2.1 Iwasawa algebras and modules

Here, we record some basic facts and notation for completed group rings and modules over them, using [Reference Neukirch, Schmidt and WingbergNSW08] and [Reference VenjakobVen02] as our guides. Throughout, we fix an infinite pro-p group $\Gamma$ equipped with a countable basis $\Gamma=\Gamma_0\supsetneq \Gamma_1\supsetneq\cdots$ of the identity consisting of open normal subgroups. As in § 1, we write $W:=W(k)$ for the Witt ring of k.

Definition 2.1. For each n, we give $\Lambda_n:=W[\Gamma/\Gamma_n]$ (respectively, $\Omega_n:=k[\Gamma/\Gamma_n]$ ) the p-adic (respectively, discrete) topology, and equip

\begin{equation*} \Lambda:=\varprojlim_{n} W[\Gamma/\Gamma_n]\quad\text{and}\quad \Omega:=\varprojlim_n \Omega_n \end{equation*}

with the inverse limit topology. We set $I_{n}:=\ker(\Lambda\twoheadrightarrow \Lambda_n)$ , and simply write $I:=I_0$ for the augmentation ideal of $\Lambda$ .

The $I_{n}$ form a decreasing chain of two-sided, closed ideals of $\Lambda$ , and each $I_n$ is generated (as either a left or right ideal) by all expressions of the form $\gamma-1$ with $\gamma\in \Gamma_n$ . Furthermore, the collection of ideals $\{p^m\Lambda+ I_n\}_{n,m\geqslant 0}$ forms a fundamental system of neighborhoods of $0\in \Lambda$ .

Definition 2.2. Assume that $\Gamma$ is topologically finitely generated. The lower central p-series of $\Gamma$ is defined recursively by

\begin{align*}P_0:=\Gamma\quad\text{and}\quad P_{n+1}:=P_n^p[P_n,\Gamma],\end{align*}

where $P_n^p:=\langle g^p\ :\ g\in P_n\rangle$ is the subgroup of $P_n$ generated by all pth powers, and $[P_n,\Gamma]$ is the commutator subgroup of $\Gamma$ (see [Reference Dixon, du Sautoy, Mann and SegalDdSM⁺99, Definition 1.15 and Corollary 1.20]).

We note that the $P_n$ form a basis of the identity in $\Gamma$ consisting of open normal subgroups [Reference Dixon, du Sautoy, Mann and SegalDdSM⁺99, Proposition 1.16].

Proof. As a $\Lambda$ -module, $I_n$ is generated by elements of the form $u:=x-1$ with $x\in \Gamma_n$ , so it suffices to prove that pu lies in $I_nI+I_{n+1}$ . We have

$$x^p -1= (1+u)^p - 1 = pu + u^2 w$$

for some $w\in \Lambda$ . Now, $x^p\in \Gamma_{n+1}$ , so $x^p - 1 \in I_{n+1}$ and clearly $u^2w \in I_n^2 \subseteq I_n I$ ; the result follows.

By a left (respectively, right) $\Lambda$ -module, we will always mean a separated, topological left (respectively, right) $\Lambda$ -module. If M is any left $\Lambda$ -module, we define $M^{\operatorname{op}}$ to be the right $\Lambda$ -module with the same underlying abelian group as M and right $\Gamma$ -action given by $m\cdot\gamma:=\gamma^{-1}m$ for $m\in M$ . This enables us to define a duality functor from the category of left $\Lambda$ -modules to itself.

Definition 2.6. If M is any left $\Lambda$ -module, We define the dual of M to be the left $\Lambda$ -module

\begin{equation*} M^{\vee}:=\operatorname{Hom}_{\Lambda}(M^{\operatorname{op}},\Lambda). \end{equation*}

This is the abelian group of all continuous homomorphisms of right $\Lambda$ -modules $\varphi:M^{\operatorname{op}}\rightarrow \Lambda$ with the left $\Lambda$ -module structure induced by the canonical $\Lambda$ - $\Lambda$ -bimodule structure on $\Lambda$ : $(\lambda\cdot\varphi)(m):=\lambda\varphi(m)$ .

2.2 Towers of modules

In this section, we generalize the commutative algebra formalism developed in [Reference CaisCai18a, § 3.1] for certain projective systems of $\Lambda$ -modules when $\Gamma=\mathbf Z_p$ to the case of arbitrary pro-p groups $\Gamma$ . This machinery is at the heart of our proofs of Theorems A and C.

Given a $\Gamma$ -tower of W-modules $\{M_n,\rho_{n,m}\}$ , we simply write $M:=\varprojlim_{n} M_n$ for the projective limit taken with respect to the maps $\rho_{n,m}$ ; it is naturally a left $\Lambda$ -module.

Proposition 2.9. Let $\{M_n, \rho_{n,m}\}$ be a $\Gamma$ -tower of W-modules with $M_n$ a finite and free W-module for all n. Assume that the following conditions hold for all $n \geqslant 0$ :

1. (i) $\overline{M}_n:=k\otimes_W M_n$ is a free $\Omega_n$ -module of rank d that is independent of n; and

2. (ii) for all $m\leqslant n$ , the map

induced from $\rho_{n,m}$ by reduction modulo p is surjective.

Then, for all $n\geqslant 0$ :

1. (i) $M_n$ is a free $\Lambda_n$ -module of rank d; and

2. (ii) the induced maps of left $\Lambda_m$ -modules,

(2.1)

are isomorphisms for all $m\leqslant n$ .

Moreover, M is a finite and free $\Lambda_W(\Gamma)$ -module of rank d, and for each n the canonical map $\Lambda_n \otimes_{\Lambda} M\rightarrow M_n$ is an isomorphism of $\Lambda_n$ -modules.

Reducing the map (2.1) modulo the ideal (p) yields a map of left $\Omega_m$ -modules

(2.2)

through which the surjective map $\overline{\rho}_{n,m}$ factors, whence (2.2) is surjective. As (p) is contained in the radical of $\Lambda_m$ , it follows from Nakayama’s lemma [Reference LamLam01, (4.22) (3)] that (2.1) is surjective as well. By our assumption (i), the map (2.1) is then a surjective map of free $\Lambda_m$ -modules of the same rank, whence it must be an isomorphism.Footnote ⁵

For any $m\leqslant n$ , the kernel of the canonical surjection $\Lambda_n\twoheadrightarrow \Lambda_m$ is contained in the radical of $\Lambda_n$ . It follows from (i)–(ii) and Nakayama’s lemma that any lift to $M_n$ of a $\Lambda_m$ -basis of $M_m$ is a $\Lambda_n$ -basis of $M_n$ . We may therefore choose, for each $n\geqslant 0$ , a basis $e_{n,1},\ldots,e_{n,d}$ of the free $\Lambda_n$ -module $M_n$ with the property that $\rho_{n,m}: M_n\rightarrow M_m$ carries $e_{n,i}$ to $e_{m,i}$ for all $m\leqslant n$ and $1\leqslant i\leqslant d$ . These choices yield isomorphisms of left $\Lambda_n$ -modules $\Lambda_n^{\oplus d}\simeq M_n$ for all n that are compatible with change in n via the maps $\rho_{n,m}$ . Passing to projective limits, we conclude that $M = \varprojlim_{n} M_n$ is free of rank d over $\Lambda_W(\Gamma)$ . The canonical map $\Lambda_n\otimes_{\Lambda} M\rightarrow M_n$ is then a surjective map of free $\Lambda_n$ -modules of the same rank, whence it is an isomorphism.

