Finite group subschemes of abelian varieties over finite fields
aa r X i v : . [ m a t h . AG ] N ov FINITE GROUP SUBSCHEMES OF ABELIAN VARIETIESOVER FINITE FIELDS
SERGEY RYBAKOV
Abstract.
Let A be an abelian variety over a finite field k . The k -isogeny class of A is uniquelydetermined by the Weil polynomial f A . We assume that f A is separable. For a given primenumber ℓ = char k we give a classification of group schemes B [ ℓ ], where B runs through theisogeny class, in terms of certain Newton polygons associated to f A . As an application weclassify zeta functions of Kummer surfaces over k . Introduction.
Throughout this paper k is a finite field F q of characteristic p , and k alg is an algebraic closureof k . Let A be an abelian variety of dimension g over k . Let A [ m ] be the group subschemeof A annihilated by a natural number m . Fix a prime number ℓ = p . We say that A [ ℓ ] is the ℓ -torsion of A . In this paper we classify ℓ -torsion of abelian varieties in two cases: when theWeil polynomial is separable, and for abelian surfaces. This result is similar to the classificationof groups of k -points A ( k ) (see [Ry10]). These two problems are closely related, but the formerone seems to be easier.Denote by A m = A [ m ]( k alg ) the kernel of multiplication by m in A ( k alg ). Let T ℓ ( A ) = lim ←− A ℓ r be the ℓ -th Tate module of A , and let V ℓ ( A ) = T ℓ ( A ) ⊗ Z ℓ Q ℓ be the corresponding vector space over Q ℓ . Then T ℓ ( A ) is a free Z ℓ -module of rank 2 g . The Frobenius endomorphism F of A acts on theTate module by a semisimple linear transformation, which we also denote by F : T ℓ ( A ) → T ℓ ( A ).The characteristic polynomial f A ( t ) = det( t − F )is called the Weil polynomial of A . It is a monic polynomial of degree 2 g with rational integercoefficients independent of the choice of prime ℓ . It is well known that for isogenous varieties A and B we have f A ( t ) = f B ( t ). Tate proved that the isogeny class of an abelian variety isdetermined by its characteristic polynomial, that is f A ( t ) = f B ( t ) implies that A is isogenous to B [Ta66].This gives a nice description of isogeny classes of abelian varieties over k in terms of Weilpolynomials. It seems natural to consider classification problems concerning abelian varietiesinside a given isogeny class. Our goal is to describe the Frobenius action on ℓ -torsion of abelianvarieties in a given isogeny class in terms of corresponding Weil polynomial. Since A [ ℓ ]( k alg ) isan F ℓ -vector space, we have to describe possible matrices of the Frobenius action on such vectorspaces.In the second section we reduce the problem to a particular linear algebra question. Here is asimplified version of the question. Let N be a nilpotent d × d matrix over F ℓ , and let Q ∈ Z ℓ [ t ]be a polynomial of degree d such that Q ≡ t d mod ℓ . Is it possible to find a matrix M over Z ℓ such that the characteristic polynomial of M is Q , and M ≡ N mod ℓ ? We will refer to thisquestion as lifting of the nilpotent matrix N to Z ℓ with respect to Q . Mathematics Subject Classification.
Key words and phrases. abelian variety, finite field, Weil polynomial, Newton polygon, Young polygon.The author is partially supported by AG Laboratory GU-HSE, RF government grant, ag. 11 11.G34.31.0023,and by RFBR grants no. 11-01-12072, 11-01-00395 and 10-01-93110-CNRSLa.E-mail address: [email protected], [email protected].
The main results of the paper are proved in section 3. First we associate to a nilpotent matrix N a polygon of special type. Let m ≥ · · · ≥ m r be the dimensions of the Jordan cells of N .The numbers m , . . . , m r determine the matrix up to conjugation. The Young polygon
Yp( N )of N is the convex polygon with vertices ( P ij =1 m j , i ) for 0 ≤ i ≤ r . For a polynomial Q ∈ Z [ t ]we denote by Np ℓ ( Q ) the Newton polygon of Q with respect to ℓ (see Section 3 for a precisedefinition). Assume that Q is separable. The main result of section 3 can be reformulated asfollows: one can lift N to Z ℓ with respect to Q if and only if Np( Q ) lies on or above Yp( N ) (seeTheorems 3.1 and 3.2). This result allows one to classify ℓ -torsion of abelian varieties belongingto an isogeny class corresponding to the Weil polynomial without multiple roots (Corollaries 3.6and 3.7).In section 4 we establish a relationship between Young polygons for the Frobenius actions onan abelian variety and its dual. We also treat the following question due to B. Poonen: is it truethat for an abelian surface A the group of k -rational points A ( k ) is isomorphic to the the groupof k -rational points b A ( k ) on its dual? The answer is no, and we give a counterexample.In section 5 we prove that (generalized) matrix factorizations correspond to Tate modules.This technique turns out to be useful when Weil polynomial is not separable. In section 6 weexplicitly classify ℓ -torsion of abelian surfaces. In the final section we apply this result to theclassification of zeta functions of Kummer surfaces. Acknowledgements.
I am deeply grateful to Alexander Kuznetsov, who communicatedhis unfinished results on zeta functions of Kummer surfaces to me and provided many usefulcorrections on the early version of the paper. I thank Michael A. Tsfasman for his attention tothis work. I am grateful to Alexey Zykin and referees for suggesting many useful corrections andcomments on the paper. 2.
Preliminaries
Finite group subschemes of abelian varieties.
A finite ´etale group scheme G over k isuniquely determined by the Frobenius action on G ( k alg ) (see [De78]). If ℓ · G = 0, then G ( k alg )is an F ℓ -vector space and Frobenius action is F ℓ -linear. By definition of the Tate module, wehave A [ ℓ ]( k alg ) ∼ = T ℓ ( A ) /ℓT ℓ ( A ). Thus the structure of a group scheme on A [ ℓ ] depends only onthe module structure on T ℓ ( A ) over R = Z ℓ [ F ] ⊂ End k ( A ). Moreover, since the action of F on V ℓ ( A ) is semisimple, f A determines the R -module V ℓ ( A ) uniquely up to isomorphism.The following lemma shows what R -modules can arise as Tate modules of varieties from afixed isogeny class. Lemma 2.1. [Mil08, IV.2.3] If f : B → A is an isogeny then, T ℓ ( f ) : T ℓ ( B ) → T ℓ ( A ) is anembedding of R -modules, and if T denotes its image then (1) F ( T ) ⊂ T and T ⊗ Z ℓ Q ℓ ∼ = T ℓ ( A ) ⊗ Z ℓ Q ℓ . Conversely, if T ⊂ T ℓ ( A ) is a Z ℓ -submodule such that (1) holds, then there exists an abelianvariety B defined over k and an isogeny f : B → A such that T ℓ ( f ) induces an isomorphism T ℓ ( B ) ∼ = T . (cid:3) Generalized Jordan form.
