ f ∗ of ( M ⊗ ¯ M ) C ∞ [ T ] ) (see the proof for thenotations). Proof.
We consider a map α : Hom C ∞ [ T ] ( M, Z ) τ ⊗ A Hom C ∞ [ T ] ( ¯ M , Z ) τ → Hom C ∞ [ T ] ( M ⊗ ¯ M , Z ) τ defined as follows: for ϕ ∈ Hom C ∞ [ T ] ( M, Z ) τ , ¯ ϕ ∈ Hom C ∞ [ T ] ( ¯ M , Z ) τ we let[ α ( ϕ ⊗ ¯ ϕ )]( f ⊗ ¯ f ) = ϕ ( f ) · ¯ ϕ ( ¯ f ) (it is obvious that α ( ϕ ⊗ ¯ ϕ ) is τ -stable). Since e , ..., e r (resp. ¯ e , ..., ¯ e ¯ r ) is a basis of Hom C ∞ [ T ] ( M, Z ) τ (resp. Hom C ∞ [ T ] ( ¯ M , Z ) τ ; weidentify L T ( M ), resp. L T ( ¯ M ) with Hom C ∞ [ T ] ( M, Z ) τ (resp. Hom C ∞ [ T ] ( ¯ M , Z ) τ )we get (using Lemma 5.4.1) that Ψ( M ), Ψ( ¯ M ) are non-degenerate. Since theirproduct is also non-degenerate, we get α ( e i ⊗ ¯ e ¯ i ) are linearly independent andhence a basis of Hom C ∞ [ T ] ( M ⊗ ¯ M , Z ) τ . Applying once again Lemma 5.4.1 we getthe lemma. (cid:3) If A , B are two matrices then columns of A ⊗ B are indexed by pairs ( k, l ) where k (resp. l ) is the number of a column of A (resp. B ). We denote by A k , B l ,22 ⊗ B ( k,l ) the respective columns. Obviosly we have: A ⊗ B ( k,l ) = A k ⊗ B l (tensorproduct of column matrices).Let us prove that for i = 1 , ..., r − n , ¯ i = 1 , ..., ¯ r − ¯ n the element b n + i ⊗ ¯ b ¯ n +¯ i ∈ q ( M ⊗ ¯ M ). According [A], Proposition 3.3.2, it is sufficient to prove that thecorresponding linear combination (see 6.3 below) of the columns of the matrixΨ − M ⊗ ¯ M is 0. Since b n + i ⊗ ¯ b ¯ n +¯ i = X j, ¯ j z ij ¯ z ¯ i ¯ j e j ⊗ ¯ e ¯ j − X j z ij e j ⊗ ¯ e ¯ n +¯ i − X ¯ j ¯ z ¯ i ¯ j e n + i ⊗ ¯ e ¯ j + e n + i ⊗ ¯ e ¯ n +¯ i we get the explicit form of this linear combination: it is sufficient to prove that forall i , ¯ i we have X j, ¯ j z ij ¯ z ¯ i ¯ j (Ψ − M ⊗ ¯ M ) ( j, ¯ j ) − X j z ij (Ψ − M ⊗ ¯ M ) ( j, ¯ n +¯ i ) − X ¯ j ¯ z ¯ i ¯ j (Ψ − M ⊗ ¯ M ) ( n + i, ¯ j ) + (Ψ − M ⊗ ¯ M ) ( n + i, ¯ n +¯ i ) = 0 (6 . − M ⊗ ¯ M ) ( k, ¯ k ) = A − ,k ⊗ ¯ A − , ¯ k ( T − θ ) + A − ,k ⊗ ¯ A , ¯ k + A ,k ⊗ ¯ A − , ¯ k T − θ hence 6.3 becomes X j, ¯ j z ij ¯ z ¯ i ¯ j ( A − ,j ⊗ ¯ A − , ¯ j ( T − θ ) + A − ,j ⊗ ¯ A , ¯ j + A ,j ⊗ ¯ A − , ¯ j T − θ ) − X j z ij ( A − ,j ⊗ ¯ A − , ¯ n +¯ i ( T − θ ) + A − ,j ⊗ ¯ A , ¯ n +¯ i + A ,j ⊗ ¯ A − , ¯ n +¯ i T − θ ) − X ¯ j ¯ z ¯ i ¯ j ( A − ,n + i ⊗ ¯ A − , ¯ j ( T − θ ) + A − ,n + i ⊗ ¯ A , ¯ j + A ,n + i ⊗ ¯ A − , ¯ j T − θ )+ A − ,n + i ⊗ ¯ A − , ¯ n +¯ i ( T − θ ) + A − ,n + i ⊗ ¯ A , ¯ n +¯ i + A ,n + i ⊗ ¯ A − , ¯ n +¯ i T − θ = 0 (6 . A − ,n + i = X j z ij A − ,j (6 . A − , ¯ n +¯ i = X ¯ j ¯ z ¯ i ¯ j ¯ A − , ¯ j For example, the left hand side of (6.4) has 2 terms containing ¯ A , ¯ j (in the middleof the first and the third lines of (6.4)). Multiplying (6.5) by ¯ z ¯ i ¯ j ¯ A , ¯ j we get that23he sum of these 2 terms of (6.4) is 0. For other pairs of terms of (6.4) the situationis the same.The proof that for i = 1 , ..., r − n , ¯ i = 1 , ..., ¯ n the element b n + i ⊗ ¯ b ¯ i ∈ q ( M ⊗ ¯ M )is analogous but simpler. We have b n + i ⊗ ¯ b ¯ i = ( T − θ )( − X j z ij e j ⊗ ¯ e ¯ i + e n + i ⊗ ¯ e ¯ i )The analog of (6.3)) is( T − θ )( − X j z ij (Ψ − M ⊗ ¯ M ) ( j, ¯ i ) + (Ψ − M ⊗ ¯ M ) ( n + i, ¯ i ) ) = 0and the analog of (6.4)) is − X j z ij A − ,j ⊗ ¯ A − , ¯ i T − θ + A − ,n + i ⊗ ¯ A − , ¯ i T − θ = 0This equality follows immediately from (6.5).Finally, elements b i ⊗ ¯ b ¯ i ( i = 1 , ..., n , ¯ i = 1 , ..., ¯ n ) obviously belong to q ( M ⊗ ¯ M ).So, we proved that q ( M ) ⊗ C ∞ [[ T − θ ]] q ( ¯ M ) ⊂ q ( M ⊗ ¯ M ). Since the C ∞ -codimensionof both subspaces in L T ( M ) ⊗ A L T ( ¯ M ) ⊗ A C ∞ [[ T − θ ]] is n ¯ n , they are equal. (cid:3)
7. Self-dual t-motives.Case A = F q [ T ] . A uniformizable t-motive M is called self-dual if there existsan isogeny α : M → M ′ . It defines an A -valued, A -bilinear form < ∗ , ∗ > α on L T ( M ′ ) as follows: < ϕ , ϕ > α = < L T ( α )( ϕ ) , ϕ > M α ′ = − α (resp. α ′ = α ) then < ∗ , ∗ > α is skew symmetric(resp. symmetric). M is called positively (resp. negatively) self-dual if α satisfies α ′ = α (resp. α ′ = − α ). Hence, we have an Analogy 7a.
The number field case analog of a pair: { negatively self-dualt-motive of rank 2 n , dimension n ; negative α : M → M ′ } is a (generic) abelianvariety of dimension n with a fixed polarization form.For example, like in the number field case, we can define the Rosati involution I α on End ( M ) := End( M ) ⊗ F q ( T ) by the same formula I α ( ϕ ) = α − ◦ ϕ ′ ◦ α .Further, we have a Conjecture 7b.
The dimension of the moduly variety of negatively self-dualt-motives (if it exists) is n ( n + 1) / Examples.
Let e ∗ be from 1.9, and let M = M ( A ) given by the equation (here A ∈ M n ( C ∞ ) is A of 1.9.1) T e ∗ = θe ∗ + Aτ e ∗ + τ e ∗ (7 . n and rank 2 n . Elements f i = e i , f n + i = τ e i ( i = 1 , ..., n )form a C ∞ [ T ]-basis of M . We have (see, for example, Section 11): M ′ is given bythe equation T e ′∗ = θe ′∗ − A t τ e ′∗ + τ e ′∗ and if we define f ′ i = τ e ′ i , f ′ n + i = e ′ i (7 . f ∗ , f ′∗ are dual in the meaning of Lemma 1.10.Let α : M → M ′ be given by the formula α ( e ∗ ) = De ′∗ where D ∈ M n ( C ∞ ) (weimpose this essential restriction only in order to simplify exposition. In the generalcase D ∈ M n ( C ∞ [ τ ]), D f ∈ M n ( C ∞ [ T ]), D f from 7.4). Condition that α is a C ∞ [ T, τ ]-map is equivalent to D (2) = D, AD (1) = − DA t (7 . α ( f ∗ ) = D f f ′∗ (7 . D f = (cid:18) DD (1) (cid:19) , hence α ′ = ± α ⇐⇒ D tf = ± D f ⇐⇒ D (1) = ± D t (7 . ε ∈ F q satisfying ε q − = −
1. Then D = ε I n satisfies 7.5 with the signminus, and the set of A satisfying 7.3 with this D is the set of symmetric matrices.This justifies 7b, because the set of A ∈ M n ( C ∞ ) such that M ( A ) = M ( A ) isconjecturally discrete.For D = I n the sign in 7.5 is plus and hence a skew symmetric A defines apositively self-dual M ( A ). Remark 7.6.
The below statements are conjectures based on arguments similarto the ones which justify the below Conjecture 11.8.3. Since they are of secondaryimportance, we do not give any details of justification here. If n ≥ A we have:End( M ( A )) = A . Conjecture 7.6.1 implies that the ”minimal” α : M → M ′ isdefined uniquely up to an element of F ∗ q , and hence the symmetric pairing < ∗ , ∗ > α is also defined uniquely up to an element of F ∗ q . If n = 2, α ′ = α then End( M ) is strictly larger than A .Other examples of a self-dual t-motive are M ⊕ M ′ where M is any t-motive,but they do not give interesting examples of pairings. There exist other (distinct from the ones defined by 7.1)self-dual t-motives M having End( M ) = A (we can use a version of standardt-motives of Section 11). Example 7.7.
