aa r X i v : . [ m a t h . N T ] A ug FULL LEVEL STRUCTURE ON SOME GROUP SCHEMES
CHUANGTIAN GUAN
Abstract.
We give a definition of full level structure on group schemes of theform G × G , where G is a finite flat commutative group scheme of rank p overa Z p -scheme S or, more generally, a truncated p -divisible group of height 1.We show that there is no natural notion of full level structure over the stackof all finite flat commutative group schemes. Introduction
Throughout this paper, all group schemes are assumed to be finite flat andcommutative over the base. Let H be such a group scheme of rank p l over a Z p -scheme S (“rank p l ” means that O H is a locally free O S -algebra of rank p l ).Suppose that H [ p ] := H × S S [ p ] is ´etale-locally isomorphic to the constant groupscheme ( Z /p r Z ) g for some r and g . This happens, for example, when H is the p r -torsion of some abelian variety of dimension g/ S (( Z /p r Z ) g , H ) be the functor from the category of S -schemes Sch S tothe category of abelian groups Ab , defined byHom S (( Z /p r Z ) g , H )( T ) := Hom gp (( Z /p r Z ) g , H ( T )) . We will use Hom S (( Z /p r Z ) g , H ) to denote the representing scheme, which in factis just H g . The general linear group GL g ( Z /p r Z ) has a natural right action onHom S (( Z /p r Z ) g , H ) by precomposing with the linear maps in GL g ( Z /p r Z ).The problem we consider is to give a notion of full level structure on H . Weexpect it to be a closed subscheme of Hom S (( Z /p r Z ) g , H ), which we denote byHom ∗ S (( Z /p r Z ) g , H ), satisfying:(1) Hom ∗ S (( Z /p r Z ) g , H ) is flat over S and of rank | GL g ( Z /p r Z ) | .(2) Hom ∗ S (( Z /p r Z ) g , H ) is GL g ( Z /p r Z )-invariant under the right GL g ( Z /p r Z )-action on Hom S (( Z /p r Z ) g , H ). Away from characteristic p we have anequation Hom ∗ S [ p ] (( Z /p r Z ) g , H [ p ]) = Isom S [ p ] (( Z /p r Z ) g , H [ p ]) as closedsubschemes of Hom S [ p ] (( Z /p r Z ) g , H [ p ]).(3) When identifying Hom S (( Z /p r Z ) g , H ) × S T with Hom T (( Z /p r Z ) g , H T ) inthe natural way, we have Hom ∗ S (( Z /p r Z ) g , H ) × S T = Hom ∗ T (( Z /p r Z ) g , H T )as closed subschemes, for any S -scheme T .We also expect our definition to coincide with the intuitive definition for somefamiliar group schemes. For example, for H = µ p r , we expect Hom ∗ Z ( Z /p r Z , H ) tobe the closed subscheme of µ p r defined by the cyclotomic polynomialΦ p r ( x ) := x p r − x p r − − x ( p − p r − + x ( p − p r − + · · · + 1 . Date : August 31, 2020.2010
Mathematics Subject Classification.
When H is the constant group scheme ( Z /p r Z ) g , the resulting full level structureHom ∗ Z (( Z /p r Z ) g , H ) should be GL g ( Z /p r Z ) ⊂ Mat g ( Z /p r Z ).The motivation of giving a well-behaved notion of full level structure comes fromthe study of integral models of Shimura varieties. For example, for modular curves,finding an integral model of the modular curve X ( p r ) essentially amounts to findinga flat model of full level structure on the p r -torsion of elliptic curves. This is doneby Katz and Mazur in their book [5]: Following an idea of Drinfeld in [3], Katz andMazur consider the case when H can be embedded into a curve. In this case a setof sections { P , . . . , P r } of H is defined to be a “full set of sections”, if the pointsgenerate the group H as Cartier divisors. Using this notion, the full level structureon H is defined to be the maps in Hom S (( Z /p r Z ) g , H ) whose image forms a full setof sections. As a scheme, Hom ∗ S (( Z /p r Z ) g , H ) can be also described as the closedsubscheme of Hom S (( Z /p r Z ) g , H ) cut out by the Cartier divisor equation H = X x ∈ ( Z /p r Z ) g [ h ( x )]where h is the universal homomorphism. Katz and Mazur’s construction, for ex-ample, gives a definition of full level structure on Z /p Z × µ p , as it is the p -torsion ofan ordinary elliptic curve. They also suggest a natural generalization of their con-struction, given by “ × -homomorphisms” [5, Appendix of Chapter 1], that can bedefined for general group schemes. Unfortunately, the notion of × -homomorphismsis deficient because the resulting closed subscheme is generally not flat over thebase. Such a negative result has been observed by Chai and Norman in [1, Ap-pendix.2]. For example, the nonflatness for × -homomorphisms even happens on µ p × µ p .As an improvement, Wake gives in [11] a good definition in the case of H = µ p × µ p over Spec Z . By using a notion of “primitive elements”, he defines the fulllevel structure, called “scheme of full homomorphisms”, to be cut out by the con-dition that all nontrivial linear combinations of rows and columns of the universalhomomorphism are primitive. Alternatively, Wake also gives another level struc-ture, called “KM+D” level structure, short for Katz-Mazur + Dual. The notion ofKM+D level structure is defined by requiring both universal homomorphism andits dual being × -homomorphisms as defined by Katz and Mazur. Wake proves thatin the case µ p × µ p , the KM+D level structure coincides with his original notion offull homomorphisms. Unfortunately, in general the “KM+D” level structure doesnot give a flat scheme over the base. For example, it is observed in [11, Example4.8] that Hom KM+D F (( Z / Z ) , ( α ) ) has larger rank than expected.In this paper, we give a definition of full level structure for H of the form H = G × G , where G is a rank p group scheme over a Z p -scheme S . When G is µ p ,our definition coincides with the one in [11]. The idea of our construction is togeneralize Wake’s “rows-and-columns” construction to a general group scheme G using Kottwitz-Wake’s notion of primitive elements ([6]). In [6] the authors givea notion of primitive elements which is well-behaved, even for general p -divisiblegroups. Using this notion, our full level structure will be cut out by the conditionthat rows and columns of the universal homomorphism are linearly independent,as in Wake’s construction. The precise description and properties are discussed inSection 3. The main point is that this construction gives a flat model. We showthis by using Oort-Tate theory to reduce to Wake’s result.One might also expect the following naturality condition: ULL LEVEL STRUCTURE ON SOME GROUP SCHEMES 3 (4) For any group scheme isomorphism H ∼ −→ H ′ , the induced isomorphismHom S (( Z /p r Z ) g , H ) → Hom S (( Z /p r Z ) g , H ′ ) restricts to an isomorphismHom ∗ S (( Z /p r Z ) g , H ) → Hom ∗ S (( Z /p r Z ) g , H ′ ).This condition (4) can be interpreted as saying the notion of full level structureis defined over the stack. Unfortunately, it turns out that in general there is nolevel structure on H satisfying all conditions (1) − (4). After I wrote a first draftof this paper, Wake informed me that he and Kottwitz had proven a similar resultin unpublished work. He pointed out that the construction cannot extend to thestack. We discuss this result in Section 6 and include their example there.I would like to thank my advisor George Pappas for his support and encourage-ment. I would also like to thank Preston Wake for very helpful conversations anduseful suggestions. 2. Review of the Oort-Tate theorem
In [10], Oort and Tate determine the structure of all finite flat commutative groupschemes of rank p over a Z p -scheme S . They prove that such a group scheme is givenby a triple ( L , u, v ) where L ∈
Pic( S ), u ∈ Γ( S, L ⊗ ( p − ) and v ∈ Γ( S, L ⊗ (1 − p ) )satisfying u ⊗ v = w p . Specifically, when S = Spec A where A is a Noetherianlocal ring, the line bundle L is trivial. Therefore to give an rank p group schemeover such S , it suffices to give two elements u, v ∈ A satisfying uv = w p . Forsuch a pair ( u, v ), the corresponding Hopf algebra is Spec A [ x ] / ( x p − ux ) with thecomultiplication m ∗ ( x ) = 1 ⊗ x + x ⊗ − p p − X i =1 vx i ⊗ x p − i w i w p − i . Here w i ’s are constants in Z p with w , · · · , w p − ∈ Z × p and w p ∈ p Z × p .Haines and Rapoport express this result using stack language in [4, Theorem3.3.1]. For convenience, we give the result here: Theorem 2.1 ([4]) . The Z p -stack OT of finite flat commutative group schemes ofrank p , satisfies the following properties:(i) OT is an Artin stack isomorphic to [(Spec Z p [ s, t ] / ( st − w p )) / G m ] . The action of G m is given by λ · ( s, t ) = ( λ p − s, λ − p t ) and w p is a constantin p Z × p .(ii) The universal group scheme G over OT is G = [(Spec OT O [ x ] / ( x p − tx )) / G m ] . The action of G m is given by λ · x = λx . Full level structure