Let K be a field, and let λ be an algebraic number over K . Put L = K ( λ ). Take a vector space L r with a natural basis v , . . . , v r . Let M : L r → L r bea linear transformation such that its matrix is a sum of Jordan cells with eigenvalue λ , i.e. M = λI r + N , where I r is the identity matrix and N is a nilpotent matrix of dimension r .The set { λ j v i | ≤ i ≤ r, ≤ j ≤ n − } is a basis of L r as a K -vector space. Denote by J ( λ, N ) the matrix of M in this basis. It is called generalized Jordan cell . We have the followinggeneralization of the Jordan decomposition theorem. Theorem 2.2.
Let M be a linear transformation of a K -vector space V with the characteristicpolynomial P . Suppose that any irreducible divisor of P is separable. Let ∆ be the set of roots of INITE GROUP SUBSCHEMES OF ABELIAN VARIETIES 3 P , and Λ ⊂ ∆ be the image of a section of the natural map ∆ → ∆ / Gal( K sep /K ) , i.e. for anyroot δ ∈ ∆ there exists a unique λ ∈ Λ which is conjugate to δ . Then there exists a basis of V such that the matrix of M is a direct sum of generalized Jordan cells J ( λ, N λ ) for λ ∈ Λ . Thisdata determines M uniquely up to isomorphism over K .Proof. Let P = Q λ ∈ Λ P d λ λ be the decomposition of P into a product of monic irreducible separablepolynomials P λ ∈ K [ t ] such that P λ ( λ ) = 0 for any λ ∈ Λ. Then by the Chinese remaindertheorem R = K [ t ] /P ( t ) K [ t ] ∼ = Y λ ∈ Λ K [ t ] /P λ ( t ) d λ K [ t ] . The vector space V is an R -module such that the image of t in R acts on V as M . Put L λ = K [ t ] /P λ ( t ) K [ t ], and R λ = K [ t ] /P λ ( t ) d λ K [ t ]. It follows that V ∼ = ⊕ V λ , where V λ = R λ V is an R λ -module. For any λ ∈ Λ the polynomial P λ is separable, thus Spec L λ is smooth overSpec K . By [EGA4, 17.5.1] (see also [Ha77, II. exercise 8.6]) we can find a section ψ λ of thenatural morphism ϕ λ : R λ → L λ , i.e., R λ is an L λ -algebra, and V λ has a structure of an L λ -vector space. Denote by t λ ∈ R λ the image of t under the natural projection K [ t ] → R λ . Then λ = ϕ λ ( t λ ), and n λ = t λ − ψ λ ( λ ) ∈ R λ is in the kernel of ϕ λ . Thus n d λ λ = 0, i.e. n λ acts on V λ asa nilpotent matrix N λ . We see that t λ acts on V λ as a generalized Jordan cell J ( λ, N λ ).Finally, we have to prove that if ⊕ λ ∈ Λ J ( λ, N λ ) is conjugate to ⊕ λ ∈ Λ J ( λ, N ′ λ ) over K , thenfor all λ ∈ Λ the matrix N λ is conjugate to N ′ λ over L λ . Indeed, these matrices have the samedimension d λ , and are conjugate by the Jordan decomposition theorem over L λ . (cid:3) Remark 2.3.
We proved that R λ ∼ = L λ [ t ] / ( t − λ ) d λ L λ [ t ]. We use this isomorphism later. Reduction step 1.
For a polynomial P ∈ Z ℓ [ t ] denote by ¯ P ∈ F ℓ [ t ] its reduction modulo ℓ , and by P ∈ Z ℓ [ t ] the unitary separable polynomial with the same set of roots as P . Wecall P the minimal polynomial of P . The Frobenius action on V ℓ ( A ) is semisimple, thus theminimal polynomial f of f is the minimal polynomial of the Frobenius action. It follows that R ∼ = Z ℓ [ t ] /f ( t ) Z ℓ [ t ].The Galois group Gal( F ℓ / F ℓ ) acts on the set ∆ of roots of ¯ f . Let Λ ⊂ ∆ be the image ofa section of the natural map ∆ → ∆ / Gal( F ℓ / F ℓ ). By Theorem 2.2 applied to the action ofFrobenius on T ℓ ( A ) /ℓT ℓ ( A ) the matrix of F is conjugate to the sum of J ( λ, N λ ) for λ ∈ Λ. Wegeneralize this result to Tate modules.By the Hensel lemma [EGA4, 18.5.13], we can decompose f into the product of monic poly-nomials f λ ∈ Z ℓ [ t ] such that ¯ f λ is a power of an irreducible monic polynomial corresponding to λ ∈ Λ. We have a natural homomorphism of rings ϕ : R → Y λ ∈ Λ Z ℓ [ t ] /f λ ( t ) Z ℓ [ t ] . Since P λ is monic, R λ = Z ℓ [ t ] /f λ ( t ) Z ℓ [ t ] is free and finitely generated Z ℓ -module. By the Chineseremainder theorem ϕ is an isomorphism modulo ℓ . On the other hand, ϕ is a homomorphism offinitely generated free Z ℓ -modules. It follows that ϕ is an isomorphism.The module T ℓ ( A ) is an R -module such that the image of t in R acts as Frobenius. Put T λ = R λ T ℓ ( A ), then T ℓ ( A ) = ⊕ T λ . By Theorem 2.2, the matrix of the action of t on T λ /ℓT λ is ofthe form J ( λ, N λ ) in some basis. We now sum up our observations. Proposition 2.4.