Case A = 0, D = I n . 25n this case we can find explicitly the matrix of the symmetric form < ∗ , ∗ > α insome basis of L T ( M ′ ). Let C be the Carlitz module over the field F q consideredas a rank 2 Drinfeld module over F q given by the equation T e = θe + τ e We have M = C ⊕ n . Let T T ( C ) be the convergent T -Tate module of C , i.e. theset of elements { z i } ∈ E ( C ) = C ∞ ( i ≥ − , z − = 0) such that T z i = z i − for i ≥ z q i + θz i = z i − ) and z i → F q [ T ]. We choose and fix its generator; its { z i } satisfy (like in 5.3.4) | z | > | z i | ∀ i >
0. We denote P ∞ k =0 z k T k by Z .Let c be a fixed element of F q − F q . Formulas (5.3.3) show that the followingelements ϕ i , ϕ ′ i ( i = 1 , ..., n ) form bases of L ( M ), L ( M ′ ) respectively ( j = 1 , ..., n ;clearly that thanks to 7.2 we have ϕ ′ i ( f ′ j ) = ϕ i ( f n + j ), n + j mod 2 n ): i ≤ n : ϕ i ( f j ) = Z δ ji , ϕ i ( f n + j ) = Z (1) δ ji i > n : ϕ i ( f j ) = c Z δ ji − n , ϕ i ( f n + j ) = c q Z (1) δ ji − n i ≤ n : ϕ ′ i ( f ′ j ) = Z (1) δ ji , ϕ ′ i ( f ′ n + j ) = Z δ ji i > n : ϕ ′ i ( f ′ j ) = c q Z (1) δ ji − n , ϕ ′ i ( f ′ n + j ) = c Z δ ji − n (by the way, it is clear that the same relation between elements of T T ( M ) andHom C ∞ [ T,τ ] ( M, Z ) holds for all M ). Formula 7.4 shows that α ′ ( ϕ ′ i ) = ϕ i + n , where i + n mod 2 n . Let us denote Ξ · Z · Z (1) ∈ F ∗ q by γ . The above definitions andformulas show that the matrix of < ∗ , ∗ > α in the basis ϕ , ϕ n +1 , ..., ϕ n , ϕ n consistsof n (2 × F q / F q ) γ (cid:18) tr (1) tr ( c )tr ( c ) tr ( N ( c )) (cid:19) = γ (cid:18) c + c q c + c q c q +1 (cid:19) The determinant of this block is − ( c − c q ) γ ; it belongs to F ∗ q ⇐⇒ q ≡ q is even. Since we have n blocks, we have:det < ∗ , ∗ > α F ∗ q ⇐⇒ q ≡ n is odd. Remark 7.8 (Jorge Morales).
There is a theorem of Harder (see e.g. W.Scharlau, ”Quadratic and Hermitian forms”, Springer-Verlag, Berlin, 1985, Chapter6, Theorem 3.3) that states that a unimodular form over k [ X ] — k being anyfield of characteristic not 2 — is the extension of a form over k , i.e. there is abasis in which all the entries of the associated symmetric matrix are constant. Thismeans that the classification of the above quadratic forms over F q [ T ] ( q odd) isvery simple. 26 emark 7.9. Let M be a t-motive which is both negatively and positivelyself-dual. There is a natural idea 7.9.2 to define an analog of Hodge structure on M . Nevertheless, this idea fails. Namely, the exact sequence0 → Ker ϕ → L ( M ) ⊗ C ∞ ϕ → Lie( M ) → A :0 → H , − ( A ) → H − ( A ) → ( H , ) ∗ ( A ) → H , − ( M ) := Ker ϕ , and the problem is to define an analogof H − , ( M ).Let us fix a negative isogeny α : M → M ′ , and let us extend the skew form < ∗ , ∗ > α to L ( M ) ⊗ C ∞ by C ∞ -linearity. It is easy to check that Ker ϕ is isotropicwith respect to this form (there is an analogy with the number field case). Let usconsider the following elementary lemma of linear algebra: Lemma 7.9.1.
Let W be a vector space of dimension 2 n over a field of charac-teristic = 2, B + (resp. B − ) a symmetric (resp. skew symmetric) non-degeneratebilinear form on W , and W ⊂ W a subspace of dimension n which is isotropicwith respect to both B + , B − . Then almost always there exists the only W ⊂ W of dimension n having properties: W ∩ W = 0; W is isotropic with respect to both B + , B − where almost always means that entries of the matrices of B + , B − in a basis of W must not satisfy (at least one of) polynomial relations. (cid:3) If End ( M ) = F q ( T ) and the action of I α on End ( M ) is not identical, then thereexists a positive isogeny β : M → M ′ and hence the symmetric form < ∗ , ∗ > β on L ( M ) ⊗ C ∞ . Ker ϕ is isotropic with respect to < ∗ , ∗ > β . Let us fix β . Idea 7.9.2.
To apply Lemma 7.9.1 to this situation ( W = L ( M ) ⊗ C ∞ , W =Ker ϕ , B + = < ∗ , ∗ > β , B − = < ∗ , ∗ > α ) in order to get a canonical subspace of L ( M ) ⊗ C ∞ which is complementary to Ker ϕ and hence can be considered as ananalog of H − , ( M ).Clearly there is no complete analogy with the number field case. But the situa-tion is even worse: Proposition 7.9.3.
For all M , α , β the ”almost always” condition of Lemma7.9.1 is not satisfied. (cid:3)
8. Relations between lattices and t-motives.
We have Theorem 8.1. ([H], Theorem 3.2). The dimension of the moduli set of puret-motives of dimension n and rank r is n ( r − n ). (cid:3) Remark.
A tuple ( e , ..., e r ) of integers entering in the statement of this theoremin [H] is (0,...,0,1,...,1) with 0 repeated r − n times and 1 repeated n times for thecase under consideration. I am grateful to Urs Hartl who indicated me this reference. n ( r − n ) is equal to the dimension of the set of lattices of rank r and dimension n , we can state an Open question 8.2.
Let r , n be given. Let us consider the lattice map fromthe set of the pure uniformizable t-motives of rank r and dimension n to the set oflattices of rank r and dimension n . Is it true that its image is open and the fibreat a generic point is discrete? If yes, what is the fibre? Remark.
Results of [GL17] give some evidence that for the case r = 2 n in a”neighborhood” of the n -th power of the rank 2 Carlitz module the fibre consistsof 1 point.Theorem 5 implies that for n = r − Corollary 8.4.
All pure t-motives of dimension r − r having N = 0are uniformizable. There is a 1 – 1 functorial correspondence between pure t-motives of dimension r − r having N = 0 ( r ≥ r in C r − ∞ having dual. Proof.
Let L be a lattice of rank r in C r − ∞ having dual L ′ . There exists theonly Drinfeld module M ′ such that L ( M ′ ) = L ′ , and let M be its dual. Theorem5 implies that L ( M ) = L . If there exists another pure t-motive M of dimension r − r having N = 0 such that L ( M ) = L then by Corollary 10.4 (itsproof is logically independent: there is no vicious circle) the dual M ′ is a Drinfeldmodule, according Theorem 5 it satisfies L ( M ′ ) = L ′ , hence M ′ = M ′ and hence M = M . (cid:3) Remark 8.5.
Recall that lattices of rank r in C r − ∞ having dual are describedin 3.5 (formulas 3.6, 3.7). We see that for the case n = r − N = 0 purity impliesuniformizability. We have Question 8.5a.
Do exist non-uniformizable t-motives having n = r − N = 0? Question 8.5b.
Do exist uniformizable t-motives having n = r − N = 0 suchthat its lattice has no dual? (Clearly this is a subquestion of 8.2). Remark 8.6.
Clearly for any r , n we have: if a lattice L of rank r and dimension n has no dual then L = L ( M ) for any pure uniformizable M . I do not know whetherTheorem 6 (which is an analog of Theorem 5 for another tensor operation) imposesa more strong similar restriction on the property of L to be the L ( M ) of some pureuniformizable M , or not.Further, for any uniformizable t-motive M we have a Corollary 8.7.
If the dual of ( L ( M ) , Lie( M )) does not exist then the dual of M does not exist. Example: the Carlitz module.
9. Main theorem in terms of Hodge-Pink structure.
Let us consider a version of a special case of the general definition of Hodge-Pinkstructure ([P], 0.2; 9.1).
Definition.
A Hodge-Pink structure of constant weight and complete dimensionis a pair H = ( H, q H ) where H is a free finite dimensional A -module and q H is a28 ∞ [[ T − θ ]]-lattice in H ⊗ A C ∞ [[ T − θ ]] such that the dimension of q H over C ∞ [[ T − θ ]]is equal to the dimension of H over A (condition of complete dimension).Let ϕ : L ֒ → C n ∞ be a lattice. It defines a Hodge-Pink structure H = H ( L ) ofconstant weight and complete dimension. Firstly, instead of a F q [ θ ]-module L weconsider an isomorphic A -module H formally defined by the property H ⊗ A F q [ θ ] = L where the map A → F q [ θ ] is ι . We denote the isomorphism H → L by ι as well;the composition ϕ ◦ ι : H → C n ∞ is a map of A -modules where T ∈ A acts on C n ∞ by multipication by θ . Further, ϕ ◦ ι extends to a surjection of C ∞ [[ T − θ ]]-modules H ⊗ A C ∞ [[ T − θ ]] → C n ∞ denoted by ϕ ◦ ι as well. Finally, q H is defined as Ker ϕ ◦ ι .If M is a pure uniformizable t-motive then we associate it a Hodge-Pink structureof constant weight and complete dimension H ( M ) = H ( L ( M )).Let m = m ( H ) be the minimal number such that q H ⊃ ( T − θ ) m H ⊗ A C ∞ [[ T − θ ]].For µ ≥ m we define the µ -dual structure H ′ µ = ( H ′ µ , q H ′ µ ) as follows: H ′ µ = H ∗ , q H ′ µ = { χ ∈ H ∗ ⊗ A C ∞ [[ T − θ ]]such that ∀ y ∈ q H we have χ ( y ) ∈ ( T − θ ) µ C ∞ [[ T − θ ]] } It is obvious that it is really a Hodge-Pink structure of constant weight and completedimension.If H = H ( L ) for a lattice L then m = 1 and if L has dual then H ′ = H ( L ′ ) (9 . Remark 9.2.
And if L has no dual? Really, H ( L ) exists even if L does notsatisfy Definition 2.1 (b). If L is a lattice having no dual this means that L ′ doesnot satisfy Definition 2.1 (b). Nevertheless, equality H ′ = H ( L ′ ) is meaningfuland holds. We are not interested in these lattices because they cannot be latticesof uniformizable t-motives having dual.Proof of the duality theorem for M having N = 0 is given in [GL18].
10. Duals of pures, and other elementary results.
We consider in this section the case of arbitrary N (i.e. not necessarily N = 0),and A = F q [ T ]. The definition 1.8 extends to the case of pr´e-t-motives, and remarks1.11 hold for this case. Lemma 10.2.
Let M be a pr´e-t-motive, m = m ( M ) from its (1.3.1), and µ ≥ m . Then M ′ — the µ -dual of M — exists as a pr´e-t-motive, and m ( M ′ ) ≤ µ .If M ′ is a t-motive then dim M ′ = rµ − dim M ( r is the rank of M ). Proof.
