2.1 Infinite-level Igusa towers and p-adic modular forms

Set $G = \operatorname{GL}_2$ and let $P \subset G$ denote the upper triangular Borel subgroup with diagonal torus T. Fix a prime number p and a neat compact open subgroup $K^p \subset G(\mathbb{A}^p_f)$ .

In this section, we will define the relevant spaces of p-adic modular forms. In order to do this, we need to introduce the infinite-level Igusa varieties as constructed in [Reference Caraiani and ScholzeCS17, §4.3] (see also [Reference HoweHow20] and [Reference Boxer and PilloniBP23, §3.4] for some complementary details). Consider the p-divisible group over $\operatorname{Spf}\mathbb{Z}_p$ given by $\mathbb{X}_{\operatorname{ord}} = \mu_{p^{\infty}} \oplus \mathbb{Q}_p/\mathbb{Z}_p$ and define the following group schemes over $\operatorname{Spf}\mathbb{Z}_p$ :

\[J_{\operatorname{ord}} = \mathrm{QIsog}( \mathbb{X}_{\operatorname{ord}}) = \begin{pmatrix} \mathbb{Q}_p^\times & \widetilde{\mu_{p^\infty}} \\ 0 & \mathbb{Q}_p^\times \\ \end{pmatrix}\!, \quad J_{\operatorname{ord}}^{\operatorname{int}} = \operatorname{Aut}(\mathbb{X}_{\operatorname{ord}}) = \begin{pmatrix} \mathbb{Z}_p^\times & T_p(\mu_{p^\infty}) \\ 0 & \mathbb{Z}_p^\times \\ \end{pmatrix}\!,\]

where $\operatorname{QIsog}(-)$ (respectively, $\operatorname{Aut}(-)$ ) denotes the self quasi-isogenies (respectively, automorphisms) of a p-divisible group and $\widetilde{\mu_{p^\infty}} = \operatorname{lim}_{\times p} \mu_{p^{\infty}}$ is the universal cover of $\mu_{p^{\infty}}$ . Recall that the universal cover sits in an exact sequence

\begin{equation*} 0 \to T_p(\mu_{p^\infty}) \to \widetilde{\mu_{p^\infty}} \to \mu_{p^\infty} \to 0,\end{equation*}

and note that $\mathbb{Q}_p^\times = \mathrm{QIsog}( \mu_{p^\infty}) = \mathrm{QIsog}( \mathbb{Q}_p/{\mathbb{Z}}_p)$ (see [Reference Scholze and WeinsteinSW13, §3.1]). We let

\[P' = \begin{pmatrix} \mathbb{Q}_p^\times & \mathbb{Q}_p(1) \\ 0 & \mathbb{Q}_p^\times \\ \end{pmatrix}\!, \quad (P')^{\operatorname{int}} = J_{\operatorname{ord}}^{\operatorname{int}} \times_{\operatorname{Spa}(\mathbb{Z}_p, \mathbb{Z}_p)} \operatorname{Spa}(\mathbb{Q}_p, \mathbb{Z}_p) = \begin{pmatrix} \mathbb{Z}_p^\times & \mathbb{Z}_p(1) \\ 0 & \mathbb{Z}_p^\times \\ \end{pmatrix}\!,\]

which are (locally) profinite proétale group schemes over $\mathrm{Spa}(\mathbb{Q}_p,{\mathbb{Z}}_p)$ . We let $U_{P'} \subset P'$ and $U_{P'}^{\operatorname{int}} \subset (P')^{\operatorname{int}}$ denote the unipotent radicals. There is a natural map $P' \hookrightarrow J_{\operatorname{ord}}^{\operatorname{an}} := J_{\operatorname{ord}} \times_{\mathrm{Spa} ({\mathbb{Z}}_p, {\mathbb{Z}}_p)} \mathrm{Spa} (\mathbb{Q}_p, {\mathbb{Z}}_p)$ from P’ to the adic generic fibre of $J_{\operatorname{ord}}$ , induced by $\mathbb{Q}_p(1) \rightarrow \widetilde{ \mu_{p^\infty}}$ (but note that $J_{\operatorname{ord}}^{\operatorname{an}}$ is a one-dimensional group scheme, and therefore much larger than P’). We will often write $\widehat{\mathbb{G}}_m$ for the p-divisible group $\mu_{p^{\infty}}$ over $\operatorname{Spf}\mathbb{Z}_p$ to emphasise the fact that $\widehat{\mathbb{G}}_m(\operatorname{Spf}R) = 1 + R^{00}$ , where $R^{00} \subset R$ denotes the ideal of topologically nilpotent elements (cf. [Reference HoweHow20, Remark 2.1.3]).

We now recall the construction of the perfect Igusa tower $\mathfrak{IG}_{K^p}$ . Let ${X}_{\mathrm{GL}_2({\mathbb{Z}}_p)K^p} \rightarrow \textrm{Spec} {\mathbb{Z}}_p $ denote the (compact) modular curve of level $\mathrm{GL}_2({\mathbb{Z}}_p)K^p$ , and $\mathfrak{X}_{\mathrm{GL}_2({\mathbb{Z}}_p)K^p} \rightarrow \operatorname{Spf} {\mathbb{Z}}_p$ its completion along the special fibre. Set $\mathfrak{IG}_{(P')^{\operatorname{int}}K^p} := \mathfrak{X}^{\operatorname{ord}}_{\mathrm{GL}_2({\mathbb{Z}}_p)K^p}$ to be the ordinary locus. We also let $\mathfrak{IG}_{U_{P'}^{\operatorname{int}}K^p} \to \mathfrak{IG}_{(P')^{\operatorname{int}}K^p}$ denote the proétale $T(\mathbb{Z}_p)$ -torsor parameterising trivialisations $\mu_{p^{\infty}} \xrightarrow{\sim} E[p^\infty]^{\circ}$ and $\mathbb{Q}_p/{\mathbb{Z}}_p \xrightarrow{\sim} E[p^\infty]^{\mathrm{et}}$ of the connected and étale parts of the p-divisible group associated with the universal (ordinary) elliptic curve (since $E[p^\infty]^{\mathrm{et}} \cong (E[p^\infty]^{\circ})^D$ the étale part extends to a p-divisible group over the boundary). Note that there is a natural lift of Frobenius $\varphi$ on $\mathfrak{IG}_{U_{P'}^{\operatorname{int}}K^p}$ .

