aa r X i v : . [ m a t h . N T ] J u l ON p -ADIC INTERPOLATION OF MOTIVIC EISENSTEIN CLASSES GUIDO KINGS
Abstract.
In this paper we prove that the motivic Eisenstein classes associated to polyloga-rithms of commutative group schemes can be p -adically interpolated in ´etale cohomology. Thisconnects them to Iwasawa theory and generalizes and strengthens the results for elliptic curvesobtained in our former work. In particular, degeneration questions can be treated easily. To John Coates, on the occasion of his 70th birthday
Contents
Introduction 1Acknowledgements 31. Notations and set up 32. The logarithm sheaf 63. The Q p -polylogarithm and Eisenstein classes 94. Sheaves of Iwasawa algebras 115. The integral logarithm sheaf 156. Interpolation of the Q p -Eisenstein classes 20References 24 Introduction
In this paper we prove that the motivic Eisenstein classes associated to polylogarithms ofcommutative group schemes can be p -adically interpolated in ´etale cohomology. This gener-alizes the results for elliptic curves obtained in our former paper [Kin15]. Already in the onedimensional case the results obtained here are stronger and much more flexible as they allow totreat degenerating elliptic curves easily.The interpolation of motivic Eisenstein classes connects them with Iwasawa theory and isessential for many applications. In the elliptic case for example, the interpolation was used in[Kin01] to prove a case of the Tamagawa number conjecture for CM elliptic curves and it wasone of the essential ingredients in the proof of an explicit reciprocity law for Rankin-convolutionsin [KLZ15]. We hope that the general case will find similar applications.Before we explain our results, we have to introduce the motivic Eisenstein classes (for theconstruction we refer to Section 3.2).Let π : G → S be a smooth commutative and connected group scheme of relative dimension d (for example a semi-abelian scheme) and denote by H := R π ! Z p (1)the first ´etale homology of G/S , which is just the sheaf of relative p -adic Tate modules of G/S .We write H Q p for the associated Q p -adic sheaf. Note that this is not a lisse sheaf in general.Evaluating the motivic polylogarithm at a non-zero N -torsion section t : S → G one definesmotivic Eisenstein classes α Eis k mot ( t ) ∈ H d − ( S, Sym k H Q ( d )) , This research was supported by the DFG grant: SFB 1085 “Higher invariants”. epending on some auxiliary data α , where Sym k H Q (1) is the k -th symmetric tensor power ofthe motivic sheaf H Q which underlies H Q p .In the case of an elliptic curve, the de Rham realization of α Eis k mot ( t ) is the cohomology classof a holomorphic Eisenstein series, which justifies the name. These motivic Eisenstein classesin the elliptic case play a major role in Beilinson’s proof of his conjectures on special values of L -functions for modular forms.In this paper we consider the ´etale regulator r ´et : H d − ( S, Sym k H Q ( d )) → H d − ( S, Sym k H Q p ( d ))which gives rise to the ´etale Eisenstein classes α Eis k Q p ( t ) := r ´et (Eis k mot ( t )) ∈ H d − ( S, Sym k H Q p ( d )) . In the elliptic case these classes were used by Kato in his seminal work to construct Eulersystems for modular forms.It is a natural question, whether these ´etale Eisenstein classes enjoy some p -adic interpolationproperties, in a similar way as one can p -adically interpolate the holomorphic Eisenstein series.At first sight, this seems to be a completely unreasonable question, as for varying k the differentmotivic cohomology groups H d − ( S, Sym k H Q (1)) are not related at all. Nevertheless, thisquestion was answered affirmatively in the elliptic case in [Kin15] and in this paper we willgeneralize this result to commutative group schemes.To explain our answer to this question we need the sheaf of Iwasawa-algebras Λ( H ), whichis defined as follows: One first defines a sheaf of ”group rings” Z /p r Z [ H r ] on S , where H r isthe ´etale sheaf associated to the [ p r ]-torsion subgroup G [ p r ] or alternatively the first homologywith Z /p r Z -coefficients (see section 4.4 for more details). These group rings form an inversesystem for varying r and hence define a pro-sheafΛ( H ) := ( Z /p r Z [ H r ]) r ≥ . Moreover, it is also possible to sheafify the classical moments of a measure to a morphism ofpro-sheaves mom k : Λ( H ) → Γ k ( H ) , where Γ k ( H ) is the k -th graded piece of the divided power algebra Γ Z p ( H ). Thus the sheafΛ( H ) p -adically interpolates the Γ k ( H ). For the Q p -sheaf H Q p the natural map Sym k H Q p → Γ k ( H Q p ) is an isomorphism and the moment map gives rise to morphismsmom k : H d − ( S, Λ( H )( d )) → H d − ( S, Sym k H Q p ( d )) . To understand this better, it is instructive to consider the case of an abelian scheme π : A → S over a scheme S which is of finite type over Spec Z (see also Section 5.5). Then H d − ( S, Λ( H )( d )) = lim ←− r H d − ( A [ p r ] , Z /p r Z ( d ))where the inverse limit is taken with respect to the trace maps along A [ p r ] → A [ p r − ]. Theright hand side is obviously an Iwasawa theoretic construction. In the one dimensional case d = 1, the right hand side has an interpretation as an inverse limit of units via Kummer theory.Our main result can now be formulated as follows: Main Theorem (see Theorem 6.3.3) . There exists a cohomology class α EI ( t ) N ∈ H d − ( S, Λ( H )( d )) called the Eisenstein-Iwasawa class , such that mom k ( α EI ( t ) N ) = N kα Eis k Q p ( t ) . his interpolation result in the elliptic case is one of the key ingredients in the proof of anexplicit reciprocity law for Rankin-convolutions of modular forms in [KLZ15].The use of this theorem also considerably simplifies the computations of the degeneration ofpolylogarithm in [Kin15]. We hope to treat this at another occasion.We would also like to point out an important open problem: In the one-dimensional torusor the elliptic curve case, the Eisenstein-Iwasawa class has a direct description in terms ofcyclotomic units or Kato’s norm compatible elliptic units respectively. Unfortunately, we donot have a similar description of the Eisenstein-Iwasawa class in the higher dimensional case. Acknowledgements
This paper is a sequel to [Kin15], which was written as a response to John Coates’ wish tohave an exposition of the results in [HK99] at a conference in Pune, India. This triggered arenewed interest of mine in the p -adic interpolation of motivic Eisenstein classes which goesback to [Kin01]. In this sense the paper would not exist without the persistence of John Coates.The paper [HK15] with Annette Huber created the right framework to treat these questions. Itis a pleasure to thank them both and to dedicate this paper to John Coates on the occasion ofhis seventieth birthday. 1. Notations and set up
The category of Z p -sheaves. All schemes will be separated of finite type over a noether-ian regular scheme of dimension 0 or 1. Let X be such a scheme and let p be a prime invertibleon X . We work in the category of constructible Z p -sheaves S ( X ) on X in the sense of [SGA77,Expos´e V].Recall that a constructible Z p -sheaf is an inverse system F = ( F n ) n ≥ where F n is a con-structible Z /p n Z -sheaf and the transition maps F n → F n − factor into isomorphisms F n ⊗ Z /p n Z Z /p n − Z ∼ = F n − . The Z p -sheaf is lisse , if each F n is locally constant. If X is connected and x ∈ X is a geometricpoint, then the category of lisse sheaves is equivalent to the category of finitely generated Z p -modules with a continous π ( X, x )-action. For a general Z p -sheaf there exists a finite partitionof X into locally closed subschemes X i , such that F | X i is lisse (see [Del77, Rapport, Prop.2.4., 2.5.]).For a Z p -sheaf F we denote by F ⊗ Q p its image in the category of Q p -sheaves, i.e., thequotient category modulo Z p -torsion sheaves.We also consider the “derived” category D ( X ) of S ( X ) in the sense of Ekedahl [Eke90].This is a triangulated category with a t -structure whose heart is the category of constructible Z p -sheaves. By loc. cit. Theorem 6.3 there is a full 6 functor formalism on these categories.Recall that an inverse system A := ( A r ) r ≥ (in any abelian category A ) satisfies the Mittag-Leffler condition (resp. is Mittag-Leffler zero), if for each r the image of A r + s → A r is constantfor all sufficiently big s (is zero for some s ≥ A satisfies the Mittag-Leffler condition and A satisfies AB ∗ (i.e. products exists and products of epimorphisms are epimorphisms) thenlim ←− r A r = 0 ( see [Roo61, Proposition 1]). If A is Mittag-Leffler zero, then for each left exactfunctor h : A → B one has R i lim ←− r h ( A r ) = 0 for all i ≥ F = ( F r ) r ≥ on X we work with Jannsen’s continuous´etale cohomology H i ( X, F ) which is the i -th derived functor of F lim ←− r H ( X, F n ). By[Jan88, 3.1] one has an exact sequence(1.1.1) 0 → lim ←− r H i − ( X, F r ) → H i ( X, F ) → lim ←− r H i ( X, F r ) → . Note in particular, that if H i − ( X, F r ) is finite for all r , one has(1.1.2) H i ( X, F ) = lim ←− r H i ( X, F r ) . or F = ( F r ) Mittag-Leffler zero, one has for all i ≥ H i ( X, F ) = 0 . The divided power algebra.