We next wish to investigate duality for towers of $\Gamma$ -modules.

Proposition 2.10. Let $\{M_n, \rho_{n,m}\}$ and $\{M'_n, \rho'_{n,m}\}$ be two $\Gamma$ -towers of W-modules satisfying the hypotheses of Proposition 2.9. Suppose that for each n, there are W-bilinear perfect pairings

(2.3)

with respect to which $\gamma$ and $\gamma^{-1}$ are adjoint for all $\gamma\in \Gamma$ , and which satisfy

\begin{equation*} \langle \rho_{n,m}x , \rho'_{n,m}y\rangle_m = \sum_{\gamma \in \Gamma_m / \Gamma_n} \langle x , \gamma^{-1} y\rangle_n \end{equation*}

for all $m\leqslant n$ . Then for each n, the pairings defined by

(2.4) \begin{equation} (x,y)_n := \sum_{\gamma\in \Gamma/\Gamma_n} \langle x,\gamma^{-1}y\rangle_n \cdot \gamma \end{equation}

compile to induce isomorphisms of left $\Lambda$ -modules

(2.5)

We end this section with a variant of the Mittag–Leffler criterion [Sta22, Tag0598], which will enable us to show that certain (derived) inverse limits of $\Gamma$ -towers vanish.

(2.6)

We claim that $\tau$ is surjective: given any $(\beta_n)_n\in \prod_n M_n$ , for each fixed n, we set $\alpha_n:=\sum_{\ell\geqslant n} \rho^{\ell-n}(\beta_{\ell})$ . By hypothesis, $\rho^{\ell-n}(\beta_{\ell}) \in J^{\ell-n} M_n$ , so as $M_n$ is J-adically complete, this sum converges for each n, and by construction we have $\tau((\alpha_n)_n)=(\beta_n)_n$ . Thus, $\varprojlim^1_n M_n = 0$ .

2.3 Generalized Jacobians

In this section, we fix a smooth, proper, and geometrically connected curve X over k.

When $\deg_k(\mathfrak m)\gt0$ , we note that $X_{\mathfrak m}$ has a distinguished k-rational point $x:\operatorname{Spec} k\rightarrow X_{\mathfrak m}$ with the property that $X_{\mathfrak m}$ is k-smooth outside of x, the divisor $\mathfrak m$ on X is the pullback of x along $\nu$ , and $\nu:X\rightarrow X_{\mathfrak m}$ is initial among all k-maps $X\rightarrow Y$ to k-schemes Y such that $\mathfrak m$ scheme-theoretically factors through Y(k). It follows at once that for any $\mathfrak m' \geqslant \mathfrak m$ , the normalization map $\nu': X\rightarrow X_{\mathfrak m'}$ factors uniquely through $\nu:X\rightarrow X_{\mathfrak m}$ .

The k-group structure on $\operatorname{Pic}_{X_{\mathfrak m}/k}$ (functorially induced by tensor product of line bundles) makes $J_{X,\mathfrak m}$ into a commutative k-group scheme and $J_{X,\mathfrak m}^n$ into a principal homogeneous space for $J_{X,\mathfrak m}$ for all n. Moreover, for any $\mathfrak m' \geqslant \mathfrak m$ , pullback of line bundles along $X_{\mathfrak m}\rightarrow X_{\mathfrak m'}$ yields a canonical surjection of k-groups $J_{X,\mathfrak m'}\rightarrow J_{X,\mathfrak m}$ with affine kernel. The structure of $J_{X,\mathfrak m}$ can be made quite explicit [Reference Bosch, Lütkebohmert and RaynaudBLR90, § 9.2].

We now turn to the functoriality of $J_{X,\mathfrak m}$ in X.

Proposition 2.16. Let $\pi:Y\rightarrow X$ be a finite and generically étale map of proper, smooth, and geometrically connected curves over k. For any modulus $\mathfrak m$ on X with support a finite set S of closed points in X, there exists a unique maximal (respectively, minimal) modulus $\mathfrak n$ (respectively, $\mathfrak n'$ ) on Y that is supported on $\pi^{-1}S$ , and canonical homomorphisms of k-group schemes

making the diagrams

(2.7)

commute, where $\pi^*=\operatorname{Pic}^0(\pi) : J_X\rightarrow J_Y$ and $\pi_*=\operatorname{Alb}(\pi): J_Y\rightarrow J_X$ are the usual maps on Jacobians associated to the finite (flat!) map $\pi: Y\rightarrow X$ by Picard (pullback of line bundles) and Albanese (norm of line bundles) functoriality. Considering S and $\pi^{-1}S$ as reduced Cartier divisors on X and Y, these maps induce morphisms $\pi^*: J_{X,S}\rightarrow J_{Y,\pi^{-1}S}$ and $\pi_*: J_{Y,\pi^{-1}S}\rightarrow J_{X,S}$ of semiabelian varieties whose composition $\pi_* \pi^* = \deg(\pi)$ is multiplication by $\deg(\pi)$ . If, moreover, $\pi$ is generically Galois with group G, then $\pi^*\pi_* = \sum_{g\in G} g^*$ , where $g^*$ is the automorphism of $J_{Y,\pi^{-1}S}$ induced by applying the preceding construction to $g:Y\rightarrow Y$ and the modulus $\pi^{-1}S=g^{-1}(\pi^{-1}S)$ .

The respective restrictions of $\pi^*$ and $\pi_*$ to the tori $T_{\mathfrak m}$ and $T_{\mathfrak n'}$ are induced by the unique descents to k of the $\overline{k}$ maps

given on the factor with index $s\in S_{\overline{k}}$ by the diagonal $\pi^* = \Delta: \mathbf{G}_m \rightarrow \prod_{t\in \pi_{\overline{k}}^{-1}(s)} \mathbf{G}_m$ and the norm map $\pi_* = \operatorname{Nm}: \prod_{t\in \pi_{\overline{k}}^{-1}(s)} {\mathbf{G}_m} \rightarrow \mathbf{G}_m$ functorially determined on R-valued points for any $\overline{k}$ -algebra R by $\operatorname{Nm}((a_t)_t):=\prod_t a_t^{e_t}$ , where $e_t$ is the ramification degree of $\mathcal{O}_{X,s}\rightarrow \mathcal{O}_{Y,t}$ .

