Abstract
In this paper we prove that the motivic Eisenstein classes associated to polylogarithms of commutative group schemes can be p-adically interpolated in étale cohomology. This connects them to Iwasawa theory and generalizes and strengthens the results for elliptic curves obtained in our former work. In particular, degeneration questions can be treated easily.
To John Coates, on the occasion of his 70th birthday
This research was supported by the DFG grant: SFB 1085 “Higher invariants”.
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1 Introduction
In this paper we prove that the motivic Eisenstein classes associated to polylogarithms of commutative group schemes can be p-adically interpolated in étale cohomology. This generalizes the results for elliptic curves obtained in our former paper [12]. Already in the one dimensional case the results obtained here are stronger and much more flexible as they allow to treat degenerating elliptic curves easily.
The interpolation of motivic Eisenstein classes connects them with Iwasawa theory and is essential for many applications. In the elliptic case for example, the interpolation was used in [11] to prove a case of the Tamagawa number conjecture for CM elliptic curves and it was one of the essential ingredients in the proof of an explicit reciprocity law for Rankin-convolutions in [13]. We hope that the general case will find similar applications.
Before we explain our results, we have to introduce the motivic Eisenstein classes (for the construction we refer to Sect. 4.2).
Let \(\pi :G\rightarrow S\) be a smooth commutative and connected group scheme of relative dimension d (for example a semi-abelian scheme) and denote by
the first étale homology of G / S, which is just the sheaf of relative p-adic Tate modules of G / S. We write \(\mathscr {H}_{\mathbb {Q}_p}\) for the associated \({\mathbb {Q}_p}\)-adic sheaf. Note that this is not a lisse sheaf in general. Evaluating the motivic polylogarithm at a non-zero N-torsion section \(t:S\rightarrow G\) one defines motivic Eisenstein classes
depending on some auxiliary data \(\alpha \), where \({{\mathrm{Sym}}}^{k}\mathscr {H}_\mathbb {Q}(1)\) is the k-th symmetric tensor power of the motivic sheaf \(\mathscr {H}_\mathbb {Q}\) which underlies \(\mathscr {H}_{\mathbb {Q}_p}\).
In the case of an elliptic curve, the de Rham realization of \(_\alpha \mathrm {Eis}_\mathrm {mot}^{k}(t)\) is the cohomology class of a holomorphic Eisenstein series, which justifies the name. These motivic Eisenstein classes in 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 étale regulator
which gives rise to the étale Eisenstein classes
In the elliptic case these classes were used by Kato in his seminal work to construct Euler systems for modular forms.
It is a natural question, whether these étale Eisenstein classes enjoy some p-adic interpolation properties, 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 different motivic cohomology groups \(H^{2d-1}_\mathrm {mot}(S, {{\mathrm{Sym}}}^{k}\mathscr {H}_\mathbb {Q}(1))\) are not related at all. Nevertheless, this question was answered affirmatively in the elliptic case in [12] and in this paper we will generalize this result to commutative group schemes.
To explain our answer to this question we need the sheaf of Iwasawa-algebras \(\Lambda (\mathscr {H})\), which is defined as follows: One first defines a sheaf of “group rings” \(\mathbb {Z}/p^{r}\mathbb {Z}[\mathscr {H}_r]\) on S, where \(\mathscr {H}_r\) is the étale sheaf associated to the \([p^{r}]\)-torsion subgroup \(G[p^{r}]\) or alternatively the first homology with \(\mathbb {Z}/p^{r}\mathbb {Z}\)-coefficients (see Sect. 5.4 for more details). These group rings form an inverse system for varying r and hence define a pro-sheaf
Moreover, it is also possible to sheafify the classical moments of a measure to a morphism of pro-sheaves
where \(\Gamma _k(\mathscr {H})\) is the k-th graded piece of the divided power algebra \(\Gamma _{\mathbb {Z}_p}(\mathscr {H})\). Thus the sheaf \(\Lambda (\mathscr {H})\) p-adically interpolates the \(\Gamma _{k}(\mathscr {H})\). For the \({\mathbb {Q}_p}\)-sheaf \(\mathscr {H}_{\mathbb {Q}_p}\) the natural map \({{\mathrm{Sym}}}^{k}\mathscr {H}_{\mathbb {Q}_p}\rightarrow \Gamma _k(\mathscr {H}_{\mathbb {Q}_p})\) is an isomorphism and the moment map gives rise to morphisms
To understand this better, it is instructive to consider the case of an abelian scheme \(\pi :A\rightarrow S\) over a scheme S which is of finite type over \({{\mathrm{Spec}}}\mathbb {Z}\) (see also Sect. 6.5). Then
where the inverse limit is taken with respect to the trace maps along \(A[p^{r}]\rightarrow A[p^{r-1}]\). The right 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 7.3.3) There exists a cohomology class
called the Eisenstein–Iwasawa class, such that
This interpolation result in the elliptic case is one of the key ingredients in the proof of an explicit reciprocity law for Rankin-convolutions of modular forms in [13].
The use of this theorem also considerably simplifies the computations of the degeneration of polylogarithm in [12]. We hope to treat this at another occasion.
We would also like to point out an important open problem: In the one-dimensional torus or the elliptic curve case, the Eisenstein–Iwasawa class has a direct description in terms of cyclotomic units or Kato’s norm compatible elliptic units respectively. Unfortunately, we do not have a similar description of the Eisenstein–Iwasawa class in the higher dimensional case.
2 Notations and Set up
2.1 The Category of \({\mathbb {Z}_p}\)-sheaves
All schemes will be separated of finite type over a noetherian regular scheme of dimension 0 or 1. Let X be such a scheme and let p be a prime invertible on X. We work in the category of constructable \({\mathbb {Z}_p}\)-sheaves \(\mathscr {S}(X)\) on X in the sense of [15, Exposé V].
Recall that a constructible \({\mathbb {Z}_p}\)-sheaf is an inverse system \(\mathscr {F}=(\mathscr {F}_n)_{n\geqslant 1}\) where \(\mathscr {F}_n\) is a constructible \(\mathbb {Z}/p^{n}\mathbb {Z}\)-sheaf and the transition maps \(\mathscr {F}_n\rightarrow \mathscr {F}_{n-1}\) factor into isomorphisms
The \({\mathbb {Z}_p}\)-sheaf is lisse, if each \(\mathscr {F}_n\) is locally constant. If X is connected and \(x\in X\) is a geometric point, then the category of lisse sheaves is equivalent to the category of finitely generated \({\mathbb {Z}_p}\)-modules with a continous \(\pi _1(X,x)\)-action. For a general \({\mathbb {Z}_p}\)-sheaf there exists a finite partition of X into locally closed subschemes \(X_i\), such that \(\mathscr {F}\mid _{X_i}\) is lisse (see [4, Rapport, Proposition 2.4., 2.5.]).
For a \({\mathbb {Z}_p}\)-sheaf \(\mathscr {F}\) we denote by \(\mathscr {F}\otimes {\mathbb {Q}_p}\) its image in the category of \({\mathbb {Q}_p}\)-sheaves, i.e., the quotient category modulo \({\mathbb {Z}_p}\)-torsion sheaves.
We also consider the “derived” category \(\mathbf {D}(X)\) of \(\mathscr {S}(X)\) in the sense of Ekedahl [5]. This is a triangulated category with a t-structure whose heart is the category of constructible \({\mathbb {Z}_p}\)-sheaves. By loc. cit. Theorem 6.3 there is a full 6 functor formalism on these categories.
Recall that an inverse system \(A:=(A_r)_{r\geqslant 0}\) (in any abelian category \(\mathcal {A}\)) satisfies the Mittag-Leffler condition (resp. is Mittag-Leffler zero), if for each r the image of \(A_{r+s}\rightarrow A_r\) is constant for all sufficiently big s (is zero for some \(s\geqslant 1\)). If A satisfies the Mittag-Leffler condition and \(\mathcal {A}\) satisfies \(AB4^{*}\) (i.e. products exists and products of epimorphisms are epimorphisms) then \({\varprojlim }_r^{1}A_r=0\) (see [14, Proposition 1]). If A is Mittag-Leffler zero, then for each left exact functor \(h:\mathcal {A}\rightarrow \mathcal {B}\) one has \(R^{i}{\varprojlim }_r h(A_r)=0\) for all \(i\geqslant 0\) ([9, Lemma 1.11.]).