The first step in defining the full level structure on G × G is defining “primitiveelements”. This is done by Kottwitz and Wake in [6]. Definition 3.1 ([6]) . Let H be a finite flat group scheme a base scheme S . Thescheme of primitive elements H × is the closed subscheme of H with the definingideal sheaf given as the annihilator of the augmentation ideal sheaf. CHUANGTIAN GUAN
An important example is the Oort-Tate group scheme G = Spec A [ x ] / ( x p − tx ),where A is a Z p -algebra. The augmentation ideal is ( x ). Thus G × is defined by theideal ( x p − − t ), coinciding with the scheme of generators defined in [4].Another example is G × G . Its underlying algebra is A [ x, y ] / ( x p − tx, y p − ty )with the augmentation ideal ( x, y ). The annihilator of ( x, y ) is the intersectionof the annihilator of x , which is ( x p − − t ), and the annihilator of y , which is( y p − − t ). Suppose f ∈ ( x p − − t ) ∩ ( y p − − t ). Since f ∈ ( x p − − t ), we mayassume f = ( x p − − t ) g with g ∈ A [ y ] of degree less than p . Similarly f = ( y p − − t ) h with h ∈ A [ x ] and deg h < p . Note that in this ring A [ x, y ] / ( x p − tx, y p − ty ), everyelement can be uniquely written as a polynomial with the degrees of x and y beingless than p . Because of the unique expression, we have ( x p − − t ) g = ( y p − − t ) h as polynomials in A [ x, y ]. By comparing the coefficients, we can get ( y p − − t ) | g .Therefore we have( x p − − t ) ∩ ( y p − − t ) = (cid:0) ( x p − − t )( y p − − t ) (cid:1) and thus the scheme of primitive elements in G is( G ) × = Spec A [ x, y ] / (cid:0) ( x p − − t )( y p − − t ) (cid:1) . See also ([6, Section 3.8]).Now we consider the operation on the points of Hom S (( Z /p Z ) , G ) = G (asfunctors). We will identify G ( T ) with Mat ( G ( T )), the additive group of 2 × G ( T ). On each entry Hom S ( Z /p Z , G )( T ) = G ( T ), there isa natural ad dition arising from the group structure of G . We denote this additionby ˙+, to distinguish it from the addition on O G . For simplicity, for any f ∈ G ( T ),let [ m ] f be f ˙+ f ˙+ · · · ˙+ f , the sum of m copies of f . Example 3.2.
Let S = Spec Z p and G = Spec Z p [ x ] / ( x p − x ) with comultiplication m ∗ ( x ) = 1 ⊗ x + x ⊗ − p P p − i =1 w p x i ⊗ x p − i w i w p − i . This is obtained by taking u = 1 and v = w p from Section 2. Let T = Spec Z p . In G ( T ), let χ ( j ) ∈ Hom S ( Z /p Z , G )( T ) = G ( T ) be the map sending x to χ ( j ), where χ is the Teichm¨uller character and let χ (0) = 0. Since the elements in G ( T ) is closed under the group action, we have([ j ](1)) p − [ j ](1) = 0. On the other hand, by the definition of the comultiplicationof G , we have [ j ](1) ≡ j mod p . Therefore [ j ](1) = χ ( j ). From [ j ](1) ˙+[ k ](1) =[ j + k ](1), we get a useful equation:(3.1) χ ( j + k ) = χ ( j ) + χ ( k ) + 11 − p p − X i =1 w p χ ( j i ) χ ( k p − i ) w i w p − i . In fact, G is isomorphic to the constant group scheme Z /p Z S = Spec Z Z /p Z p .The Hopf algebra isomorphism between Z p [ x ] / ( x p − x ) and Z Z /p Z p is given by x P χ ( i ) e i and e i λ ( i ) Q j = i ( x − χ ( j )), where λ (0) = − λ ( i ) = p − otherwise.To see this, a priori it is easy to see the maps give an algebra isomorphism. Tosee that it preserves the comultiplication, we can check straightforwardly usingEquation (3.1). We will skip the detailed calculation here.Now we define Hom ∗ S (( Z /p Z ) , G ) to be the subfunctor of Hom S (( Z /p Z ) , G )as follows: Definition 3.3.
Define Hom ∗ S (( Z /p Z ) , G ) to be the functor whose T -valuedpoints are the elements in Hom S (( Z /p Z ) , G )( T ) = Mat ( G ( T )) so that all nonzero ULL LEVEL STRUCTURE ON SOME GROUP SCHEMES 5 F p -linear combinations of rows and columns are in ( G ) × ( T ). For nonzero F p -linearcombinations, we mean elements like [ m ] f ˙+[ n ] g where m and n are not both zero. Remark . It is easy to see that the functor Hom ∗ S (( Z /p Z ) , G ) we defined aboveis representable. Indeed, each linear combination being primitive is a closed con-dition and thus gives a subscheme of Hom ∗ S (( Z /p Z ) , G ) = G . Therefore thefunctor Hom ∗ S (( Z /p Z ) , G ) is represented by the scheme-theoretical intersectionof those subschemes. We use Hom ∗ S (( Z /p Z ) , G ) for the representing scheme.Here is the main result: Theorem 3.5.