There is an isomorphism of R -modules T ℓ ( A ) ∼ = ⊕ λ ∈ Λ T λ such that F acts on T λ /ℓT λ with matrix J ( λ, N λ ) in some basis. Reduction step 2.
Let L λ be an unramified extension of Q ℓ with residue field F ℓ ( λ ). Denoteby S λ the ring of integers of L λ . Proposition 2.5.
There is an isomorphism R λ ∼ = S λ [ t ] /gS λ [ t ] for some g ∈ S λ [ t ] such that g ≡ ( t − λ ) d mod ℓS λ , where d is the multiplicity of λ in ¯ f λ . SERGEY RYBAKOV
Proof.
Let ∆ λ be the set of roots of ¯ f λ . Then f λ ≡ Y δ ∈ ∆ λ ( t − δ ) d δ mod ℓS λ , for some natural numbers d δ . By the Hensel lemma, f λ equals to a product of monic polynomials g δ ∈ S λ [ t ] such that g δ ≡ ( t − δ ) d δ mod ℓS λ , where δ ∈ ∆ λ . Define the homomorphism of rings R λ → R λ ⊗ Z ℓ S λ by r r ⊗
1. By the Chinese remainder theorem, R λ ⊗ Z ℓ S λ ∼ = Y δ ∈ ∆ λ Z δ where Z δ ∼ = S λ [ t ] /g δ S λ [ t ]. Take the projection R λ ⊗ Z ℓ S λ → Z λ . We get a homomorphism ϕ : R λ → Z λ . It is a homomorphism of free Z ℓ -modules and an isomorphism modulo ℓ byRemark 2.3. We conclude that ϕ is an isomorphism. Put g = g λ . (cid:3) Choose an element α ∈ S λ such that ¯ α = λ . The polynomial Q λ ( t ) = g ( t − α ) is the minimalpolynomial of F − α acting on T λ . Clearly, Q λ ≡ t d mod ℓ , where d = deg Q λ . We have reducedour task to the following linear algebra problem. The problem.
Let L be an unramified extension of Q ℓ , and let S be its ring of integers.Suppose we are given a polynomial Q ∈ S [ t ] such that Q ≡ t d mod ℓ , where d = deg Q . Let V be an L -vector space of dimension d , and let E be a semisimple linear transformation on V withcharacteristic polynomial Q . Denote by Q the minimal polynomial of E . Put R = S [ t ] /Q ( t ) S [ t ].We give a structure of an R -module on V such that t acts as E . Describe all isomorphism classesof finite R -modules of the form T /ℓT , where T is an arbitrary R -invariant S -lattice in V .If we choose a basis of T , the problem can be reformulated as follows. Let N be the matrixof the action of E on T /ℓT in some basis over the finite field
S/ℓS . It is a nilpotent matrix over
S/ℓS , since Q ≡ t d mod ℓ . Is it possible to find a matrix M over S such that Q ( t ) = det( t − M ),and M ≡ N mod ℓ ? We will refer to this question as lifting of nilpotent matrix N to S withrespect to Q . 3. ℓ -torsion of abelian varieties Let S be the ring of integers in an unramified extension L of Q ℓ . Assume we are given afinitely generated free S -module T endowed with an S -linear injective endomorphism E whichinduces on T /ℓT a nilpotent endomorphism N . Let Q ( t ) = det( t − E ). In this section we give apartial answer to the question: when is it possible to lift N to S with respect to Q ? Using thisresult we get a classification of group schemes of the form A [ ℓ ] for A from a fixed isogeny classsuch that f A is separable.We associate to N a polygon of special type. For a sequence of natural numbers m ≥· · · ≥ m r > the Young polygon Yp( m , . . . , m r ) as the convex polygon with vertices( P ij =1 m j , i ) for 0 ≤ i ≤ r . The dimension of Y = Yp( m , . . . , m r ) is dim Y = P rj =1 m j . The height of Y is Ht( Y ) = r .There is a basis of T /ℓT such that the matrix of N is a sum of Jordan cells of dimensions m , . . . , m r . Clearly, numbers m , . . . , m r determine N uniquely up to conjugation. We associateto N the Young polygon Yp( N ) given by the sequence m , . . . , m r . We also denote this Youngpolygon by Yp( E | T ).The Young polygon has (0 ,
0) and ( d, r ) as its endpoints, and its slopes are 1 /m , . . . , /m r .For example, the following picture shows Young polygons for the zero matrix (Pic. 1) and theJordan cell of dimension two (Pic. 2). INITE GROUP SUBSCHEMES OF ABELIAN VARIETIES 5 ν the normalized valuation on L , i.e. ν ( ℓ ) = 1. Suppose that Q ( t ) = P i Q i t d − i .Take the lower convex hull of the points ( i, ν ( Q i )) for 0 ≤ i ≤ deg Q in R . The boundary of thisregion is called the Newton polygon Np( Q ) of Q . Its vertices have integer coefficients, and (0 , d, ν ( Q d )) are its endpoints. The slopes of Q are the slopes of this polygon. Note that eachslope has a multiplicity. Theorem 3.1.
The Newton polygon
Np( Q ) lies on or above Young polygon Yp( E | T ) .Proof. Let Q ∈ S [ t ] be the minimal polynomial of E , and let R = S [ t ] /Q ( t ) S [ t ]. Let x ∈ R bethe image of t under the natural projection. The module T is naturally an R -module such that x acts as E .Suppose Yp( E | T ) = Yp( m , . . . , m r ). Take generators v ′ , . . . , v ′ r of T /ℓT over R such that v ′ , xv ′ . . . , x m − v ′ , . . . , v ′ r , . . . , x m r − v ′ r is a Jordan basis for x . By the Nakayama lemma there exist generators v , . . . , v r of T over R which lift v ′ , . . . , v ′ r . Let H be a matrix of x in the basis v , xv . . . , x m − v , . . . , v r , . . . , x m r − v r , and let H i ,...,i m be the determinant of the submatrix of H cut by the columns and rows with thenumbers i , . . . , i m . The characteristic polynomial of x acting on T is Q ( t ) = d X m =0 Q m t d − m , and Q m = ( − m X i < ··· m + · · · + m s − , then the set { i , . . . , i m } contains no lessthan s blocks, and ν ( H i ,...,i m ) ≥ s . Thus if ( m, ν ( Q m )) is a vertex of Np( Q ) then ν ( Q m ) ≥ s ,and Np( Q ) lies on or above Yp( E | T ). (cid:3) Theorem 3.2.