For a pro-system of étale sheaves \(\mathscr {F}=(\mathscr {F}_r)_{r\geqslant 0}\) on X we work with Jannsen’s continuous étale cohomology \(H^{i}(X,\mathscr {F})\) which is the i-th derived functor of \(\mathscr {F}\mapsto {\varprojlim }_r H^{0}(X,\mathscr {F}_n)\). By [9, 3.1] one has an exact sequence
Note in particular, that if \(H^{i-1}(X,\mathscr {F}_r)\) is finite for all r, one has
For \(\mathscr {F}=(\mathscr {F}_r)\) Mittag-Leffler zero, one has for all \(i\geqslant 0\)
2.2 The Divided Power Algebra
Let A be a commutative ring and M be an A-module. Besides the usual symmetric power algebra \({{\mathrm{Sym}}}_A(M)\) we need also the divided power algebra \(\Gamma _A(M)\) (see [1, Appendix A] for more details).
The A-algebra \(\Gamma _A(M)\) is a graded augmented algebra with \(\Gamma _0(M)=A\), \(\Gamma _1(M)=M\) and augmentation ideal \(\Gamma ^{+}(M):=\bigoplus _{k\geqslant 1}\Gamma _k(M)\). For each element \(m\in M\) one has the divided power \(m^{[k]}\in \Gamma _k(M)\) with the property that \(m^{k}=k! m^{[k]}\) where \(m^{k}\) denotes the k-th power of m in \(\Gamma _A(M)\). Moreover, one has the formula
In the case where M is a free A-module with basis \(m_1,\ldots ,m_r\) the A-module \(\Gamma _k(M)\) is free with basis \(\{m_1^{[i_1]}\cdots m_r^{[i_r]}\mid \sum i_j=k\}\). Further, for M free, there is an A-algebra isomorphism
with the algebra of symmetric tensors (\({{\mathrm{TSym}}}^{k}_A(M)\subset {{\mathrm{Sym}}}^{k}_A(M)\) are the invariants of the symmetric group), which maps \(m^{[k]}\) to \(m^{\otimes k}\). Also, by the universal property of \({{\mathrm{Sym}}}_A(M)\), one has an A-algebra homomorphism
which maps \(m^{k}\) to \(k!m^{[k]}\). In particular, if A is a \(\mathbb {Q}\)-algebra, this map is an isomorphism.
If M is free and \(M^{\vee }:={{\mathrm{Hom}}}_A(M,A)\) denotes the A-dual one has in particular
The algebra \(\Gamma _A(M)\) has the advantage over \({{\mathrm{TSym}}}_A(M)\) of being compatible with arbitrary base change
and thus sheafifies well. Recall from [8, I 4.2.2.6.] that if \(\mathscr {F}\) is an étale sheaf of \({\mathbb {Z}_p}\)-modules, then \(\Gamma _{{\mathbb {Z}_p}}(\mathscr {F})\) is defined to be the sheaf associated to the presheaf
Definition 2.2.1
We denote by
the completion of \(\Gamma _A(M)\) with respect to the divided powers of the augmentation ideal.
Note that \(\Gamma ^{+}(M)^{[r]}=\bigoplus _{k\geqslant r}\Gamma _k(M)\) so that as A-module one has \(\widehat{\Gamma }_A(M)\cong \prod _{k\geqslant 0}\Gamma _k(M)\).
In the same way we define the completion of \({{\mathrm{Sym}}}_A(M)\) with respect to the augmentation ideal \({{\mathrm{Sym}}}^{+}_A(M)\) to be
2.3 Unipotent Sheaves
Let \(\Lambda =\mathbb {Z}/p^{r}\mathbb {Z}, {\mathbb {Z}_p}\) or \({\mathbb {Q}_p}\) and let \(\pi :X\rightarrow S\) be a separated scheme of finite type, with X, S as in Sect. 2.1. A \(\Lambda \)-sheaf \(\mathscr {F}\) on X is unipotent of length n, if it has a filtration \(0=\mathscr {F}^{n+1}\subset \mathscr {F}^n\subset \ldots \subset \mathscr {F}^0=\mathscr {F}\) such that \(\mathscr {F}^i/\mathscr {F}^{i+1}\cong \pi ^*\mathscr {G}^i\) for a \(\Lambda \)-sheaf \(\mathscr {G}^i\) on S.
The next lemma is taken from [7], where it is stated in the setting of \({\mathbb {Q}_p}\)-sheaves.
Lemma 2.3.1
Let \(\Lambda =\mathbb {Z}/p^{r}\mathbb {Z}, {\mathbb {Z}_p}\) or \({\mathbb {Q}_p}\) and let \(\pi _1:X_1\rightarrow S\) and \(\pi _2:X_2\rightarrow S\) be smooth of constant fibre dimension \(d_1\) and \(d_2\). Let \(f:X_1\rightarrow X_2\) be an S-morphism. Let \(\mathscr {F}\) be a unipotent \(\Lambda \)-sheaf. Then
Proof
Put \(c=d_1-d_2\) the relative dimension of f. We start with the case \(\mathscr {F}=\pi _2^*\mathscr {G}\). In this case
In particular, \(f^!\Lambda =\Lambda (c)[2c]\) and we may rewrite the formula as
There is always a map from the left to right via adjunction from the projection formula
Hence we can argue on the unipotent length of \(\mathscr {F}\) and it suffices to consider the case \(\mathscr {F}=\pi ^*\mathscr {G}\). This case was settled above. \(\square \)
The next lemma is also taken from [7]. Let \(X\rightarrow S\) be a smooth scheme with connected fibres and \(e:S\rightarrow X\) a section. Homomorphisms of unipotent sheaves are completely determined by their restriction to S via \(e^{*}\):
Lemma 2.3.2
Let \(\pi :X\rightarrow S\) be smooth with connected fibres and \(e:S\rightarrow X\) a section of \(\pi \). Let \(\Lambda =\mathbb {Z}/p^{r}\mathbb {Z}, {\mathbb {Z}_p}\) or \({\mathbb {Q}_p}\) and \(\mathscr {F}\) a unipotent \(\Lambda \)-sheaf on X. Then
is injective.
Proof
Let \(0\rightarrow \mathscr {F}_1\rightarrow \mathscr {F}_2\rightarrow \mathscr {F}_3\rightarrow 0\) be a short exact sequence of unipotent \(\Lambda \)-sheaves on G. As \(e^*\) is exact and \({{\mathrm{Hom}}}\) left exact, we get a commutative diagram of exact sequences
Suppose that the left and right vertical arrows are injective, then the middle one is injective as well and it is enough to show the lemma in the case where \(\mathscr {F}=\pi ^{*}\mathscr {G}\). But the isomorphism
factors through
where the last map is induced by the trace map \(R\pi _!\pi ^{!}\Lambda \rightarrow \Lambda \). This proves the claim. \(\square \)
2.4 The Geometric Situation
We recall the geometric set up from [7] using as much as possible the notations from loc. cit. Let
be a smooth separated commutative group scheme with connected fibres of relative dimension d. We denote by \(e:S\rightarrow G\) the unit section and by \( \mu :G\times _SG\rightarrow G\) the multiplication. Let \(j:U\rightarrow G\) be the open complement of e(S).
Let \(\iota _D:D\rightarrow G\) be a closed subscheme with structural map \(\pi _D:D\rightarrow S\). Typically \(\pi _D\) will be étale and contained in the c-torsion of G for some \(c\geqslant 1\). We note in passing, that for c invertible on S the c-torsion points of G, i.e. the kernel of the c-multiplication G[c], is quasi-finite and étale over S. Denote by \(j_D:U_D=G\setminus D\rightarrow G\) the open complement of D. We summarize the situation in the basic diagram
We will also consider morphisms \(\phi :G_1\rightarrow G_2\) of S-group schemes as above. In this case we decorate all notation with an index 1 or 2, e.g., \(d_1\) for the relative dimension of \(G_1/S\).
3 The Logarithm Sheaf
3.1 Homology of G
The basic sheaf in our constructions is the relative first \({\mathbb {Z}_p}\)-homology \(\mathscr {H}_G\) of G / S, which we define as follows:
Definition 3.1.1
For the group scheme \(\pi :G\rightarrow S\) we let
We write \(\mathscr {H}_r:=\mathscr {H}\otimes \mathbb {Z}/p^{r}\mathbb {Z}\) and \(\mathscr {H}_{{\mathbb {Q}_p}}:=\mathscr {H}\otimes {\mathbb {Q}_p}\) for the associated \({\mathbb {Q}_p}\)-sheaf.
Note that \(\mathscr {H}\) is not a lisse \({\mathbb {Z}_p}\)-sheaf in general, but the stalks are free \({\mathbb {Z}_p}\)-modules of finite rank, which follows for example from Lemma 3.1.2 below.