Let S be a Z p -scheme and let G, G ′ be finite flat commutative groupschemes of rank p over S. Let GL ( F p ) act on Hom S (( Z /p Z ) , G ) by precomposi-tion and let Hom ∗ S (( Z /p Z ) , G ) be as defined above. Then we have:(i) Hom ∗ S (( Z /p Z ) , G ) is flat over S of rank | GL ( F p ) | .(ii) Hom ∗ S (( Z /p Z ) , G ) is GL ( F p ) -invariant. When being away from charac-teristic p we have Hom ∗ S [ p ] (( Z /p Z ) , G [ p ] ) = Isom S [ p ] (( Z /p Z ) , G [ p ] ) asclosed subschemes of Hom S [ p ] (( Z /p Z ) , G [ p ] ) .(iii) By identifying Hom S (( Z /p Z ) , G ) × S T = Hom T (( Z /p Z ) , G T ) for any S -scheme T , we have Hom ∗ S (( Z /p Z ) , G ) × S T = Hom ∗ T (( Z /p Z ) , G T ) as closedsubschemes.(iv) Let φ : G → G ′ be an isomorphism and let Φ : G → ( G ′ ) be the isomor-phism given by (cid:18) φ φ (cid:19) . Then the isomorphism
Hom S (( Z /p Z ) , G ) → Hom S (( Z /p Z ) , ( G ′ ) ) induced by Φ restricts to an isomorphism on the fulllevel structures Hom ∗ S (( Z /p Z ) , G ) → Hom ∗ S (( Z /p Z ) , ( G ′ ) ) . Proof of the main theorem
Firstly we will prove Theorem 3.5 (iv). As before, we identify Hom(( Z /p Z ) , G ( T ) )with Mat ( G ( T )). The isomorphism Φ( T ) : Mat ( G ( T )) → Mat ( G ′ ( T )) applies φ ( T ) on all the entries. Therefore for (cid:18) a a a a (cid:19) ∈ Mat ( G ( T )) with([ m ] a ˙+[ n ] a , [ m ] a ˙+[ n ] a ) ∈ ( G ) × ( T ) , we have (cid:0) [ m ] φ ( a ) ˙+[ n ] φ ( a ) , [ m ] φ ( a ) ˙+[ n ] φ ( a ) (cid:1) = (cid:0) φ ([ m ] a ˙+[ n ] a ) , φ ([ m ] a ˙+[ n ] a ) (cid:1) ∈ Φ(( G ) × ( T )) = (( G ′ ) ) × ( T ) . The last equation is from Definition 3.1 since a group scheme isomorphism willpreserve the augmentation ideal. The case for linear combinations of columns issimilar and thus Theorem 3.5 (iv) follows.Now we move to prove Theorem 3.5 (iii), saying that the full level structure iscompatible with base change:
Lemma 4.1.
Let S , G be as in Theorem 3.5. Let T be an S -scheme. Then wehave Hom ∗ S (( Z /p Z ) , G ) × S T = Hom ∗ T (( Z /p Z ) , G T ) . Here they are both regarded as closed subschemes of
Hom T (( Z /p Z ) , G T ) . CHUANGTIAN GUAN
Proof.
It suffices to show the left and right side represent the same subfunctor ofHom T (( Z /p Z ) , G T ) . Let W be a T -scheme. Note that elements in Hom ∗ S (( Z /p Z ) , G ) × S T ( W ) (resp.in Hom ∗ T (( Z /p Z ) , G T )( W )) are 2 × G ( W ) so that thelinear combinations of rows and columns are in ( G ) × ( W ) (resp. in ( G T ) × ( W )).Then they are same immediately from ( G T ) × ( W ) = (( G ) × ) T ( W ) = ( G ) × ( W ).Here the first equality is from [6, Section 3.5]. (cid:3) Now we prove Theorem 3.5 (ii). Note that the identity Hom S (( Z /p Z ) , G ) × S T = Hom T (( Z /p Z ) , G T ) is GL ( F p )-equivalent. By Lemma 4.1, the GL ( F p )-invariance of Hom ∗ S (( Z /p Z ) , G ) can be checked affine-locally on S . Recall fromSection 2 that the group scheme G/S is determined by a triple ( L , u, v ). We mayrestrict S so that L is trivial on S . Let A = Z p [ s, t ] / ( st − w p ) and S = Spec A . Let G = Spec A [ x ] / ( x p − tx ) be the group scheme over S with comultiplication(4.1) m ∗ ( x ) = 1 ⊗ x + x ⊗ − p p − X i =1 sx i ⊗ x p − i w i w p − i . Then
G/S will be the pull back of G / S through a morphism S → S determined by u and v . Then it suffices to show the GL ( F p )-invariance for Hom ∗S (( Z /p Z ) , G ).For the second part of Theorem 3.5 (ii), note that G [ p ] is ´etale locally isomorphic to Z /p Z and by definition Hom ∗ (( Z /p Z ) , ( Z /p Z ) ) = Isom(( Z /p Z ) , ( Z /p Z ) ). Thenthis part follows from Lemma 4.1 immediately.For Theorem 3.5 (i), from Lemma 4.1, since being flat is a local property, we canreduce to the case where S = Spec A with A being a local Z p -algebra. As above,the line bundle on S is trivial and therefore G/S is a pullback of G / S . ApplyingLemma 4.1 again, we can see that it suffices to show the flatness of the full levelstructure for G / S .From the discussion above, it remains to look at G / S and check that the fulllevel structure Hom ∗S (( Z /p Z ) , G ) is GL ( F p )-invariant and flat of rank | GL ( F p ) | over S .We first look at Hom ∗S (( Z /p Z ) , G ) over the two pieces Spec Z p [ s, s − ] andSpec Z p [ t, t − ]. It is easy to check that after applying ´etale base changes by addingthe p − s, s − , t, t − , we get G × S Spec Z p [ s p − , s − p − ] ∼ = µ p and G × S Spec Z p [ t p − , t − p − ] ∼ = Z /p Z . In these cases, the following lemma is as ex-pected: Lemma 4.2.