Let R = S [ t ] /Q ( t ) S [ t ] , and let V = R ⊗ Z ℓ Q ℓ . Let Y be a Young polygon suchthat Np( Q ) lies on or above Y . Then there exists an R -lattice T in V such that Yp( x | T ) = Y . SERGEY RYBAKOV
Proof.
Recall that Q = det( t − x | V ) is the minimal polynomial of the action of x on V . Let Y = Y ( m , . . . , m r ). First, we find a lattice T in V over S such that R ⊂ T ⊂ V . After that weprove that T is an R -module.Let m = m + · · · + m s , and let Q ( t ) = P di =0 Q i t d − i . For 1 ≤ s ≤ r we put v s +1 = x m + P mj =1 Q j x m − j ℓ s . In addition, let v = 1, and let v r +1 = 0. Note that v , xv . . . , x m − v , . . . , v r , . . . , x m r − v r have different degrees viewed as polynomials in x , and hence generate a lattice T over S .Now we prove that T is an R -module. The point ( m − m s , s −
1) is a vertex of Y . Byassumption, Np( Q ) lies on or above Y , thus ( m − m s , s −
1) is not higher than Np( Q ). It followsthat ℓ s divides Q j for all j > m − m s . Thus u s = P mj = m − m s +1 Q j x m − j ℓ s ∈ S · ⊂ T. Moreover, x m s v s = ℓ ( v s +1 − u s ) ∈ ℓT. This proves that xT ⊂ T , and that Yp( x | T ) = Y . (cid:3) Example 3.3.
Let Q ( t ) = t − ℓt − ℓ . Its Newton polygon is drawn on Picture 2. Then we canlift the nonzero nilpotent Jordan cell (its Young polygon is equal to Np( Q )). For example, take M = (cid:18) ℓ ℓ (cid:19) . Clearly, Q ( t ) = det( t − M ). We can not lift the zero matrix, because its Young polygon (seePic. 1) is higher than Np( Q ).Note that if Q is not separable, then the action of x on V is not semisimple. By Theorems 3.1and 3.2 we get Corollary 3.4.
Suppose that Q is separable. One can lift N to S with respect to Q if and onlyif Np( Q ) lies on or above Yp( N ) . Suppose that Q is not separable, and N is a nilpotent matrix such that Np( Q ) lies on or aboveYp( N ). Then in general it is not possible to lift N to S with respect to Q . We discuss a partialsolution of this problem in section 5. Now we prove the following simple result. Proposition 3.5.
Suppose Q = P r , where deg P = 2 , and P is separable. Let R = S [ t ] /P ( t ) S [ t ] ,and let V = ( R ⊗ Z ℓ Q ℓ ) r . There exists an S -lattice T in V such that x acts on T /ℓT with Youngpolygon Y if and only if Np( P r ) lies on or above Y , and all slopes of Y are equal to / or . (cid:3) Proof.
Let N be a nilpotent matrix with Young polygon Y . Suppose such a lattice T exists.Since deg P = 2, any Jordan cell of N has dimension at most 2, thus all slopes of Yp( N ) areequal to 1 / N = ⊕ N i such that dim Yp( N i ) = 2. ByTheorem 3.2 for any i there exists an S -lattice T i such that x acts on T i /ℓT i with the matrix N i .Put T = ⊕ T i . (cid:3) We call a polynomial f ∈ Z ℓ [ t ] distinguished if ¯ f is a power of an irreducible polynomial. Wenow use notation of section 2. Let f ∈ Z ℓ [ t ] be the minimal polynomial of f . Choose a root λ of ¯ f and its lifting α λ ∈ S λ . By Proposition 2.5, R λ = Z ℓ [ t ] /f Z ℓ [ t ] ∼ = S λ [ t ] /gS λ [ t ] , INITE GROUP SUBSCHEMES OF ABELIAN VARIETIES 7 and g ≡ ( t − λ ) d mod ℓS λ . Put Q ( t ) = g ( t − α λ ). Note that Q ( t ) divides f ( t − α λ ) over L λ . Takea unitary polynomial Q = Q λ ∈ S λ [ t ] of maximal degree such that Q ( t ) divides f ( t − α λ ) over L λ ,and the minimal polynomial of Q is Q . We could define Q in other way. Let V be a Q ℓ –vectorspace endowed with semisimple linear transformation F such that f ( t ) = det( t − F ). Then V is an L λ –vector space, and Q ( t ) is the characteristic polynomial of the L λ –linear transformation F − α λ .Recall that an ´etale group scheme is uniquely determined by the linear Frobenius action on thegroup of k alg -points. Let Y be a Young polygon of dimension deg Q such that Np( Q ) lies on orabove Y , and let N be a nilpotent matrix such that Y = Yp( N ). A distinguished group scheme is a finite ´etale group scheme A ( f, Y ) over k such that dim F ℓ A ( f, Y )( k alg ) = deg f , and F actson A ( f, Y )( k alg ) with the matrix J ( λ, N ) in some basis. Note that for a given f the polynomial Q is uniquely determined modulo ℓ . Thus A ( f, Y ) is uniquely determined by f and Y up to anisomorphism. Corollary 3.6.
Let A be an abelian variety over k . Then A [ ℓ ] is isomorphic to a sum ofdistinguished group schemes.Proof. Let f A = Q λ ∈ Λ f λ be a product of pairwise coprime distinguished polynomials. By Propo-sition 2.4, T ℓ ( A ) ∼ = ⊕ λ ∈ Λ T λ . By Proposition 2.5, R λ ∼ = S λ [ t ] /gS λ [ t ]. This gives a structure ofan S λ -module on T λ . As before, let α λ ∈ S λ be a lifting of λ , and let Q λ be the characteristicpolynomial of the action of F − α λ on T λ . By Theorem 3.1, F acts on T λ /ℓT λ with the matrix J ( λ, N λ ), where N λ is a nilpotent matrix such that Np( Q λ ) lies on or above Yp( N λ ). Thus A [ ℓ ] ∼ = ⊕ λ ∈ Λ A ( f λ , Yp( N λ )). (cid:3) Corollary 3.7.