Let A be a commutative ring and M be an A -module.Besides the usual symmetric power algebra Sym A ( M ) we need also the divided power algebraΓ A ( M ) (see [BO78, Appendix A] for more details).The A -algebra Γ A ( M ) is a graded augmented algebra with Γ ( M ) = A , Γ ( M ) = M andaugmentation ideal Γ + ( M ) := L k ≥ Γ k ( M ). For each element m ∈ M one has the dividedpower m [ k ] ∈ Γ k ( M ) with the property that m k = k ! m [ k ] where m k denotes the k -th power of m in Γ A ( M ). Moreover, one has the formula( m + n ) [ k ] = X i + j = k m [ i ] n [ j ] . In the case where M is a free A -module with basis m , . . . , m r the A -module Γ k ( M ) is free withbasis { m [ i ]1 · · · m [ i r ] r | P i j = k } . Further, for M free, there is an A -algebra isomorphismΓ A ( M ) ∼ = TSym A ( M )with the algebra of symmetric tensors (TSym kA ( M ) ⊂ Sym kA ( M ) are the invariants of thesymmetric group), which maps m [ k ] to m ⊗ k . Also, by the universal property of Sym A ( M ), onehas an A -algebra homomorphism(1.2.1) Sym A ( M ) → Γ A ( M )which maps m k to k ! m [ k ] . In particular, if A is a Q -algebra, this map is an isomorphism.If M is free and M ∨ := Hom A ( M, A ) denotes the A -dual one has in particularSym k ( M ∨ ) ∼ = Γ k ( M ) ∨ ∼ = TSym kA ( M ) ∨ . The algebra Γ A ( M ) has the advantage over TSym A ( M ) of being compatible with arbitrary basechange Γ A ( M ) ⊗ A B ∼ = Γ B ( M ⊗ A B )and thus sheafifies well. Recall from [Ill71, I 4.2.2.6.] that if F is an ´etale sheaf of Z p -modules,then Γ Z p ( F ) is defined to be the sheaf associated to the presheaf(1.2.2) U Γ Z p ( U ) ( F ( U )) . Definition 1.2.1.
We denote by b Γ A ( M ) := lim ←− r Γ A ( M ) / Γ + ( M ) [ r ] the completion of Γ A ( M ) with respect to the divided powers of the augmentation ideal. Note that Γ + ( M ) [ r ] = L k ≥ r Γ k ( M ) so that as A -module one has b Γ A ( M ) ∼ = Q k ≥ Γ k ( M ).In the same way we define the completion of Sym A ( M ) with respect to the augmentationideal Sym + A ( M ) to be(1.2.3) d Sym A ( M ) := lim ←− k Sym A ( M ) / (Sym + A ( M )) k Unipotent sheaves.
Let Λ = Z /p r Z , Z p or Q p and let π : X → S be a separated schemeof finite type, with X, S as in 1.1. A Λ-sheaf F on X is unipotent of length n , if it has afiltration 0 = F n +1 ⊂ F n ⊂ . . . ⊂ F = F such that F i / F i +1 ∼ = π ∗ G i for a Λ-sheaf G i on S .The next lemma is taken from [HK15], where it is stated in the setting of Q p -sheaves. Lemma 1.3.1.
Let
Λ = Z /p r Z , Z p or Q p and let π : X → S and π : X → S be smooth ofconstant fibre dimension d and d . Let f : X → X be an S -morphism. Let F be a unipotent Λ -sheaf. Then f ! F = f ∗ F ( d − d )[2 d − d ] . roof. Put c = d − d the relative dimension of f . We start with the case F = π ∗ G . In thiscase f ! F = f ! π ∗ G = f ! π !2 G ( − d )[ − d ] = π !1 G ( − d )[ − d ]= π ∗ G ( c )[2 c ] = f ∗ π ∗ G ( c )[2 c ] = f ∗ F ⊗ Λ( c )[2 c ] . In particular, f ! Λ = Λ( c )[2 c ] and we may rewrite the formula as f ∗ F ⊗ f ! Λ = f ! ( F ⊗ Λ) . There is always a map from the left to right via adjunction from the projection formula Rf ! ( f ∗ F ⊗ f ! Λ) = F ⊗ Rf ! f ! Λ → F ⊗ Λ . Hence we can argue on the unipotent length of F and it suffices to consider the case F = π ∗ G .This case was settled above. (cid:3) The next lemma is also taken from [HK15]. Let X → S be a smooth scheme with connectedfibres and e : S → X a section. Homomorphisms of unipotent sheaves are completely determinedby their restriction to S via e ∗ : Lemma 1.3.2.
Let π : X → S be smooth with connected fibres and e : S → X a section of π .Let Λ = Z /p r Z , Z p or Q p and F a unipotent Λ -sheaf on X . Then e ∗ : Hom X (Λ , F ) → Hom S (Λ , e ∗ F ) is injective.Proof. Let 0 → F → F → F → G .As e ∗ is exact and Hom left exact, we get a commutative diagram of exact sequences0 / / Hom X (Λ , F ) / / (cid:15) (cid:15) Hom X (Λ , F ) / / (cid:15) (cid:15) Hom X (Λ , F ) (cid:15) (cid:15) / / Hom S (Λ , e ∗ F ) / / Hom S (Λ , e ∗ F ) / / Hom S (Λ , e ∗ F ) . Suppose that the left and right vertical arrows are injective, then the middle one is injective aswell and it is enough to show the lemma in the case where F = π ∗ G . But the isomorphismHom X (Λ , π ∗ G ) ∼ = Hom X ( π ! Λ , π ! G ) ∼ = Hom S ( Rπ ! π ! Λ , G )factors through Hom X ( π ! Λ , π ! G ) e ! −→ Hom S (Λ , G ) → Hom S ( Rπ ! π ! Λ , G )where the last map is induced by the trace map Rπ ! π ! Λ → Λ. This proves the claim. (cid:3)
The geometric situation.
We recall the geometric set up from [HK15] using as much aspossible the notations from loc. cit. Let π : G → S be a smooth separated commutative group scheme with connected fibres of relative dimension d . We denote by e : S → G the unit section and by µ : G × S G → G the multiplication . Let j : U → G be the open complement of e ( S ).Let ι D : D → G be a closed subscheme with structural map π D : D → S . Typically π D will be ´etale and contained in the c -torsion of G for some c ≥
1. We note in passing, thatfor c invertible on S the c -torsion points of G , i.e. the kernel of the c -multiplication G [ c ], isquasi-finite and ´etale over S . Denote by j D : U D = G \ D → G the open complement of D . Wesummarize the situation in the basic diagram U D := G \ D j D / / % % ▲▲▲▲▲▲▲▲▲▲▲ G π (cid:15) (cid:15) D ι D o o π D (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ S e will also consider morphisms φ : G → G of S -group schemes as above. In this case wedecorate all notation with an index 1 or 2, e.g., d for the relative dimension of G /S .2. The logarithm sheaf
Homology of G . The basic sheaf in our constructions is the relative first Z p -homology H G of G/S , which we define as follows:
Definition 2.1.1.
For the group scheme π : G → S we let H := H G := R d − π ! Z p ( d ) = R − π ! π ! Z p . We write H r := H ⊗ Z /p r Z and H Q p := H ⊗ Q p for the associated Q p -sheaf. Note that H is not a lisse Z p -sheaf in general, but the stalks are free Z p -modules of finiterank, which follows for example from Lemma 2.1.2 below.The sheaf H and more generally R i π ! Z p is covariant functorial for any map of S -schemes f : G → X using the adjunction f ! f ! Z p → Z p . In particular, the group multiplication µ : G × S G → G induces a product R i π ! Z p ( d ) ⊗ R j π ! Z p ( d ) → R i + j − d π ! Z p ( d )and the diagonal ∆ : G → G × S G induces a coproduct R i π ! Z p ( d ) → M j R j π ! Z p ( d ) ⊗ R d + i − j π ! Z p ( d )on R · π ! Z p , which gives it the structure of a Hopf algebra and one has(2.1.1) R i π ! Z p ( d ) ∼ = d − i ^ H (this follows by base change to geometric points and duality from [BS13, Lemma 4.1.]). Thesame result holds for Z /p r Z -coefficients. Lemma 2.1.2.
Let G [ p r ] be the kernel of the p r -multiplication [ p r ] : G → G . Then there is acanonical isomorphism of ´etale sheaves G [ p r ] ∼ = R − π ! π ! Z /p r Z = H r . In particular, H G is the p -adic Tate-module of G .Proof. This is standard and we only sketch the proof: Consider G [ p r ] as an ´etale sheaf on S .The Kummer sequence is a G [ p r ]-torsor on G , hence gives a class in H ( G, π ∗ G [ p r ]) ∼ = Ext G ( π ∗ Z /p r Z , π ∗ G [ p r ]) ∼ = Ext G ( π ! Z /p r Z , π ! G [ p r ]) ∼ = ∼ = Ext S ( Rπ ! π ! Z /p r Z , G [ p r ]) ∼ = Hom S ( R − π ! π ! Z /p r Z , G [ p r ]) . Thus the Kummer torsor induces a map R − π ! π ! Z /p r Z → G [ p r ] and one can perform a basechange to geometric points s ∈ S to show that this is an isomorphism. But this follows thenfrom Poincar´e-duality and the isomorphism Hom s ( G [ p r ] , µ p r ) ∼ = H ( G, µ p r ) shown in [BS13,Lemma 4.2.]. (cid:3) The first logarithm sheaf.