Denote by $\theta_{X,\mathfrak m} : X - S\rightarrow J^1_{X,\mathfrak m}$ the canonical map of k-schemes functorially determined by sending a Z-valued point P of $X_Z-S_Z$ to the inverse ideal sheaf $\mathcal{O}(P)$ associated to the closed subscheme P of $(X_{\mathfrak m})_Z$ . Consider the map of k-schemes

(2.8)

By the universal mapping property of generalized Jacobians [Reference SerreSer88, V § 4 No. 22, Proposition 13], there is a modulus $\mathfrak n'$ on Y with support $\pi^{-1}S$ , a map of k-schemes $\rho^1_{\mathfrak n'}: J_{Y,\mathfrak n'}^1\rightarrow J_{X,\mathfrak m}^1$ , and a map of k-group schemes $\rho_{\mathfrak n'}: J_{Y,\mathfrak n'}\rightarrow J_{X,\mathfrak m}$ such that $\rho^1_{\mathfrak n'}$ is equivariant with respect to $\rho_{\mathfrak n'}$ and $\rho = \rho^1_{\mathfrak n'}\circ \theta_{Y,\mathfrak n'}$ . Each such modulus uniquely determines $\rho_{\mathfrak n'}$ and $\rho^1_{\mathfrak n'}$ , and there is a unique minimal modulus verifying these conditions. We simply write $\pi_*: J_{Y,\mathfrak n'}\rightarrow J_{X,\mathfrak m}$ for the map $\rho_{\mathfrak n'}$ associated to the minimal modulus $\mathfrak n'$ . As pullback of line bundles along $\nu$ preserves degree (see, e.g., [Reference Bosch, Lütkebohmert and RaynaudBLR90, § 9.1]), for any $n\in \mathbf Z$ , we have a canonical map of k-schemes $J_{Y,\mathfrak n'}^n\rightarrow J_{Y}^n$ whose fibers are affine; as $J_X^n$ is proper, it follows that the composite

(2.9)

factors through $J_{Y,\mathfrak n'}^n \rightarrow J_Y^n$ via a unique map $\mu^n: J_Y^n\rightarrow J_X^n$ which must, moreover, be a map of k-groups when $n=0$ . Consider the resulting diagram

in which the middle square commutes by the very definition of $\mu^1$ and the outer squares are readily seen to commute from the definition of $\theta_{\star}$ and the fact that the normalization maps are isomorphisms over the smooth locii. As the top region commutes by the very definition of $\rho^1_{\mathfrak n'}$ , we conclude that $\mu^1\circ \theta_Y$ and $\theta_X\circ \pi$ agree on the dense open subscheme $Y-\pi^{-1}S$ of Y; since $J_X^1$ is separated, we deduce that they agree on Y. By the uniqueness aspect of $\rho^1$ , it follows that $\rho^1=\mu^1$ , giving the commutativity of the second diagram in (2.7).

To prove that the restrictions of the maps that we have constructed to the tori $T_{\mathfrak m}$ and $T_{\mathfrak n'}$ have the specified descriptions, we may assume that k is algebraically closed. As two maps from a reduced k-scheme to a separated k-scheme agree if and only if they agree on k-points, it suffices to check that the proposed descriptions hold at the level of k-points, which we now do.

Let P,Q be two k-points of Y not in $\pi^{-1}S$ and let $\mathscr{L}_Y$ (respectively, $\mathscr{L}_X$ ) be the line bundle on $Y_{\mathfrak n'}$ (respectively, $X_{\mathfrak m})$ corresponding to the degree-zero Cartier divisor $Q-P$ (respectively, $\pi(Q)-\pi(P)$ ). It follows easily from the universal property of $\rho^1_{\mathfrak n'}$ and $\rho_{\mathfrak n'}$ that $\pi_* \mathscr{L}_Y = \mathscr{L}_X$ , and one concludes that if $D = \sum n_Q Q$ is any degree-zero Cartier divisor on Y supported away from $\mathfrak n'$ , then $\pi_*\mathscr{L}(D)= \mathscr{L}(\pi(D))$ , where $\pi(D)=\sum n_Q \pi(Q)$ is the direct image of D. Thus, if $g\in k(Y)$ is any function, we conclude that

\begin{equation*} \pi_*(\mathscr{L}(\operatorname{div}(g))) = \mathscr{L}(\pi(\operatorname{div}(g))) = \mathscr{L}(\operatorname{div}(\operatorname{Nm}_{\pi}(g))), \end{equation*}

where $\operatorname{Nm}_{\pi}: k(Y)\rightarrow k(X)$ is the norm function associated to the extension of function fields $k(Y)/k(X)$ (cf. the proof of [Reference SerreSer88, III § 1 No. 2, Proposition 4]). Choose a k-valued point $\xi$ of $T_{\mathfrak n'} = \prod_{s\in S}\prod_{t\in \pi^{-1}(s)} \mathbf{G}_m/\mathbf{G}_m$ , and let $(a_t)\in \prod_{s\in S}\prod_{t\in \pi^{-1}(s)} k^{\times} $ be any representative of it. Let $g \in k(Y)$ be any function with $\operatorname{ord}_t(a_t - g) \gt 0$ at every point $t\in \pi^{-1}S$ . The image of $\xi$ under the inclusion $T_{\mathfrak n'}(k)\rightarrow J_{Y,\mathfrak n'}(k)$ is the line bundle $\mathscr{L}(\operatorname{div}(g))$ on $Y_{\mathfrak n'}$ , with image under $\pi_*$ the line bundle $\mathscr{L}(\operatorname{div}(\operatorname{Nm}_{\pi}(g)))$ on $X(\mathfrak m')$ . For any point $s\in S$ , the function g is a local unit at every $t\in \pi^{-1}(s)$ by construction, and we have

\begin{equation*} \operatorname{Nm}_{\pi}(g) = \prod_{t\in \pi^{-1}(s)} \operatorname{Nm}_{\widehat{\mathcal{O}}_{Y,t}/\widehat{\mathcal{O}}_{X,s}}(g) \equiv \prod_{t\in \pi^{-1}(s)} a_t^{e_t} \pmod{\mathfrak{p}_s}, \end{equation*}

with $\mathfrak{p}_s$ the maximal ideal of $\mathcal{O}_{X,s}$ and $e_t$ the ramification index of $\mathcal{O}_{X,s}\rightarrow \mathcal{O}_{Y,t}$ . The given description of $\pi_*$ follows. Likewise, if $h\in k(X)$ , then $\pi^*\mathscr{L}(\operatorname{div}_X(h))=\mathscr{L}(\pi^*\operatorname{div}_X(h))=\mathscr{L}(\operatorname{div}_Y(\pi^*h))$ , so if $a_s\in k^{\times}$ and h satisfies $\operatorname{ord}_s(a_s-h)\gt0$ , then $\operatorname{ord}_t(a_s-\pi^*h)\gt0$ for every $t\in \pi^{-1}s$ since the pullback map $\pi^*: k(X)\rightarrow k(Y)$ is a map of k-algebras. The given description of $\pi^*$ follows.

Finally, $\pi^*$ and $\pi_*$ induce maps on maximal semiabelian quotients as any k-homomorphism from a unipotent group to a semiabelian variety is zero, and the fact that these induced maps satisfy $\pi_*\pi^*=\deg(\pi)$ and (when $\pi$ is generically Galois with group G) $\pi^*\pi_* = \sum_{g\in G} g^*$ follows from the corresponding fact on (usual) Jacobians and the given descriptions of these maps on torii.

2.4 Dieudonné modules and de Rham cohomology

We begin with a brief recall of Dieudonné theory for finite k-group schemes of p-power order and p-divisible groups. The standard reference for this material is [Reference FontaineFon77, II–III]. For a modern synopsis, we recommend [Reference Chai, Conrad and OortCCO14, § 1.4]. Throughout, we will simply write finite k-group for a finite, commutative k-group scheme of p-power order.