Let A be an abelian variety over k . Suppose f A is separable, and f A = Q λ ∈ Λ f λ is a product of coprime distinguished polynomials. Then for any family of distin-guished group schemes A ( f λ , Y λ ) there exists an abelian variety B isogenous to A such that B [ ℓ ] ∼ = ⊕ λ ∈ Λ A ( f λ , Y λ ) .Proof. By Proposition 2.4, V ℓ ( A ) ∼ = ⊕ λ ∈ Λ V λ , where V λ = R λ V ℓ ( A ) is an R λ -module. Let α λ be alift of λ . By Theorem 3.2, there exists an S λ -lattice T λ ⊂ V λ such that F − α λ acts on T λ /ℓT λ with Young polygon Y λ . Put T = ⊕ T λ . By Lemma 2.1, there exists a variety B such that T ∼ = T ℓ ( B ). (cid:3) Young polygons and duality. By b A we denote the dual variety of an abelian variety A . Suppose f is a distinguishedpolynomial such that f divides f A , and polynomials f and f A /f have no common roots modulo ℓ . By Proposition 2.4, there exists a direct summand T of T ℓ ( A ) such that F acts on T withcharacteristic polynomial f . The polynomial ˆ f ( t ) = ( tq ) deg f f ( qt ) divides f A . Clearly, ˆ f ( t ) isdistinguished. Denote the corresponding direct summand of T ℓ ( b A ) by b T .Let λ be a root of ¯ f , and let S be the ring of integers in an unramified extension of Q ℓ withresidue field F ℓ ( λ ). By Proposition 2.5, T is an S -module. Clearly, q/λ is a root of ˆ f ( t ), and b T is an S -module too. Proposition 4.1.
Let α ∈ S be a lift of λ . Then Yp( F − α | T ) = Yp( F − q/α | b T ) .Proof. The Weil pairing e : T ℓ ( A ) × T ℓ ( b A ) → Z ℓ is non-degenerate, and e ( F x, F y ) = qe ( x, y ),where x ∈ T ℓ ( A ) and y ∈ T ℓ ( b A ) [Mum70]. This yields that its restriction to T × b T is non-degenerate. By an integral version of Deligne trick [BGK06, Lemma 3.1], there exists an S -linearpairing e S : T × b T → S such that e S ( F x, F y ) = qe S ( x, y ), and e = T r L/ Q ℓ ◦ e S , where L is thefraction field of S . We have e S ( F x, y ) = e S ( F x, F ( F − y )) = e S ( x, ( qF − ) y ) . SERGEY RYBAKOV
Let M = J ( λ, N ) be the matrix of the action of F on T /ℓT in some basis over
S/ℓS , and let c M be the matrix of the action of F on b T /ℓ b T in the dual basis. It follows that c M t = qM − , where · t means transpose. One easily proves that for any cell of M corresponding to the Jordan cellof dimension d there exists a cell of c M corresponding to the same Jordan cell. The propositionfollows. (cid:3) We now give an example of an abelian surface A such that the group of points A ( k ) is notisomorphic to the group of points on the dual surface b A ( k ). Recall that A ( k ) is a kernel of1 − F : A → A , and the ℓ -component A ( k ) ℓ = ker(1 − F ) : T ℓ ( A ) → T ℓ ( A ). Example 4.2.
Let q = 7, and let ℓ = 5. Suppose f a ( t ) = t + 2 t + 7 and f b ( t ) = t − t + 7 areWeil polynomials of two elliptic curves. The polynomial f = f a f b is the Weil polynomial of anabelian surface. Note that f a ( t ) ≡ f b ( t ) ≡ ( t − t − q ) mod 5. Thus we have a decomposition f = f f over Z , where f ≡ ( t − mod 5, and f ≡ ( t − q ) mod 5. For any abelian surface B with Weil polynomial f we have a decomposition T ( B ) ∼ = T ⊕ T , where F acts on T i withcharacteristic polynomial f i for i = 1 ,
2. By Theorem 3.1, F − T / T trivially and byTheorem 3.2, there exists a lattice T in T ⊗ Q such that F − q acts on T / T non-trivially. Inthe first case the Young polygon of F − F − q is Yp(1 , A such that T ( A ) ∼ = T ⊕ T .By the previous proposition, A ( F ) ∼ = Z / Z ⊕ Z / Z , and b A ( F ) ∼ = Z / Z .5. Matrix factorizations.
In this section we turn our attention to the case when the Weil polynomial is not separable.First we establish a connection between matrix factorizations and Tate modules. Given a Tatemodule T with the Weil polynomial f and the minimal polynomial f one can produce the unique(up to isomorphism) Tate module T ′ with the Weil polynomial g = f r /f for some r . If we arelucky, g do not have multiple roots. In this case, one can get some information on T from T ′ using theorem 3.1. Moreover, one can reverse the construction and produce a Tate module T with a given Young polygon starting from T ′ constructed using theorem 3.2.Let S be the ring of integers in a finite unramified extension L of Q ℓ . Fix a pair of polynomials f, f ∈ S [ t ] and a positive integer r . Let R = S [ t ] /f S [ t ], and let ¯ S = S/ℓS . Denote by x ∈ R the image of t under the natural projection from S [ t ]. We assume that f ≡ t d mod ℓ , anddeg f = d . Definition 5.1.
A matrix factorization (with respect to f, f and r ) is a pair ( X, Y ) of r × r matrices with coefficients in S [ t ] such that Y X = f · I r and det X = f .Suppose we are given a matrix factorization ( X, Y ). The matrix X defines a map of free S [ t ]modules: S [ t ] r X −→ S [ t ] r . Its cokernel T is annihilated by f . It is equivalent to say that T is an R -module. We see thatthe matrix factorization ( X, Y ) corresponds to a finitely generated R -module T given by thepresentation:(2) S [ t ] r X −→ S [ t ] r → T → . Proposition 5.2.