Consider the complex Rπ ! π ! Z p calculating the homology of π : G → S and its canonical filtration whose associated graded pieces are the R i π ! π ! Z p . Weapply this to R Hom G ( π ! Z p , π ! H ) ∼ = R Hom S ( Rπ ! π ! Z p , H ) . Then the resulting hypercohomology spectral sequence gives rise to the five term sequence0 → Ext S ( Z p , H ) π ! −→ Ext G ( π ! Z p , π ! H ) → Hom S ( H , H ) →→ Ext S ( Z p , H ) π ! −→ Ext G ( π ! Z p , π ! H ) nd the maps π ! are injective because they admit the splitting e ! induced by the unit section e .This gives(2.2.1) 0 → Ext S ( Z p , H ) π ! −→ Ext G ( π ! Z p , π ! H ) → Hom S ( H , H ) → . Note that Ext G ( π ! Z p , π ! H ) ∼ = Ext G ( Z p , π ∗ H ). The same construction is also possible with thebase ring Λ r := Z /p r Z and H r and gives the exact sequence(2.2.2) 0 → Ext S (Λ r , H r ) π ! −→ Ext G ( π ! Λ r , π ! H r ) → Hom S ( H r , H r ) → . Definition 2.2.1.
The first logarithm sheaf ( L og (1) , (1) ) on G consists of an extension class → π ∗ H → L og (1) → Z p → such that its image in Hom S ( H , H ) is the identity together with a fixed splitting (1) : e ∗ Z p → e ∗ L og (1) . In exactly the same way one defines L og (1)Λ r . We denote by L og (1) Q p the associated Q p -sheaf. The existence and uniqueness of ( L og (1) , (1) ) follow directly from (2.2.1). The automor-phisms of L og (1) form a torsor under Hom G ( Z p , π ∗ H ). In particular, the pair ( L og (1) , (1) )admits no automorphisms except the identity.It is obvious from the definition that one has(2.2.3) L og (1) ⊗ Z p Λ r ∼ = L og (1)Λ r so that L og (1) = ( L og (1)Λ r ) r ≥ . Moreover, L og (1) is compatible with arbitrary base change. If(2.2.4) G T f T −−−−→ G π T y y π T f −−−−→ S is a cartesian diagram one has f ∗ T L og (1) G ∼ = L og (1) G T and f ∗ T ( (1) ) defines a splitting.Let ϕ : G → G be a homomorphism of group schemes of relative dimension d , d , respectively and write c := d − d . Theorem 2.2.2.
For ϕ : G → G as above, there is a unique morphism of sheaves ϕ : L og (1) G → ϕ ∗ L og (1) G ∼ = ϕ ! L og (1) G ( − c )[ − c ] such that ϕ ( (1) G ) = (1) G . Moreover, if ϕ is an isogeny of degree prime to p , then ϕ is anisomorphism.Proof. Pull-back of L og (1) G gives an exact sequence0 → π ∗ H G → ϕ ∗ L og (1) G → Z p → L og (1) G by π ∗ H G → π ∗ H G induces a map(2.2.5) 0 −−−−→ π ∗ H G −−−−→ L og (1) G −−−−→ Z p −−−−→ y y h (cid:13)(cid:13)(cid:13) −−−−→ π ∗ H G −−−−→ ϕ ∗ L og (1) G −−−−→ Z p −−−−→ . If ϕ is an isogeny and deg ϕ is prime to p , then π ∗ H G → π ∗ H G is an isomorphism, hence also h .By uniqueness there is a unique isomorphism of the pair ( L og (1) G , e ∗ ( h ) ◦ (1) G ) with ( L og (1) G , (1) G ).The composition of this isomorphism with h is the desired map. If h ′ : L og (1) G → ϕ ! L og (1) G is nother map with this property, the difference h − h ′ : Z p → π ∗ H G is uniquely determinedby its pull-back e ∗ ( h − h ′ ) : Z p → e ∗ L og (1) G according to Lemma 1.3.2. If both, h and h ′ arecompatible with the splittings, then e ∗ ( h − h ′ ) = 0 and hence h = h ′ . (cid:3) Corollary 2.2.3 (Splitting principle) . Let ϕ : G → G be an isogeny of degree prime to p .Then if t : S → G is in the kernel of ϕ , then t ∗ L og (1) G ∼ = t ∗ ϕ ∗ L og (1) G ∼ = e ∗ ϕ ∗ L og (1) G ∼ = e ∗ L og (1) G . Proof.
Apply t ∗ to ϕ . (cid:3) The Q p -logarithm sheaf. We are going to define the Q p -logarithm sheaf, which has beenstudied extensively in [HK15]. Definition 2.3.1.
We define L og ( k ) Q p := Sym k ( L og (1) Q p ) and denote by ( k ) := 1 k ! Sym k ( (1) ) : Q p → L og ( k ) Q p the splitting induced by (1) . We note that L og ( k ) Q p is unipotent of length k and that the splitting ( k ) induces an isomor-phism(2.3.1) e ∗ L og ( k ) Q p ∼ = k Y i =0 Sym i H Q p . To define transition maps(2.3.2) L og ( k ) Q p → L og ( k − Q p consider the morphism L og (1) Q p → Q p ⊕ L og (1) Q p given by the canonical projection and the identity.Then we have L og ( k ) Q p = Sym k ( L og (1) Q p ) → Sym k ( Q p ⊕ L og (1) Q p ) ∼ = M i + j = k Sym i ( Q p ) ⊗ Sym j ( L og (1) Q p ) →→ Sym ( Q p ) ⊗ Sym k − ( L og (1) Q p ) ∼ = L og ( k − Q p . A straightforward computation shows that ( k ) ( k − under this transition map.2.4. Main properties of the Q p -logarithm sheaf. The logarithm sheaf has three mainproperties: functoriality, vanishing of cohomology and a universal mapping property for unipo-tent sheaves. Functoriality follows trivially from Theorem 2.2.2. We review the others briefly,referring for more details to [HK15].Let ϕ : G → G be a homomorphism of group schemes of relative dimension d , d , respec-tively and let c := d − d be the relative dimension of the homomorphism. Theorem 2.4.1 (Functoriality) . For ϕ : G → G as above there is a unique homomorphismof sheaves ϕ : L og Q p ,G → ϕ ∗ L og Q p ,G ∼ = ϕ ! L og Q p ,G ( − c )[ − c ] such that G maps to G . Moreover, if ϕ is an isogeny, the ϕ is an isomorphism.Proof. This follows directly from Theorem 2.2.2 and the fact that deg ϕ is invertible in Q p . (cid:3) orollary 2.4.2 (Splitting principle) . Let ϕ : G → G be an isogeny. Then if t : S → G isin the kernel of ϕ , one has ̺ t : t ∗ L og Q p ,G ∼ = t ∗ ϕ ∗ L og Q p ,G ∼ = e ∗ ϕ ∗ L og Q p ,G ∼ = e ∗ L og Q p ,G ∼ = Y k ≥ Sym k H Q p ,G . More generally, if ι : ker ϕ → G is the closed immersion, one has ι ∗ L og G ∼ = π | ∗ ker ϕ Y k ≥ Sym k H Q p ,G , where π | ker ϕ : ker ϕ → S is the structure map.Proof. Apply t ∗ to both sides of the isomorphism ϕ and use (2.3.1). For the second statementmake the base change to ker ϕ and apply the first statement to the tautological section ofker ϕ . (cid:3) Theorem 2.4.3 (Vanishing of cohomology) . One has R i π ! L og Q p ∼ = ( Q p ( − d ) if i = 2 d if i = 2 d. More precisely, the transition maps R i π ! L og ( k ) Q p → R i π ! L og ( k − Q p are zero for i < d and one hasan isomorphism R d π ! L og ( k ) Q p ∼ = Q p ( − d ) compatible with the transition maps.Proof. This is Theorem 3.3.1. in [HK15]. (cid:3)
Let F be a unipotent sheaf of finite length n on G . Consider the homomorphism(2.4.1) π ∗ Hom G ( L og Q p , F ) → e ∗ F defined as the composition of π ∗ Hom G ( L og Q p , F ) → π ∗ e ∗ e ∗ Hom G ( L og Q p , F ) → Hom S ( e ∗ L og Q p , e ∗ F )with Hom S ( e ∗ L og Q p , e ∗ F ) ( ) ∗ −−→ Hom S ( Q p , e ∗ F ) ∼ = e ∗ F . Theorem 2.4.4 (Universal property) . Let F be a unipotent sheaf of finite length. Then themap (2.4.1) induces an isomorphism π ∗ Hom( L og Q p , F ) ∼ = e ∗ F . Proof.