We must check that Q ′ has no denominators, and the condition (1.3.1).The module τ M is a C ∞ [ T ]-submodule of M (because aτ x = τ a /q x for x ∈ M ),hence there are C ∞ [ T ]-bases f ∗ = ( f , ...f r ) t , g ∗ = ( g , ...g r ) t of M , τ M respectivelysuch that g i = P i f i , where P | P | ... | P r , P i ∈ C ∞ [ T ]. Condition (1.3.1) means29hat ∀ i ( T − θ ) m f i ∈ τ M , i.e. P i | ( T − θ ) m , i.e. ∀ i P i = ( T − θ ) m i where0 ≤ m i ≤ m i +1 ≤ m . There exists a matrix Q = { q ij } ∈ M r ( C ∞ [ T ]) such that τ f i = r X j =1 q ij g j = r X j =1 q ij P j f j (10 . . τ is not a linear operator, it is easy to see that Q ∈ GL r ( C ∞ [ T ]) (really,there exists C = { c ij } ∈ M r ( C ∞ [ T ]) such that g i = P i f i = τ ( P rj =1 c ij f j ), we have C (1) Q = I r ).We denote the matrix diag ( P , P , ..., P r ) by P , so (10.2.1) means that Q = QP (10 . . Remark 10.2.3.
Since QP ∈ GL r ( C ∞ ( T )), we get that the action of τ on i ( M ) is invertible.It is clear that if M is a t-motive thendim M = r X j =1 m j (10 . . M = dim C ∞ ( M/τ M ). Further, (10.2.2) implies that for Q ′ = Q ( M ′ )we have Q ′ = Q t − diag (( T − θ ) µ − m , ..., ( T − θ ) µ − m r ) (10 . . Q ′ have no denominators. The condition (1.3.1) for M ′ follows easily from (10.2.5) (because Q t − ∈ GL r ( C ∞ [ T ])), and the dimensionformula (for the case M ′ is a t-motive) follows immediately from (10.2.4) appliedto M ′ . (cid:3) A definition of a pure t-motive can be found in [G] ((5.5.2), (5.5.6) of [G] +formula (1.3.1) of the present paper).
Theorem 10.3.
Let M be a pure t-motive and m = m ( M ) from (1.3.1). Then(if rm − n >
0) its m -dual M ′ exists, and it is pure. Proof.
The definition of pure ([G], (5.5.2)) is valid for pr´e-t-motives. We useits following matrix form. We denote T − by S and for any C we let C [ i ] = C ( i − · C ( i − · ... · C (1) · C Lemma 10.3.1.
Let Q ∈ M r ( C ∞ [ T ]) be a matrix such that formula (1.9.3)defines an t-motive M . Then it is pure iff there exists C ∈ GL r ( C ∞ (( S )) ) suchthat for some q , s > S q C ( s ) Q [ s ] C − ∈ GL r ( C ∞ [[ S ]])i.e. iff S q C ( s ) Q [ s ] C − is S -integer and its inicial coefficient is invertible. Proof.
Elementary matrix calculations. We take C as a matrix of base changeof f ∗ to a C ∞ [[ S ]]-basis of W of (5.5.2) of [G]. (cid:3) emma 10.3.2. Let µ = m . We have: M ′ = M ′ µ of Lemma 10.2 is a purepr´e-t-motive. Proof.
Let q , s and C be from Lemma 10.3.1. We have Q ′ [ s ] = (( T − θ ) [ s ] ) µ Q [ s ] t − (we use (1.2)). We take C ′ = C t − . We have S sµ − q C ′ ( s ) Q ′ [ s ] C ′− == S sµ − q C ( s ) t − Q [ s ] t − (( 1 S − θ ) [ s ] ) µ C t = ((1 − Sθ ) [ s ] ) µ S − q C ( s ) t − Q [ s ] t − C t = ((1 − Sθ ) [ s ] ) µ ( S q C ( s ) Q [ s ] C − ) t − We have: q /s = n/r ([G], (5.5.6)), hence ( sµ − q ) /s = ( rµ − n ) /r and sµ − q > − Sθ ) [ s ] ) µ ∈ GL r ( C ∞ [[ S ]]), and the result follows from Lemma 10.3.1. (cid:3) Remark.
This result holds also for µ > m .The theorem 10.3 follows from Lemma 10.2, the above lemmas and the proposi-tion that a pure pr´e-t-motive satisfying (1.3.1) is a t-motive ([G], (5.5.6), (5.5.7)). (cid:3)
Corollary 10.4.
Let M be a t-motive such that m = 1, n = r −
1. Then M has dual ⇐⇒ M is pure ⇐⇒ M is dual to a Drinfeld module. Proof.
Dimension formula shows that M ′ (if it exists) is a Drinfeld module, andthey are all pure. (cid:3) Example 10.5.
Let M be given by (notations of 1.9.1) A = θI , A = (cid:18) a a (cid:19) , A = (cid:18) (cid:19) This M has m = 1, n = 2, r = 3, and it is easy to see that it has no dual. Really,for this M we have (notations of 1.9) f = e , f = τ e , f = e , Q = T − θ − a − a t − θ , Q ′ = a t − θ a The last line of Q ′ means that τ f ′ = f ′ . This is a contradiction to the propertythat M ′ C ∞ [ τ ] is free. It is possible also to show (Proposition 11.3.4) that M is notpure, and to use 10.4 in order to prove that it has no dual.Later (Section 11) we shall construct examples of non-pure abelian t-motiveswhich have dual. Considerations of 11.8 predict that there is enough such t-motives.31 heorem 10.6. For any t-motive M there exists µ such that for all µ ≥ µ the object M ′ µ exists as a t-motive. For these µ we have M ′ µ +1 = M ′ µ ⊗ C (10 . . Proof. (10.6.1) holds at the level of pr´e-t-motives, because Q ( C ) = ( T − θ ) I .According [G], Lemma 5.4.10 it is sufficient to prove that M ′ µ is finitely generatedas a C ∞ [ τ ]-module. We shall use notations of Lemma 10.2. We take µ = 1 + { the maximum of the degrees of entries of Q ( M ) as polynomials in T } + max( m k )Let f ′ , ...f ′ r be the basis of M ′ µ over C ∞ [ T ] dual to f , ...f r . It is sufficient to provethe Lemma 10.6.2.
Let i = µ − min( m k ). Then elements T i f ′ j , i < i , j = 1 , ...., r ,generate M ′ µ as a C ∞ [ τ ]-module. Proof of the lemma.
By induction, it is sufficient to show that for all α ≥ i the equation τ x = ( T − θ ) α f ′ j (10 . . M ′ µ ) has a solution x = r X k =1 C k f ′ k where C k ∈ C ∞ [ T ], deg( C k ) < α . According (10.2.5), the solution to (10.6.3) isgiven by ( C (1)1 , ..., C (1) r ) = (0 , ... , ( T − θ ) α − µ + m j , , ... Q t (the non-0 element of the row matrix is at the j -th place). Unequalities satisfiedby µ and α show that all C (1) k are polynomials of degree < α . Since c c q issurjective on C ∞ , we get the desired. (cid:3) We need two elementary lemmas.
Lemma 10.7.0. If M is a t-motive then M ⊗ C is also a t-motive. Proof.
Let f j ( j = 1 , ..., r ) be a C ∞ [ T ]-basis of M C ∞ [ T ] and f from 1.10.2, so f j ⊗ f is a C ∞ [ T ]-basis of ( M ⊗ C ) C ∞ [ T ] . It is sufficient to prove that ( M ⊗ C ) C ∞ [ τ ] is finitely generated. Since M C ∞ [ τ ] is finitely generated, it is easy to see that thereexists a such that elements( T − θ ) i f j , i = 0 , ..., a, j = 1 , ..., r generate M C ∞ [ τ ] . This means that ∀ j = 1 , ..., r there exist c ijkl ∈ C ∞ such that( T − θ ) a +1 f j = a X i =0 γ X k =0 r X l =1 c ijkl ( T − θ ) i τ k f l (10 . . . This notion was indicated me by Taguchi. Anderson proved (not published) that the tensor product of any t-motives is also a t-motive. γ is a number.Let us multiply (10.7.0.1) by ( T − θ ) γ . Taking into consideration the formula ofthe action of τ on M ⊗ C we get that the result gives us the following formula in M ⊗ C :( T − θ ) a + γ +1 f j ⊗ f = a X i =0 γ X k =0 r X l =1 c ijkl ( T − θ ) i + γ − k τ k · ( f l ⊗ f ) (10 . . . j the element ( T − θ ) a + γ +1 f j ⊗ f is a linear combination of( T − θ ) i f l ⊗ f , i = 0 , ..., a + γ, l = 1 , ..., r (10 . . . M ⊗ C ) C ∞ [ τ ] . Multiplying (10.7.0.2) by consecutive powers of T − θ we get byinduction that elements of 10.7.0.3 generate ( M ⊗ C ) C ∞ [ τ ] . (cid:3) Lemma 10.7.1. If M ⊗ C is isomorphic to M ⊗ C then M is isomorphic to M . Proof.
Let f i ∗ ( i = 1 ,
2) be a C ∞ [ T ]-basis of ( M i ) C ∞ [ T ] , Q i from 1.9.3, α : M ⊗ C → M ⊗ C an isomorphism and C ∈ GL r ( C ∞ [ T ]) the matrix of α in f ∗ ⊗ f , f ∗ ⊗ f . The matrix of the action of τ on M i ⊗ C in the base f i ∗ ⊗ f is ( T − θ ) Q i ,and the condition that α commutes with multiplication by τ is( T − θ ) Q C = C (1) ( T − θ ) Q Dividing this equality by T − θ we get that the map α from M to M having thesame matrix C in the bases f i ∗ , commutes with τ , i.e. defines an isomorphism from M to M . (cid:3) Using Lemma 10.7.1 we can state the following
Definition.
A virtual t-motive is an object M ⊗ C ⊗ µ where M is a t-motiveand µ ∈ Z , with the standard equivalence relation (here µ ≥ µ ): M ⊗ C ⊗ µ = M ⊗ C ⊗ µ ⇐⇒ M = M ⊗ C ⊗ ( µ − µ ) ⇐⇒ ∃ µ such that µ + µ ≥ µ + µ ≥ M ⊗ C ⊗ ( µ + µ ) = M ⊗ C ⊗ ( µ + µ ) Lemma 10.7.1 shows that these conditions are really equivalent.
Corollary 10.7.2.
The µ -dual of a virtual t-motive is well-defined and alwaysexists as a virtual t-motive. (cid:3) Proposition 10.8.
The following formula is valid at the level of pr´e-t-motives:for any µ , µ , if M ′ µ , M ′ µ exist then ( M ⊗ M ) ′ ( µ + µ ) exists and( M ⊗ M ) ′ ( µ + µ ) = M ′ µ ⊗ M ′ µ Proof.
This is a functorial equality; also we can check it by means of elementarymatrix calculations. (cid:3)
Proposition 10.9.
Let P ∈ A be an irreducible element. The Tate module T P ( M ′ µ ) is equal to T P ( C ) ⊗ µ ⊗ \ T P ( M )33equality of Galois modules) where \ T P ( M ) is the dual Galois module. Proof.