As k is perfect, any finite k-group or p-divisible group G admits a functorial decomposition

(2.10) \begin{equation} G = G^{\mathrm{\acute{e}t}} \times G^{\mathrm{m}} \times G^{\mathrm{ll}},\end{equation}

with $G^{\mathrm{\acute{e}t}}$ étale (reduced with connected dual), $G^{\mathrm{m}}$ multiplicative (connected with reduced dual), and $G^{\mathrm{ll}}$ local–local (connected with connected dual). If G is a finite k-group, then $G=\operatorname{Spec} R$ for some k-algebra R that is finite dimensional as a k-vector space, and the order of G is $|G| = \dim_k R;$ this recovers the usual notion when G is the constant group scheme associated to a finite abelian group. If $G=\{G[p^n]\}_n$ is a p-divisible group, then there is an integer h so that $|G[p^n]|=p^{nh}$ for all n, and we say that G is of height h, and write $\operatorname{ht}(G):=h$ .

The Dieudonné ring relative to k is the (associative but noncommutative if $k\neq \mathbf F_p$ ) W-algebra $\mathcal{D}_k:=W[F,V]$ generated by variables F (Frobenius) and V (Verschiebung) subject to the following relations:

• – $FV = VF = p$ ;

• – $F\lambda = \sigma(\lambda) F$ ; and

• – $V\lambda = \sigma^{-1}(\lambda) V$ .

for all $\lambda\in W$ . By definition, a Dieudonné module is a (left) $\mathcal{D}_k$ -module that is finitely generated as a W-module; equivalently, it is a finite type W-module M with additive maps $F,V:M\rightarrow M$ satisfying the three conditions above. These form an abelian category, with morphisms $\mathcal{D}_k$ -module homomorphisms, i.e. W-linear maps that are compatible with F and V. This category is equipped with an involutive duality functor: if M is a Dieudonné module of finite W-length, the dual of M is the Pontryagin dual $M^{\vee}:=\operatorname{Hom}_W(M,W[1/p]/W)$ as a W-module, with $F_{M^{\vee}}(f)(x):=\sigma f(Vx)$ and $V_{M^{\vee}}(f)(x):=\sigma^{-1}f(Fx)$ for $f\in M^{\vee}$ and $x\in M$ . If M is finite and free as a W-module, we set $M^{\vee}:=\operatorname{Hom}_W(M,W)$ with Frobenius and Verschiebung defined in the same way.

A crucial ingredient in the proofs of our main theorems is Oda’s theorem [Reference OdaOda69, Corollary 5.11], which provides an explicit description of Dieudonné modules of p-torsion group schemes of abelian varieties in terms of de Rham cohomology. We will only need this result for Jacobians of curves, where it can be stated in terms of the de Rham cohomology of the curves themselves.

Associated to any smooth and proper curve X over k is the short exact ‘Hodge filtration’ sequence of its de Rham cohomology:

(2.11)

This sequence is functorial in finite morphisms of smooth and proper curves $\pi:Y\rightarrow X$ in two ways: we write $\pi^*$ for the contravariant pullback map associated to $\pi$ , and $\pi_*$ for the covariant trace map; we have $\pi_*\pi^*=\deg(\pi)$ and when $Y\rightarrow X$ is generically Galois with group G, we moreover have $\pi^*\pi_* = \sum_{g\in G} g^*$ . Cup-product pairing induces a canonical autoduality pairing $\langle \cdot,\cdot\rangle_X$ on (2.11), via which $\pi_*$ and $\pi^*$ are adjoint, for any finite $\pi:Y\rightarrow X$ . In particular, as the absolute Frobenius morphism $\operatorname{F}_X:X\rightarrow X$ is finite, the exact sequence (2.11) is equipped with semilinear endomorphisms $F := \operatorname{F}_X^*$ and $V:=(\operatorname{F}_X)_*$ , which are (semilinearly) adjoint under $\langle \cdot,\cdot\rangle_X$ , and satisfy $VF=\deg \operatorname{F}_X=0$ . The map V is called the Cartier operator, and admits a more explicit description, as in [Reference OdaOda69, Definition 5.5] (for a proof that our definition of V coincides with that of Oda, see [Reference CaisCai18a, § 2.1]; also cf. [Reference SerreSer58, § 10, Proposition 9]). In this way, (2.11) becomes a short exact sequence of Dieudonné modules: note that $F=0$ on $H^0(X,\Omega^1_{X/k})$ as pullback of any differential form by the p-power map in characteristic p is zero, and by duality $V=0$ in $H^1(X,\mathcal{O}_X)$ .

On the other hand, writing $J_X$ for the Jacobian of X, the relative Verschiebung of $J_X[p]^{(p^{-1})}$ sits in a short exact sequence of finite group schemes

(2.12)

This short exact sequence is contravariantly functorial in finite morphisms of smooth and proper curves $\pi:Y\rightarrow X$ via Picard functoriality $\pi^*:=\operatorname{Pic}^0(\pi)$ (pullback of line bundles) and covariantly functorial via Albanese functoriality $\pi_*:=\operatorname{Alb}(\pi)$ (norm of line bundles). Applying the Dieudonné functor to (2.12) yields a short exact sequence of Dieudonné modules,

(2.13)

that is covariantly functorial in finite morphisms $\pi:Y\rightarrow X$ via $\mathbf{D}(\pi^*)$ and contravariantly functorial via $\mathbf{D}(\pi_*)$ . The principal polarization of $J_X$ yields a canonical isomorphism $J_X[p]^{\vee}\simeq J_X[p]$ intertwining $(\pi^*)^{\vee}$ with $\pi_*$ and $(\pi_*)^{\vee}$ with $\pi^*$ ; via the compatibility of $\mathbf{D}(\cdot)$ with duality, the dual of (2.13) is then functorially identified with (2.13).

Theorem 2.21 (Oda). Let X be a smooth and proper curve over k with Jacobian $J_X$ . There is a canonical isomorphism of short exact sequences of Dieudonné modules

wherein the top row is (2.13) and the bottom row is (2.11). This isomorphism is compatible with the autodualities of each row, and for any morphism $\pi:Y\rightarrow X$ of smooth and proper curves, intertwines $\mathbf{D}(\pi_*)$ on Dieudonné modules with $\pi^*$ on de Rham cohomology, and $\mathbf{D}(\pi^*)$ with $\pi_*$ .