We have the following two isomorphisms of group schemes:(i)
Hom ∗S (( Z /p Z ) , G ) × S Spec Z p [ s p − , s − p − ] ∼ = Hom fullSpec Z p [ s p − ,s − p − ] (( Z /p Z ) , µ p ) . Here the
Hom full is the full level structure for µ p × µ p defined by Wake in [11] .(ii) Hom ∗S (( Z /p Z ) , G ) × S Spec Z p [ t p − , t − p − ] ∼ = GL ( Z /p Z ) . Proof.
Note that from the definition of Hom ∗ , we have Hom ∗ (( Z /p Z ) , µ p ) =Hom full (( Z /p Z ) , µ p ) as they are defined in the same way. For the ´etale part,note that sections of constant group schemes being primitive exactly means be-ing nonzero. So Hom ∗ (( Z /p Z ) , ( Z /p Z ) ) consists of the matrices satisfying that ULL LEVEL STRUCTURE ON SOME GROUP SCHEMES 7 nonzero linear combinations of rows and columns are nonzero, thus invertible ma-trices. Hence Hom ∗ (( Z /p Z ) , ( Z /p Z ) ) = GL ( Z /p Z ) and the claim is immediatefrom Lemma 4.1. (cid:3) To make the full level structure explicit for G / S , it is helpful to use the univer-sal homomorphism for description. Consider the universal base S univ = Spec A univ where A univ = A [ a, b, c, d ] / ( a p − ta, b p − tb, c p − tc, d p − td ). Then we have S univ =Hom S (( Z /p Z ) , G ). Let h ∈ Hom S univ (( Z /p Z ) , G S univ ) be the universal homo-morphism defined over S univ , given by (1 , ( a, b ), (0 , ( c, d ). ThenHom ∗S (( Z /p Z ) , G ), as a subscheme of the universal base S univ , is cut out by thecondition h ∈ Hom ∗S univ (( Z /p Z ) , G S univ ). Therefore by definition Hom ∗S (( Z /p Z ) , G )is given by the ideal I ⊂ A univ generated by n(cid:16)(cid:0) [ m ] a ˙+[ n ] b (cid:1) p − − t (cid:17) (cid:16)(cid:0) [ m ] c ˙+[ n ] d (cid:1) p − − t (cid:17) , (cid:16)(cid:0) [ m ] a ˙+[ n ] c (cid:1) p − − t (cid:17) (cid:16)(cid:0) [ m ] b ˙+[ n ] d (cid:1) p − − t (cid:17)o ( m,n ) ∈ F p \{ (0 , } . Recall that the notion [ m ] a ˙+[ n ] b are coming from regarding a, b, c, d as elementsin G ( S univ ), corresponding to the homomorphisms A [ x ] / ( x p − tx ) → A [ a, b, c, d ] / ( a p − ta, b p − tb, c p − tc, d p − td )sending x to a, b, c, d . As an abstract group, G ( S univ ) is given by { x ∈ A [ a, b, c, d ] / ( a p − ta, b p − tb, c p − tc, d p − td ) (cid:12)(cid:12) x p = tx } with the group structure given by x ˙+ y = x + y + 11 − p p − P i =1 sx i y p − i w i w p − i . Therefore[2] a = 2 a + 11 − p p − X i =1 sa p w i w p − i = 2 a + 11 − p p − X i =1 staw i w p − i = − p p − X i =1 w p w i w p − i ! a. Using Equation (3.1), we get [2] a = χ (2) a and in general by induction we have[ m ] a = χ ( m ) a . Therefore the full level structure on G / S has the following expres-sion: Hom ∗S (( Z /p Z ) , G ) ∼ = Spec Z p [ s, t, a, b, c, d ] , st − w p ,a p − ta,b p − tb,c p − tc,d p − td, n(cid:16) ( χ ( m ) a ˙+ χ ( n ) b ) p − − t (cid:17)(cid:16) ( χ ( m ) c ˙+ χ ( n ) d ) p − − t (cid:17) , (cid:16) ( χ ( m ) a ˙+ χ ( n ) c ) p − − t (cid:17)(cid:16) ( χ ( m ) b ˙+ χ ( n ) d ) p − − t (cid:17)o . (4.2)The action is given by (cid:18) a bc d (cid:19) g t (cid:18) a bc d (cid:19) for any g ∈ GL ( F p ). Here the scalar product in matrix multiplication is given by[ · ] and the addition is ˙+. To see that Hom ∗S (( Z /p Z ) , G ) is invariant under theaction of GL ( F p ), by decomposing g using elementary matrices, the only nontrivial CHUANGTIAN GUAN case is for g = (cid:18) k (cid:19) . Consider (cid:16)(cid:0) χ ( m ) (cid:0) a ˙+ χ ( k ) c (cid:1) ˙+ χ ( n ) (cid:0) b ˙+ χ ( k ) d (cid:1)(cid:1) p − − t (cid:17) (cid:16)(cid:0) χ ( m ) c ˙+ χ ( n ) d (cid:1) p − − t (cid:17) = (cid:16)(cid:0)(cid:0) χ ( m ) a ˙+ χ ( n ) b (cid:1) ˙+ χ ( k ) (cid:0) χ ( m ) c ˙+ χ ( n ) d (cid:1)(cid:1) p − − t (cid:17) (cid:16)(cid:0) χ ( m ) c ˙+ χ ( n ) d (cid:1) p − − t (cid:17) = (cid:16)(cid:0) χ ( m ) a ˙+ χ ( n ) b (cid:1) p − − t (cid:17) (cid:16)(cid:0) χ ( m ) c ˙+ χ ( n ) d (cid:1) p − − t (cid:17) . Here the last equality is from the definition of ˙+ and note that (cid:0) χ ( m ) c ˙+ χ ( n ) d (cid:1) (cid:16)(cid:0) χ ( m ) c ˙+ χ ( n ) d (cid:1) p − − t (cid:17) = 0since G ( S univ ) is a group. This proves the GL ( F p )-invariance.Now we only need to prove the flatness. We will use the lemma below: Lemma 4.3 ([9] Page 51 Lemma 1) . Let Y be a reduced scheme and F a coherentsheaf on Y such that dim k ( y ) F ⊗ O y k ( y ) = r , for all y ∈ Y . Then F is a locallyfree of rank r on Y . We will apply Lemma 4.3 to Y = S . Note that for y ∈ Spec Z p [ t, t − ], weknow that dim k ( y ) O Hom ∗S ⊗ O y k ( y ) = | GL ( F p ) | from Lemma 4.2 (ii) and ´etaledescent. For y ∈ Spec Z p [ s, s − ], we can get dim k ( y ) O Hom ∗S ⊗ O y k ( y ) = | GL ( F p ) | by combining Lemma 4.2 (i) together with Wake’s result on Hom full and ´etaledescent. The only remaining point is y for s = t = p = 0.Consider the base change of S to A F p by setting t = p = 0. The concerningpoint y is the origin of A F p . Note that χ ( m ) ≡ m mod p . Therefore after the basechange from (4.2) we get(4.3) Hom ∗ A F p (( Z /p Z ) , G A F p ) = Spec F p [ s, a, b, c, d ] . (cid:18) ap,bp,cp,dp n(cid:0) ma ˙+ nb (cid:1) p − (cid:0) mc ˙+ nd (cid:1) p − , (cid:0) ma ˙+ nb (cid:1) p − (cid:0) mc ˙+ nd (cid:1) p − o (cid:19) . Here the “ ˙+” operation is given as x ˙+ y = x + y + p − P i =1 sx i y p − i i !( p − i )! (note that w i ≡ i ! mod p, see [10, Page 9]). Lemma 4.4.
In the coordinate ring (4 . , we have ma ˙+ nb = u ( ma + nb ) for someunit u .Proof. When p = 2, we simply have ma ˙+ nb = ma + nb + smnab = ( ma + nb )(1 + sma ) . Here 1 + sma is a unit as a is nilpotent.Now suppose that p >
2. Let g ( x, y ) = p − P i =1 x i y p − i i !( p − i )! be a polynomial in F p [ x, y ].Note this polynomial g ( x, y ) is divisible by x + y as g ( x, − x ) = 0 (note that p is odd).Assume g ( x, y ) = ( x + y ) g ′ ( x, y ). Then [ m ] a ˙+[ n ] b = ( ma + nb )(1 + sg ′ ( ma, nb )).Note that g ′ has no constant term and a, b are nilpotent in the coordinate ring.Therefore 1 + sg ′ ( ma, nb ) is a unit, as the claim. (cid:3) Applying Lemma 4.4 to Equation 4.3, we get that(4.4) Hom ∗ A F p (( Z /p Z ) , G A F p ) = Spec F p [ s, a, b, c, d ] , a p ,b p ,c p ,d p { ( ma + nb ) p − ( mc + nd ) p − , ( ma + nb ) p − ( mc + nd ) p − } ! . ULL LEVEL STRUCTURE ON SOME GROUP SCHEMES 9
A key observation is that quotient ideal is independent on s . Therefore in par-ticular, for any point y ∈ A F p away from y , we have dim k ( y ) O Hom ∗S ⊗ O y k ( y ) =dim k ( y ) O Hom ∗S ⊗ O y k ( y ) = | GL ( F p ) | . Applying Lemma 4.3, we finish proving theflatness.5. Full level structure on truncated height-one p -divisible groups In this section we let G be a truncated p -divisible group of height 1 of rank p r over S . Suppose G [ p ] is ´etale locally isomorphic to Z /p r Z . Therefore G [ p r ] is flatof rank p r over S for r ≤ l and G [ p ] × S S [ p ] is ´etale locally isomorphic to Z /p Z .As an easy application of our result, we may give a notion of full level structureon G as follows: Definition 5.1.