It is completely analogous to the proof of the corresponding theorem fortensor products ([G], Proposition 5.7.3, p. 157). All modules in the below proofwill be the Galois modules, and equalities of modules will be equalities of Galoismodules. Recall that E = E ( M ). Since T P ( M ) = invlim n E P n , it is sufficient toprove that for any a ∈ A we have E ( M ′ µ ) a = E ( C ⊗ µ ) a ⊗ ˆ E a , where ˆ E a is thedual of E a in the meaning of [T], Definition 4.1. We have the following sequence ofequalities of modules: M ′ µ /aM ′ µ = Hom C ∞ [ T ] ( M/aM, C ⊗ µ /a C ⊗ µ ) (10 . . τ on both sides of this equality coincide (to define the actionof τ on the right and side of (10.9.2) we need the action of τ − on M/aM ; it iswell-defined, because the determinant of the action of τ on M is a power of T − θ ,hence its image in C ∞ [ T ] /a C ∞ [ T ] is invertible). 10.9.2 follows immediately fromthe definition of M ′ µ ;( M ′ µ /aM ′ µ ) τ = Hom F q [ T ] (( M/aM ) τ , ( C ⊗ µ /a C ⊗ µ ) τ ) (10 . . M /a M = ( M /a M ) τ ⊗ F q [ T ] /a F q [ T ] C ∞ [ T ] /a C ∞ [ T ]applied to both M = M , M = M ′ µ (we use that both M , M ′ µ are free C ∞ [ T ]-modules). Finally, we have a formula E ( M ) a = Hom F q (( M /a M ) τ , F q )([G], p. 152, last line of the proof of Proposition 5.6.3). Applying this formula to10.9.3 we get the desired. (cid:3)
11. An explicit formula.
We return to the case N = 0. Let e ∗ , A , A i , l , n be from (1.9). We consider inthe present section two simple types of t-motives (called standard-1 and standard-2t-motives respectively) whose A i have a row echelon form, and we give an explicitformula for the dual of some standard-1 t-motives. Analogous formula can be easilyobtained for more general types of t-motives. These results are the first step of theproblem of description of all t-motives having duals. For the reader’s convenience, we give here the definition of standard-1t-motives for the case n = 2 (here λ and λ satisfying λ = l , l > λ ≥ A = θI , for 0 < i < λ A i is arbitrary, A λ = (cid:18) ∗ ∗ (cid:19) , for λ < i < l A i = (cid:18) ∗ ∗ (cid:19) , A l = (cid:18) (cid:19) To define standard-2 t-motives of dimension n , we need to fix34. A permutation ϕ ∈ S n , i.e. a 1 – 1 map ϕ : (1 , ..., n ) → (1 , ..., n );2. A function k : (1 , ..., n ) → Z + where Z + is the set of integers ≥ Definition.
A standard-2 t-motive of the type ( ϕ, k ) is an abelian t-motive ofdimension n given by the formulas ( i = 1 , ..., n ): T e ϕ ( i ) = θe ϕ ( i ) + n X α =1 k ( α ) − X j =1 a j,ϕ ( i ) ,α τ j e α + τ k ( i ) e i (11 . . a j,ϕ ( i ) ,α ∈ C ∞ is the ( ϕ ( i ) , α )-th entry of the matrix A j . Proposition 11.2.2.
Formula 11.2.1 really defines a t-motive denoted by M = M ( ϕ, k ) = M ( ϕ, k, a ∗∗∗ ). Its rank is P nα =1 k ( α ) and elements X αj := τ j e α , α =1 , ..., n , j = 0 , ..., k ( α ) −
1, form its C ∞ [ T ]-basis. (cid:3) The group S n acts on the set of types ( ϕ, k ) and on the set of the above M ;clearly for any ψ ∈ S n we have ψ ( M ) is isomorphic to M . Particularly, we canconsider only ϕ of the following form of the product of i cycles ( α = 0 , α i = n ): ϕ = ( α + 1 , ..., α )( α + 1 , ..., α ) ... ( α i − + 1 , ..., α i ) (11 . . γ = α j we have ϕ ( γ ) = γ + 1,for γ = α j we have ϕ ( α j ) = α j − + 1). Example 11.2.4.
Let ϕ be defined by 11.2.3, the quantity of cycles i is equal to 1and all a ∗∗∗ = 0. Then the corresponding M is of complete multiplication by a CM-field F q r ( T ) and its CM-type Φ is { Id , fr k (1) , fr k (1)+ k (2) , ..., fr k (1)+ k (2)+ ... + k ( n − } where fr is the Frobenuis homomorphism F q r → ¯ F q (see 13.3, first case: formulas13.3.1, 13.3.2 coinside with 11.2.1 for the given ϕ and a ∗∗∗ = 0; i j of 13.3.0 is k (1) + k (2) + ... + k ( j −
1) of the present notations).
Definition 11.3.
A standard-1 t-motive is a standard-2 t-motive whose ϕ isthe identical permutation Id . Let M = M ( Id, k ) be a standard-1 t-motive. Acting by ψ ∈ S n we canconsider only the case of non-increasing k ( j ). We introduce a number m ≥ k ( j ), and two sequences0 = γ < γ < ... < γ m = n (sequence of arguments of points of jumps of the function k ) and0 = λ m +1 < λ m < ... < λ < λ = l (sequence of values of k on segments [ γ i − + 1 , ..., γ i ]) by the formulas k (1) = ... = k ( γ ) = λ k ( γ + 1) = ... = k ( γ ) = λ ...k ( γ m − + 1) = ... = k ( γ m ) = λ m (11 . . xample 11.3.2. The t-motive M of 11.1 is a standard-1 having m = 2, γ = 1, γ = 2 and λ , λ as in 11.1. Its rank r = λ + λ . Conjecture 11.3.3.
A standard-2 t-motive of the type ( ϕ, k ) (notations of11.2.3) is pure iff ∀ j = 1 , ..., i we have: α j − α j − P α j γ = α j − +1 k ( γ ) = nr This conjecture is obviously true if all a ∗∗∗ are 0.To simplify exposition, we prove here only the following particular case of thisconjecture. Proposition 11.3.4.
Let M be a standard-1 t-motive having m >
1, definedover F q ( θ ), having a good reduction at a point of degree 1 of F q ( θ ) (i.e. a point θ + c , c ∈ F q ). Then M is not pure. Proof.
Let M be defined by 11.2.1, we use notations of 11.3.1. We consider theaction of Frobenius on ˜ M — the reduction of M at θ + c . According [G], Theorem5.6.10, it is sufficient to prove that orders of the roots of the characteristic polyno-mial of Frobenius over A are not equal. More exactly, we consider the valuationinfinity on A (defined by the condition ord( T ) = − M ). The action of Frobenius on ˜ M coincideswith multiplication by τ , because the degree of the reduction point is 1.A basis f ∗ of M C ∞ [ T ] is the set of X αj := τ j e α of 11.2.2. The matrix Q ( M ) isdefined by the following formulas for the action of τ on X αj : τ ( X αj ) = X α,j +1 if j < k ( α ) − . . . τ ( X α,k ( α ) − ) = T X α, − m X δ =1 λ δ − X d = λ δ +1 γ δ X c =1 a dαc X cd (11 . . . X αj in lexicographic order ( X α j precedes to X α j if α < α ) then the matrix Q ( M ) has the block form: Q ( M ) = ( C ij ) ( i, j = 1 , ..., n )where C ij is a k ( i ) × k ( j )-matrix of the form C ii = ...
00 0 1 ... ... ... ... ... ... ... T − θ ∗ ∗ ... ∗ , C ij = ... ... ... ... ... ... ∗ ... ∗ ( i = j )where asterisks mean elements a ∗∗∗ (in some order). We consider the characteristicpolynomial P ( X ) ∈ ( C ∞ [ T ])[ X ] of Q ( M ). We have C ii − X I k ( i ) = − X ... − X ... ... ... ... ... ... ... t − θ ∗ ∗ ... ∗ − X
36 subset of the set of entries of a matrix is called (following N.N.Luzin) a light-ning if each row and each column of the matrix contains exactly one element of thissubset. The product of elements of a lightning is called the value of this lightning(i.e. the determinant is the alternating sum of the values of all lightnings).
Lemma 11.3.4.3.
If a non-zero lightning of C ii − X I k ( i ) contains the term T − θ , then it does not contain any term containing X . (cid:3) Let J be a subset of the set 1 , ..., n and J ′ its complement. Corollary 11.3.4.4.
If a non-zero lightning of Q ( M ) − X I r contains terms T − θ of blocks C α , j ∈ J , then its value is a polynomial in X of degree ≤ P j ′ ∈ J ′ k ( j ′ ),and there exists exactly one such lightning (called the principal J -lightning) whosevalue is a polynomial in X of degree P j ′ ∈ J ′ k ( j ′ ). (cid:3) Since the characteristic polynomial of Frobenius of ˜ M is ˜ P (respectively thevaluation infinity of C ∞ [ T ]), it is sufficient to prove that the Newton polygon of P ( X ) is not reduced to the segment ((0 , − n ); ( r, T − θ ) n and X r . To do it, it is sufficient to find a point on its Newton polygonwhich is below this segment. We consider J min = the set of all γ m − γ m − diagonalblocks C ii ( i = γ m − + 1 , ..., γ m ) of Q ( M ) of minimal size λ m . The value of theprincipal J min -lightning is ( T − θ ) γ m − γ m − times polynomial in X of degree d := r − ( γ m − γ m − ) λ m . Corollary 11.3.4.4 implies that if the value of any other lightningof Q ( M ) − X I r contains a term whose X -degree is equal to d , then the T -degreeof this term is strictly less than γ m − γ m − . This means that if we write P ( X ) = P ri =0 C i X i , C i ∈ C ∞ [ T ], then ord ∞ ( C d ) = − ( γ m − γ m − ), i.e. the point withcoordinates [ − ( γ m − γ m − ) , d ] belongs to the Newton diagram of P ( X ), i.e. it isabove (really, at) the Newton polygon of P ( X ). This point is below the segment((0 , − n ); ( r, (cid:3) Remark 11.3.4.5.
It is easy to see that the Newton polygon of P ( X ) coincideswith the Newton polygon of the direct sum of trivial Drinfeld modules of ranks λ ∗ ,i.e. with the Newton polygon of the polynomial m Y i =1 ( X λ i − T ) γ i − γ i − To formulate the below theorem 11.5 we need some notations. Let M bea standard-1 t-motive defined by formulas 11.2.1, 11.3.1. We impose the condition λ m ≥
3. Theorem 11.5 affirms that it has dual. To find explicitly the dual of M ,we need to choose an arbitrary function ν : ( i, j ) → ν ( i, j ) which is a 1 - 1 mapfrom the set of pairs ( i, j ) such that1 ≤ i ≤ n ; 1 ≤ j ≤ k ( i ) − . . n + 1 , ..., r − n ] (recall that r = P ni =1 k ( i ) = P m i =1 ( γ i − γ i − ) λ i ).Let the ( r − n ) × ( r − n )-matrices B , B be defined by the following formulas(here and until the end of the proof of 11.5 we have i, α = 1 , ..., n ; b βγδ is the( γδ )-th entry of B β , all entries of B , B that are not in the below list are 0): b iα = − a k ( i ) − ,α,i ; 37 ,ν ( i,j ) ,α = − a j,α,i for 1 ≤ j ≤ k ( i ) − b ,ν ( i,j +1) ,ν ( i,j ) = 1 for 1 ≤ j ≤ k ( i ) − b ,i,ν ( i,k ( i ) − = 1; b ,ν ( i, ,i = 1.We let B = θI r − n + B τ + B τ and consider a t-motive M ( B ) (see 11.5.1 below).Formulas 11.4.2 mean that M ( B ) is standard-2, its ϕ = ϕ B is a product of n cycles i ϕ B → ν ( i, ϕ B → ν ( i, ϕ B → ... ϕ B → ν ( i, k ( i ) − ϕ B → i and its k = k B is defined by the formulas k B ( γ ) = 2 for γ ∈ [1 , ..., n ], k B ( γ ) = 1 for γ ∈ [ n + 1 , ..., r − n ]. Theorem 11.5.