We will also need the following variant of Oda’s theorem for reduced but possibly non-smooth curves. If X is any reduced proper curve over k, its Jacobian $J_X:=\operatorname{Pic}^0_{X/k}$ is a smooth commutative group scheme over k. While the kernel of multiplication by p on $J_X$ may not be finite, the kernel of relative Frobenius $J_X\rightarrow J_X^{(p)}$ is finite, so we may consider its Dieudonné module. On the other hand, writing $\omega_{X/k}$ for the relative dualizing sheaf of X as in [Reference ConradCon00, § 5.2], Grothendieck’s theory of the trace map attached to any finite map of proper reduced curves $\pi:Y\rightarrow X$ yields a map on dualizing sheaves $\pi_* : \pi_*\omega_{X/k}\rightarrow \omega_{Y/k}$ that is dual to the pullback map $\pi^*: \mathcal{O}_Y\rightarrow \pi_*\mathcal{O}_X$ via Grothendieck duality (see [Reference CaisCai18a, Appendix A]). If $\pi$ is in addition flat, $\pi_*\mathcal{O}_Y$ is a finite locally free $\mathcal{O}_X$ -module, so one has a trace mapping $\pi_*:\pi_*\mathcal{O}_Y\rightarrow \mathcal{O}_X$ on functions, whose dual is a pullback map $\pi^*:\omega_{X/k}\rightarrow \pi_*\omega_{Y/k}$ on dualizing sheaves that coincides with pullback of differential forms when X and Y are smooth. In particular, the finite absolute Frobenius map induces a semilinear mapping

which recovers the previously defined Cartier operator when X is smooth. In this way, $H^0(X,\omega_{X/k})$ is naturally a Dieudonné module with $F=0$ .

Lemma 2.22. Let X be a proper curve over k, and $J_X:=\operatorname{Pic}^0_{X/k}$ its Jacobian. There is a natural isomorphism of Dieudonné modules

(2.14) \begin{equation} \mathbf{D}(J_X[F]) \simeq H^0(X,\omega_{X/k}). \end{equation}

If $\pi:Y\rightarrow X$ is a finite map of reduced curves, then (2.14) intertwines $\mathbf{D}(\pi^*)$ with $\pi_*$ , and if $\pi$ is in addition flat, it intertwines $\mathbf{D}(\pi_*)$ with $\pi^*$ .

Besides Oda’s theorem, there is one other crucial tool in our proofs, Nakajima’s equivariant Deuring–Shafarevich formula [Reference NakajimaNak85]. To state it in the form that we will need, we first note that any Dieudonné module M admits a functorial decomposition

\begin{equation*} M = M^{V-\mathrm{bij}} \oplus M^{F-\mathrm{bij}} \oplus M^{F,V-\mathrm{nil}},\end{equation*}

where $M^{V-\mathrm{bij}}$ (respectively, $M^{F-\mathrm{bij}}$ ) is the maximal $\mathcal{D}_k$ -stable submodule on which V (respectively, F) is bijective and F (respectively, V) is topologically nilpotent and $M^{F,V-\mathrm{nil}}$ is the maximal submodule on which both F and V are topologically nilpotent. While the existence of this decomposition is a fact of ‘pure’ semilinear algebra, it corresponds to (2.10) via the equivalence of Theorem 2.19.

In the remainder of this section, we record some additional results on the behavior of the trace mapping attached to finite maps of curves.

Proposition 2.26. Let $\pi:Y\rightarrow X$ be a finite étale map of proper, smooth, and geometrically connected curves over k. Let $S\subseteq X(\overline{k})$ be a finite and nonempty set of geometric points of X and $T:=\pi^{-1}S$ , and consider S and T as (reduced and effective) divisors on X and Y, respectively. The trace mapping

(2.15)

is surjective.

Proof. As the formation of (2.15) is compatible with base change, by passing to a finite extension of k if need be, we may assume that $S\subseteq X(k)$ , and we will induct on $|S|$ . First, suppose that $S=\{s\}$ is a single point, and denote by $Y_T$ the singular curve associated to the modulus T. Since T is reduced and $\pi(T) = s\in X(k)$ , the restriction of $\pi: Y \rightarrow X $ to T scheme-theoretically factors through X(k) [Sta22, Tag 0356]. By the universal property of $Y\rightarrow Y_T$ , we obtain a unique map $\rho: Y_T\rightarrow X$ through which $\pi$ factors. Now the curve $Y_T$ is by construction geometrically integral, and hence (geometrically) reduced and therefore Cohen–Macaulay, and X is by hypothesis smooth. It follows from [Reference MatsumuraMat89, Corollary to Theorem 23.1] that $\rho$ is flat, and necessarily totally ramified over s. By Proposition 2.25, the map $\rho^*: \operatorname{Pic}^0_{X/k} \rightarrow \operatorname{Pic}^0_{Y_T/k}$ has trivial scheme-theoretic kernel. Using the fact that $\operatorname{Lie}(\cdot)$ is left exact, together with [Reference Liu, Lorenzini and RaynaudLLR04, Proposition 1.3], we deduce that $\rho^*: H^1(X,\mathcal{O}_X)\rightarrow H^1(Y_T,\mathcal{O}_{Y_T})$ is injective, so by Grothendieck duality the trace map

$$\rho_*: H^0(Y_T,\omega_{Y_T})\rightarrow H^0(X,\Omega^1_X)= H^0(X,\Omega^1_X(S))$$

is surjective; the final equality uses the fact that the sum of the residues at all singular points of a meromorphic differential on a proper curve is zero. On the other hand, by Lemma 2.13, the normalization map $\nu:Y\rightarrow Y_T$ induces an identification $\nu_*: H^0(Y,\Omega^1_Y(T))\simeq H^0(Y_T,\omega_{Y_T/k})$ . As $\pi = \rho \circ \nu$ , we conclude that $\pi_*$ is surjective when $S=\{s\}$ .

Now supposing that the result holds for a given finite and nonempty set $S\subseteq X(k)$ , let $s\in X(k)$ be a point not in S, and set $S':=S\cup \{s\}$ and $T':=T \cup \pi^{-1}s$ . We then have a commutative diagram of short exact sequences.

(2.16)

Indeed, exactness of the rows is a straightforward consequence of the Riemann–Roch theorem, and the (right square of the) diagram commutes due to [Reference CaisCai18a, Remark 2.2]. The left vertical arrow of (2.16) is now surjective by our inductive hypothesis, while surjectivity of the right vertical arrow is clear. We conclude that the middle vertical arrow is surjective as well.

3.1 The ramified case

By hypothesis, $\pi_{n,0}:X_n\rightarrow X_0$ is totally ramified at every point of $S_n$ , and so induces an identification of étale k-schemes $S_n\simeq S_0=S$ for all n. We will henceforth make these identifications, which induce identifications $\mathscr{T}_n\simeq \mathscr{T}_0$ and $\mathscr{Q}_n\simeq\mathscr{Q}_0$ for all n, which we likewise make, denoting these common p-divisible groups simply by $\mathscr{T}_S$ and $\mathscr{Q}_S$ , respectively. Similarly, we often write $\mathscr{G}_{n,S}$ for $\mathscr{G}_{n,S_n}$ , and $\mathbf{D}_{n,S}^{\star}$ in place of $\mathbf{D}_{n,S_n}^{\star}$ .

In order to prove Theorem C, we apply Propositions 2.9 and 2.10 to the $\Gamma$ -towers of W-modules $\{\mathbf{D}_{n,S}^{\mathrm{m}},\rho\}$ and $\{\mathbf{D}_{n,S}^{\mathrm{\acute{e}t}},\rho'\}$ . Recalling the notation of Proposition 2.9(i), it suffices to prove the following.