Let
G/S be as above. We define the full level structure on G asthe fiber product:Hom ∗ S (( Z /p Z ) , G ]) Hom S (( Z /p Z ) , G ]) = G Hom ∗ S (( Z /p Z ) , G [ p ] ]) Hom S (( Z /p Z ) , G [ p ] ]) = G [ p ] (cid:3) Here the bottom horizontal map is the closed immersion and the right verticalmap is a quadruple of p l − : G → G [ p ]. Theorem 5.2.
Let
G/S be as above. The full level structure
Hom ∗ S (( Z /p r Z ) , G ) defined above satisfies the conditions (1)-(3) in Section 1. In particular it is flatover S of rank | GL ( Z /p r Z ) | .Proof. By general theory of p -divisible groups (for example, [8, Lemma 1.5 (b)]), G → G [ p ] is (faithfully) flat of rank p l − . Therefore by definition the full level struc-ture Hom ∗ S (( Z /p r Z ) , G ) is flat over Hom ∗ S (( Z /p Z ) , G [ p ] ]) of rank p l − . Theorem3.5 implies that Hom ∗ S (( Z /p Z ) , G [ p ] ]) is flat of rank | GL ( Z /p Z ) | over S . Notethat | GL ( Z /p r Z ) | = p l − | GL ( Z /p Z ) | . Therefore Hom ∗ S (( Z /p r Z ) , G ) is flat over S of rank | GL ( Z /p r Z ) | . The conditions (2) and (3) are immediate from the defi-nition. (cid:3) Nonexistence of full level structure over the stack
Let C be a stack of group schemes of certain type over Sch Z p . (By a stackhere we simply mean a category fibered in groupoids over Sch Z p as in [2].) So,we assume that the objects in C are group schemes G/S of certain fixed type(for example, finite flat commutative and of certain rank) and the morphisms areCartesian squares. By a full level structure over C , we mean a fibered functor F : C → Sch , such that F ( G/S ) is a closed subscheme of Hom S (( Z /p r Z ) g , G ) andsuch that for f : G/S → G ′ /S ′ , the morphism F ( f ) : F ( G/S ) → F ( G ′ /S ′ ) is therestriction of the morphism Hom S (( Z /p r Z ) g , G ) → Hom S ′ (( Z /p r Z ) g , G ′ ) inducedby f , and satisfies conditions as in (1)-(3) of Section 1. We can see that the condition(4) is automatic by this definition.In this section, we observe the lack of a good notion of full level structure overthe stack of finite flat commutative group schemes. In fact, consider the substack OT × OT , whose objects are G × G ′ where G, G ′ are Oort-Tate schemes. We willsee that even on OT × OT , there is no good notion of full level structure: Theorem 6.1.