Let M be from 11.4 (i.e. a standard-1 t-motive having λ m ≥ M ′ = M ( B ). Proof. Let e ′∗ = ( e ′ , ...e ′ r − n ) t be the vector column of elements of a basis of M ( B ) over C ∞ [ τ ] satisfying T e ′∗ = Be ′∗ (11 . . j, k ) such that either j = 1 , ..., n , k = 0 , j = n + 1 , ..., r − n , k = 0. For each pair ( j, k ) of this set we let (as in [T], p. 580) Y j k = τ k e ′ j . Formulas (11.4.2) show that these Y ∗∗ form a basis of M ( B ) C ∞ [ T ] ,and the action of τ on this basis is given by the following formulas (here j =1 , ..., k ( i ) − τ ( Y i, ) = Y i, (11 . . . τ ( Y i, ) = ( T − θ ) Y ν ( i, , + n X γ =1 a γi Y γ, (11 . . . τ ( Y ν ( i,j ) , ) = ( T − θ ) Y ν ( i,j +1) , + n X γ =1 a j +1 ,γ,i Y γ, if j < k ( i ) − . . . τ ( Y ν ( i,k ( i ) − , ) = ( T − θ ) Y i, + n X γ =1 a k ( i ) − ,γ,i Y γ, (11 . . . X ′∗∗ be the dual basis to the basis X ∗∗ of 11.2.2. Let us consider the following correspondence between X ′∗∗ and Y ∗∗ : X ′ ij corresponds to Y ν ( i,j ) , for the pair ( i, j ) like in (11.4.1), X ′ i corresponds to Y i for 1 ≤ i ≤ n ; X ′ i,k ( i ) − corresponds to Y i for 1 ≤ i ≤ n .Therefore, in order to prove the Theorem 11.5 we must check that matrices de-fined by the dual to (11.3.4.*) and by (11.5.2.*) satisfy (1.10.1) under identification(11.5.3). This is an elementary exercise. (cid:3) This proof is a generalization of the corresponding proof of Taguchi; we keep his notations. emark 11.6. Clearly it is possible to generalize the Theorem 11.5 to a largerclass of t-motives — some subclass of standard-3 t-motives, see Definition 11.8.1.The below example of the proof of Proposition 11.8.7 shows that probably thecondition λ m ≥ λ m ≥
2: it is necessaryto modify slightly formulas 11.4.2. From another side, a standard-1 t-motive of theExample 2.5 shows that this condition cannot be changed to λ m ≥ To formulate the proposition 11.7.3,we change slightly notations in 1.9.1, namely, instead of A = P li =0 A i τ i we considerpolynomials P k ( M ) of x , ..., x n ( k = 1 , ..., n ) defined by the formula P k ( M ) = l X i =0 n X j =1 a ikj x q i j (11 . . E is the t-module associated to M (see [G], 5.4.5), x ∗ = ( x , ..., x n ) t an element of E then 11.7.1 is equivalent to T x ∗ = P ∗ ( x ∗ ) where P ∗ =( P ( M ) , ..., P n ( M )) t is the vector column. For a standard-1 t-motive M (weuse notations of 11.3.0) having m ≥ P ( M ) =( P ( M ) , ..., P γ ( M )) t , P ( M ) = ( P γ +1 ( M ) , ..., P γ ( M )) t . We use similar nota-tions for M ′ . Let M be as above, we consider the case λ = λ −
1. Let C be a fixed γ × ( γ − γ )-matrix. We define a transformed t-motive M by the formulas P ( M ) = P ( M ) + C P ( M ) q P i ( M ) = P i ( M ) for i > γ Proposition 11.7.3.
For M , C , M of 11.7.2 the dual M ′ of M is describedby the following formulas: P ( M ′ ) = P ( M ′ ) − C t P ( M ′ ) q P i ( M ′ ) = P i ( M ′ ) for i [ γ + 1 , ..., γ ] Proof is similar to the proof of the Theorem 11.5, it is omitted. (cid:3)
Most results of this subsection are conditional.We shall show that under some natural conjecture the condition of purity in 8.2and 8.4 is essential, and that for non-pure t-motives the notion of algebraic dualityis richer than the notion of analytic duality.We generalize slightly the definition 11.2.1 as follows. Let ≻ be a linear orderingon the set [1 , ..., n ], and let ϕ , k be as in 11.2. Definition 11.8.1.
A standard-3 t-motive of the type ( ϕ, k, ≻ ) is a t-motive ofdimension n given by the formulas T e ϕ ( i ) = θe ϕ ( i ) + n X j =1 k ( j ) − X l =1 a l,ϕ ( i ) ,j τ l e j + X j ≻ i a k ( j ) ,ϕ ( i ) ,j τ k ( j ) e j + τ k ( i ) e i (11 . . a ∗∗∗ ∈ C ∞ are coefficients (the only difference with 11.2.1 is the term P j ≻ i a k ( j ) ,ϕ ( i ) ,j τ k ( j ) e j ). We denote it by M ( a ∗∗∗ ).Let M = M ( a ∗∗∗ ), M = M ( a ∗∗∗ ) be two isomorphic standard-3 t-motivesof the same type ( ϕ, k, ≻ ) with C ∞ [ τ ]-bases e ∗ , e ∗ respectively (we use notationsof 11.8.2 for both M , M ). There exists C ∈ M n ( C ∞ [ τ ]) such that the formuladefining an isomorphism between M and M is the following: e ∗ = Ce ∗ . Conjecture 11.8.3.
For a generic set of a ∗∗∗ there exists only a countable setof a ∗∗∗ such that M is isomorphic to M .This conjecture is based on calculations in some explicit cases. Particularly, itis proved if M , M are given by the below formula 11.8.5.1 and entries of C arepolynomials in τ of degree ≤ M u ( r, n ) the moduli space of uniformizable t-motives of the rank r and dimension n , by L ( r, n ) the moduli space of lattices of the rank r and di-mension n and by L : M u ( r, n ) → L ( r, n ) the functor of lattice associated to anuniformizable t-motive. Proposition 11.8.5.
Conjecture 11.8.3 implies that the dimension of the fibersof L is > r = 3, n = 2. Particularly, we cannot omit condition of purity inthe statement of 8.2. Proof.
We consider standard-3 t-motives of the type n = 2, ϕ = Id , k (1) = 2, k (2) = 1, 2 ≻
1. Such M = M ( a , a , a ) is given by A = θI , A = (cid:18) a a a (cid:19) , A = (cid:18) (cid:19) (11 . . . r = 3, it is not pure, hence it has no dual.Conjecture 11.8.3 implies that the dimension of the moduli space of these t-motivesis 3 (because there are 3 coefficients a , a , a ). Uniformizable t-motives forman open subset of this moduli space, while the moduli space of lattices of n = 2and r = 3 has dimension 2. (cid:3) Remark.
Similar calculations are valid for any sufficiently large r , n .Standard-3 t-motives of the above type have not dual. The following propositionshows that the same phenomenon holds for t-motives having dual. We denote by M u,d ( r, n ) the moduli space of uniformizable t-motives of the rank r and dimension n having dual, by L d ( r, n ) the moduli space of lattices of the rank r and dimension n having dual, by L d : M u,d ( r, n ) → L d ( r, n ) the functor of lattice and by D M : M u,d ( r, n ) → M u,d ( r, r − n ), D L : L d ( r, n ) → L d ( r, r − n ) the functors of duality ont-motives and lattices respectively. Practically, Theorem 5 means that the followingdiagram is commutative: M u,d ( r, n ) D M → M u,d ( r, r − n ) L d ↓ L d ↓L d ( r, n ) D L → L d ( r, r − n ) (11 . . Proposition 11.8.7.
Conjecture 11.8.3 implies that the dimension of the fibersof L d in the diagram (11.8.6) is > r = 5, n = 2.40ractically, this means that the notion of algebraic duality is ”richer” than thenotion of analytic duality. Proof.
We consider standard-3 t-motives of the type n = 2, ϕ = Id , k (1) = 3, k (2) = 2, 2 ≻ r = 5. Such M is given by A = θI , A = (cid:18) a a a a (cid:19) , A = (cid:18) a a a (cid:19) , A = (cid:18) (cid:19) (notations of Example 10.5). It has dual. Really, we denote by A i ∗ j the j -th columnof A i , and we denote by ( C | C ) the matrix formed by union of columns C , C .Then M ′ = M ( B ) is also a standard-3 t-motive, where B = − det A − a − det( A ∗ | A ∗ ) − a − det( A ∗ | A ∗ ) − a , B = − a q The same arguments as in the proof of Proposition 11.8.5 show that the conjec-ture 11.8.3 implies that the dimension of the moduli space of these t-motives is 7,while the moduli space of lattices of n = 2 and r = 5 has dimension 6. (cid:3) As above, similar calculations are valid for any sufficiently large r , n ; clearly thedimension of fibers of L d becomes larger as r , n grow.Let us mention two open questions related to the functor L . Firstly, let L be aself-dual lattice such that L ∈ L ( M u,d (2 n, n )). This means that D M : L − d ( L ) → L − d ( L ) is defined. Open question 11.8.8.
What can we tell on this functor, for example, whatis the dimension of its stable elements?Secondly, let us consider M , M of CM-type with CM-field F q r ( T ), see 13.3. Open question 11.8.9.
Let the CM-types Φ , Φ of the above M , M sat-isfy Φ = α Φ , where α ∈ Gal ( F q r ( T ) / F q ( T )). Are lattices L ( M ), L ( M ) non-isomorphic?Clearly the negative answer to this question implies the negative answer to theQuestion 8.2.For any given M , M the answer can be easily found by computer calcula-tion. Really, let M be one of M , M , c , ..., c r a basis of F q r / F q and α , ..., α n ⊂ Gal ( F q r ( θ ) / F q ( θ )) the CM-type of M . We define matrices M , N as follows:( M ) ij = α j ( c i ) ( i, j = 1 , ..., n ), ( N ) ij = α j ( c n + i ), j = 1 , ..., n , i = 1 , ..., r − n .The Siegel matrix Z ( M ) is obviously N M − . So, we can find explicitly Z ( M ), Z ( M ) for both M , M . To check whether Z ( M ), Z ( M ) are equivalent or not,it is sufficient to find a solution to 3.8.1 such that the entries of A , B , C , D are in M ∗ , ∗ ( F q ) (this is obvious: the condition ∃ γ ∈ GL r ( Z ∞ ) is equivalent to the con-dition ∃ γ ∈ GL r ( F q ), because entries of Z ( M ), Z ( M ) are in F q r ). The equation3.8.1 is linear with respect to A , B , C , D , and we can check whether its solutionsatisfying det γ = 0 exists or not.For the case q = 2, r = 4, n = 2, CM-types of M , M are ( Id, F r ), (
Id, F r )respectively, a calculation shows that the answer is positive: lattices L ( M ), L ( M )are not isomorphic. 41
2. t-motives having multiplications.