This is Theorem 3.3.2. in [HK15]. (cid:3) The Q p -polylogarithm and Eisenstein classes Construction of the Q p -polylogarithm. Fix an auxiliary integer c > S and consider the c -torsion subgroup D := G [ c ] ⊂ G . We write U D := G \ D and consider U D j D −→ G ι D ←− D. We also write π D : D → S for the structure map.For any sheaf F the localization triangle defines a connecting homomorphism(3.1.1) Rπ ! Rj D ∗ j ∗ D F [ − → Rπ ! ι D ! ι ! D F . As L og ( k ) Q p ( d )[2 d ] is unipotent we may use Lemma 1.3.1 to replace ι ! D by ι ∗ D . Using Corollary2.4.2 one gets π D ! ι ! D L og ( k ) Q p ( d )[2 d ] ∼ = k Y i =0 π D ! π ∗ D Sym i H Q p . utting everything together and taking the limit over the transition maps L og ( k ) Q p → L og ( k − Q p gives the residue map (3.1.2) res : H d − ( S, Rπ ! Rj D ∗ j ∗ D L og Q p ( d )) → H ( S, Y k ≥ π D ! π ∗ D Sym k H Q p ) . Proposition 3.1.1.
The localization triangle induces a short exact sequence → H d − ( S, Rπ ! Rj D ∗ j ∗ D L og Q p ( d )) res −−→ H ( S, Y k ≥ π D ! π ∗ D Sym k H Q p ) → H ( S, Q p ) → . Proof.
This is an immediate consequence from the localization triangle and the computation of Rπ ! L og in Theorem 2.4.3. (cid:3) Definition 3.1.2.
Let Q p [ D ] := ker( H ( S, π D ! Q p ) → H ( S, Q p )) where the map is induced by the trace π D ! Q p → Q p . Note that Q p [ D ] ⊂ ker H ( S, Y k ≥ π D ! π ∗ D Sym k H Q p ) → H ( S, Q p ) . Definition 3.1.3.
Let α ∈ Q p [ D ] . Then the unique class α pol Q p ∈ H d − ( S, Rπ ! Rj D ∗ j ∗ D L og Q p ( d )) with res( α pol Q p ) = α is called the polylogarithm class associated to α . We write α pol k Q p for theimage of α pol Q p in H d − ( S, Rπ ! Rj D ∗ j ∗ D L og ( k ) Q p ( d )) . Eisenstein classes.
Recall that D = G [ c ] and fix an integer N > S , suchthat ( N, c ) = 1 and let t : S → U D = G \ D be an N -torsion section. Consider the composition(3.2.1) Rπ ! Rj D ∗ j ∗ D L og ( d ) → Rπ ! Rj D ∗ j ∗ D Rt ∗ t ∗ L og ( d ) ∼ = Rπ ! Rt ∗ t ∗ L og ( d ) ∼ = t ∗ L og ( d )induced by the adjunction id → Rt ∗ t ∗ , the fact that Rt ∗ = Rt ! and because π ◦ t = id. Togetherwith the splitting principle from Corollary 2.4.2 and the projection to the k -th component onegets an evaluation map(3.2.2) H d − ( S, Rπ ! Rj D ∗ j ∗ D L og ( k ) Q p ( d )) ̺ t ◦ t ∗ −−−→ H d − ( S, k Y i =0 Sym i H Q p ) pr k −−→ H d − ( S, Sym k H Q p ) . Definition 3.2.1.
Let α ∈ Q p [ D ] . The image of α pol Q p under the evaluation map (3.2.2) α Eis k Q p ( t ) ∈ H d − ( S, Sym k H Q p ) is called the k -th ´etale Q p -Eisenstein class for G .Remark . The normalization in [Kin15, Definition 12.4.6] is different. There we had anadditional factor of − N k − in front of α Eis k Q p ( t ). This has the advantage to make the residues of α Eis k Q p ( t ) at the cusps integral, but is very unnatural from the point of view of the polylogarithm.Recall from [HK15, Theorem 5.2.1] that the polylogarithm α pol k Q p is motivic, i.e., there existsa class in motivic cohomology α pol k mot ∈ H d − ( S, Rπ ! Rj D ∗ j ∗ D L og ( k )mot ( d )) , the motivic polylogarithm , which maps to α pol k Q p under the ´etale regulator r ´et : H d − ( S, Rπ ! Rj D ∗ j ∗ D L og ( k )mot ( d )) → H d − ( S, Rπ ! Rj D ∗ j ∗ D L og ( k ) Q p ( d )) . ith the motivic analogue of the evaluation map (3.2.2) one can define exactly in the same wayas in the ´etale case motivic Eisenstein classes for α ∈ Q [ D ] (3.2.3) α Eis k mot ( t ) ∈ H d − ( S, Sym k H Q ) . The next proposition is obvious from the fact that the evaluation map is compatible with the´etale regulator.
Proposition 3.2.3.
For α ∈ Q [ D ] the image of the motivic Eisenstein class α Eis k mot ( t ) underthe ´etale regulator r ´et : H d − ( S, Sym k H Q ) → H d − ( S, Sym k H Q p ) is the ´etale Q p -Eisenstein class α Eis k Q p ( t ) . Sheaves of Iwasawa algebras
Iwasawa algebras.
Let X = lim ←− r X r be a profinite space with transition maps λ r : X r +1 → X r and Λ r [ X r ] := Map( X r , Z /p r Z )the Z /p r Z -module of maps from X r to Z /p r Z . For each x r we write δ x r ∈ Λ r [ X r ] for themap which is 1 at x r and 0 else. It is convenient to interpret Λ r [ X r ] as the space of Z /p r Z -valued measures on X r and δ x r as the delta measure at x r . Then the push-forward along λ r : X r +1 → X r composed with reduction modulo p r induces Z p -module maps(4.1.1) λ r ∗ : Λ r +1 [ X r +1 ] → Λ r [ X r ]which are characterized by λ r ∗ ( δ x r +1 ) = δ λ r ( x r ) . Definition 4.1.1.
The module of Z p -valued measures on X is the inverse limit Λ( X ) := lim ←− r Λ r [ X r ] of Λ r [ X r ] with respect to the transition maps from (4.1.1) . Let x = ( x r ) r ≥ ∈ X . We define δ x := ( δ x r ) r ≥ ∈ Λ( X ), which provides a map δ : X → Λ( X ) . For each continuous map ϕ : X → Y of profinite spaces we get a homomorphism(4.1.2) ϕ ∗ : Λ( X ) → Λ( Y )”push-forward of measures” with the property ϕ ∗ ( δ x ) = δ ϕ ( x ) . Obviously, one has Λ r [ X r × Y r ] ∼ =Λ r [ X r ] ⊗ Λ r [ Y r ] so thatΛ( X × Y ) ∼ = Λ( X ) b ⊗ Λ( Y ) := lim ←− r Λ r [ X r ] ⊗ Λ r [ Y r ] . In particular, if X = G = lim ←− r G r is a profinite group, the group structure µ : G × G → G induces a Z p -algebra structure on Λ( G ), which coincides with the Z p -algebra structure inducedby the inverse limit of group algebras lim ←− r Λ r [ G r ]. Definition 4.1.2. If G = lim ←− r G r is a profinite group, we call Λ( G ) := lim ←− r Λ r [ G r ] the Iwasawa algebra of G . More generally, if G acts continuously on the profinite space X , one gets a mapΛ( G ) b ⊗ Λ( X ) → Λ( X )which makes Λ( X ) a Λ( G )-module. If X is a principal homogeneous space under G , then Λ( X )is a free Λ( G )-module of rank 1. .2. Properties of the Iwasawa algebra.
In this section we assume that H is a finitelygenerated free Z p -module. We let H r := H ⊗ Z p Z p /p r Z p so that H = lim ←− r H r with the natural transition maps H r +1 → H r .In the case H = Z p , the so called Amice transform of a measure µ ∈ Λ( Z p ) A µ ( T ) := ∞ X n =0 T n Z Z p (cid:18) xn (cid:19) µ ( x )induces a ring isomorphism A : Λ( Z p ) ∼ = Z p [[ T ]] (see [Col00, Section 1.1.]). A straightforwardgeneralization shows that Λ( H ) is isomorphic to a power series ring in rk H variables. On theother hand one has the so called Laplace transform of µ (see loc. cit.) L µ ( t ) := ∞ X n =0 t n n ! Z Z p x n µ ( x ) . This map is called the moment map in [Kat76] and we will follow his terminology. In the nextsection, we will explain this map from an abstract algebraic point of view. For this we interpret t n n ! as t [ n ] in the divided power algebra Γ Z p ( Z p ).4.3. The moment map.
We return to the case of a free Z p -module H of finite rank. Proposition 4.3.1.
Let H be a free Z p -module of finite rank and H r := H ⊗ Z p Z /p r Z . Then b Γ Z p ( H ) ∼ = lim ←− r b Γ Z /p r Z ( H r ) . Proof.