Lemma 3.5. For each pair of positive integers $m\leqslant n$ , the maps

are surjective. Moreover, $\overline{\mathbf{D}}_{n,S}^{\mathrm{m}}$ and $\overline{\mathbf{D}}_{n,S}^{\mathrm{\acute{e}t}}$ are free $\Omega_n$ -modules of rank $d:=\gamma+\deg S -1$ for all n.

Proof. We may assume that k is algebraically closed.Footnote ⁶ On the one hand, Theorem 2.19 provides identifications of Dieudonné modules

$$ \mathbf{D}(\mathscr{G}_{n,S}^{\mathrm{m}})\otimes_W k \simeq \mathbf{D}(J_{X_n,S}[p]^{\mathrm{m}}) \simeq \mathbf{D}(J_{X_n,S}[F])^{V-\mathrm{bij}}. $$

For the final isomorphism, note that if a finite commutative k-group scheme G is killed by p, then one has $FV=0$ on $\mathbf{D}(G)$ , so that G is of multiplicative type if and only if V is bijective and $F=0$ on $\mathbf{D}(G)$ due to Theorem 2.19(iv). On the other hand, Lemmas 2.13 and 2.22 provide a natural identification

$$\mathbf{D}(J_{X_n,S}[F])^{V-\mathrm{bij}} \simeq H^0(X_n,\Omega^1_{X_n/k}(S))^{V-\mathrm{bij}}.$$

These identifications and the duality isomorphism of (3.2) yield natural isomorphisms of $\Omega_n$ -modules

(3.4a) \begin{equation}\overline{\mathbf{D}}_{n,S}^{\mathrm{m}} := \mathbf{D}(\mathscr{G}_{n,S}^{\mathrm{m}})\otimes_W k \simeq \mathbf{D}(J_{X_n,S}[p]^{\mathrm{m}}) \simeq H^0(X_n,\Omega^1_{X_n/k}(S))^{V-\mathrm{bij}},\end{equation}

(3.4b) \begin{equation} \overline{\mathbf{D}}_{n,S}^{\mathrm{\acute{e}t}} := \mathbf{D}((\mathscr{G}_{n,S}^{\mathrm{m}})^{\vee})\otimes_W k \simeq \mathbf{D}(J_{X_n,S}[p]^{\mathrm{m}})^{\vee} \simeq (H^0(X_n,\Omega^1_{X_n/k}(S))^{V-\mathrm{bij}})^{\vee}. \end{equation}

It follows from this and Nakajima’s Theorem 2.24(i) that $\overline{\mathbf{D}}^{\mathrm{m}}_{n,S}$ and $\overline{\mathbf{D}}^{\mathrm{\acute{e}t}}_{n,S}$ are each free $\Omega_n$ -modules of rank d. Now, (3.4a ) and (3.4b ) are compatible with change in n, in the sense that for all $m\leqslant n$ , each of the two diagrams

commutes. We are thereby reduced to the claim that the map $\pi^*$ (respectively, $\pi_*$ ) is injective (respectively, surjective) on differential forms. Pullback of differential forms is injective as $\pi_{n,m}$ is generically étale, and to see that trace is surjective, we may dualize, apply Serre duality, and argue that the pullback map

(3.5)

is injective. This pullback map fits into the following commutative diagram with exact rows.

Since $X_n$ is (geometrically) connected for all n, we have $H^0(X_n,\mathcal{O}_{X_n})\simeq k$ and the first vertical map is an isomorphism. As S is reduced, we have $H^0(X_n, \mathcal{O}_S) \simeq k^{\deg S}$ for all n, and the second vertical map is likewise an isomorphism. The injectivity of the final vertical map follows from Proposition 2.25, arguing as in the proof of Proposition 2.26. A diagram chase then implies that (3.5) is injective as well.

Proof of Theorem C. Due to Lemma 3.5 and our calculations immediately below (3.3), the $\Gamma$ -towers of W-modules $\{\mathbf{D}_{n,S}^{\mathrm{m}},\rho\}$ and $\{\mathbf{D}_{n,S}^{\mathrm{\acute{e}t}},\rho'\}$ satisfy the hypotheses of Propositions 2.9 and 2.10, whose conclusions immediately give Theorem C.

Lemma 3.6. For each pair of positive integers $m\leqslant n$ , the map $\pi_{n,m}: X_n\rightarrow X_m$ induces commutative diagrams of p-divisible groups

(3.6a)

and

(3.6b)

compatibly with change in m,n. Moreover, (3.6a) and (3.6b) are equivariant for the $(\cdot)^*$ -action and the $(\cdot)_*$ -action of $\Gamma$ , respectively.

Passing to multiplicative parts on (3.6a ) and applying the (contravariant) Dieudonné module functor yields, for each $m\leqslant n$ , the following commutative diagram of Dieudonné modules, whose rows are short exact.

(3.7)

Similarly, passing to multiplicative parts on (3.6b ), dualizing, and applying $\mathbf{D}(\cdot)$ yields

(3.8)

where we have used the canonical identifications $\mathscr{G}_n^{\vee}\simeq \mathscr{G}_n$ coming from principal polarizations on Jacobians. As duality of Jacobians interchanges Picard and Albanese functoriality, these identifications are equivariant for the action of $(\gamma_*)^{\vee}$ on $\mathscr{G}_n^{\vee}$ and $\gamma^*$ on $\mathscr{G}_n$ , and identify the transition maps $\rho'$ and $\rho$ on $\mathbf{D}_n^{\mathrm{\acute{e}t}}$ as in Remark 3.3. These diagrams give short exact sequences of projective systems.

To prove Theorem D, we will need to apply the functor $\varprojlim$ to the exact sequences of inverse systems given by (3.7) and (3.8). To analyze the result, we will need to know that certain derived limits vanish, which is the content of the next lemma.

Lemma 3.7. For each pair of positive integers $m\leqslant n$ , the map

is surjective. In particular, the map $\rho: \mathbf{D}_{n}^{\star}\rightarrow \mathbf{D}_{m}^{\star}$ is surjective for any $\star \in \{\mathrm{\acute{e}t},\mathrm{m}, \mathrm{ll},\operatorname{null}\}$ .

Therefore, it suffices to prove that the trace map on de Rham cohomology is surjective. This map fits into the following commutative diagram with short exact rows.

(3.9)

Therefore, it is enough to prove that the flanking vertical maps are surjective. Dualizing and using Serre duality, we equivalently wish to show that the pullback maps

(3.10)

are injective. As in the proof of Lemma 3.5, the second map of (3.10) is injective as $\pi_{n,m}$ is generically étale, while the first map is injective due to Proposition 2.25, as in the proof of Proposition 2.26.