Let K be a separable extension of F q ( T ) such that K C := K ⊗ F q C ∞ is also a field, π : X → P ( C ∞ ) the projection of curves over C ∞ corresponding to C ∞ ( T ) ⊂ K C .Let K , X satisfy the condition: ∞ ∈ X is the only point on X over ∞ ∈ P ( C ∞ ).Let A K be the subring of K consisting of elements regular outside of infinity. Wedenote g = dim K / F q ( T ) and α , ..., α g : K → C ∞ — inclusions over ι : F q ( T ) → C ∞ (recall that ι ( T ) = θ ). Let W be a central simple algebra over K of dimension q .Each α i : K → C ∞ can be extended to a representation χ i : W → M q ( C ∞ ). Let (
L, V ) be as in Section 2 (recall that A = F q [ T ]) such that there exists an inclusion i : W →
End ( L, V ), whereEnd ( L, V ) = End(
L, V ) ⊗ A F q ( T ). It defines a representation of W on V de-noted by Ψ which is isomorphic to P gi =1 r i χ i where { r i } are some multiplicities(the CM-type of the action of W on ( L, V )). [Proof: restriction of Ψ on K is a sumof one-dimensional representations, i.e. V = ⊕ gi =1 V i where k ∈ K acts on V i bymultiplication by α i ( k ). Spaces V i are Ψ-invariant. We consider an isomorphism W ⊗ K C ∞ = M q ( C ∞ ) where the inclusion of K in C ∞ is α i . We extend Ψ | V i to W ⊗ K C ∞ by C ∞ -linearity using the inclusion α i of K in C ∞ . It remains to showthat a representation of M q ( C ∞ ) is a direct sum of its q -dimensional standard rep-resentations. We consider the corresponding representation of Lie algebra sl q ( C ∞ ).It is a sum of irreducible representations. Let ω be the highest weight of any ofthese irreducible representations. ω is extended uniquely to the set of diagonalmatrices of M q ( C ∞ ), because ω is identical on scalars. Since our representation isnot only of Lie algebra but of algebra M q ( C ∞ ), we get that ω is a ring homomor-phism Diag ( M q ( C ∞ )) → C ∞ . There exists the only such ω corresponding to the q -dimensional standard representation].Further, we denote m = dim W L ⊗ F q ( T ) ( g , q , Ψ, r i , m are analogs of g , q , Φ, r i , m of [Sh63] respectively). Clearly we have n = q g X i =1 r i , r = mg q (12 . i ′ : W op → End ( L ′ , V ′ ) where W op isthe opposite algebra. Remark.
A construction of Hilbert-Blumental modules ([A], 4.3, p. 498) practi-cally is a particular case of the present construction: for Hilbert-Blumental moduleswe have q = 1, i.e. K = W , and all r i = 1. Anderson considers the case when ∞ splits completely; this difference with the present case is not essential. Proposition 12.3.
If the dual pair ( L ′ , V ′ ) exists then the CM-type of the dualinclusion is { m q − r i } , i = 1 , ..., g . Proof.
We have L ⊗ Z ∞ C ∞ is isomorphic to ( W ⊗ F q ( θ ) C ∞ ) m as a W -module. Sincethe natural representation of W on W ⊗ F q ( θ ) C ∞ is isomorphic to q P gi =1 χ i we getthat L ⊗ Z ∞ C ∞ is isomorphic to m q P gi =1 χ i as a W -module. Consideration of theexact sequence 0 → V ′∗ → L ⊗ Z ∞ C ∞ → V → (cid:3) emark 12.4.1. This result is an analog of the corresponding theorem in thenumber field case. We use notations of [Sh63], Section 2. Let A be an abelianvariety having endomorphism algebra of type IV, and ( r ν , s ν ) = ( r ν ( A ) , s ν ( A )) arefrom [Sh63], Section 2, (8). Then r ν ( A ′ ) = mq − r ν ( A ) = s ν ( A ) , s ν ( A ′ ) = mq − s ν ( A ) = r ν ( A )By the way, Shimura writes that the CM-types of A and A ′ coincide ([Sh98], 6.3,second line below (5), case A of CM-type). We see that his affirmation is notnatural: he considers the complex conjugate action of the endomorphism ring on A ′ . It is necessary to take into consideration this difference of notations comparingformulas of 12.3 and 13.2 with the corresponding formulas of Shimura. Remark 12.4.2.
According [L09], a t-motive M is an analog of an abelianvariety A with multiplication by an imaginary quadratic field K . The above con-sideration shows that this analogy holds for M and A having more multiplications.Really, if A has more multiplications then (we use notations of [Sh63], Section2) F = F K , and numbers ( r ν ( A ) , s ν ( A )) satisfy n ( A ) = q P gi =1 r ν ( A ), where( n ( A ) , dim( A ) − n ( A )) is the signature of A treated as an abelian variety withmultiplication by K . This is an analog of 12.2. Here we consider the case q = m = 1, i.e. K = W and g = r . Lemma 12.5.1.
In this case the condition N = 0 implies that the CM-type r X i =1 r i α i (12 . . K on on ( L, V ) has the property: all r i are 0 or 1. Proof. N = 0 means that the action of T ∈ A on V is simply multiplication by θ . We write the CM-type P ri =1 r i χ i in the form P ni =1 χ α i where α , ..., α n ∈ [1 , ..., r ]are not necessarily distinct. Let l be an (only) element of a basis of L ⊗ A K K over K and e , ..., e n a basis of V over C ∞ such that the action of K on V is given bythe formulas k ( e i ) = χ α i ( k ) e i , k ∈ K Multiplying e i by scalars if necessary, we can assume that l = P e i . Therefore,if α i = α j (i.e. not all r ∗ in (12.5.2) are 0, 1) then the e α i -th coordinate of anyelement of L coincide with its e α j -th coordinate, hence L does not C ∞ -generate V — a contradiction. (cid:3) Let M be a t-motive of rank r and dimension n having multiplication by A K . Recall that we consider only the case N = 0. This means that the characterof the action of K on M/τ M is isomorphic to P ri =1 r i α i . Since E ( M ) = ( M/τ M ) ∗ we get that the character of the action of K on E ( M ) is the same. Ifall r i are 0 or 1 (12 . . M has the CM-type Φ ⊂ { α , ..., α r } where Φ isdefined by the condition α i ∈ Φ ⇐⇒ r i = 1.43t is easy to see that this case occurs for uniformizable M . Really, if M isuniformizable then the action of K can be prolonged on ( L ( M ) , V ( M )), and thecharacter of the action of K on V ( M ) coincides with the one on E ( M ). The resultfollows from Lemma 12.5.1. Lemma 12.5.5.
There exists a canonical isomorphism γ from the set of inclu-sions α , ..., α r to the set of points θ α , ..., θ α r of X over θ ∈ P ( C ∞ ). Proof.
A point t ∈ X over θ ∈ P ( C ∞ ) defines a function ϕ t : K C → P ( C ∞ )— the value of an element f ∈ K C treated as a function on X at the point t . Thisfunction must satisfy the standard axioms of valuation and the condition ϕ t ( T ) = θ .Let α i be an inclusion of K to C ∞ over ι . It defines a valuation ϕ α i : K C → P ( C ∞ )by the formula ϕ α i ( k ⊗ f ) = α i ( k ) f ( θ ), where k ∈ K , f ∈ C ∞ ( T ). We define γ ( α i )by the condition ϕ γ ( α i ) = ϕ α i ; it is easy to see that γ is an isomorphism. (cid:3) Theorem 12.6.
For any above { K , Φ } there exists an t-motive ( M, τ ) withcomplete multiplication by K having CM-type Φ. Proof (Drinfeld).
We denote the divisor P α i ∈ Φ γ ( α i ) by θ Φ . We construct a F -sheaf F of dimension 1 over K which will give us M . Let fr be the Frobeniusmap on Pic ( X ). It is an algebraic map, and the fr − Id : Pic ( X ) → Pic ( X ) isan algebraic map as well. Since the action of fr on the tangent space of Pic ( X )at 0 is the zero map, the action of fr − Id on the tangent space of Pic ( X ) at 0 isthe minus identical map and hence fr − Id is an isogeny of Pic ( X ). Particularly,there exists a divisor D of degree 0 on X such that we have the following equalityin Pic ( X ): fr( D ) − D = − θ Φ + n ∞ (12 . . F = F Φ = O ( D ) then there exists a rational map τ X = τ X, Φ : F (1) → F such that Div( τ X ) = θ Φ − n ∞ (12 . . F Φ , τ X, Φ ) is the desired F -sheaf. Remark.
It is easy to see that if the genus of X is > , Φ give us different sheaves F Φ , F Φ , while if the genus of X is 0 then F Φ = F Φ = O , but the maps τ X, Φ , τ X, Φ are clearly different.Let U = X − {∞} be an open part of X . We denote F ( U ) by M , hence F (1) ( U ) = M (1) . Since the support of the negative part of the right hand sideof 12.6.1 is {∞} , we get that the (a priory rational) map τ X ( U ) : M (1) → M isreally a map of A K -modules.Let M be a C ∞ [ T ]-module obtained from M by restriction of scalars from A K to C ∞ [ T ]. Construction F M is functorial, and we denote this functor by δ .Further, we denote by α the tautological isomorphism M → M . M is a free r -dimensional C ∞ [ T ]-module, and (because M (1) is isomorphic to M ) the samerestriction of scalars of τ X ( U ) defines us a C ∞ [ T ]-skew map from M to M denotedby τ (skew means that τ ( zm ) = z q τ ( m ), z ∈ C ∞ ). τ is defined by the formula τ ( m ) = α ◦ τ X (( α − ( m )) (1) ).It is easy to check that ( M, τ ) is the required t-motive. Really, M is a A K -module, and τ commutes with this multiplication. The fact that the positive partof the right hand side of 12.6.1 is θ Φ means that 1.13.2 holds for M and that theCM-type of the action of A K is Φ. 44 emark 12.6.2. It is easy to prove for this case that M is a free C ∞ [ τ ]-module.Really, it is sufficient to prove (see [G], Lemma 12.4.10) that M is finitely generatedas a C ∞ [ τ ]-module. We choose D such that ∞ 6∈ Supp ( D ). There exists P ∈ K ∗ C such that τ X ( U ) : M (1) → M is multiplication by P (recall that both M (1) , M are A K -submodules of K ). 12.6.0 implies that − ord ∞ ( P ) = n . There exists a number n such that (a) h ( X, O ( D + n ∞ )) >
0; (b) for any k ≥ h ( X, O ( D + ( n + k ) ∞ )) = h ( X, O ( D + n ∞ )) + k (12 . . h ( X, O ( D (1) + ( n + k ) ∞ )) = h ( X, O ( D (1) + n ∞ )) + k (12 . . g , ..., g k are elements of a basis of H ( X, O ( D + ( n + n ) ∞ )), then for any Q ∈ M the element α ( Q ) ∈ M is generated by α ( g ) , ..., α ( g k )over C ∞ [ τ ]. We prove it by induction by n := − ord ∞ ( Q ). If n ≤ n + n the resultis trivial. If not then 12.6.3, 12.6.4 imply that the multiplication by P defines anisomorphism H ( X, O ( D (1) + ( n − n ) ∞ )) /H ( X, O ( D (1) + ( n − n − ∞ )) →→ H ( X, O ( D + n ∞ )) /H ( X, O ( D + ( n − ∞ ))This means that ∃ Q ∈ H ( X, O ( D (1) + ( n − n ) ∞ )), − ord ∞ ( Q ) = n − n suchthat − ord ∞ ( Q − P Q ) ≤ n −
1. An element Q ( − ∈ M exists; since α ( Q ) = τ ( α ( Q ( − )) + α ( Q − P Q ), the result follows by induction. (cid:3) If K and Φ are given then the construction of the Theorem 12.6 defines F uniquelyup to tensoring by O ( D ) where D ∈ Div( X ( K )). We denote the set of these F by F ( { K , Φ } ), and we denote by M ( { K , Φ } ) the set δ ( F ( { K , Φ } )). Further, we denoteby Φ ′ = { α , ..., α r } − Φ the complementary CM-type.