As each Γ Z p ( H ) / Γ + ( H ) [ k ] is a finitely generated free Z p -module, this follows by thecompatibility with base change of Γ Z p ( H ) and the fact that one can interchange the inverselimits. (cid:3) By the universal property of the finite group ring Λ r [ H r ], the group homomorphism H r → b Γ Z /p r Z ( H r ) × h r X k ≥ h [ k ] r induces a homomorphism of Z /p r Z -algebrasmom r : Λ r [ H r ] → b Γ Z /p r Z ( H r ) . Corollary 4.3.2.
The maps mom r induce in the inverse limit a Z p -algebra homomorphism mom : Λ( H ) → b Γ Z p ( H ) . which is functorial in H . Definition 4.3.3.
We call mom : Λ( H ) → b Γ Z p ( H ) the moment map and the composition withthe projection to Γ k ( H ) mom k : Λ( H ) → Γ k ( H ) the k -th moment map . .4. Sheafification of the Iwasawa algebras.
Let X be a separated noetherian scheme offinite type as in section 1.1 and X := ( p r : X r → X ) r be an inverse system of quasi-finite ´etaleschemes over X with ´etale transition maps λ r : X r → X r − . We often write(4.4.1) Λ r := Z /p r Z . The adjunction λ r ! λ ! r → id defines a homomorphism p r +1! Λ r +1 = p r ! λ r ! λ ! r Λ r +1 → p r ! Λ r +1 , because λ r is ´etale. If one composes this with reduction modulo p r , one gets a trace map(4.4.2) Tr r +1 : p r +1 , ! Λ r +1 → p r, ! Λ r . Definition 4.4.1.
We define an ´etale sheaf on X by Λ r [ X r ] := p r ! Λ r . With the trace maps Tr r +1 : Λ r +1 [ X r +1 ] → Λ r [ X r ] as transition morphisms we define thepro-sheaf Λ( X ) := (Λ r [ X r ]) r ≥ . This definition is functorial in X . If ( ϕ r ) r : ( X r ) r → ( Y r ) r is a morphism of inverse systemof quasi-finite ´etale schemes over X , then the adjunction ϕ r ! ϕ ! r → id defines a morphism ϕ r ! : Λ r [ X r ] → Λ r [ Y r ]compatible with the transition maps, and hence a morphism of pro-sheavesΛ( X ) → Λ( Y ) . Moreover, the formation of Λ( X ) is compatible with base change: if X r,T := X r × S T for an S -scheme f : T → S , then by proper base change one has f ∗ Λ r [ X r ] ∼ = Λ[ X r,T ] . By the K¨unneth formula, one hasΛ r [ X r × X Y r ] ∼ = Λ r [ X r ] ⊗ Λ r [ Y r ]and hence Λ( X × X Y ) ∼ = Λ( X ) b ⊗ Λ( Y ) by taking the inverse limit. In particular, in the casewhere X = G is an inverse system of quasi-finite ´etale group schemes G r , the group structure µ r : G r × X G r → G r induces a ring structureΛ( G ) b ⊗ Λ( G ) → Λ( G )on Λ( G ). Similarly, if G × X X → X is a group action of inverse systems, i.e., a compatible family of actions G r × X X r → X r , thenΛ( X ) becomes a Λ( G )-module.The next lemma shows that the above construction indeed sheafifies the Iwasawa algebrasconsidered before. Lemma 4.4.2.
Let x ∈ X be a geometric point and write p r,x : X r,x → x for the base changeof X r to x considered as a finite set. Then Λ r [ X r ] x ∼ = Λ r [ X r ] . Proof.
This follows directly from the base change property of Λ r [ X r ] and the fact that p r,x, ! Λ r ∼ =Λ r [ X r ] over an algebraically closed field. (cid:3) We return to our basic set up, where π : G → S is a separated smooth commutative groupscheme with connected fibres. Recall from 2.1.2 that H r is the sheaf associated to G [ p r ], whichis quasi-finite and ´etale over S . Definition 4.4.3.
Define the sheaf of Iwasawa algebras Λ( H ) on S to be the pro-sheaf Λ( H ) := (Λ r [ H r ]) r ≥ . .5. Sheafification of the moment map.
We keep the notation of the previous section. Inparticular, we consider the ´etale sheaf H r and the sheaf Λ r [ H r ].Over G [ p r ] the sheaf [ p r ] ∗ H r has the tautological section τ r ∈ Γ( G [ p r ] , [ p r ] ∗ H r ) correspondingto the identity map G [ p r ] → H r . This gives rise to the section(4.5.1) τ [ k ] r ∈ Γ( G [ p r ] , [ p r ] ∗ Γ k ( H r ))of the k -th divided power of H r . Using the chain of isomorphisms (note that [ p r ] ∗ = [ p r ] ! as[ p r ] is ´etale)Γ( G [ p r ] , [ p r ] ∗ Γ k ( H r )) ∼ = Hom G [ p r ] ( Z /p r Z , [ p r ] ∗ Γ k ( H r )) ∼ = Hom S ([ p r ] ! Z /p r Z , Γ k ( H r )) , the section τ [ k ] r gives rise to a morphism of sheaves(4.5.2) mom kr : Λ r [ H r ] → Γ k ( H r ) . Lemma 4.5.1.
There is a commutative diagram Λ r [ H r ] mom kr −−−−→ Γ k ( H r ) Tr r y y Λ r − [ H r − ] mom kr − −−−−−→ Γ k ( H r − ) where the right vertical map is given by the reduction map Γ k ( H r ) → Γ k ( H r ) ⊗ Z /p r Z Z /p r − Z ∼ = Γ k ( H r − ) . Proof.
Denote by λ r : H r → H r − the transition map. Reduction modulo p r − gives a commu-tative diagram [ p r ] ! Z /p r Z mom kr −−−−→ Γ k ( H r ) y y [ p r ] ! λ ∗ r Z /p r − Z mom kr ⊗ Z /p r − Z −−−−−−−−−−→ Γ k ( H r − ) . As the image of the tautological class τ [ k ] r ∈ Γ( G [ p r ] , [ p r ] ∗ Γ k ( H r )) under the reduction mapgives the the pull-back of the tautological class λ ∗ r τ [ k ] r − ∈ Γ( G [ p r ] , [ p r ] ∗ Γ k ( H r − )) ∼ = Hom G [ p r ] ( λ ∗ r Z /p r − Z , [ p r ] ∗ Γ k ( H r − )) ∼ = Hom S ([ p r ] ! λ ∗ r Z /p r − Z , Γ k ( H r − ))one concludes that mom kr ⊗ Z /p r − Z coincides with the map given by λ ∗ r τ [ k ] r − . This means thatmom kr ⊗ Z /p r − Z has to factor through Tr r , i.e., the diagram[ p r − ] ! λ r ! λ ∗ r Z /p r − Z mom kr ⊗ Z /p r − Z / / Tr r ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ Γ k ( H r − )[ p r − ] ! Z /p r − Z mom kr − ♥♥♥♥♥♥♥♥♥♥♥♥ commutes, which gives the desired result. (cid:3) With this result we can now define the moment map for the sheaf of Iwasawa algebras Λ( H ). Definition 4.5.2.
We define the k -th moment map to be the map of pro-sheaves mom k : Λ( H ) → Γ k ( H ) defined by (mom kr ) r ≥ and mom : Λ( H ) → b Γ Z p ( H ) by taking mom k in the k -th component. emark . In each stalk the the map mom k coincides with the map mom k defined in 4.3.3(see [Kin15, Lemma 12.2.14]).5. The integral logarithm sheaf
Definition of the integral logarithm sheaf.
We now define a pro-sheaf L on G ofmodules over π ∗ Λ( H ), which will give a Z p -structure of the logarithm sheaf L og Q p . For thiswrite G r := G considered as a quasi-finite ´etale G -scheme via the p r -multiplication(5.1.1) [ p r ] : G r = G → G. Note that this is a G [ p r ]-torsor 0 → G [ p r ] → G r [ p r ] −−→ G → G . Let λ r : G r → G r − be the transition map, which is just the [ p ]-multiplication in thiscase. Then, as in (4.4.2), we have trace mapsTr r : Λ r [ G r ] → Λ r − [ G r − ] . We will also need the following variant. Let Λ s := Z /p s Z and write(5.1.2) Λ s [ G r ] := [ p r ] ! Λ s . Then the adjunction λ r ! λ ! r → id defines transition morphisms(5.1.3) λ r ! : Λ s [ G r ] → Λ s [ G r − ] . Definition 5.1.1.
With the above transition maps we can define the pro-sheaves L := (Λ r [ G r ]) r ≥ and L Λ s := (Λ s [ G r ]) r ≥ . We call L the integral logarithm sheaf . Note that the reduction modulo p s − gives transition maps L Λ s → L Λ s − and that we havean isomorphism of pro-sheaves(5.1.4) L ∼ = ( L Λ s ) s ≥ . By the general theory outlined above, L is a module over π ∗ Λ( H ) which is free of rank 1.Let t : S → G be a section and denote by G [ p r ] h t i the G [ p r ]-torsor defined by the cartesiandiagram(5.1.5) G [ p r ] h t i −−−−→ G r y y [ p r ] S t −−−−→ G. We denote by H r h t i the ´etale sheaf defined by G [ p r ] h t i and by H h t i := ( H r h t i ) the pro-systemdefined by the trace maps. We writeΛ( H h t i ) := (Λ r [ H r h t i ]) r ≥ for the sheaf of Iwasawa modules defined by H h t i . Lemma 5.1.2.