The sheaf \(\mathscr {H}\) and more generally \(R^{i}\pi _!{\mathbb {Z}_p}\) is covariant functorial for any map of S-schemes \(f:G\rightarrow X\) using the adjunction \(f_!f^{!}{\mathbb {Z}_p}\rightarrow {\mathbb {Z}_p}\). In particular, the group multiplication \(\mu :G\times _SG \rightarrow G\) induces a product
and the diagonal \(\Delta :G\rightarrow G\times _SG\) induces a coproduct
on \(R^{\cdot }\pi _!{\mathbb {Z}_p}\), which gives it the structure of a Hopf algebra and one has
(this follows by base change to geometric points and duality from [2, Lemma 4.1.]). The same result holds for \(\mathbb {Z}/p^{r}\mathbb {Z}\)-coefficients.
Lemma 3.1.2
Let \(G[p^{r}]\) be the kernel of the \(p^{r}\)-multiplication \([p^{r}]:G\rightarrow G\). Then there is a canonical isomorphism of étale sheaves
In particular, \(\mathscr {H}_G\) is the p-adic Tate-module of G.
Proof
This is standard and we only sketch the proof: Consider \(G[p^{r}]\) as an étale sheaf on S. The Kummer sequence is a \(G[p^{r}]\)-torsor on G, hence gives a class in
Thus the Kummer torsor induces a map \(R^{-1}\pi _!\pi ^!\mathbb {Z}/p^{r}\mathbb {Z}\rightarrow G[p^{r}]\) and one can perform a base change to geometric points \(\overline{s}\in S\) to show that this is an isomorphism. But this follows then from Poincaré-duality and the isomorphism \({{\mathrm{Hom}}}_{\overline{s}}(G[p^{r}],\mu _{p^{r}})\cong H^{1}(G,\mu _{p^{r}})\) shown in [2, Lemma 4.2.]. \(\square \)
3.2 The First Logarithm Sheaf
Consider the complex \(R\pi _!\pi ^!{\mathbb {Z}_p}\) calculating the homology of \(\pi :G\rightarrow S\) and its canonical filtration whose associated graded pieces are the \(R^{i}\pi _!\pi ^!{\mathbb {Z}_p}\). We apply this to
Then the resulting hypercohomology spectral sequence gives rise to the five term sequence
and the maps \(\pi ^{!}\) are injective because they admit the splitting \(e^{!}\) induced by the unit section e. This gives
Note that \({{\mathrm{Ext}}}^1_G(\pi ^!{\mathbb {Z}_p},\pi ^!\mathscr {H})\cong {{\mathrm{Ext}}}^1_G({\mathbb {Z}_p},\pi ^*\mathscr {H})\). The same construction is also possible with the base ring \(\Lambda _r:=\mathbb {Z}/p^{r}\mathbb {Z}\) and \(\mathscr {H}_r\) and gives the exact sequence
Definition 3.2.1
The first logarithm sheaf \((\mathcal {L}og^{(1)},\mathbf {1}^{(1)})\) on G consists of an extension class
such that its image in \({{\mathrm{Hom}}}_S(\mathscr {H},\mathscr {H})\) is the identity together with a fixed splitting \(\mathbf {1}^{(1)}:e^*{\mathbb {Z}_p}\rightarrow e^*\mathcal {L}og^{(1)}\). In exactly the same way one defines \(\mathcal {L}og^{(1)}_{\Lambda _r}\). We denote by \(\mathcal {L}og^{(1)}_{{\mathbb {Q}_p}}\) the associated \({\mathbb {Q}_p}\)-sheaf.
The existence and uniqueness of \((\mathcal {L}og^{(1)},\mathbf {1}^{(1)})\) follow directly from (8). The automorphisms of \(\mathcal {L}og^{(1)}\) form a torsor under \({{\mathrm{Hom}}}_G({\mathbb {Z}_p},\pi ^{*}\mathscr {H})\). In particular, the pair \((\mathcal {L}og^{(1)},\mathbf {1}^{(1)})\) admits no automorphisms except the identity.
It is obvious from the definition that one has
so that \(\mathcal {L}og^{(1)}=(\mathcal {L}og^{(1)}_{\Lambda _r})_{r\geqslant 0}\). Moreover, \(\mathcal {L}og^{(1)}\) is compatible with arbitrary base change. If
is a cartesian diagram one has \(f_T^{*}\mathcal {L}og^{(1)}_G\cong \mathcal {L}og^{(1)}_{G_T}\) and \(f_T^{*}(\mathbf {1}^{(1)})\) defines a splitting.
Let
be a homomorphism of group schemes of relative dimension \(d_1\), \(d_2\), respectively and write \(c:=d_1-d_2\).
Theorem 3.2.2
For \(\varphi :G_1\rightarrow G_2\) as above, there is a unique morphism of sheaves
such that \(\varphi _\#(\mathbf {1}^{(1)}_{G_1})=\mathbf {1}^{(1)}_{G_2}\). Moreover, if \(\varphi \) is an isogeny of degree prime to p, then \(\varphi _\#\) is an isomorphism.
Proof
Pull-back of \(\mathcal {L}og^{(1)}_{G_2}\) gives an exact sequence
and push-out of \(\mathcal {L}og^{(1)}_{G_1}\) by \(\pi _1^{*}\mathscr {H}_{G_1}\rightarrow \pi _1^{*}\mathscr {H}_{G_2}\) induces a map
If \(\varphi \) is an isogeny and \(\deg \varphi \) is prime to p, then \(\pi _1^{*}\mathscr {H}_{G_1}\rightarrow \pi _1^{*}\mathscr {H}_{G_2}\) is an isomorphism, hence also h. By uniqueness there is a unique isomorphism of the pair \((\mathcal {L}og^{(1)}_{G_2},e_1^{*}(h)\circ \mathbf {1}^{(1)}_{G_1})\) with \((\mathcal {L}og^{(1)}_{G_2},\mathbf {1}^{(1)}_{G_2})\). The composition of this isomorphism with h is the desired map. If \(h':\mathcal {L}og^{(1)}_{G_1}\rightarrow \varphi ^{!}\mathcal {L}og^{(1)}_{G_2}\) is another map with this property, the difference \(h-h':{\mathbb {Z}_p}\rightarrow \pi _1^{*}\mathscr {H}_{G_2}\) is uniquely determined by its pull-back \(e^{*}(h-h'):{\mathbb {Z}_p}\rightarrow e_2^{*}\mathcal {L}og^{(1)}_{G_2}\) according to Lemma 2.3.2. If both, h and \(h'\) are compatible with the splittings, then \(e^{*}(h-h')=0\) and hence \(h=h'\). \(\square \)
Corollary 3.2.3
(Splitting principle) Let \(\varphi :G_1\rightarrow G_2\) be an isogeny of degree prime to p. Then if \(t:S\rightarrow G_1\) is in the kernel of \(\varphi \), then
Proof
Apply \(t^{*}\) to \(\varphi _\#\). \(\square \)
3.3 The \({\mathbb {Q}_p}\)-logarithm Sheaf
We are going to define the \({\mathbb {Q}_p}\)-logarithm sheaf, which has been studied extensively in [7].
Definition 3.3.1
We define
and denote by
the splitting induced by \(\mathbf {1}^{(1)}\).
We note that \(\mathcal {L}og^{(k)}_{\mathbb {Q}_p}\) is unipotent of length k and that the splitting \(\mathbf {1}^{(k)}\) induces an isomorphism
To define transition maps
consider the morphism \(\mathcal {L}og^{(1)}_{\mathbb {Q}_p}\rightarrow {\mathbb {Q}_p}\oplus \mathcal {L}og^{(1)}_{\mathbb {Q}_p}\) given by the canonical projection and the identity. Then we have
A straightforward computation shows that \(\mathbf {1}^{(k)}\mapsto \mathbf {1}^{(k-1)}\) under this transition map.
3.4 Main Properties of the \({\mathbb {Q}_p}\)-logarithm Sheaf
The logarithm sheaf has three main properties: functoriality, vanishing of cohomology and a universal mapping property for unipotent sheaves. Functoriality follows trivially from Theorem 3.2.2. We review the others briefly, referring for more details to [7].
Let \(\varphi :G_1\rightarrow G_2\) be a homomorphism of group schemes of relative dimension \(d_1\), \(d_2\), respectively and let \(c:=d_1-d_2\) be the relative dimension of the homomorphism.
Theorem 3.4.1
(Functoriality) For \(\varphi :G_1\rightarrow G_2\) as above there is a unique homomorphism of sheaves
such that \(\mathbf {1}_{G_1}\) maps to \(\mathbf {1}_{G_2}\). Moreover, if \(\varphi \) is an isogeny, the \(\varphi _\#\) is an isomorphism.
Proof
This follows directly from Theorem 3.2.2 and the fact that \(\deg \varphi \) is invertible in \({\mathbb {Q}_p}\). \(\square \)
Corollary 3.4.2
(Splitting principle) Let \(\varphi :G_1\rightarrow G_2\) be an isogeny. Then if \(t:S\rightarrow G_1\) is in the kernel of \(\varphi \), one has
More generally, if \(\iota :\ker \varphi \rightarrow G_1\) is the closed immersion, one has
where \(\pi \mid _{\ker \varphi }:\ker \varphi \rightarrow S\) is the structure map.