Theorem 12.7.
Let M ∈ M ( { K , Φ } ). Then M ′ exists, and M ′ ∈ M ( { K , Φ ′ } ).More exactly, if F ∈ F ( { K , Φ } ) then F − ⊗ D − ∈ F ( { K , Φ ′ } ) where D is thedifferent sheaf on X , and if M = δ ( F ) then M ′ = δ ( F − ⊗ D − ). Proof.
Let G be any invertible sheaf on X . We have a Lemma 12.7.0.
There exists the canonical isomorphism ϕ G : π ∗ ( G − ⊗D − ) → Hom P ( π ∗ ( G ) , O ). Proof.
At the level of affine open sets ϕ G comes from the trace bilinear form offield extension K / F q ( T ). Concordance with glueing is obvious. (cid:3) We need the relative version of this lemma. Let G , G be invertible sheaveson X , ρ : G → G any rational map. Obviously there exists a rational map ρ − : G − → G − . Recall that we denote by ρ inv : G → G the rational mapwhich is inverse to ρ respectively the composition. The map π ∗ ( ρ − ⊗ D − ) : π ∗ ( G − ⊗ D − ) → π ∗ ( G − ⊗ D − ) is obviously defined. The map (denoted by β ( ρ ))from Hom P ( π ∗ ( G ) , O ) to Hom P ( π ∗ ( G ) , O ) is defined as follows at the level ofaffine open sets: let γ ∈ Hom P ( π ∗ ( G ) , O )( U ) where U is a sufficiently small affine45ubset of P , such that we have a map γ ( U ) : π ∗ ( G )( U ) → O ( U ). Then ( β ( γ ))( U )is the composition map γ ( U ) ◦ π ∗ ( ρ inv )( U ): π ∗ ( G )( U ) π ∗ ( ρ inv )( U ) −→ π ∗ ( G )( U ) γ ( U ) → O ( U ) Lemma 12.7.1.
The above maps form a commutative diagram: π ∗ ( G − ⊗ D − ) π ∗ ( ρ − ⊗D − ) −→ π ∗ ( G − ⊗ D − ) ϕ G ↓ ϕ G ↓ Hom P ( π ∗ ( G ) , O ) β ( ρ ) −→ Hom P ( π ∗ ( G ) , O ) (cid:3) We apply this lemma to the case { ρ : G → G } = { τ X, Φ : F (1) → F } . We have:Div( τ − X, Φ ⊗ D − ) = − Div( τ X, Φ ) = − θ Φ + n ∞ Futher, we multiply τ − X, Φ ⊗ D − by T − θ . We have:Div(( T − θ ) τ − X, Φ ⊗ D − ) = Div( T − θ ) + Div( τ − X, Φ ⊗ D − ) = θ Φ ′ − ( r − n ) ∞ i.e. ( T − θ ) τ − X, Φ ⊗ D is one of τ X, Φ ′ , i.e. F − ⊗ D − ∈ F ( { K , Φ ′ } ). Further,( T − θ ) β ( τ X, Φ ) is the map which is used in the definition of duality of M . Thismeans that the lemma 12.7.1 implies the theorem. (cid:3) Remark 12.8.
There exists a simple proof of the second part of the Theorem5 for uniformizable abelian t-motives M with complete multiplication by A K ⊂ K .Recall that this second part is the proof of 2.7 for M . Really, let us consider thediagram 2.5. The CM-types of action of K on Lie( M ) and on E ( M ) coincide, andthe CM-types of action of K on a vector space and on its dual space coincide. Thismeans that the CM-type of V ∗ is Φ and the CM-type of V ′ is Φ ′ . Further, γ D of2.5 commutes with complete multiplication: this follows immediately for examplefrom a description of γ D given in Remark 5.2.8. Really, all homomorphisms of5.2.9 commute with complete multiplication. For example, this condition for δ of1.11.1 is written as follows: if k ∈ K , m k ( M ), resp. m k ( M ′ ) is the map of completemultiplication by k of M , resp. M ′ , then ( m k ( M ) ⊗ Id ) ◦ δ = ( Id ⊗ m k ( M ′ )) ◦ δ —see any textbook on linear algebra.Finally, since Φ ∩ Φ ′ = ∅ and the map ϕ ′ ◦ γ D ◦ ϕ ∗ commutes with completemultiplication, we get that it must be 0.
13. Miscellaneous.
Let now (
L, V ) be from 12.1, case q = m = 1, i.e. K = W and r = g , and letthe ring of complete multiplication be the maximal order A K . We identify A and Z ∞ via ι , i.e. we consider K as an extension of F q ( θ ). Let Φ be the CM-type of theaction of K on V . This means that — as an A K -module — L is isomorphic to I where I is an ideal of A K . The class of I in Cl( A K ) is defined by L and Φ uniquely;we denote it by Cl( L, Φ).
Remark.
Cl( L, Φ) depends on Φ, because the action of A K on V depends onΦ. Really, let a ∈ L ⊂ V , a = ( a , ..., a n ) its coordinates, Φ = { α i , ..., α i n } ⊂ α , ..., α r } and k ∈ A K . Then ka has coordinates ( α i ( k ) a , ..., α i n ( k ) a n ), i.e.depends de Φ. Particularly, the A K -module structure on L depends on Φ, andhence Cl( L, Φ) depends on Φ. For example, if n = 1, r = 2, Φ = { α } , Φ = { α } ,then Cl( L, Φ ) is the conjugate of Cl( L, Φ ). Theorem 13.1.
Cl( L ′ , Φ ′ ) = (Cl( d )) − (Cl( L, Φ)) − where d is the differentideal of the ring extension A K / A . Proof.
This theorem follows from the above results; nevertheless, I give here anexplicit elementary proof. Let a ∗ = ( a , ..., a r ) t be a basis (considered as a vectorcolumn) of K over F q ( θ ) and b ∗ = ( b , ..., b r ) t the dual basis. Recall that it satisfies2 properties:(1) ∀ i = j α i ( a ∗ ) t α j ( b ∗ ) = 0 ( i.e. r X k =1 α i ( a k ) α j ( b k ) = 0) (13 . . x ∈ K let m x,a ∗ (resp m x,b ∗ ) be the matrix of multiplication by x in thebasis a ∗ (resp. b ∗ ). Then for all x ∈ K we have m x,a ∗ = m tx,b ∗ (13 . . I n,r − n as an r × r block matrix (cid:18) I r − n − I n (cid:19) , and we define a newbasis ˜ b ∗ = (˜ b , ..., ˜ b r ) t by ˜ b ∗ = I n,r − n b ∗ (13 . . b , ..., ˜ b r ) = ( b n +1 , ..., b r , − b , ..., − b n )).We can assume that Φ = { α , ..., α n } . Since L has multiplication by A K and theCM-type of this multiplication is Φ, it is possible to choose a ∗ such that L ⊂ C n ∞ is generated over Z ∞ by e , ..., e r where e i = ( α ( a i ) , ..., α n ( a i )) (13 . . L ⊂ C r − n ∞ be generated over Z ∞ by ˆ e , ..., ˆ e r whereˆ e i = ( α n +1 (˜ b i ) , ..., α r (˜ b i )) (13 . . Lemma 13.1.6. L ′ = ˆ L . Proof.