There is an canonical isomorphism t ∗ L ∼ = Λ( H h t i ) . In particular, for the unit section e : S → G one has e ∗ L ∼ = Λ( H ) and hence a section : Z p → e ∗ L given by mapping to .Proof. This follows directly from the fact that L is compatible with base change and the defi-nitions. (cid:3) .2. Basic properties of the integral logarithm sheaf.
The integral logarithm sheaf enjoysthe same properties as its Q p -counterpart, namely functoriality, vanishing of cohomology and auniversal property for unipotent sheaves.Let ϕ : G → G be a homomorphism of group schemes of relative dimension d and d over S . Denote by L and L the integral logarithm sheaves on G and G respectively. Theorem 5.2.1 (Functoriality) . Let c := d − d . Then there is a canonical map ϕ : L → ϕ ∗ L ∼ = ϕ ! L ( − c )[ − c ] . Moreover, if ϕ is an isogeny of degree prime to p , then ϕ : L ∼ = ϕ ∗ L is an isomorphism.Proof. The homomorphism ϕ induces a homomorphism of group schemes over G (5.2.1) ϕ : G ,r → G ,r × G G which induces by adjunction ϕ ! ϕ ! → id and the base change property of Λ r [ G ,r ] a morphismof sheaves ϕ : Λ r [ G ,r ] → ϕ ∗ Λ r [ G ,r ] = ϕ ! Λ r [ G ,r ]( − c )[ − c ] . Passing to the limit gives the required map. If ϕ is an isogeny of degree prime to p , then themap in (5.2.1) is an isomorphism. Hence this is also true for ϕ . (cid:3) Corollary 5.2.2 (Splitting principle) . Let c be an integer prime to p and let t : S → G be a c -torsion section. Then there is an isomorphism [ c ] : t ∗ L ∼ = Λ( H ) . More generally, if D := G [ c ] with ( c, p ) = 1 then ι ∗ D L ∼ = π ∗ D Λ( H ) , where ι D : D → G and π D : D → S is the structure map.Proof. Apply t ∗ respectively, ι ∗ D to the isomorphism [ c ] : L → [ c ] ∗ L . (cid:3) Theorem 5.2.3 (Vanishing of cohomology) . Recall that d is the relative dimension of π : G → S . Then the pro-sheaves R i π ! L for i < d are Mittag-Leffler zero (see 1.1) and R d π ! L ( d ) ∼ = Z p . We start the proof of this theorem with a lemma:
Lemma 5.2.4.
The endomorphism [ p r ] ! : R i π ! Z /p s Z → R i π ! Z /p s Z is given by multiplicationwith p r (2 d − i ) .Proof. By Lemma 2.1.2 we see that [ p r ] ! is given by p r -multiplication on H s . The result followsfrom this and the Z /p s Z -version of the isomorphism (2.1.1) (cid:3) Proof of Theorem 5.2.3.
Consider the transition map Λ s [ G r + j ] → Λ s [ G r ]. If we apply R i π ! weget the homomorphism [ p j ] ! : R i π r + j, ! Λ s → R i π r, ! Λ s , where π r = π : G r → S is the structure map of G r = G . By Lemma 5.2.4, the map [ p j ] ! acts bymultiplication with p j (2 d − i ) on R i π r + j, ! Λ s . In particular, this is zero for i = 2 d and j ≥ s andthe identity for i = 2 d . This proves the theorem, because R d π ! Λ s ( d ) ∼ = Λ s . (cid:3) The sheaf L satisfies also a property analogous to Theorem 2.4.4. To formulate this properly,we first need a property of unipotent Z /p s Z -sheaves. emma 5.2.5. Let F be a unipotent Λ s = Z /p s Z -sheaf of length n on G . Then [ p ns ] ∗ F istrivial on G ns in the sense that there exists a Λ s -sheaf G on S such that [ p ns ] ∗ F ∼ = π ∗ ns G , where π ns : G ns → S is the structure map.Proof. We show this by induction. For n = 0 there is nothing to show. So let0 → F ′ → F → π ∗ G ′′ → F ′ unipotent of length n −
1, so that by induction hypothe-ses [ p ( n − s ] ∗ F ′ ∼ = π ∗ G ′ on G ( n − s . Thus it suffices to show that for an extension F ∈ Ext G ( π ∗ G ′′ , π ∗ G ′ ), the sheaf [ p s ] ∗ F is trivial on G s . One hasExt G ( π ∗ G ′′ , π ∗ G ′ ) ∼ = Ext G ( π ! G ′′ , π ! G ′ ) ∼ = Ext S ( Rπ ! π ! G ′′ , G ′ )and the pull-back by [ p s ] ∗ on the first group is induced by the trace map [ p s ] ! : Rπ ! [ p s ] ! [ p s ] ! π ! G ′′ → Rπ ! π ! G ′′ on the last group. By the projection formula we have Rπ ! π ! G ′′ ∼ = Rπ ! Λ s ( d )[2 d ] ⊗ G ′′ and the triangle τ < d Rπ ! Λ s ( d )[2 d ] → Rπ ! Λ s ( d )[2 d ] → R d π ! Λ s ( d ) ∼ = Λ s gives rise to a long exact sequence of Ext-groups . . . → Ext S ( G ′′ , G ′ ) → Ext S ( Rπ ! Λ s ( d )[2 d ] ⊗ G ′′ , G ′ ) → Ext S ( τ < d Rπ ! Λ s ( d )[2 d ] ⊗ G ′′ , G ′ ) → . . . If we pull-back by [ p s ] ∗ and use Lemma 5.2.4 the resulting map on Ext S ( τ < d Rπ ! Λ s ( d )[2 d ] ⊗ G ′′ , G ′ ) is zero, which shows that [ p s ] ∗ F is in the image ofExt S ( G ′′ , G ′ ) [ p s ] ∗ π ∗ −−−−→ Ext G ([ p s ] ∗ π ∗ G ′′ , [ p s ] ∗ π ∗ G ′ ) . This is the desired result. (cid:3)
Exactly as in (2.4.1) one can define for each Λ s -sheaf F and each r a homomorphism(5.2.2) π ∗ Hom G (Λ s [ G r ] , F ) → e ∗ F as the composition π ∗ Hom G ( L Λ s ,r , F ) → π ∗ e ∗ e ∗ Hom G ( L Λ s ,r , F ) → Hom S ( e ∗ L Λ s ,r , e ∗ F ) ∗ −→ Hom S (Λ s , e ∗ F )The next theorem corrects and generalizes [Kin15, Proposition 4.5.3], which was erroneouslystated for all Z /p s Z -sheaves and not just for unipotent ones. Theorem 5.2.6 (Universal property) . Let F be a unipotent Λ s -sheaf of length n . Then thehomomorphism (5.2.2) π ∗ Hom G (Λ s [ G ns ] , F ) ∼ = e ∗ F is an isomorphism.Proof. Let F be unipotent of length n . Then we know from Lemma 5.2.5 that there is a Λ s -sheaf G on S such that [ p ns ] ∗ F ∼ = π ∗ ns G , where π ns : G ns → S is the structure map. Similarly,we write e ns for the unit section of G ns . Then one has e ∗ F ∼ = e ∗ ns [ p ns ] ∗ F ∼ = e ∗ ns π ∗ ns G ∼ = G . Further, one has the following chain of isomorphisms π ∗ Hom G (Λ s [ G ns ] , F ) = π ∗ Hom G ([ p ns ] ! Λ s , F ) ∼ = π ns ∗ Hom G ns (Λ s , [ p ns ] ∗ F ) ∼ = π ns ∗ Hom G ns (Λ s , π ∗ ns G ) ∼ = Hom S ( Rπ ns ! Λ s ( d )[2 d ] , G ) ∼ = Hom S ( R d π ns ! Λ s ( d ) , G ) ∼ = G ∼ = e ∗ F , which prove the theorem. (cid:3) .3. The integral ´etale poylogarithm.
In this section we define in complete analogy withthe Q p -case the integral ´etale polylogarithm.We recall the set-up from section 3.1. Denote by c > S and primeto p and let D := G [ c ] be the c -torsion subgroup. Then the localization triangle for j D : U D ⊂ G and ι D : D → G reads Rπ ! L ( d )[2 d − → Rπ ! Rj D ∗ j ∗ D L ( d )[2 d − → π D ! ι ! D L ( d ) . By relative purity and the splitting principle ι ! D L ( d )[2 d ] ∼ = ι ∗ D L ∼ = π ∗ D Λ( H ). We apply thefunctor H j ( S, − ) to this triangle. As the R i π ! L are Mittag-Leffler zero for i = 2 d by Theorem5.2.3 one gets with (1.1.3): Proposition 5.3.1.