Proof
Apply \(t^{*}\) to both sides of the isomorphism \(\varphi _{\#}\) and use (13). For the second statement make the base change to \(\ker \varphi \) and apply the first statement to the tautological section of \(\ker \varphi \). \(\square \)
Theorem 3.4.3
(Vanishing of cohomology) One has
More precisely, the transition maps \(R^i\pi _!\mathcal {L}og^{(k)}_{\mathbb {Q}_p}\rightarrow R^i\pi _!\mathcal {L}og^{(k-1)}_{\mathbb {Q}_p}\) are zero for \(i<2d\) and one has an isomorphism \(R^{2d}\pi _!\mathcal {L}og^{(k)}_{\mathbb {Q}_p}\cong {\mathbb {Q}_p}(-d)\) compatible with the transition maps.
Proof
This is Theorem 3.3.1. in [7]. \(\square \)
Let \(\mathscr {F}\) be a unipotent sheaf of finite length n on G. Consider the homomorphism
defined as the composition of
with
Theorem 3.4.4
(Universal property) Let \(\mathscr {F}\) be a unipotent sheaf of finite length. Then the map (15) induces an isomorphism
Proof
This is Theorem 3.3.2. in [7]. \(\square \)
4 The \({\mathbb {Q}_p}\)-polylogarithm and Eisenstein Classes
4.1 Construction of the \({\mathbb {Q}_p}\)-polylogarithm
Fix an auxiliary integer \(c>1\) invertible on S and consider the c-torsion subgroup \(D:=G[c]\subset G\). We write \(U_D:=G\setminus D\) and consider
We also write \(\pi _D:D\rightarrow S\) for the structure map.
For any sheaf \(\mathscr {F}\) the localization triangle defines a connecting homomorphism
As \(\mathcal {L}og^{(k)}_{\mathbb {Q}_p}(d)[2d]\) is unipotent we may use Lemma 2.3.1 to replace \(\iota _D^{!}\) by \(\iota _D^{*}\). Using Corollary 3.4.2 one gets
Putting everything together and taking the limit over the transition maps \(\mathcal {L}og^{(k)}_{\mathbb {Q}_p}\rightarrow \mathcal {L}og^{(k-1)}_{\mathbb {Q}_p}\) gives the residue map
Proposition 4.1.1
The localization triangle induces a short exact sequence
Proof
This is an immediate consequence from the localization triangle and the computation of \(R\pi _!\mathcal {L}og\) in Theorem 3.4.3. \(\square \)
Definition 4.1.2
Let
where the map is induced by the trace \(\pi _{D!}{\mathbb {Q}_p}\rightarrow {\mathbb {Q}_p}\).
Note that
Definition 4.1.3
Let \(\alpha \in {\mathbb {Q}_p}[D]^{0}\). Then the unique class
with \(\mathrm {res}({_\alpha }\mathrm {pol}_{\mathbb {Q}_p})=\alpha \) is called the polylogarithm class associated to \(\alpha \). We write \(_{\alpha }\mathrm {pol}_{\mathbb {Q}_p}^{k}\) for the image of \(_{\alpha }\mathrm {pol}_{\mathbb {Q}_p}\) in \(H^{2d-1}(S,R\pi _!Rj_{D*}j_D^{*}\mathcal {L}og^{(k)}_{\mathbb {Q}_p}(d))\).
4.2 Eisenstein Classes
Recall that \(D=G[c]\) and fix an integer \(N>1\) invertible on S, such that \((N,c)=1\) and let \(t:S\rightarrow U_D=G\setminus D\) be an N-torsion section. Consider the composition
induced by the adjunction \(\mathrm {id}\rightarrow Rt_*t^{*}\), the fact that \(Rt_*=Rt_!\) and because \(\pi \circ t=\mathrm {id}\). Together with the splitting principle from Corollary 3.4.2 and the projection to the k-th component one gets an evaluation map
Definition 4.2.1
Let \(\alpha \in {\mathbb {Q}_p}[D]\). The image of \(_\alpha \mathrm {pol}_{\mathbb {Q}_p}\) under the evaluation map (19)
is called the k -th étale \({\mathbb {Q}_p}\) -Eisenstein class for G.
Remark 4.2.2
The normalization in [12, Definition 12.4.6] is different. There we had an additional factor of \(-N^{k-1}\) in front of \(_\alpha \mathrm {Eis}^{k}_{{\mathbb {Q}_p}}(t)\). This has the advantage to make the residues of \(_\alpha \mathrm {Eis}^{k}_{{\mathbb {Q}_p}}(t)\) at the cusps integral, but is very unnatural from the point of view of the polylogarithm.
Recall from [7, Theorem 5.2.1] that the polylogarithm \({_\alpha }\mathrm {pol}^{k}_{\mathbb {Q}_p}\) is motivic, i.e., there exists a class in motivic cohomology
the motivic polylogarithm, which maps to \(_\alpha \mathrm {pol}^{k}{_{\mathbb {Q}_p}}\) under the étale regulator
With the motivic analogue of the evaluation map (19) one can define exactly in the same way as in the étale case motivic Eisenstein classes for \(\alpha \in \mathbb {Q}[D]^{0}\)
The next proposition is obvious from the fact that the evaluation map is compatible with the étale regulator.
Proposition 4.2.3
For \(\alpha \,{\in }\, \mathbb {Q}[D]^{0}\) the image of the motivic Eisenstein class \(_\alpha \mathrm {Eis}_\mathrm {mot}^{k}(t)\) under the étale regulator
is the étale \({\mathbb {Q}_p}\)-Eisenstein class \(_\alpha \mathrm {Eis}^{k}_{{\mathbb {Q}_p}}(t)\).
5 Sheaves of Iwasawa Algebras
5.1 Iwasawa Algebras
Let \(X={\varprojlim }_r X_r\) be a profinite space with transition maps \(\lambda _r:X_{r+1}\rightarrow X_r\) and
the \(\mathbb {Z}/p^{r}\mathbb {Z}\)-module of maps from \(X_r\) to \(\mathbb {Z}/p^{r}\mathbb {Z}\). For each \(x_r\) we write \(\delta _{x_r}\in \Lambda _r[X_r]\) for the map which is 1 at \(x_r\) and 0 else. It is convenient to interpret \(\Lambda _r[X_r]\) as the space of \(\mathbb {Z}/p^{r}\mathbb {Z}\)-valued measures on \(X_r\) and \(\delta _{x_r}\) as the delta measure at \(x_r\). Then the push-forward along \(\lambda _r:X_{r+1}\rightarrow X_r\) composed with reduction modulo \(p^{r}\) induces \({\mathbb {Z}_p}\)-module maps
which are characterized by \(\lambda _{r*}(\delta _{x_{r+1}})=\delta _{\lambda _r(x_r)}\).
Definition 5.1.1
The module of \({\mathbb {Z}_p}\) -valued measures on X is the inverse limit
of \(\Lambda _r[X_r]\) with respect to the transition maps from (21).
Let \(x=(x_r)_{r\geqslant 0}\in X\). We define \(\delta _x:=(\delta _{x_r})_{r\geqslant 0}\in \Lambda (X)\), which provides a map
For each continuous map \(\varphi :X\rightarrow Y\) of profinite spaces we get a homomorphism
“push-forward of measures” with the property \(\varphi _*(\delta _x)=\delta _{\varphi (x)}\). Obviously, one has \(\Lambda _r[X_r\times Y_r]\cong \Lambda _r[X_r]\otimes \Lambda _r[Y_r]\) so that
In particular, if \(X=G={\varprojlim }_r G_r\) is a profinite group, the group structure \(\mu :G\times G\rightarrow G\) induces a \({\mathbb {Z}_p}\)-algebra structure on \(\Lambda (G)\), which coincides with the \({\mathbb {Z}_p}\)-algebra structure induced by the inverse limit of group algebras \({\varprojlim }_r \Lambda _r[G_r]\).
Definition 5.1.2
If \(G={\varprojlim }_r G_r\) is a profinite group, we call
the Iwasawa algebra of G.
More generally, if G acts continuously on the profinite space X, one gets a map
which makes \(\Lambda (X)\) a \(\Lambda (G)\)-module. If X is a principal homogeneous space under G, then \(\Lambda (X)\) is a free \(\Lambda (G)\)-module of rank 1.
5.2 Properties of the Iwasawa Algebra
In this section we assume that H is a finitely generated free \({\mathbb {Z}_p}\)-module. We let
so that \(H={\varprojlim }_r H_r\) with the natural transition maps \(H_{r+1}\rightarrow H_r\).