The module T is free of finite rank d over S , and characteristic polynomialof the action of x on T is equal to f .Proof. Since f ≡ t d mod ℓ , and deg f = d , the ring R is generated as S -module by theelements 1 , x, . . . , x d − . By definition, T is a finitely generated R -module, thus it is finitelygenerated over S . INITE GROUP SUBSCHEMES OF ABELIAN VARIETIES 9
Take the tensor product of the presentation (2) with ¯ S [ t ]:¯ S [ t ] r ¯ X −→ ¯ S [ t ] r → T ⊗ S ¯ S → . The ring ¯ S [ t ] is a principal ideal domain, thus there exist matrices M and M over ¯ S [ t ] such thatdet M = det M = 1 and M ¯ XM is the diagonal matrix with determinant t d . It follows that M ¯ XM is the diagonal matrix diag( t m , . . . , t m r ) for some m . . . , m r ∈ N such that P m i = d .We get T ⊗ S ¯ S ∼ = ⊕ ri =1 ¯ S [ t ] /t m i ¯ S [ t ] . By the Nakayama lemma, T is generated by d elements over S .Now take the presentation of T ⊗ S L : L [ t ] r X −→ L [ t ] r → T ⊗ S L → . As before, there exist matrices M and M over L [ t ] such that det M = det M = 1, and M XM = diag( g , . . . , g r ). Clearly, T ⊗ S L ∼ = ⊕ ri =1 L [ t ] /g i L [ t ] , and rk T = d . This proves that T is free over S . To conclude the proof we note that thecharacteristic polynomial of the action of x on L [ t ] /g i L [ t ] is equal to g i . (cid:3) The following proposition shows that modules over R give rise to matrix factorizations. Proposition 5.3.
Let T be an R -module which is free of finite rank over S . Suppose that T isgenerated over R by r elements, and that Yp( x | T ) = Yp( m , . . . , m r ) . Then there exists a matrixfactorization ( X, Y ) such that T has presentation (2) , and X ≡ diag( t m , . . . , t m r ) mod ℓ. Proof.
Let v , . . . , v r be generators of T over R . Then x m i v i = P j a ji ( x ) v j , where a ji ∈ S [ t ] anddeg a ji < m j . Let X be the matrix with the entries t m i δ ji − a ji . Define an R -module T ′ by thepresentation: S [ t ] r X −→ S [ t ] r → T ′ → . Put m = P i m i . Then det X ≡ t m mod ℓ , and from the inequalities deg a ji < m j it follows thatdet X is a polynomial of degree m . By Proposition 5.2, T ′ is a free S module of rank m . Bydefinition of T ′ , we have a surjective map of S -modules T ′ → T . Since they have the same rankas S -modules, this map is an isomorphism, and, by Proposition 5.2, det X = f .Multiplying presentation (2) by f we get the commutative diagram S [ t ] r X / / f (cid:15) (cid:15) S [ t ] rf (cid:15) (cid:15) / / T / / (cid:15) (cid:15) S [ t ] r X / / S [ t ] r / / T / / S [ t ] r is free, there exists a matrix Y such that the diagram S [ t ] r X / / S [ t ] rY | | ①①①①①①①① f (cid:15) (cid:15) / / T / / (cid:15) (cid:15) S [ t ] r X / / S [ t ] r / / T / / Y X = f I r . Thus, the pair ( X, Y ) is a matrix factorization. (cid:3)
Example 5.4.
Let deg f = 3, and let f = f . Suppose f is separable. When there exists an R -module T such that x acts with r = 3 Jordan cells of dimension 2? By Proposition 5.3, sucha module exists iff there exists a matrix factorization ( X, Y ) such that X ≡ diag( t , t , t ) mod ℓ. The matrix factorization (
Y, X ) gives a module T ′ over R which is generated by 3 elements andthe characteristic polynomial of x is equal to det Y = f /f = f . Moreover, Y ≡ diag( t, t, t )mod ℓ. It follows that Yp( x | T ′ ) = Yp(1 , , T ′ existsiff Np( f ) lies on or above Yp( x | T ′ ). Thus T exists iff Np( f ) lies on or above Yp(1 , , ℓ -torsion of abelian surfaces. In this section we classify isomorphism classes of ℓ -torsion subschemes of abelian surfaces.We use the following notation. Let P = Q λ ∈ Λ P λ be the decomposition of a polynomial P ( t ) ∈ Z ℓ [ t ] into a product of distinguished polynomials. Then A ( P,
0) is the group scheme ⊕ λ ∈ Λ A ( P λ , Y deg P λ ), where Y n is the Young polygon of the zero matrix of dimension n . Theorem 6.1.