In the above situation there is a short exact sequence → H d − ( S, Rπ ! Rj D ∗ j ∗ D L ( d )) res −−→ H ( S, π D ! π ∗ D Λ( H )) → H ( S, Z p ) → . As in the Q p -case we define Z p [ D ] := ker (cid:0) H ( S, π D ! π ∗ D Z p ) → H ( S, Z p ) (cid:1) so that one has Z p [ D ] ⊂ ker (cid:0) H ( S, π D ! π ∗ D Λ( H )) → H ( S, Z p ) (cid:1) . With these preliminaries we can define the integral polylogarithm.
Definition 5.3.2.
The integral ´etale polylogarithm associated to α ∈ Z p [ D ] is the unique class α pol ∈ H d − ( S, Rπ ! Rj D ∗ j ∗ D L ( d )) such that res( α pol) = α . The Eisenstein-Iwasawa class.
Recall that D = G [ c ] and let t : S → U D = G \ D be an N -torsion section with ( N, c ) = 1 but N not necessarily prime to p . The same chain of mapsas in (3.2.1) gives a map(5.4.1) H d − ( S, Rπ ! Rj D ∗ j ∗ D L ( d )) → H d − ( S, t ∗ L ( d )) ∼ = H d − ( S, Λ( H h t i )( d )) . By functoriality the N -multiplication induces a homomorphism[ N ] : Λ( H h t i ) → Λ( H ) . Definition 5.4.1.
Let α ∈ Z p [ D ] and t : S → U D be an N -torsion section. Then the image α EI ( t ) ∈ H d − ( S, Λ( H h t i )( d )) of α pol under the map (5.4.1) is called the Eisenstein-Iwasawa class . We write α EI ( t ) N := [ N ] ( α EI ( t )) ∈ H d − ( S, Λ( H )( d )) . Remark . Note that α EI ( t ) N depends on N and not on t alone. The class α EI ( t ) NM differs from α EI ( t ) N .The k -th moment map induces a homomorphism of cohomology groups(5.4.2) mom k : H d − ( S, Λ( H )( d )) → H d − ( S, Γ k ( H )( d )) . Definition 5.4.3.
The class α Eis kN ( t ) := mom k ( α EI N ) ∈ H d − ( S, Γ k ( H )( d )) is called the integral ´etale Eisenstein class . These Eisenstein classes are interpolated by the Eisenstein-Iwasawa class by definition. Wewill see later how they are related to the Q p -Eisenstein class, which are motivic, i.e., in theimage of the ´etale regulator from motivic cohomology. .5. The Eisenstein-Iwasawa class for abelian schemes.
It is worthwhile to consider thecase of abelian schemes in more detail. In this section we let G = A be an abelian scheme over S , so that in particular π : A → S is proper and we can write Rπ ∗ instead of Rπ ! .The first thing to observe is the isomorphism H d − ( S, Rπ ! Rj D ∗ j ∗ D L og ( d )) ∼ = H d − ( U D , L og ( d )) , so that the Q p -polylogarithm is a class α pol Q p ∈ H d − ( U D , L og ( d )) . Evaluation at the N -torsion section t : S → U D is just the pull-back with t ∗ t ∗ α pol Q p ∈ H d − ( S, t ∗ L og ( d )) ∼ = H d − ( S, Y k ≥ Sym k H Q p ( d ))and the k -th component of t ∗ α pol Q p is α Eis k Q p ( t ).There is one specific choice of α which is particularly important, which we define next.Consider the finite ´etale morphism π D : G [ c ] → S and the unit section e : S → G [ c ]. Theseinduce e ∗ : H ( S, Q p ) → H ( S, π D ∗ Q p )(coming from π D ∗ e ! e ! Q p → π D ∗ Q p ) and π ∗ D : H ( S, Q p ) → H ( S, π D ∗ Q p ) . One checks easily that e ∗ (1) − π ∗ D (1) is in the kernel of H ( S, π D ∗ Q p ) → H ( S, Q p ). Definition 5.5.1.
Let α c ∈ Q p [ D ] be the class α c := e ∗ (1) − π ∗ D (1) . We write c pol Q p and c Eis k Q p ( t ) for the polylogarithm and the Eisenstein class defined with α c . We now assume that S is of finite type over Spec Z . Then H d − ( A r \ A r [ cp r ] , Z /p r Z ( d )) isfinite, so that one has by (1.1.2) H d − ( S, Rπ ! Rj D ∗ j ∗ D L ( d )) ∼ = H d − ( A \ A [ c ] , L ( d )) ∼ = lim ←− r H d − ( A r \ A r [ cp r ] , Z /p r Z ( d ))where, as before, [ p r ] : A r = A → A is the p r -multiplication and the transition maps are givenby the trace maps. The integral ´etale polylogarithm is then a class α pol ∈ lim ←− r H d − ( A r \ A r [ cp r ] , Z /p r Z ( d )) . In the special case where A = E is an elliptic curve over S it is shown in [Kin15, Theorem12.4.21] that c pol ∈ lim ←− r H ( E r \ E r [ cp r ] , Z /p r Z ( d ))is given by the inverse limit of Kato’s norm compatible elliptic units c ϑ E . Unfortunately, we donot have such a description even in the case of abelian varieties of dimension ≥
2. If we write A [ p r ] h t i for the A [ p r ]-torsor defined by diagram (5.1.5), then α EI ( t ) ∈ H d − ( S, t ∗ L ( d )) = lim ←− r H d − ( A [ p r ] h t i , Z /p r Z ( d ))where the inverse limit is again over the trace maps. . Interpolation of the Q p -Eisenstein classes An integral structure on L og ( k ) Q p . For the comparison between the integral L and the Q p -polylogarithm L og Q p we need an intermediate object, which we define in this section. Thisis purely technical. The reason for this is as follows: In general a unipotent Q p -sheaf does notnecessarily have a Z p -lattice which is again a unipotent sheaf. In the case of L og ( k ) Q p however, itis even possible to construct a Z p -structure L og ( k ) such that L og ( k )Λ r := L og ( k ) ⊗ Z p Λ r is a unipotent Λ r = Z /p r Z -sheaf.Let L og (1) be the Z p -sheaf defined in 2.2.1(6.1.1) 0 → H → L og (1) → Z p → (1) : Z p → e ∗ L og (1) a fixed splitting. Definition 6.1.1.
We define L og ( k ) := Γ k ( L og (1) ) as the k -th graded piece of the divided power algebra Γ Z p ( L og (1) ) . We further denote by ( k ) := Γ k ( (1) ) : Z p → L og ( k ) the splitting induced by (1) . As Z p and H are flat Z p -sheaves (all stalks are Z p -free), the k -th graded piece of the dividedpower algebra Γ k ( L og (1) ) has a filtration with graded pieces π ∗ Γ i ( H ) ⊗ Γ k − i ( Z p ) (see [Ill71, V4.1.7]). In particular, the Γ k ( L og (1) ) are unipotent Z p -sheaves of length k . By base change thesame is true for the Λ r -sheaf(6.1.2) L og ( k )Λ r := L og ( k ) ⊗ Z p Λ r . To define transition maps(6.1.3) L og ( k ) → L og ( k − we proceed as in Section 2.3. Consider L og (1) → Z p ⊕ L og (1) given by the canonical projectionand the identity. Then we define L og ( k ) = Γ k ( L og (1) ) → Γ k ( Z p ⊕ L og (1) ) ∼ = M i + j = k Γ i ( Z p ) ⊗ Γ j ( L og (1) ) →→ Γ ( Z p ) ⊗ Γ k − ( L og (1) ) ∼ = L og ( k − where we identify Γ ( Z p ) ∼ = Z p . A straightforward computation shows that ( k ) ( k − underthe transition map. Definition 6.1.2.
We denote by L og the pro-sheaf ( L og ( k ) ) k ≥ with the above transition mapsand let : Z p → e ∗ L og be the splitting defined by ( ( k ) ) k ≥ .Remark . We would like to point out that, contrary to the Q p -situation, the pro-sheaf( L og ( k ) ) k ≥ is not the correct definition of the Z p -logarithm sheaf. In fact, the correct integrallogarithm sheaf is L . Proposition 6.1.4.
Denote by L og ( k ) ⊗ Q p the Q p -sheaf associated to L og ( k ) . Then there is acanonical isomorphism L og ( k ) Q p ∼ = L og ( k ) ⊗ Q p which maps ( k ) Q p to ( k ) . roof. First note that the canonical map Sym k L og (1) Q p → Γ k ( L og (1) Q p ) is an isomorphism. Thiscan be checked at stalks, where it follows from (1.2.1) as L og (1) Q p is a sheaf of Q p -modules. Theclaim in the proposition then follows from the isomorphisms L og ( k ) Q p = Sym k L og (1) Q p ∼ = Γ k ( L og (1) Q p ) ∼ = Γ k ( L og (1) ) ⊗ Q p = L og ( k ) ⊗ Q p and the claim about the splitting follows from the explicit formula for the map Sym k L og (1) Q p → Γ k ( L og (1) Q p ) given after (1.2.1). (cid:3) Corollary 6.1.5.