In the case \(H={\mathbb {Z}_p}\), the so called Amice transform of a measure \(\mu \in \Lambda ({\mathbb {Z}_p})\)
induces a ring isomorphism \(\mathcal {A}:\Lambda ({\mathbb {Z}_p})\cong {\mathbb {Z}_p}[[T]]\) (see [3, Sect. 1.1.]). A straightforward generalization shows that \(\Lambda (H)\) is isomorphic to a power series ring in \({{\mathrm{rk}}}H\) variables. On the other hand one has the so called Laplace transform of \(\mu \) (see loc. cit.)
This map is called the moment map in [10] and we will follow his terminology. In the next section, we will explain this map from an abstract algebraic point of view. For this we interpret \( \frac{t^{n}}{n!}\) as \( t^{[n]} \) in the divided power algebra \(\Gamma _{\mathbb {Z}_p}({\mathbb {Z}_p})\).
5.3 The Moment Map
We return to the case of a free \({\mathbb {Z}_p}\)-module H of finite rank.
Proposition 5.3.1
Let H be a free \({\mathbb {Z}_p}\)-module of finite rank and \(H_r:=H\otimes _{\mathbb {Z}_p}\mathbb {Z}/p^{r}\mathbb {Z}\). Then
Proof
As each \(\Gamma _{\mathbb {Z}_p}(H)/\Gamma ^{+}(H)^{[k]}\) is a finitely generated free \({\mathbb {Z}_p}\)-module, this follows by the compatibility with base change of \(\Gamma _{\mathbb {Z}_p}(H)\) and the fact that one can interchange the inverse limits. \(\square \)
By the universal property of the finite group ring \(\Lambda _r[H_r]\), the group homomorphism
induces a homomorphism of \(\mathbb {Z}/p^{r}\mathbb {Z}\)-algebras
Corollary 5.3.2
The maps \( {{\mathrm{mom}}}_r \) induce in the inverse limit a \({\mathbb {Z}_p}\)-algebra homomorphism
which is functorial in H.
Definition 5.3.3
We call \({{\mathrm{mom}}}:\Lambda (H)\rightarrow \widehat{\Gamma }_{{\mathbb {Z}_p}}(H)\) the moment map and the composition with the projection to \(\Gamma _k(H)\)
the k -th moment map.
5.4 Sheafification of the Iwasawa Algebras
Let X be a separated noetherian scheme of finite type as in Sect. 2.1 and \(\mathscr {X}:=(p_r:\mathscr {X}_r\rightarrow X)_r\) be an inverse system of quasi-finite étale schemes over X with étale transition maps \(\lambda _r:\mathscr {X}_r\rightarrow \mathscr {X}_{r-1}\). We often write
The adjunction \(\lambda _{r!}\lambda _r^{!}\rightarrow \mathrm {id}\) defines a homomorphism
because \(\lambda _r\) is étale. If one composes this with reduction modulo \(p^{r}\), one gets a trace map
Definition 5.4.1
We define an étale sheaf on X by
With the trace maps \({{\mathrm{Tr}}}_{r+1}:\Lambda _{r+1}[\mathscr {X}_{r+1}]\rightarrow \Lambda _{r}[\mathscr {X}_{r}]\) as transition morphisms we define the pro-sheaf
This definition is functorial in \(\mathscr {X}\). If \((\varphi _r)_r:(\mathscr {X}_r)_r\rightarrow (\mathscr {Y}_r)_r\) is a morphism of inverse system of quasi-finite étale schemes over X, then the adjunction \(\varphi _{r!}\varphi _r^{!}\rightarrow \mathrm {id}\) defines a morphism
compatible with the transition maps, and hence a morphism of pro-sheaves
Moreover, the formation of \(\Lambda (\mathscr {X})\) is compatible with base change: if \(\mathscr {X}_{r,T}:=\mathscr {X}_r\times _ST\) for an S-scheme \(f:T\rightarrow S\), then by proper base change one has
By the Künneth formula, one has
and hence \(\Lambda (\mathscr {X}\times _X\mathscr {Y})\cong \Lambda (\mathscr {X})\widehat{\otimes }\Lambda (\mathscr {Y})\) by taking the inverse limit. In particular, in the case where \(\mathscr {X}=\mathscr {G}\) is an inverse system of quasi-finite étale group schemes \(\mathscr {G}_r\), the group structure \(\mu _r:\mathscr {G}_r\times _X\mathscr {G}_r\rightarrow \mathscr {G}_r\) induces a ring structure
on \(\Lambda (\mathscr {G})\). Similarly, if
is a group action of inverse systems, i.e., a compatible family of actions \(\mathscr {G}_r\times _X\mathscr {X}_r\rightarrow \mathscr {X}_r\), then \(\Lambda (\mathscr {X})\) becomes a \(\Lambda (\mathscr {G})\)-module.
The next lemma shows that the above construction indeed sheafifies the Iwasawa algebras considered before.
Lemma 5.4.2
Let \(\overline{x}\in X\) be a geometric point and write \(p_{r,\overline{x}}:\mathscr {X}_{r,\overline{x}}\rightarrow \overline{x}\) for the base change of \(\mathscr {X}_r\) to \(\overline{x}\) considered as a finite set. Then
Proof
This follows directly from the base change property of \(\Lambda _r[\mathscr {X}_r]\) and the fact that \(p_{r,\overline{x},!}\Lambda _r \cong \Lambda _r[X_r]\) over an algebraically closed field. \(\square \)
We return to our basic set up, where \(\pi :G\rightarrow S\) is a separated smooth commutative group scheme with connected fibres. Recall from Lemma 3.1.2 that \(\mathscr {H}_r\) is the sheaf associated to \(G[p^{r}]\), which is quasi-finite and étale over S.
Definition 5.4.3
Define the sheaf of Iwasawa algebras \(\Lambda (\mathscr {H})\) on S to be the pro-sheaf
5.5 Sheafification of the Moment Map
We keep the notation of the previous section. In particular, we consider the étale sheaf \(\mathscr {H}_r\) and the sheaf \(\Lambda _r[\mathscr {H}_r]\).
Over \(G[p^{r}]\) the sheaf \([p^{r}]^{*}\mathscr {H}_r\) has the tautological section \(\tau _r\in \Gamma (G[p^{r}],[p^{r}]^{*}\mathscr {H}_r)\) corresponding to the identity map \(G[p^{r}]\rightarrow \mathscr {H}_r\). This gives rise to the section
of the k-th divided power of \(\mathscr {H}_r\). Using the chain of isomorphisms (note that \([p^{r}]^{*}=[p^{r}]^{!}\) as \([p^{r}]\) is étale)
the section \(\tau _r^{[k]}\) gives rise to a morphism of sheaves
Lemma 5.5.1
There is a commutative diagram
where the right vertical map is given by the reduction map
Proof
Denote by \(\lambda _r:\mathscr {H}_r\rightarrow \mathscr {H}_{r-1}\) the transition map. Reduction modulo \(p^{r-1}\) gives a commutative diagram
As the image of the tautological class \(\tau _r^{[k]}\in \Gamma (G[p^{r}],[p^{r}]^{*}\Gamma _k(\mathscr {H}_r))\) under the reduction map gives the the pull-back of the tautological class
one concludes that \({{\mathrm{mom}}}^{k}_r\otimes \mathbb {Z}/p^{r-1}\mathbb {Z}\) coincides with the map given by \(\lambda _r^{*}\tau ^{[k]}_{r-1}\). This means that \({{\mathrm{mom}}}^{k}_r\otimes \mathbb {Z}/p^{r-1}\mathbb {Z}\) has to factor through \({{\mathrm{Tr}}}_r\), i.e., the diagram
commutes, which gives the desired result. \(\square \)
With this result we can now define the moment map for the sheaf of Iwasawa algebras \(\Lambda (\mathscr {H})\).
Definition 5.5.2
We define the k -th moment map to be the map of pro-sheaves
defined by \(({{\mathrm{mom}}}_r^{k})_{r\geqslant 0}\) and
by taking \({{\mathrm{mom}}}^{k}\) in the k-th component.
Remark 5.5.3
In each stalk the the map \({{\mathrm{mom}}}^{k}\) coincides with the map \({{\mathrm{mom}}}^{k}\) defined in Definition 5.3.3 (see [12, Lemma 12.2.14]).