Let A be an abelian surface over k with the Weil polynomial f A ( t ) = t + a t + a t + qa t + q . Suppose first that f A is separable, then we have the following five cases: (1): if f A ( t ) is separable modulo ℓ , then A [ ℓ ] ∼ = A ( f A , ; (2): if f A ( t ) ≡ f f mod ℓ , where f ≡ ( t − α ) mod ℓ , and f is separable modulo ℓ , then A [ ℓ ] ∼ = A ( f , ⊕ A ( f , Y ) , where dim Y = 2 ; (3): if f A ( t ) ≡ f f mod ℓ , where f i ≡ ( t − α i ) mod ℓ , for i = 1 , and α α mod ℓ ,then A [ ℓ ] ∼ = A ( f , Y ) ⊕ A ( f , Y ) , where dim Y = dim Y = 2 ; (4): if f A ( t ) ≡ h ( t ) mod ℓ , where h is irreducible modulo ℓ , then A [ ℓ ] ∼ = A ( f A , Y ) , where dim Y = 2 . If ℓ = 2 , and ℓ does not divide a − a + 8 q , or ℓ = 2 , and does not divide a + a + 1 − q , then Y = Yp(2) ; (5): if f A ( t ) ≡ ( t − α ) mod ℓ , then A [ ℓ ] ∼ = A ( f, Y ) , where dim Y = 4 .Suppose that f A is not separable, then we have the following three cases: (6): f A = P , where P is separable. Then (a): if P is separable modulo ℓ , then A [ ℓ ] ∼ = A ( P, ⊕ A ( P, ; (b): if P ( t ) ≡ ( t − α ) mod ℓ , then A [ ℓ ] ∼ = A ( P, Y ) ⊕ A ( P, Y ) , where dim Y = dim Y =2 . (7): f A ( t ) = ( t ±√ q ) ( t − bt + q ) , where P ( t ) = t − bt + q is separable. Let P ( t ) = ( t ±√ q ) . (a): If P P mod ℓ , and P is separable modulo ℓ , then A [ ℓ ] ∼ = A ( P , ⊕ A ( P , . (b): If P P mod ℓ , and P ( t ) ≡ ( t − α ) mod ℓ , then A [ ℓ ] ∼ = A ( P , Y ) ⊕ A ( P , ,where dim Y = 2 . (c): If P ≡ P mod ℓ , then (i): either A [ ℓ ] ∼ = A ( P ( t )( t ± √ q ) , Y ) ⊕ A ( t ± √ q, , where dim Y = 3 ; or (ii): if ℓ divides P ( ∓√ q ) , then A [ ℓ ] ∼ = A ( P , Y ) ⊕ A (( t ± √ q ) , Y ) , where Y =Yp(2) . (8): If f A ( t ) = ( t ± √ q ) , then A [ ℓ ] ∼ = A ( f A , .Conversely, for any group scheme G described above there exists an abelian variety B in theisogeny class of A such that B [ ℓ ] ∼ = G .Proof. Assume that f A is separable. Note that if f A ( t ) ≡ ( t − α ) ( t − β ) mod ℓ , then α ≡ β mod ℓ .Thus by Corollaries 3.6 and 3.7 the cases (1) − (3) and (5) follow. In the case (4) we have A [ ℓ ] ∼ = A ( f A , Y ), where dim Y = 2, and Y is the Young polygon of the zero matrix if and onlyif R = Z ℓ [ t ] /f A ( t ) Z ℓ [ t ] is not a DVR. Indeed, if R is regular, then T ℓ ( A ) is free, and hence theaction of t on T ℓ ( A ) /ℓT ℓ ( A ) is non-trivial. If R is not regular, then the integral closure O of R isan example of an R -module such that O /ℓ O ∼ = A ( f A , k alg ). By the Dedekind lemma [PZ97, INITE GROUP SUBSCHEMES OF ABELIAN VARIETIES 11 R is regular if and only if ( f A − h ) /ℓ is prime to h modulo ℓ . An easy computation showsthat the two polynomials are coprime if and only if ℓ does not divide a − a + 8 q for ℓ = 2,and 4 does not divide a + a + 1 − q for ℓ = 2.Assume now that f A is not separable. It follows from the classification of Weil polynomi-als (see [MN02]), that only the cases (6) − (8) are possible. The case (6) follows from theProposition 3.5, and the case (8) is obvious since Frobenius acts as multiplication by ∓√ q . ByCorollary 3.6, the conditions of (7 a ), (7 b ) and (7 c ( i )) are necessary. Let us prove that they aresufficient. We have to construct Tate module T with the prescribed Frobenius action. Then byLemma 2.1, there exists an abelian variety B in the isogeny class of A such that T ∼ = T ℓ ( B ).We give a construction for the case (7 b ). Put T = T ⊕ T , where T i is a torsion-free module ofrank 1 over R i = Z ℓ [ t ] /P i Z ℓ [ t ]. The module T is uniquely determined, and T can be constructedusing Theorem 3.2. The case (7 a ) is similar.In the case (7 c ( i )) we construct the Tate module as the sum T = T ⊕ T , where T is amodule over R = Z ℓ [ t ] / ( t ± √ q ) Z ℓ [ t ], and T is a module over R = Z ℓ [ t ] /P ( t )( t ± √ q ) Z ℓ [ t ].By Theorem 3.2, for any 3 × N such that Np( P ( t ∓ √ q ) t ) lies on or aboveYp( N ) there exists an R -module T such that t acts on T /ℓT with the matrix N ∓ √ qI . Then T = R ⊕ T is the desired Tate module.Suppose now that we have a module T from the case (7 c ( ii )). Let P ( t ) = tP ( t ∓ √ q ). ByProposition 5.3, there exists a matrix factorization ( X, Y ) such that det X = f A ( t ∓ √ q ) and Y X = P ( t ). Moreover, X ≡ diag( t , t ) mod ℓ. Define T ′ by the presentation:(3) Z ℓ [ t ] Y −→ Z ℓ [ t ] → T ′ → , Note that det Y = P ( t ) /f A ( t ∓ √ q ) = P ( t ∓ √ q ), and Y ≡ diag( t, t ) mod ℓ. Thus T ′ is amodule over R ′ = Z ℓ [ t ] /P Z ℓ [ t ]. By Theorem 3.1, such a module exists iff Np( P ( t ∓ √ q )) lies onor above the Young polygon Yp(1 , T exists then ℓ divides P ( ∓√ q ). Onthe other hand if ℓ divides P ( ∓√ q ), then we can construct a module T ′ over R ′ such that F acts on T ′ /ℓT ′ with the matrix ∓√ qI . By Proposition 5.3, there exists a matrix factorization( Y, X ) such that det Y = P ( t ∓ √ q ) and XY = P ( t ). Then the matrix factorization ( X, Y )corresponds to a desired module T . (cid:3) Kummer surfaces
Suppose p = 2. Let A be an abelian surface, and let τ : A → A be the involution a
7→ − a . Let p A : A → A/τ be the quotient map. The variety
A/τ is singular, and p A ( A [2]) is the singularlocus. Let σ : S → A/τ be the blow up of