For all i there are isomorphisms H i ( S, Rπ ! Rj D ∗ j ∗ D L og ( k ) ( d )) ⊗ Z p Q p ∼ = H i ( S, Rπ ! Rj D ∗ j ∗ D L og ( k ) Q p ( d )) H i ( S, π D ! π ∗ D k Y i =0 Γ i ( H )) ⊗ Z p Q p ∼ = H i ( S, π D ! π ∗ D k Y i =0 Sym i H Q p ) H i ( S, Rπ ! L og ( k ) ( d )[2 d ]) ⊗ Z p Q p ∼ = H i ( S, Rπ ! L og ( k ) Q p ( d )[2 d ]) Proof.
The first and the third follow directly from the proposition and the definition of thecohomology of a Q p -sheaf. For the second one observes that the canonical mapSym k H Q p ∼ = Sym k H ⊗ Q p → Γ k ( H ) ⊗ Q p ∼ = Γ k ( H Q p )is an isomorphism. This can be checked on stalks, where it follows again from (1.2.1). (cid:3) Comparison of integral and Q p -polylogarithm. In this section we want to compare L and L og Q p . We first compare L with the sheaves L og ( k ) defined in 6.1.1.Define a comparison map comp k : L → L og ( k ) as follows. By Theorem 5.2.6 one has for the sheaves L og ( k )Λ r from (6.1.2) the isomorphismHom G (Λ r [ G rk ] , L og ( k )Λ r ) ∼ = H ( S, e ∗ L og ( k )Λ r ) , so that the splitting ( k ) ⊗ Λ r : Λ r → e ∗ L og ( k )Λ r defines a morphism of sheaves on G (6.2.1) comp kr : Λ r [ G rk ] → L og ( k )Λ r , which is obviously compatible with the transition maps and functorial in G . Passing to thepro-systems over r ≥
0, this defines a homomorphism(6.2.2) comp k : L → L og ( k ) . Taking also the pro-system in the k -direction leads to a comparison map(6.2.3) comp : L → L og.
For each k applying comp k to the localization triangle for D ֒ → G ← ֓ U D gives(6.2.4) Rπ ! Rj D ∗ j ∗ D L ( d )[2 d − −−−−→ π D ! π ∗ D Λ( H ) −−−−→ Rπ ! L ( d )[2 d ] y comp k y comp k y comp k Rπ ! Rj D ∗ j ∗ D L og ( k ) ( d )[2 d − −−−−→ π D ! π ∗ D L og ( k ) −−−−→ Rπ ! L og ( k ) ( d )[2 d ]compatible with the transition maps L og ( k ) → L og ( k − . Proposition 6.2.1.
There is a commutative diagram of short exact sequences / / H d − ( S, Rπ ! Rj D ∗ j ∗ D L ( d )) res / / comp (cid:15) (cid:15) H ( S, π D ! π ∗ D Λ( H )) / / e ∗ comp (cid:15) (cid:15) H ( S, Z p ) / / (cid:15) (cid:15) / / H d − ( S, Rπ ! Rj D ∗ j ∗ D L og Q p ( d )) res / / H ( S, π D ! π ∗ D Q k ≥ Sym k H Q p ( d )) / / H ( S, Q p ) / / . roof. Take the long exact cohomology sequence of the commutative diagram in (6.2.4), ten-sor the lower horizontal line with Q p and then pass to the inverse limit over k . Using theisomorphisms in Corollary 6.1.5 gives the commutative diagram as stated. (cid:3) Corollary 6.2.2.
Let α ∈ Z p [ D ] , with D = G [ c ] as before. Then one has comp( α pol) = α pol Q p in H d − ( S, Rπ ! Rj D ∗ j ∗ D L og Q p ( d )) . In particular, for every N -torsion section t : S → U D onehas comp( α EI ( t )) = t ∗ ( α pol Q p ) . Proof.
Immediate from the definition of α pol and α pol Q p and the commutative diagram inthe proposition. The second statement follows from the first as comp is compatible with theevaluation map at t . (cid:3) Interpolation of the Q p -Eisenstein classes. For our main result, we first have to relatethe comparison map comp k with the moment map mom k . Proposition 6.3.1.
The composition Λ( H ) e ∗ (comp k ) −−−−−−→ e ∗ L og ( k ) pr k −−→ Γ k ( H ) coincides with the moment map mom k .Proof. By the definitions of mom k and comp k it suffices to prove this statement for Λ r -coefficients.Consider comp kr : Λ r [ G rk ] → L og ( k )Λ r from (6.2.1). This comes by adjunction from a map β r : Λ r → [ p rk ] ∗ L og ( k )Λ r , on G rk which has by definition the property that its pull-back e ∗ rk ( β r ) coincides with ( k ) : Λ r → e ∗ L og ( k )Λ r . By Lemma 1.3.2 the map β r is uniquely determined by this property. As L og ( k )Λ r isunipotent of length k , the pull-back [ p rk ] ∗ L og ( k )Λ r is trivial by Lemma 5.2.5 and is hence of theform [ p rk ] ∗ L og ( k )Λ r ∼ = π ∗ rk e ∗ L og ( k )Λ r ∼ = π ∗ rk k Y i =0 Γ i ( H r ) , where the last isomorphism is obtained by the splitting ( k ) . Thus the mapΛ r → [ p rk ] ∗ L og ( k )Λ r ∼ = π ∗ rk k Y i =0 Γ i ( H r ) 1 k X i =0 τ [ i ] r , where τ [ i ] r is the i -th divided power of the tautological section from (4.5.1), has the property thatits pull-back by e ∗ rk coincides with ( k ) . It follows that this map equals β r and by definitionof the moment map in (4.5.2) the projection to the k -th component coincides also with themoment map. (cid:3) Let t : S → U D be an N -torsion section. We need a compatibility between the compositionmom kN := mom k ◦ [ N ] : Λ( H h t i ) → Λ( H h t i ) → Γ k ( H )and the map ̺ t in the splitting principle 2.4.2 composed with the projection onto the k -thcomponent pr k ◦ ̺ t : t ∗ L og ( k ) Q p ∼ = k Y i =0 Sym k H Q p → Sym k H Q p . roposition 6.3.2. There is a commutative diagram H d − ( S, Λ( H h t i )( d )) mom kN −−−−→ H d − ( S, Γ k ( H )( d )) t ∗ comp k y y H d − ( S, t ∗ L og ( k ) Q p ( d )) N k pr k ◦ ̺ t −−−−−−→ H d − ( S, Sym k H Q p ( d ) , where mom kN = mom k ◦ [ N ] and ̺ t = [ N ] − ◦ [ N ] .Proof. The commutative diagram H d − ( S, Λ( H h t i )( d )) [ N ] −−−−→ H d − ( S, Λ( H )( d )) t ∗ comp k y y e ∗ comp k H d − ( S, t ∗ L og ( k ) Q p ( d )) [ N ] −−−−→ ∼ = H d − ( S, e ∗ L og ( k ) Q p ( d ))coming from functoriality of comp k and the isomorphisms H d − ( S, t ∗ L og ( k ) ( d )) ⊗ Z p Q p ∼ = H d − ( S, t ∗ L og ( k ) Q p ( d )) H d − ( S, e ∗ L og ( k ) ( d )) ⊗ Z p Q p ∼ = H d − ( S, e ∗ L og ( k ) Q p ( d ))reduces the proof of the proposition to show the commutativity of the diagram H d − ( S, Λ( H )( d )) mom k −−−−→ H d − ( S, Γ k ( H )( d )) e ∗ comp k y y H d − ( S, e ∗ L og ( k ) Q p ( d )) N k pr k ◦ [ N ] − −−−−−−−−−→ H d − ( S, Sym k H Q p ( d ) . The isogeny [ N ] acts by N -multiplication on H , hence by multiplication with N k on Sym k H Q p ,which means that pr k ◦ [ N ] − = [ N ] − ◦ pr k = N − k pr k . Thus it remains to show that the diagram H d − ( S, Λ( H )( d )) mom k −−−−→ H d − ( S, Γ k ( H )( d )) e ∗ comp k y y H d − ( S, e ∗ L og ( k ) Q p ( d )) pr k −−−−→ H d − ( S, Sym k H Q p ( d ))commutes, which follows from Proposition 6.3.1 and the isomorphism H d − ( S, Γ k ( H )( d )) ⊗ Z p Q p ∼ = H d − ( S, Sym k H Q p ( d ))which was obtained in Corollary 6.1.5. (cid:3) Recall from Definition 5.4.1 the Eisenstein-Iwasawa class α EI ( t ) N = [ N ] ( α EI ( t )) ∈ H d − ( S, Λ( H )( d ))and from 3.2.1 the Q p -Eisenstein class α Eis k Q p ( t ) ∈ H d − ( S, Sym k H Q p ) . We consider its image under the k -th moment mapmom k : H d − ( S, Λ( H )( d )) → H d − ( S, Γ k ( H )( d )) . The main result of this paper can now be formulated as follows: heorem 6.3.3 (Interpolation of Q p -Eisenstein classes) . The image of α EI ( t ) N under the k -thmoment map is given by mom k ( α EI ( t ) N ) = N kα Eis k Q p ( t ) . Proof.
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