Let A (resp. B ) be a matrix whose lines are the lines of coordinatesof e , ..., e n (resp. e n +1 , ..., e r ) in 13.1.4, and C (resp. D ) a matrix whose lines arethe lines of coordinates of ˆ e , ..., ˆ e r − n (resp. ˆ e r − n +1 , ..., ˆ e r ) in 13.1.5. By definitionof Siegel matrix, we have L = L ( BA − ), ˆ L = L ( DC − ) ( L is defined in 3.1, 3.2).So, it is sufficient to prove that ( BA − ) t = DC − , i.e. A t D = B t C . This followsimmediately from the definition of A, B, C, D and (13.1.1). (cid:3)
For x ∈ A K we denote by M x ( L ) the matrix of multiplication by x in the basis e ∗ (see the notations of Remark 3.8). Obviously M x ( L ) = m x,a ∗ .Let now A K acts on C r − n ∞ (the ambient space of L ′ ) by CM-type Φ ′ . According(13.1.2) and (13.1.3), the matrix of the action of x ∈ A K in the basis ˜ b ∗ is I n,r − n m tx,a ∗ I − n,r − n (13 . . M , M ′ be from Remark 3.8. Formula 3.8.4 shows that M ′ = I n,r − n M t I − n,r − n (13 . . (cid:3) The reader can think that Theorem 13.1 is incompatible with the main theoremof complex multiplication, because of the − A over a number field coincides with the CM-type of A ′ , while we see thatit is really the complement. Since an analog of even the weak form of the maintheorem of complex multiplication — Theorem 13.2.6 — for the function field caseis not proved yet, the main result of the present section — Theorem 13.2.8 — isconditional: it affirms that if this weak form of the main theorem — Conjecture13.2.7 — is true for a t-motive with complete multiplication M , then it is truefor M ′ as well. By the way, even if it will turn out that the statement of theConjecture 13.2.7 is not correct, the proof of 13.2.8 will not be affected, becausethe main ingredient of the proof is the formula 13.2.10 ”neutralizing” the − A = C n /L with complex multiplication by K . The set Hom( K, ¯ Q ) consistsof n pairs of mutually conjugate inclusions { ϕ , ¯ ϕ , ..., ϕ n , ¯ ϕ n } . Φ is a subset of theset Hom( K, ¯ Q ) such that ∀ i = 1 , ..., n we have:Φ ∩ { ϕ i , ¯ ϕ i } consists of one element. (13 . . K on C n is isomorphic to the direct sum of the elements of Φ. Let F be the Galois envelopeof K/ Q , G := Gal ( F/ Q ) , H := Gal ( F/K ) , S := [ α ∈ Φ Hα (13 . . x ∈ F from the right, i.e. by the formula x αβ = ( x α ) β ; for α ∈ Φ we denote by α also a representative in G of the coset α ).We denote H ref := { γ ∈ G | Sγ = S } (13 . . K ref be the subfield of F corresponding to H ref . We have: H ref S − = S − (13 . . S − is an union of cosets of H ref in G . We can identify these cosets withelements of Hom( K ref , ¯ Q ). Φ ref ⊂ Hom( K ref , ¯ Q ) is, by definition, the set of thesecosets. There is a map det Φ ref : K ref × → K × defined as follows:det Φ ref ( x ) := Y α ∈ Φ α ( x ) (13 . . ref ( x ) reallybelongs to K × ). It can be extended to the group of ideles and factorized to thegroup of classes of ideals, we denote this map by det Cl Φ ref : Cl( K ref ) → Cl( K ).Finally, let θ ref : Gal ( K ref Hilb /K ref ) → Cl( K ref ) be an isomorphism defined bythe Artin reciprocity law.We consider the case End( A ) = O K . In this case L is isomorphic to an idealof O K , its class is well-defined by the class of isomorphism of A , we denote it byCl( A ). Theorem 13.2.6. A is defined over K ref Hilb ;For any γ ∈ Gal ( K ref Hilb /K ref ) we haveCl( γ ( A )) = det Cl Φ ref ◦ θ ref ( γ ) − (Cl( A )). (cid:3) This is a weak form of [SH71], Theorem 5.15 — the main theorem of complexmultiplication.Now we define analogous objects for the function field case in order to formulatea conjectural analog of Theorem 13.2.6. Let K , Φ be from 12.5.3. K ref , Φ ref ,det Φ ref are defined by the same formulas 13.2.2 – 13.2.5 like in the number fieldcase ( Q must be replaced by F q ( T )). The facts that 13.2.1 has no meaning in thefunction field case and that the order of S is not necessarily the half of the orderof G do not affect the definitions.The ∞ -Hilbert class field of K (denoted by K Hilb ∞ ) is an abelian extension of K corresponding to the subgroup K ∗∞ · Y v = ∞ O ∗ K v · K ∗ of the idele group of K . We have an isomorphism θ : Gal ( K Hilb ∞ / K ) → Cl( A K ).We formulate the function field analog of Theorem 13.2.6 only for the case when (*) There exists only one point over ∞ ∈ P ( F q ) in the extension K ref / F q ( T ).In this case the field K ref Hilb ∞ and the ring A K ref are naturally defined, andwe have an isomorphism θ ref : Gal ( K ref Hilb ∞ / K ref ) → Cl( A K ref ).Let M be an uniformizable t-motive of rank r and dimension n having completemultiplication by A K , and Φ its CM-type. Cl( M ) is defined like Cl( A ) in thenumber field case, it is Cl( L, Φ) of 13.1.
Conjecture 13.2.7.
If (*) holds, then M is defined over K ref Hilb ∞ , and forany γ ∈ Gal ( K ref Hilb ∞ / K ref ) we have Cl( γ ( M )) = det Cl Φ ref ◦ θ ref ( γ ) − Cl( M ).Now we can formulate the main theorem of this section. Theorem 13.2.8.
If conjecture 13.2.7 is true for M then it is true for M ′ . Proof.
It follows immediately from the functional analogs of 13.2.2 – 13.2.4 that( K , Φ ′ ) ref = ( K ref , (Φ ref ) ′ ) (13 . . Cl Φ ′ ref = (det Cl Φ ref ) − (13 . . ref ( x ) · det(Φ ref ) ′ ( x ) = N K ref / F q ( T ) ( x ) ∈ F q ( T ) × , hence gives thetrivial class of ideals (we use here (13.2.9). Finally, for γ ∈ Gal ( K ref ) we have( γ ( M )) ′ = γ ( M ′ ) (13 . . M ) isCl( L, Φ) of 13.1). (cid:3)
We give here an elementary explicit proof ofthe theorem 12.7 in two simple cases: K = F q r ( T ) and F q ( T /r ). By the way, sincethe extension F q r ( T ) / F q ( T ) is not absolutely irreducible, formally this case is notcovered by the theorem 12.7. Case A K = F q r [ T ] . Let α i , where i = 0 , ..., r −
1, be inclusions K → C ∞ . For ω ∈ F q r we have α i ( ω ) = ω q i . Let0 ≤ i < i < ... < i n ≤ r − . . { α i j } , j = 1 , ..., n . We consider the following t-motive M = M ( K , Φ). Let e , ..., e n be a basis of M C ∞ [ τ ] such that m ω ( e j ) = ω q ij e j andthe multiplication by T is defined by formulas T e = θe + τ i − i n + r e n (13 . . T e j = θe j + τ i j − i j − e j − , j = 2 , ..., n (13 . . M has complete multiplication by A K , and its CM-typeis Φ. Remark.
It is possible to prove that M ( K , Φ) is the only t-motive having theseproperties; we omit the proof.
Proposition 13.3.3.
For A K = F q r [ T ] we have: M ( K , Φ) ′ = M ( K , Φ ′ ). Proof.
Elements τ j e k for k = 1 , ..., n , j = 0 , ..., i k +1 − i k − k < n and j = 0 , ..., i − i n + r − k = n form a basis of M C ∞ [ T ] . Let us arrange theseelements in the lexicographic order ( τ j e k precedes to τ j e k if k < k ) and makea cyclic shift of them by i denoting e by f i +1 , τ i − i − e by f i etc. until τ i − i n + r − e n = f i . Formulas 13.3.1, 13.3.2 become τ ( f i ) = f i +1 if i
6∈ { i , ..., i n } τ ( f i ) = ( T − θ ) f i +1 if i ∈ { i , ..., i n } ( i mod r , i.e. f r +1 = f ). Formula 1.10.1 shows that in the dual basis f ′∗ we have τ ( f ′ i ) = f ′ i +1 if i ∈ { i , ..., i n } τ ( f ′ i ) = ( T − θ ) f ′ i +1 if i
6∈ { i , ..., i n } which proves the proposition. (cid:3) Case A K = F q [ T /r ] , ( r, q ) = 1 . In order to define M ( K , Φ) we need morenotations. We denote θ /r and T /r by s and S respectively, and let ζ r be a50rimitive r -th root of 1. Let α i , i < i < ... < i n and Φ be the same as inthe case A K = F q r [ T ]. We have α i ( S ) = ζ ir S . Further, we consider an overring C ∞ [ S, τ ] of C ∞ [ T, τ ] ( S is in the center of this ring), and we consider the categoryof modules over C ∞ [ S, τ ] such that the condition 1.9.2 is changed by a weakenedcondition 13.3.4 (here A S, ∈ M n ( C ∞ ) is defined by the formula Se ∗ = A S e ∗ , where A S ∈ M n ( C ∞ )[ τ ], A S = P ∗ i =0 A S,i τ i ): A rS, = θI n + N (13 . . M be a C ∞ [ S, τ ]-module such that dim ¯ M C ∞ [ S ] = 1, f the only element of abasis of ¯ M C ∞ [ S ] and τ f = ( S − ζ i r s ) · ... · ( S − ζ i n r s ) f By definition, M = M ( K , Φ) is the restriction of scalars from C ∞ [ S, τ ] to C ∞ [ T, τ ]of ¯ M . Like in the case A K = F q r [ T ], it is easy to check that M has completemultiplication by A K with CM-type Φ, and it is possible to prove that it is the onlyt-motive having these properties. Proposition 13.3.5.
For A K = F q [ T /r ], ( r, q ) = 1 we have: M ( K , Φ) ′ = M ( K , Φ ′ ). Proof.
For i = 1 , ..., r we denote f i = S i − f . These f ∗ = f ∗ (Φ) form a basis of M C ∞ [ T ] , and the matrix Q = Q ( f ∗ , Φ) of multiplication of τ in this basis has thefollowing description. We denote by σ k (Φ) the elementary symmetric polynomial σ k ( ζ i r , ..., ζ i n r ).The first line of Q is σ n (Φ) s n σ n − (Φ) s n − ... σ (Φ) s ... i -th line is obtained from the first line by 2 operations:1. Cyclic shift of elements of the first line by i − i + n − r elements of the obtained line by T .We consider another basis g ∗ = g ∗ (Φ) of M C ∞ [ T ] obtained by inversion of orderof f i , i.e. g i = f r +1 − i . The elements of Q ( g ∗ ) are obtained by reflection of positionsof elements of Q ( f ∗ ) respectively the center of the matrix.The theorem for the present case follows from the formula Q ( f ∗ , Φ) Q ( g ∗ , Φ ′ ) t = ( T − θ ) I r whose proof is an elementary exercise: let Φ ′ = { j , ..., j r − n } ; we apply equality σ k ( x , ...x r ) = X l σ l ( x i , ..., x i n ) σ k − l ( x j , ..., x j r − n )to 1 , ζ r , ..., ζ r − r . (cid:3) Recall notations of 1.16. Let L be a finite extension of F q ( θ ), p a valuation of L over a valuation P = ∞ of F q ( θ ), and we denote ι − ( P ) ⊂ A by P . Let M be a t-motive defined over L having a good ordinary reduction ˜ M at p and such that the dual M ′ exists. According 1.15.1, the L -structure on M ′
51s well-defined. We denote by M P , the kernel of the reduction map M P → ˜ M P .Condition of ordinarity means that M P , = ( A / P ) n . Conjecture 13.4.1.
For the above M , M ′ we have: M P , and M ′P , are mutually dual with respect to the pairing of Remarks 4.2,5.1.6 (recall that conjecturally M ′ also has good ordinary reduction at p ). Proof for a particular case: M is a Drinfeld module, P = T .(1.9.1) for M has a form T e = θe + a τ e + ...a r − τ r − e + τ r e Condition of good ordinary reduction means a i ∈ L , ord p ( a i ) ≥
0, ord p a = 0.Let x ∈ M T , y ∈ M ′ T ; we can consider x (resp. y ) as an element of C ∞ (resp. C r − ∞ ) satisfying some polynomial equation(s). Considering Newton polygon ofthese polynomials we get immediately (1) for both M , M ′ . Let y = ( y , ..., y r − )be the coordinates of y ; explicit formula (5.3.5) for the present case has the form < x, y > M = Ξ( xy qr − + x q y + x q y + ... + x q r − y r − )The same consideration of the Newton polygon of the above polynomials shows thatfor x ∈ M T, , y ∈ M ′ T, we have ord p x , ord p y i ≥ / ( q − p Ξ = − / ( q − p ( < x, y > M ) > < x, y > M ∈ F q ) we have < x, y > M = 0. Dimensions of M T, , M ′ T, are complementary, hence they aremutually dual. (cid:3) Remark 13.4.2.