A/τ in p A ( A [2]). Then S is smooth. It is called a Kummer surface . In this section we compute zeta functions of Kummer surfaces in terms of thezeta functions of the covering abelian surfaces.Let X be a variety over a finite field F q , and let N d be the number of points of degree 1 on X ⊗ F q d . The zeta function of X is the formal power series Z X ( t ) = exp( ∞ X d =1 N d t d d ) . In fact, Z X ( t ) is rational. For an abelian variety A we have the following formula:(4) Z X ( t ) = g Y i =0 P i ( t ) ( − i +1 , where P i ( t ) = det(1 − tF | V i V ℓ ( A )). Note that if f A ( t ) = Q ( t − ω j ), then P i ( t ) = Y j < ··· Let Z A ( t ) = Y i =0 P i ( A, t ) ( − i +1 be the zeta function of an abelian surface A . Then (5) Z S ( t ) = (1 − t ) − P ( t ) − (1 − q t ) − , where (6) P ( t ) = P ( A, t ) Y a ∈ A [2] (1 − ( qt ) deg a ) . In particular (7) | S ( k ) | = f A (1) + f A ( − q | A [2]( k ) | Proof. Since S is the blow up of X , we have Z X ( t ) = Z S ( t ) Y a ∈ A [2] (1 − ( qt ) deg a ) . Let us prove that | X ( F q r ) | = f r (1) + f r ( − , where f r = det( t − F r ) is the Weil polynomial of A r = A ⊗ F q r . Put A ( r ) = A r [2]( F q r ) . By [Mum70, IV.19.4], f A ( n ) = deg( n − F ) for n ∈ Z , were deg means the degree of an isogeny.There are two types of possible fibers of the map p A over a nonsingular F q r -point of X .(1) The fiber is a union of two points of degree 1. There are f r (1) − A ( r )2 such fibers.(2) The fiber is a point of degree 2. There are f r ( − − A ( r )2 such fibers.This gives the desired equality.Let f r ( t ) = t + a ( r ) t + a ( r ) t + a ( r ) q r t + q r , then a ( r ) = tr( F r | H ( ¯ A, Q ℓ )), and Z X ( t ) = exp( ∞ X r =1 ( f r (1) + f r ( − t r r ) =exp( ∞ X r =1 t r r ) exp( ∞ X r =1 a ( r ) t r r ) exp( ∞ X r =1 q r t r r ) =(1 − t ) − P ( A, t ) − (1 − q t ) − The last equality follows from lemma C.4.1 of [Ha77]. (cid:3) Now we classify the zeta functions of A [2] in terms of the Weil polynomial f A . Let b r be thenumber of points of degree r on A [2]. Then P ( t ) = P ( A, t ) Q r (1 − ( qt ) r ) b r . We compute thenumbers b r using Theorem 6.1.Suppose first that f A is separable, and assume that f A ( t ) ≡ ( t + 1) mod 2. Note that theslopes of Np( f A ( t + 1)) may be greater than 1. This may create many unnecessary cases inthe table below. However, we can use the polynomial f ( t ) = f A ( t + λ ) instead of f A ( t + 1),where λ ≡ ℓ , satisfying the property that slopes of Np( f ( t )) are less then or equal to 1.Equivalently, we take Np( f A ( t + 1)) and change all its slopes that are greater than 1 to 1. Thisoperation simplifies the notation, and clearly, it does not change the final answer, since all theslopes of Young polygons are not greater than 1. INITE GROUP SUBSCHEMES OF ABELIAN VARIETIES 13 Table 1:Slopes of Np( f ( t )) b i (1 / b = 2 , b = 1 , b = 3(1 / , b = 2 , b = 1 , b = 3 b = 4 , b = 2 , b = 2(1 / , / b = 2 , b = 1 , b = 3 b = 4 , b = 2 , b = 2 b = 4 , b = 6(2 / , / , , b = 2 , b = 1 , b = 3or (3 / b = 4 , b = 2 , b = 2 b = 4 , b = 6 b = 8 , b = 4(1 , , , b = 2 , b = 1 , b = 3 b = 4 , b = 2 , b = 2 b = 4 , b = 6 b = 8 , b = 4 b = 16If f A ( t ) ( t + 1) mod 2, then Table 2: f A ( t ) mod 2 t + t + t + t + 1 b = 1 , b = 3 t + t + t + 1 b = 2 , b = 1 , b = 2 , b = 1and 4 does not divide f A (1) t + t + t + 1 b = 2 , b = 1 , b = 2 , b = 1and 4 divides f A (1) b = 4 , b = 4 t + t + 1 b = 1 , b = 1 , b = 2and 4 does not divide a + a + 1 − qt + t + 1 b = 1 , b = 5and 4 divides a + a + 1 − q b = 1 , b = 1 , b = 2If f A is not separable, we have three cases of theorem 6.1. Let f A ( t ) = P A ( t ) thenTable 3: P A ( t ) mod 2 t + t + 1 b = 1 , b = 5 t + 1 and 4 does not divide P A (1) b = 4 , b = 6 t + 1 b = 4 , b = 6and 4 divides P A (1) b = 8 , b = 4 b = 16If f A ( t ) = ( t ± √ q ) f ( t ), then Table 4: f ( t ) mod 2 t + t + 1 b = 4 , b = 4 t + 1 b = 8 , b = 4and 4 does not divide f (1) b = 4 , b = 2 , b = 2 t + 1 b = 16and 4 divides f (1) b = 8 , b = 4 b = 4 , b = 6 b = 4 , b = 2 , b = 2Finally, if f A ( t ) = ( t ± √ q ) , we have b = 16. References [BGK06] G. Banaszak, W. Gajda, P. Krason. On the image of l-adic Galois representations for abelian varietiesof type I and II. Doc. Math. Extra Volume Coats (2006), 35–75.[CF67] Algebraic number theory. Proceedings of an instructional conference organized by the London Mathemat-ical Society with the support of the Inter national Mathematical Union. Edited by J. W. S. Cassels and A.Fr¨olich. Academic Press, London; Thompson Book Co., Inc., Washington, D.C. 1967[De78] M. Demazure. 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Cambridge University Press, Cambridge, 1997.[Ry10] S. Rybakov. The groups of points on abelian varieties over finite fields. Cent. Eur. J. Math. 8(2), 2010,282–288. arXiv:0903.0106v4[Ta66] J. Tate. Endomorphisms of abelian varieties over finite fields. Inventiones mathematicae 1966, Volume 2,Issue 2, pp 134–144.[Wa69] W. Waterhouse. Abelian varieties over finite fields. Ann. scient. ´Ec. Norm. Sup., 4 serie , 1969, 521–560.[WM69] W. Waterhouse, J. Milne. Abelian varieties over finite fields. Proc. Sympos. Pure Math., Vol. XX, StateUniv. New York, Stony Brook, N.Y., 1969, 53–64. Poncelet laboratory (UMI 2615 of CNRS and Independent University of Moscow)Institute for information transmission problems of the Russian Academy of SciencesLaboratory of Algebraic Geometry, NRU HSE, 7 Vavilova Str., Moscow, Russia, 117312 E-mail address ::