6 The Integral Logarithm Sheaf
6.1 Definition of the Integral Logarithm Sheaf
We now define a pro-sheaf \(\mathcal {L}\) on G of modules over \(\pi ^{*}\Lambda (\mathscr {H})\), which will give a \({\mathbb {Z}_p}\)-structure of the logarithm sheaf \(\mathcal {L}og_{\mathbb {Q}_p}\). For this write \(G_r:=G\) considered as a quasi-finite étale G-scheme via the \(p^{r}\)-multiplication
Note that this is a \(G[p^{r}]\)-torsor
over G. Let \(\lambda _r:G_r\rightarrow G_{r-1}\) be the transition map, which is just the [p]-multiplication in this case. Then, as in (24), we have trace maps
We will also need the following variant. Let \(\Lambda _s:=\mathbb {Z}/p^{s}\mathbb {Z}\) and write
Then the adjunction \(\lambda _{r!}\lambda _r^{!}\rightarrow \mathrm {id}\) defines transition morphisms
Definition 6.1.1
With the above transition maps we can define the pro-sheaves
We call \(\mathcal {L}\) the integral logarithm sheaf.
Note that the reduction modulo \(p^{s-1}\) gives transition maps \(\mathcal {L}_{\Lambda _s}\rightarrow \mathcal {L}_{\Lambda _{s-1}}\) and that we have an isomorphism of pro-sheaves
By the general theory outlined above, \(\mathcal {L}\) is a module over \(\pi ^{*}\Lambda (\mathscr {H})\) which is free of rank 1.
Let \(t:S\rightarrow G\) be a section and denote by \(G[p^{r}]\langle t\rangle \) the \(G[p^{r}]\)-torsor defined by the cartesian diagram
We denote by \(\mathscr {H}_r\langle t\rangle \) the étale sheaf defined by \(G[p^{r}]\langle t\rangle \) and by \(\mathscr {H}\langle t\rangle :=(\mathscr {H}_r\langle t\rangle )\) the pro-system defined by the trace maps. We write
for the sheaf of Iwasawa modules defined by \(\mathscr {H}\langle t\rangle \).
Lemma 6.1.2
There is an canonical isomorphism
In particular, for the unit section \(e:S\rightarrow G\) one has
and hence a section \(\mathbf {1}:{\mathbb {Z}_p}\rightarrow e^{*}\mathcal {L}\) given by mapping 1 to 1.
Proof
This follows directly from the fact that \(\mathcal {L}\) is compatible with base change and the definitions. \(\square \)
6.2 Basic Properties of the Integral Logarithm Sheaf
The integral logarithm sheaf enjoys the same properties as its \({\mathbb {Q}_p}\)-counterpart, namely functoriality, vanishing of cohomology and a universal property for unipotent sheaves.
Let \(\varphi :G_1\rightarrow G_2\) be a homomorphism of group schemes of relative dimension \(d_1\) and \(d_2\) over S. Denote by \(\mathcal {L}_1\) and \(\mathcal {L}_2\) the integral logarithm sheaves on \(G_1\) and \(G_2\) respectively.
Theorem 6.2.1
(Functoriality) Let \(c:=d_1-d_2\). Then there is a canonical map
Moreover, if \(\varphi \) is an isogeny of degree prime to p, then \(\varphi _\#:\mathcal {L}_1\cong \varphi ^{*}\mathcal {L}_2\) is an isomorphism.
Proof
The homomorphism \(\varphi \) induces a homomorphism of group schemes over \(G_1\)
which induces by adjunction \(\varphi _!\varphi ^{!}\rightarrow \mathrm {id}\) and the base change property of \(\Lambda _{r}[G_{2,r}]\) a morphism of sheaves
Passing to the limit gives the required map. If \(\varphi \) is an isogeny of degree prime to p, then the map in (32) is an isomorphism. Hence this is also true for \(\varphi _\#\). \(\square \)
Corollary 6.2.2
(Splitting principle) Let c be an integer prime to p and let \(t:S\rightarrow G\) be a c-torsion section. Then there is an isomorphism
More generally, if \(D:=G[c]\) with \((c,p)=1\) then
where \(\iota _D:D\rightarrow G\) and \(\pi _D:D\rightarrow S\) is the structure map.
Proof
Apply \(t^{*}\) respectively, \(\iota _D^{*}\) to the isomorphism \([c]_\#:\mathcal {L}\rightarrow [c]^{*}\mathcal {L}\). \(\square \)
Theorem 6.2.3
(Vanishing of cohomology) Recall that 2d is the relative dimension of \(\pi :G\rightarrow S\). Then the pro-sheaves
are Mittag-Leffler zero (see Sect. 2.1) and
We start the proof of this theorem with a lemma:
Lemma 6.2.4
The endomorphism \([p^{r}]_!:R^{i}\pi _!\mathbb {Z}/p^{s}\mathbb {Z}\rightarrow R^{i}\pi _!\mathbb {Z}/p^{s}\mathbb {Z}\) is given by multiplication with \(p^{r(2d-i)}\).
Proof
By Lemma 3.1.2 we see that \([p^{r}]_!\) is given by \(p^{r}\)-multiplication on \(\mathscr {H}_s\). The result follows from this and the \(\mathbb {Z}/p^{s}\mathbb {Z}\)-version of the isomorphism (7) \(\square \)
Proof of Theorem 6.2.3. Consider the transition map \(\Lambda _s[G_{r+j}]\rightarrow \Lambda _s[G_r]\). If we apply \(R^{i}\pi _!\) we get the homomorphism
where \(\pi _r=\pi :G_r\rightarrow S\) is the structure map of \(G_r=G\). By Lemma 6.2.4, the map \([p^{j}]_!\) acts by multiplication with \(p^{j(2d-i)}\) on \(R^{i}\pi _{r+j,!}\Lambda _s\). In particular, this is zero for \(i\ne 2d\) and \(j\geqslant s\) and the identity for \(i=2d\). This proves the theorem, because \(R^{2d}\pi _!\Lambda _s(d)\cong \Lambda _s\). \(\square \)
The sheaf \(\mathcal {L}\) satisfies also a property analogous to Theorem 3.4.4. To formulate this properly, we first need a property of unipotent \(\mathbb {Z}/p^{s}\mathbb {Z}\)-sheaves.
Lemma 6.2.5
Let \(\mathscr {F}\) be a unipotent \(\Lambda _s=\mathbb {Z}/p^{s}\mathbb {Z}\)-sheaf of length n on G. Then \([p^{ns}]^{*}\mathscr {F}\) is trivial on \(G_{ns}\) in the sense that there exists a \(\Lambda _s\)-sheaf \(\mathscr {G}\) on S such that
where \(\pi _{ns}:G_{ns}\rightarrow S\) is the structure map.
Proof
We show this by induction. For \(n=0\) there is nothing to show. So let
be an exact sequence with \(\mathscr {F}'\) unipotent of length \(n-1\), so that by induction hypotheses \([p^{(n-1)s}]^{*}\mathscr {F}'\cong \pi ^{*}\mathscr {G}'\) on \(G_{(n-1)s}\). Thus it suffices to show that for an extension \(\mathscr {F}\in {{\mathrm{Ext}}}^{1}_{G}(\pi ^{*}\mathscr {G}'',\pi ^{*}\mathscr {G}')\), the sheaf \([p^{s}]^{*}\mathscr {F}\) is trivial on \(G_s\). One has
and the pull-back by \([p^{s}]^{*}\) on the first group is induced by the trace map \([p^{s}]_!:R\pi _![p^{s}]_![p^{s}]^{!}\pi ^{!}\mathscr {G}''\rightarrow R\pi _!\pi ^{!}\mathscr {G}''\) on the last group. By the projection formula we have \(R\pi _!\pi ^{!}\mathscr {G}''\cong R\pi _!\Lambda _s(d)[2d]\otimes \mathscr {G}''\) and the triangle
gives rise to a long exact sequence of \({{\mathrm{Ext}}}\)-groups
If we pull-back by \([p^{s}]^{*}\) and use Lemma 6.2.4 the resulting map on the module \({{\mathrm{Ext}}}^{1}_{S}(\tau _{<2d}R\pi _!\Lambda _s(d)[2d]\otimes \mathscr {G}'',\mathscr {G}')\) is zero, which shows that \([p^{s}]^{*}\mathscr {F}\) is in the image of
This is the desired result. \(\square \)
Exactly as in (15) one can define for each \(\Lambda _s\)-sheaf \(\mathscr {F}\) and each r a homomorphism
as the composition
The next theorem corrects and generalizes [12, Proposition 4.5.3], which was erroneously stated for all \(\mathbb {Z}/p^{s}\mathbb {Z}\)-sheaves and not just for unipotent ones.
Theorem 6.2.6
(Universal property) Let \(\mathscr {F}\) be a unipotent \(\Lambda _s\)-sheaf of length n. Then the homomorphism (33)
is an isomorphism.