There is no notion of full level structure over the stack OT × OT .Proof. Let G / S be as in Section 3. Assume there is a full level structure on OT × OT satisfying (1)-(4). Then the full level structure on G / S must be the one we defined.In fact over the generic fiber of S , the full level structure is given by the condition(2). Therefore the only way to satisfy condition (1) is defining the full level structureover S as the Zariski closure of the corresponding scheme over the generic fiber.Note that any group scheme of rank p over a local ring can be obtained from G / S by base change. Because of condition (3), the full level structure on G × G over alocal base must be the one we defined above. However, this only possible structureis not preserved under all group scheme automorphisms. Here is one examplecommunicated to the author by Wake:Consider the full level structure on α p × α p over F p with p >
2. By our definitionand Lemma 4.4,Hom ∗ F p (( Z /p Z ) , α p ) ∼ = Spec F p [ a, b, c, d ] , a p ,b p ,c p ,d p { ( ma + nb ) p − ( mc + nd ) p − , ( ma + nc ) p − ( mb + nd ) p − } ! . Note that Aut F p ( α p ) = GL ( F p ), with the action given by multiplying (cid:18) a bc d (cid:19) by elements of GL ( F p ) from the right. Since ( m, n ) ∈ F p \ { (0 , } , it is not hardto see that the ideal is not invariant under the action of GL ( F p ). (cid:3) Remark . The notion of Hom ∗ is not preserved under a general isomorphism G × G → G ′ × G ′ as observed. However, as we see in Theorem 3.5, we may restrictthe isomorphisms to those of the formΦ = (cid:18) φ φ (cid:19) where φ : G ∼ −→ G ′ is an isomorphism. Roughly speaking, we fail to have a goodfull level structure over OT × OT , but the one we give behaves well when regardedas a full level structure over OT . Remark . Although as shown, a good notion of full level structure on the stackof all finite group schemes does not exist, one might still hope to define a full levelstructure on truncated p -divisible groups. However, some new idea is needed.7. Full level structure on G Another natural question we may ask is whether we can have some similar resultsfor group schemes of the form G n , where G is an Oort-Tate group scheme. We recordsome partial result here, However, a full answer to this question requires some newidea.Let us take G = µ p over Spec Z . One intermediate step towards defining afull level structure on G is defining a “partial level structure” as a subscheme ofHom Z (( Z /p Z ) , ( µ p ) ). We will still require that the resulting scheme is flat overthe base and when inverting p we want Hom ∗ Z (( Z /p Z ) , ( µ p ) ) ∼ = Mat ∗ × ( F p ), whereMat ∗ × denote the set of all 2 × h be the universal homomorphism. ThenHom ∗ Z (( Z /p Z ) , ( µ p ) ) is cut out by the conditions:(i) All nonzero linear combinations of rows and columns are primitive. ULL LEVEL STRUCTURE ON SOME GROUP SCHEMES 11 (ii) After applying any left GL ( F p )-action and right GL ( F p )-action to h , oneof the three 2 × ∗ Z (( Z /p Z ) , ( µ p ) ).Let us make (ii) clear here. Let h = (cid:18) a a a a a a (cid:19) be the universal homomorphism. Let I , resp. I , I , be the ideal defined byrequiring that (cid:18) a a a a (cid:19) , resp. (cid:18) a a a a (cid:19) , (cid:18) a a a a (cid:19) , lies in the full level structure subscheme Hom ∗ (( Z /p Z ) , ( µ p ) ). Then the idealdefining “one of the three 2 × I I ∩ I I ∩ I I . The closed subscheme Hom ∗ Z (( Z /p Z ) , ( µ p ) ) cut out by theseconditions is flat of rank | Mat ∗ × ( F p ) | over the base. This result of “partial levelstructure” Hom ∗ Z (( Z /p Z ) , ( µ p ) ) can be extended to Hom ∗ (( Z /p Z ) , G ).One might hope to define Hom ∗ Z (( Z /p Z ) , ( µ p ) ) using the “partial level struc-ture” above, by requiring that after applying the left and right GL ( F p )-action andpossibly Cartier dual to the universal homomorphism, the resulting homomorphismis such that any 2 × µ p over F p . For p = 2, the above condition will givea closed subscheme of rank 169 over F p , while | GL ( F ) | = 168. For p = 3, theobtained subscheme has rank 11473 over F p , while | GL ( F ) | = 11232 (comparingwith 3 = 19683). So, some further conditions need to be discovered. References
1. C. Chai and P. Norman, Bad reduction of the Siegel moduli scheme of genus two with Γ ( p )-level structure, American Journal of Mathematics, Volume 112, 1990, No.6, 1003–1071.2. P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst.Hautes tudes Sci. Publ. Math. No. 36, 1969, 75–109.3. V. Drinfeld, Elliptic modules, Mat. Sb. (N.S.), Volume 94(136), 1974, 594–627, 656.4. T. Haines and M. Rapoport, Shimura varieties with Γ ( p )-level via Hecke algebra isomorphisms:the Drinfeld case, Ann. Sci. ´Ec. Norm. Sup´er. (4), Volume 45, 2012, No.5,719–785 (2013).5. N. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies,Volume 108, Princeton University Press, Princeton, NJ, 1985, xiv+514.6. R. Kottwitz and P. Wake, Primitive elements for p -divisible groups, Research in Number The-ory, Volume 3, 2017, Art. 20.7. H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, Vol-ume 8, Translated from the Japanese by M. Reid, Cambridge University Press, Cambridge,1986, xiv+320.8. W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelianschemes, Lecture Notes in Mathematics, Vol. 264, Springer-Verlag, Berlin-New York, 1972,iii+190.9. D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathemat-ics, No. 5, 1970, viii+242.10. J. Tate and F. Oort, Group schemes of prime order, Ann. Sci. ´Ecole Norm. Sup. (4), Volume3, 1970, 1–21.11. P. Wake, Full level structures revisited: pairs of roots of unity, Journal of Number Theory,Volume 168, 2016, 81–100. Department of Mathematics, Michigan State University, 619 Red Cedar Road, EastLansing, MI 48824, USA
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