Proof of Theorem D. We apply $\varprojlim$ to the exact sequences of inverse systems given by (3.7) and (3.8). By Lemma 3.7, the inverse system $\{\mathbf{D}_n^{\mathrm{m}}, \rho\}$ is Mittag–Leffler, so the derived limit $\varprojlim^1_{n,\rho} \mathbf{D}_n^{\mathrm{m}}$ vanishes, which yields the short exact sequence of Theorem D(i). Likewise, $\mathbf{D}(\mathscr{Q}_S)$ is a finite free W-module, so it follows from Lemma 2.11 that applying $\varprojlim$ to (3.8) results in an isomorphism of $\Lambda$ -modules $\mathbf{D}^{\mathrm{\acute{e}t}}_{S} \simeq \mathbf{D}^{\mathrm{\acute{e}t}}$ , as in Theorem D(i). This isomorphism and the control aspect of Theorem C yield canonical isomorphisms of $\Lambda_n$ -modules

\begin{equation*} \Lambda_n \mathop{\otimes}\limits_{\Lambda}\mathbf{D}^{\mathrm{\acute{e}t}} \simeq \Lambda_n \mathop{\otimes}\limits_{\Lambda} \mathbf{D}_S^{\mathrm{\acute{e}t}} \simeq \mathbf{D}_{n,S}^{\mathrm{\acute{e}t}} \end{equation*}

and the first short exact sequence of Theorem D(ii) follows from (3.8). To obtain the second exact sequence, we apply $\Lambda_n \otimes_{\Lambda} (\cdot)$ to the short exact sequence of (i) to get a long exact sequence that fits into the commutative diagram

whose bottom row is (3.7) and whose vertical maps are the canonical projections. The rightmost vertical map is identified with the multiplication map which, as the $\Lambda$ -module structure map of $\mathbf{D}(\mathscr{T}_S)$ factors through the augmentation map $\Lambda\twoheadrightarrow \Lambda_n \twoheadrightarrow W$ , is easily seen to be an isomorphism. The middle vertical map is an isomorphism due to Theorem C(ii), so a diagram chase yields the short exact sequence

(3.11)

On the other hand, applying $(\cdot)\otimes_{\Lambda} \mathbf{D}(\mathscr{T}_S)$ to the tautological exact sequence

gives the exact sequence

Again, as the $\Lambda$ -module structure on $\mathbf{D}(\mathscr{T}_S)$ factors through $\Lambda\rightarrow \Lambda_n \rightarrow W$ , the final map above is an isomorphism, which yields

(3.12) \begin{equation} \operatorname{Tor}^{\Lambda}_1(\Lambda_n,\mathbf{D}(\mathscr{T}_S))\simeq I_{n}\mathop{\otimes}\limits_{\Lambda}\mathbf{D}(\mathscr{T}_S) \simeq (I_{n} \mathop{\otimes}\limits_{\Lambda} W) \mathop{\otimes}\limits_W \mathbf{D}({\mathscr{T}_S}) \simeq \frac{I_{n}}{I \cdot I_{n}}\mathop{\otimes}\limits_W \mathbf{D}(\mathscr{T}_S), \end{equation}

where the final isomorphism results from the canonical identification $W\simeq \Lambda/I$ via the augmentation map. Putting (3.11) and (3.12) together gives the second exact sequence in Theorem D(ii).

It remains to establish the explicit descriptions of $\mathbf{D}(\mathscr{T}_S)$ and $\mathbf{D}(\mathscr{Q}_S)$ given in Theorem D(iii). By Corollary 2.18 and the very definition of $\mathscr{T}_S$ , we have a short exact sequence of p-divisible groups

Applying $\mathbf{D}(\cdot)$ to this gives the claimed description of $\mathbf{D}(\mathscr{T}_S)$ , and the description of $\mathbf{D}(\mathscr{Q}_S)$ then follows from the very definition $\mathscr{Q}_S:=\mathscr{T}_S^{\vee}$ and the compatibility of $\mathbf{D}(\cdot)$ with duality.

Proof of Theorem E. Let d be the dimension of the p-adic Lie group $\Gamma$ , and without loss of generality assume that $\{\Gamma_n\}_n$ is the lower central p-series of $\Gamma$ , so that $\Gamma/\Gamma_n$ is a finite p-group of order $p^{d n }$ .

The functorial decompositions $\mathscr{G}_n = \mathscr{G}_n^{\mathrm{\acute{e}t}} \times \mathscr{G}_n^{\mathrm{m}}\times \mathscr{G}_n^{\mathrm{ll}}$ yield a canonical isomorphism of $\Lambda$ -modules

\begin{align*}\mathbf{D}\simeq \mathbf{D}^{\mathrm{\acute{e}t}}\times \mathbf{D}^{\mathrm{m}}\times \mathbf{D}^{\mathrm{ll}},\end{align*}

so as $\mathbf{D}^{\star}$ is finitely generated (even as a $\Lambda$ -module) for $\star=\mathrm{\acute{e}t},\mathrm{m}$ due to Theorems C and D, it suffices to prove that $\mathbf{D}$ is not a finitely generated $\Lambda[\![F,V]\!]$ -module. Assume to the contrary that it is, and choose $\Lambda[\![F,V]\!]$ -module generators $\delta_1,\ldots,\delta_r$ . Since the transition maps $\rho:\mathbf{D}_n\rightarrow \mathbf{D}_m$ are surjective due to Lemma 3.7, the canonical projection $\mathbf{D}\rightarrow \mathbf{D}_n$ is surjective for every n, whence the images of $\delta_1,\ldots,\delta_r$ generate $\mathbf{D}_n $ as a $\Lambda_n[F,V]$ -module. It follows that $\mathbf{D}_n / (F,V)$ is generated by at most r elements as a $\Lambda_n$ -module, for all n. The canonical isomorphisms $\mathbf{D}_n/p\mathbf{D}_n \simeq H^1_{dR}(X_n/k)$ of Oda’s theorem induce isomorphisms

\begin{equation*} D_n/(F,V) \simeq \operatorname{coker} (V : H^0(\Omega^1_{X_n})\rightarrow H^0(\Omega^1_{X_n})) \end{equation*}

for each n so that the cokernel of V is generated as an $\Omega_n$ -module by at most r generators. It follows that the cokernel, and hence the kernel, of V on $H^0(\Omega^1_{X_n})$ has k-dimension at most $r\cdot |\Gamma/\Gamma_n| = r p ^{dn}$ . Writing $a_n$ for this dimension, we will prove that for any $D\gt0$ , there exists N with $a_n \gt D p^{dn}$ for all $n \gt N$ , contradicting the above bound coming from our finite generation assumption. To do so, we may, and henceforth do, assume that k is algebraically closed.