Proof
Let \(\mathscr {F}\) be unipotent of length n. Then we know from Lemma 6.2.5 that there is a \(\Lambda _s\)-sheaf \(\mathscr {G}\) on S such that \([p^{ns}]^{*}\mathscr {F}\cong \pi ^{*}_{ns}\mathscr {G}\), where \(\pi _{ns}:G_{ns}\rightarrow S\) is the structure map. Similarly, we write \(e_{ns}\) for the unit section of \(G_{ns}\). Then one has
Further, one has the following chain of isomorphisms
which prove the theorem. \(\square \)
6.3 The Integral étale Poylogarithm
In this section we define in complete analogy with the \({\mathbb {Q}_p}\)-case the integral étale polylogarithm.
We recall the set-up from Sect. 4.1. Denote by \(c>1\) an integer invertible on S and prime to p and let \(D:=G[c]\) be the c-torsion subgroup. Then the localization triangle for \(j_D:U_D\subset G\) and \(\iota _D:D\rightarrow G\) reads
By relative purity and the splitting principle \(\iota _D^{!}\mathcal {L}(d)[2d]\cong \iota _D^{*}\mathcal {L}\cong \pi _D^{*}\Lambda (\mathscr {H})\). We apply the functor \(H^{j}(S,-)\) to this triangle. As the \(R^{i}\pi _!\mathcal {L}\) are Mittag-Leffler zero for \(i\ne 2d\) by Theorem 6.2.3 one gets with (3):
Proposition 6.3.1
In the above situation there is a short exact sequence
As in the \({\mathbb {Q}_p}\)-case we define
so that one has
With these preliminaries we can define the integral polylogarithm.
Definition 6.3.2
The integral étale polylogarithm associated to \(\alpha \in {\mathbb {Z}_p}[D]^{0}\) is the unique class
such that \(\mathrm {res}(_\alpha \mathrm {pol})=\alpha \).
6.4 The Eisenstein–Iwasawa Class
Recall that \(D=G[c]\) and let \(t:S\rightarrow U_D=G\setminus D\) be an N-torsion section with \((N,c)=1\) but N not necessarily prime to p. The same chain of maps as in (18) gives a map
By functoriality the N-multiplication induces a homomorphism
Definition 6.4.1
Let \(\alpha \in {\mathbb {Z}_p}[D]^{0}\) and \(t:S\rightarrow U_D\) be an N-torsion section. Then the image
of \(_\alpha \mathrm {pol}\) under the map (34) is called the Eisenstein–Iwasawa class. We write
Remark 6.4.2
Note that \(_\alpha {{\mathrm{{\mathcal {EI}}}}}(t)_N\) depends on N and not on t alone. The class \(_\alpha {{\mathrm{{\mathcal {EI}}}}}(t)_{NM}\) differs from \(_\alpha {{\mathrm{{\mathcal {EI}}}}}(t)_N\).
The k-th moment map induces a homomorphism of cohomology groups
Definition 6.4.3
The class
is called the integral étale Eisenstein class.
These Eisenstein classes are interpolated by the Eisenstein–Iwasawa class by definition. We will see later how they are related to the \({\mathbb {Q}_p}\)-Eisenstein class, which are motivic, i.e., in the image of the étale regulator from motivic cohomology.
6.5 The Eisenstein–Iwasawa Class for Abelian Schemes
It is worthwhile to consider the case of abelian schemes in more detail. In this section we let \(G=A\) be an abelian scheme over S, so that in particular \(\pi :A\rightarrow S\) is proper and we can write \(R\pi _*\) instead of \(R\pi _!\).
The first thing to observe is the isomorphism
so that the \({\mathbb {Q}_p}\)-polylogarithm is a class
Evaluation at the N-torsion section \(t:S\rightarrow U_D\) is just the pull-back with \(t^{*}\)
and the k-th component of \(t^{*}_\alpha \mathrm {pol}_{\mathbb {Q}_p}\) is \(_\alpha \mathrm {Eis}_{\mathbb {Q}_p}^{k}(t)\).
There is one specific choice of \(\alpha \) which is particularly important, which we define next. Consider the finite étale morphism \(\pi _D:G[c]\rightarrow S\) and the unit section \(e:S\rightarrow G[c]\). These induce
(coming from \(\pi _{D*}e_!e^{!}{\mathbb {Q}_p}\rightarrow \pi _{D*}{\mathbb {Q}_p}\)) and
One checks easily that \(e_*(1)-\pi ^{*}_D(1)\) is in the kernel of \(H^{0}(S,\pi _{D*}{\mathbb {Q}_p})\rightarrow H^{0}(S,{\mathbb {Q}_p})\).
Definition 6.5.1
Let \(\alpha _c\in {\mathbb {Q}_p}[D]^{0}\) be the class
We write \(_c\mathrm {pol}_{\mathbb {Q}_p}\) and \(_c\mathrm {Eis}^{k}_{\mathbb {Q}_p}(t)\) for the polylogarithm and the Eisenstein class defined with \(\alpha _c\).
We now assume that S is of finite type over \({{\mathrm{Spec}}}\mathbb {Z}\). Then \(H^{2d-1}(A_r\setminus A_r[cp^{r}],\mathbb {Z}/p^{r}\mathbb {Z}(d))\) is finite, so that one has by (2)
where, as before, \([p^{r}]:A_r=A\rightarrow A\) is the \(p^{r}\)-multiplication and the transition maps are given by the trace maps. The integral étale polylogarithm is then a class
In the special case where \(A=E\) is an elliptic curve over S it is shown in [12, Theorem 12.4.21] that
is given by the inverse limit of Kato’s norm compatible elliptic units \(_c\vartheta _E\). Unfortunately, we do not have such a description even in the case of abelian varieties of dimension \(\geqslant 2\). If we write \(A[p^{r}]\langle t\rangle \) for the \(A[p^{r}]\)-torsor defined by diagram (31), then
where the inverse limit is again over the trace maps.
7 Interpolation of the \({\mathbb {Q}_p}\)-Eisenstein Classes
7.1 An Integral Structure on \(\mathcal {L}og^{(k)}_{\mathbb {Q}_p}\)
For the comparison between the integral \(\mathcal {L}\) and the \({\mathbb {Q}_p}\)-polylogarithm \(\mathcal {L}og_{{\mathbb {Q}_p}}\) we need an intermediate object, which we define in this section. This is purely technical. The reason for this is as follows: In general a unipotent \({\mathbb {Q}_p}\)-sheaf does not necessarily have a \({\mathbb {Z}_p}\)-lattice which is again a unipotent sheaf. In the case of \(\mathcal {L}og^{(k)}_{\mathbb {Q}_p}\) however, it is even possible to construct a \({\mathbb {Z}_p}\)-structure \(\mathcal {L}og^{(k)}\) such that
is a unipotent \(\Lambda _r=\mathbb {Z}/p^{r}\mathbb {Z}\)-sheaf.
Let \(\mathcal {L}og^{(1)}\) be the \({\mathbb {Z}_p}\)-sheaf defined in Definition 3.2.1
and denote by \(\mathbf {1}^{(1)}:{\mathbb {Z}_p}\rightarrow e^{*}\mathcal {L}og^{(1)}\) a fixed splitting.
Definition 7.1.1
We define
as the k-th graded piece of the divided power algebra \(\Gamma _{\mathbb {Z}_p}(\mathcal {L}og^{(1)})\). We further denote by
the splitting induced by \(\mathbf {1}^{(1)}\).
As \({\mathbb {Z}_p}\) and \(\mathscr {H}\) are flat \({\mathbb {Z}_p}\)-sheaves (all stalks are \({\mathbb {Z}_p}\)-free), the k-th graded piece of the divided power algebra \(\Gamma _k(\mathcal {L}og^{(1)})\) has a filtration with graded pieces \(\pi ^{*}\Gamma _i(\mathscr {H})\otimes \Gamma _{k-i}({\mathbb {Z}_p})\) (see [8, V 4.1.7]). In particular, the \(\Gamma _k(\mathcal {L}og^{(1)})\) are unipotent \({\mathbb {Z}_p}\)-sheaves of length k. By base change the same is true for the \(\Lambda _r\)-sheaf
To define transition maps
we proceed as in Sect. 3.3. Consider \(\mathcal {L}og^{(1)}\rightarrow {\mathbb {Z}_p}\oplus \mathcal {L}og^{(1)}\) given by the canonical projection and the identity. Then we define
where we identify \(\Gamma _1({\mathbb {Z}_p})\cong {\mathbb {Z}_p}\). A straightforward computation shows that \(\mathbf {1}^{(k)}\mapsto \mathbf {1}^{(k-1)}\) under the transition map.
Definition 7.1.2
We denote by \(\mathcal {L}og\) the pro-sheaf \((\mathcal {L}og^{(k)})_{k\geqslant 0}\) with the above transition maps and let \(\mathbf {1}:{\mathbb {Z}_p}\rightarrow e^{*}\mathcal {L}og\) be the splitting defined by \((\mathbf {1}^{(k)})_{k\geqslant 0}\).
Remark 7.1.3
We would like to point out that, contrary to the \({\mathbb {Q}_p}\)-situation, the pro-sheaf \((\mathcal {L}og^{(k)})_{k\geqslant 0}\) is not the correct definition of the \({\mathbb {Z}_p}\)-logarithm sheaf. In fact, the correct integral logarithm sheaf is \(\mathcal {L}\).
Proposition 7.1.4
Denote by \(\mathcal {L}og^{(k)}\otimes {\mathbb {Q}_p}\) the \({\mathbb {Q}_p}\)-sheaf associated to \(\mathcal {L}og^{(k)}\). Then there is a canonical isomorphism
which maps \(\mathbf {1}^{(k)}_{\mathbb {Q}_p}\) to \(\mathbf {1}^{(k)}\).
Proof
First note that the canonical map \({{\mathrm{Sym}}}^{k}\mathcal {L}og^{(1)}_{\mathbb {Q}_p}\rightarrow \Gamma _k(\mathcal {L}og^{(1)}_{\mathbb {Q}_p})\) is an isomorphism. This can be checked at stalks, where it follows from (4) as \(\mathcal {L}og^{(1)}_{\mathbb {Q}_p}\) is a sheaf of \({\mathbb {Q}_p}\)-modules. The claim in the proposition then follows from the isomorphisms
and the claim about the splitting follows from the explicit formula for the map \({{\mathrm{Sym}}}^{k}\mathcal {L}og^{(1)}_{\mathbb {Q}_p}\rightarrow \Gamma _k(\mathcal {L}og^{(1)}_{\mathbb {Q}_p})\) given after (4). \(\square \)
Corollary 7.1.5
For all i there are isomorphisms
Proof
The first and the third follow directly from the proposition and the definition of the cohomology of a \({\mathbb {Q}_p}\)-sheaf. For the second one observes that the canonical map
is an isomorphism. This can be checked on stalks, where it follows again from (4). \(\square \)
7.2 Comparison of Integral and \({\mathbb {Q}_p}\)-polylogarithm
In this section we want to compare \(\mathcal {L}\) and \(\mathcal {L}og_{{\mathbb {Q}_p}}\). We first compare \(\mathcal {L}\) with the sheaves \(\mathcal {L}og^{(k)}\) defined in Definition 7.1.1.
Define a comparison map
as follows. By Theorem 6.2.6 one has for the sheaves \(\mathcal {L}og^{(k)}_{\Lambda _r}\) from (37) the isomorphism
so that the splitting \(\mathbf {1}^{(k)}\otimes \Lambda _r:\Lambda _r\rightarrow e^{*}\mathcal {L}og^{(k)}_{\Lambda _r}\) defines a morphism of sheaves on G
which is obviously compatible with the transition maps and functorial in G. Passing to the pro-systems over \(r\geqslant 0\), this defines a homomorphism
Taking also the pro-system in the k-direction leads to a comparison map
For each k applying \({\text {comp}}^{k}\) to the localization triangle for \(D\hookrightarrow G\hookleftarrow U_D\) gives
compatible with the transition maps \(\mathcal {L}og^{(k)}\rightarrow \mathcal {L}og^{(k-1)}\).
Proposition 7.2.1
There is a commutative diagram with short exact columns
Proof
Take the long exact cohomology sequence of the commutative diagram in (42), tensor the lower horizontal line with \({\mathbb {Q}_p}\) and then pass to the inverse limit over k. Using the isomorphisms in Corollary 7.1.5 gives the commutative diagram as stated. \(\square \)
Corollary 7.2.2
Let \(\alpha \in {\mathbb {Z}_p}[D]^{0}\), with \(D=G[c]\) as before. Then one has
in \(H^{2d-1}(S,R\pi _!Rj_{D*}j_D^{*}\mathcal {L}og_{\mathbb {Q}_p}(d))\). In particular, for every N-torsion section \(t:S\rightarrow U_D\) one has
Proof
Immediate from the definition of \(_\alpha \mathrm {pol}\) and \(_\alpha \mathrm {pol}_{\mathbb {Q}_p}\) and the commutative diagram in the proposition. The second statement follows from the first as \({\text {comp}}\) is compatible with the evaluation map at t. \(\square \)
7.3 Interpolation of the \({\mathbb {Q}_p}\)-Eisenstein Classes
For our main result, we first have to relate the comparison map \({\text {comp}}^{k}\) with the moment map \({{\mathrm{mom}}}^{k}\).
Proposition 7.3.1
The composition
coincides with the moment map \({{\mathrm{mom}}}^{k}\).
Proof
By the definitions of \({{\mathrm{mom}}}^{k}\) and \({\text {comp}}^{k}\) it suffices to prove this statement for \(\Lambda _r\)-coefficients. Consider
from (39). This comes by adjunction from a map
on \(G_{rk}\) which has by definition the property that its pull-back \(e_{rk}^{*}(\beta _r)\) coincides with \(\mathbf {1}^{(k)}:\Lambda _r\rightarrow e^{*}\mathcal {L}og^{(k)}_{\Lambda _r}\). By Lemma 2.3.2 the map \(\beta _r\) is uniquely determined by this property. As \(\mathcal {L}og^{(k)}_{\Lambda _r}\) is unipotent of length k, the pull-back \([p^{rk}]^{*}\mathcal {L}og^{(k)}_{\Lambda _r}\) is trivial by Lemma 6.2.5 and is hence of the form
where the last isomorphism is obtained by the splitting \(\mathbf {1}^{(k)}\). Thus the map
where \(\tau _r^{[i]}\) is the i-th divided power of the tautological section from (25), has the property that its pull-back by \(e_{rk}^{*}\) coincides with \(\mathbf {1}^{(k)}\). It follows that this map equals \(\beta _r\) and by definition of the moment map in (26) the projection to the k-th component coincides also with the moment map. \(\square \)
Let \(t:S\rightarrow U_D\) be an N-torsion section. We need a compatibility between the composition
and the map \(\varrho _t\) in the splitting principle Corollary 3.4.2 composed with the projection onto the k-th component
Proposition 7.3.2
There is a commutative diagram
where \({{\mathrm{mom}}}_N^{k}={{\mathrm{mom}}}^{k}\circ [N]_\#\) and \(\varrho _t=[N]_\#^{-1}\circ [N]_\#\).
Proof
The commutative diagram
coming from functoriality of \({\text {comp}}^{k}\) and the isomorphisms
reduces the proof of the proposition to show the commutativity of the diagram
The isogeny [N] acts by N-multiplication on \(\mathscr {H}\), hence by multiplication with \(N^{k}\) on \({{\mathrm{Sym}}}^{k}\mathscr {H}_{\mathbb {Q}_p}\), which means that
Thus it remains to show that the diagram
commutes, which follows from Proposition 7.3.1 and the isomorphism
which was obtained in Corollary 7.1.5. \(\square \)
Recall from Definition 6.4.1 the Eisenstein–Iwasawa class
and from Definition 4.2.1 the \({\mathbb {Q}_p}\)-Eisenstein class
We consider its image under the k-th moment map
The main result of this paper can now be formulated as follows:
Theorem 7.3.3
(Interpolation of \({\mathbb {Q}_p}\)-Eisenstein classes) The image of \(_\alpha {{\mathrm{{\mathcal {EI}}}}}(t)_N\) under the k-th moment map is given by
Proof
This follows by combining Corollaries 7.2.2, 7.3.2 and the definition of the \({\mathbb {Q}_p}\)-Eisenstein class Definition 4.2.1. \(\square \)
Remark 7.3.4
For comparison with [12, Theorem 12.4.21] we point out again that the normalization of \({_\alpha }\mathrm {Eis}_{\mathbb {Q}_p}^{k}(t)\) in loc. cit. is different. We had there a factor of \(-N^{k-1}\) in front of the Eisenstein series.
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Acknowledgements
This paper is a sequel to [12], which was written as a response to John Coates’ wish to have an exposition of the results in [6] at a conference in Pune, India. This triggered a renewed interest of mine in the p-adic interpolation of motivic Eisenstein classes which goes back to [11]. In this sense the paper would not exist without the persistence of John Coates. The paper [7] with Annette Huber created the right framework to treat these questions. It is a pleasure to thank them both and to dedicate this paper to John Coates on the occasion of his seventieth birthday.
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Kings, G. (2016). On p-adic Interpolation of Motivic Eisenstein Classes. In: Loeffler, D., Zerbes, S. (eds) Elliptic Curves, Modular Forms and Iwasawa Theory. JHC70 2015. Springer Proceedings in Mathematics & Statistics, vol 188. Springer, Cham. https://doi.org/10.1007/978-3-319-45032-2_10
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