Since $\Gamma/\Gamma_n$ is a p-group, it is solvable, and we may find a subgroup $H_n \leqslant \Gamma/\Gamma_n$ of order p. Let $L_n\subset K_n$ be the fixed field of $H_n$ , and so $K_n/L_n$ a degree-p Galois extension, with Galois group $H_n\simeq \mathbf Z/p\mathbf Z$ , totally ramified over the finite set of places of $L_n$ lying over $S\subseteq K_0$ . For $Q\in S$ , let $d_{Q,n}$ be the unique break in the lower numbering ramification filtration of $H_n$ above Q. As the lower numbering of the ramification groups is inherited by subgroups, $d_{Q,n}$ is the largest lower break in the ramification filtration of $\Gamma/\Gamma_n$ above Q. Working locally at Q, let $K_{n,Q}$ be the localization of $K_n$ at Q, and $L_Q$ the compositum of all $K_{n,Q}$ . Then $L_Q/K_Q$ is strictly APF [Reference WintenbergerWin79, Théorème 1.2], so the Herbrand functions $\psi(x):=\int_0^x [\Gamma: \Gamma^t] dt$ and $\phi:=\psi^{-1}$ are well-defined and continuous, piecewise linear increasing bijections on $[0,\infty)$ . Let $u_n:= \phi(d_n)$ be the last upper break in the ramification filtration of $\Gamma/\Gamma_n$ at Q, so that $d_n = \psi(u_n)$ . By [Reference WintenbergerWin79, Thàreme 1.2], the ratio ${\psi(x)}/{[\Gamma: \Gamma^x]}$ tends to infinity with x. It follows that for any $D\gt0$ , we have $\psi(x) \gt D [\Gamma: \Gamma^x]$ for all x sufficiently large. Evaluating on $x= u_n+\varepsilon$ , and noting that by definition of $u_n$ we have $\Gamma^x \subseteq \Gamma_n$ for all $x \gt u_n$ , we find that for any D, there exists N so that, for all $n\gt N$ and all $\varepsilon \gt 0$ ,

(3.13) \begin{equation} \psi(u_n+\varepsilon) \gt D [\Gamma: \Gamma_n] \cdot [\Gamma_n : \Gamma^{u_n + \varepsilon}]. \end{equation}

As $\psi$ is continuous, we conclude that $d_{Q,n}=\psi(u_n) \geqslant D [\Gamma: \Gamma_n]$ for all $n \gt N$ . We may clearly arrange for this to hold for all $Q\in S$ simultaneously.

On the other hand, by [Reference Booher and CaisBC20, Theorem 1.1] we have the lower bound

\begin{equation*} a_n \gt \bigg(1 - \frac{1}{p}\bigg)\bigg\lfloor\frac{p}{2}\bigg\rfloor \sum_{Q\in S}\bigg(\bigg\lceil\frac{p}{2}\bigg\rceil\frac{d_{Q,n}}{p} - 1\! \bigg) \end{equation*}

and we conclude that for any $D\gt0$ , there exists N so that for all $n\gt N$ we have $a_n \gt D |\Gamma/\Gamma_n|$ .

To prove Theorem F, we will need the following lemma.

Lemma 3.8. Let A be an abelian variety over a finite field k of cardinality $p^r$ , let $\mathscr{G}:=A[p^{\infty}]$ be the p-divisible group of A, and let $\wp:=1-F^r$ be the Lang isogeny. There is a functorial exact sequence

where $\mathscr{G}(k)^*:=\operatorname{Hom}_{\mathbf Z_p}(\mathscr{G}(k),\mathbf Q_p/\mathbf Z_p)$ is the Pontryagin dual of $\mathscr{G}(k)$ .

from which we deduce the exact sequence

(3.14)

Indeed, for each n, multiplication by $p^n$ and the snake lemma give exact sequences

(3.15)

which, for variable n, form an inductive system in which the transition maps $A(k)/p^nA(k)\rightarrow A(k)/p^{n+1}A(k)$ are multiplication by p. Passing to étale parts (which is exact) and applying the exact functor $\varinjlim$ to (3.15) therefore yields (3.14). Now, the Dieudonné module functor for a constant group scheme $\underline{H}$ is particularly simple:

\begin{equation*} \mathbf{D}(\underline{H}) = \varinjlim_n\operatorname{Hom}_{k-\mathrm{gp}}(\underline{H}, W_n) = (\operatorname{Hom}(H, \mathbf Q_p/\mathbf Z_p)\otimes_{\mathbf Z_p} W(\overline{k}))^{\operatorname{Gal}(\overline{k}/k)} = \operatorname{Hom}(H,\mathbf Q_p/\mathbf Z_p)\otimes_{\mathbf Z_p} W, \end{equation*}

as the Galois action on $\underline{H}(\overline{k})=H$ is trivial. Applying $\mathbf{D}(\cdot)$ to (3.14) therefore gives the exact sequence of Lemma 3.8.

which yields the isomorphism of finite length W-modules

$$ \mathbf{D}(G^{\mathrm{\acute{e}t}})/\wp\mathbf{D}(G^{\mathrm{\acute{e}t}}) \simeq G(k)^*\otimes_{\mathbf Z_p} W. $$

Proof of Theorem F. By Lemma 3.8, for each n we have exact sequences of $\Lambda_n$ -modules,

(3.16)

which are compatible with change in n using the maps induced by Picard functoriality. By Lemma 3.7, applying $\varprojlim$ results in the first short exact sequence of Theorem F once we observe that the canonical map

is an isomorphism, due to the fact that $\mathscr{G}_n(k)$ is finite (and hence of finite length as a $\mathbf Z_p$ -module) and W is finitely presented as a $\mathbf Z_p$ -module. Theorem D(ii) and the exact sequence (3.16) provide the following commutative diagram with exact rows and columns.

From this, we deduce (via the snake lemma) the short exact sequence of $\Lambda_n$ -modules

As the formation of cokernels commutes with tensor product, the already established first exact sequence of Theorem F identifies the middle term above with $\Lambda_n \otimes_{\Lambda} M_W$ , while Remark 3.9 provides the identification $\mathbf{D}(\mathscr{Q}_S)/\wp \mathbf{D}(\mathscr{Q}_S)\simeq \mathscr{Q}_S(k)^*\otimes_{\mathbf Z_p} W$ ; this yields the second exact sequence of Theorem F.

Proof of Corollary 1.5. Applying $(\cdot)\otimes_W k$ to the second exact sequence of Theorem F gives an exact sequence

On the other hand, for each n we have canonical isomorphisms

\begin{equation*} (\Lambda_n \mathop{\otimes}\limits_{\Lambda}M_W )\mathop{\otimes}\limits_{W} k \simeq \Omega_n \mathop{\otimes}\limits_{\Omega} (M_W \mathop{\otimes}\limits_W k) \end{equation*}

and for any abelian group G, we have $G^*/pG^* = (G[p])^*$ , whence the exact sequence

and we deduce that for all n,

\begin{equation*} \log_p |\mathscr{G}_n[p](k)| = \dim_k (\mathscr{G}_n[p](k)\mathop{\otimes}\limits_{\mathbf F_p} k ) =\dim_{k} (\Omega_n\mathop{\otimes}\limits_{\Omega}(M_W \otimes_W k) )+ O(1). \end{equation*}

On the other hand, applying $(\cdot)\otimes_W k$ to the first exact sequence of Theorem F shows that $M_W \otimes_W k$ is a finitely generated $\Omega$ -module, and the result follows from [Reference Emerton and PaškūnasEP20, Proposition 2.18] for general $\Gamma$ and from [Reference MonskyMon83a, Theorem 1.8] when $\Gamma=\mathbf Z_p^d$ is abelian.

3.2 The étale case

We now suppose that $\Sigma=\emptyset$ – i.e. that $\pi_{n,0}:X_n\rightarrow X_0$ is étale for all n – and we recall that $S=\{x_0\}$ for a fixed closed point $x_0$ of $X_0$ , and $S_n=\pi_{n,0}^{-1}(S_0)$ is the fiber of $\pi_{n,0}$ over S. The proofs of Theorems A and B proceed along much the same lines as those of Theorems 3 and 4, though there are some key differences.