For any compact open subgroup $K_{p,P} \subseteq P'$ (i.e., one which is commensurable to $(P')^{\operatorname{int}}$ ), we let $\overline{K_{p, P}}$ denote its schematic closure in $J_{\operatorname{ord}}$ . By [Reference Boxer and PilloniBP23, Lemma 3.4.8] the group scheme $\overline{K_{p, P}}$ is a profinite flat group scheme over $\textrm{Spec} {\mathbb{Z}}_p$ with generic fibre $K_{p, P}$ . We let $\mathfrak{IG}_{K_{p,P}K^p}$ be the flat formal scheme equal to the categorical quotient of $\mathfrak{IG}_{K^p}$ by $\overline{K_{p,P}}$ . This is the affine formal scheme whose ring of functions is $(\mathcal{O}_{\mathfrak{IG}_{K^p}})^{\overline{K_{p,P}}}$ . Similarly, for any compact open

\[U_{K_{p,P}} \subseteq U_{P'} = \left( \begin{matrix} 1 & \mathbb{Q}_p(1) \\ & 1 \end{matrix} \right)\!,\]

we let $\mathfrak{IG}_{U_{K_{p,P}}K^p}$ be the flat formal scheme equal to the categorical quotient of $\mathfrak{IG}_{K^p}$ by $\overline{U_{K_{p,P}}}$ . If $K_{p,P}$ is a compact open subgroup of P’ and if we set $U_{K_{p,P}} = K_{p, P} \cap U_{P'}$ , then there is a short exact sequence

\[0 \rightarrow U_{K_{p,P}} \rightarrow K_{p,P} \rightarrow M_{K_{p,P}}\rightarrow 0.\]

The map $\mathfrak{IG}_{U_{K_{p,P}}K^p}\rightarrow \mathfrak{IG}_{{K_{p,P}}K^p}$ is a proétale torsor under the group $\overline{M_{K_{p,P}}}$ .

Recall that the space of p-adic modular forms can be defined as (isotypic parts of) sections of the classical Katz moduli space parameterising elliptic curves (modulo prime-to-p quasi-isogenies) equipped with a trivialisation of the connected part of its associated p-divisible group. Let $\overline{U_{P'}^{\operatorname{int}}}$ denote the schematic closure of $U_{P'}^{\operatorname{int}}$ in $J_{\operatorname{ord}}$ (which is equal to the unipotent part of $J_{\operatorname{ord}}^{\operatorname{int}}$ ). By [Reference HoweHow20, Lemma 5.1.1], the Igusa tower $\mathfrak{IG}_{K^p}$ is an fpqc $\overline{U_{P'}^{\operatorname{int}}}$ -torsor over the above Katz moduli space, which explains the following definition.

Definition 2.4. The space of p-adic modular formsFootnote ² of tame level $K^p$ is defined as

\[\mathscr{M}^+ := \mathrm{H}^0(\mathfrak{IG}_{U_{P'}^{\operatorname{int}} K^p}, \mathcal{O}_{{\mathfrak{IG}}_{U_{P'}^{\operatorname{int}}K^p}}).\]

The following theorem summarises some of its key structures.

Let $T^+ \subset T(\mathbb{Q}_p)$ denote the submonoid of elements $\operatorname{diag}(t_1, t_2)$ such that $v_p(t_1) \geq v_p(t_2)$ . Then we see that $T({\mathbb{Z}}_p), U_p$ and $S_p$ generate an action of $T^+$ . We let $\Theta \colon \mathscr{M}^+ \to \mathscr{M}^+$ be the operator corresponding to the action of the identity function $\mathrm{Id}_{{\mathbb{Z}}_p} \in C_{\operatorname{cont}}(\mathbb{Z}_p, \mathbb{Z}_p)$ .

Let $\mathscr{M}^+_{U_{K_{p,P}}} = \mathrm{H}^0(\mathfrak{IG}_{U_{K_{p,P}} K^p}, \mathcal{O}_{\mathfrak{IG}_{U_{K_{p,P}}K^p}})$ . Using the action of $\mathrm{diag}(p^n,1) \in J_{\operatorname{ord}}$ by conjugation, we can identify $\mathfrak{IG}_{U_{K_{p,P}} K^p}$ with $\mathfrak{IG}_{U_{P'}^{\operatorname{int}} K^p}$ and $\mathscr{M}^+$ with $\mathscr{M}^+_{U_{K_{p,P}}}$ . This conjugation action will transport the $U_p$ , $\varphi$ , $S_p$ -actions and will conjugate the action of $C_{\operatorname{cont}}({\mathbb{Z}}_p, {\mathbb{Z}}_p) = C_{\operatorname{cont}}(U_{P'}^{\operatorname{int}}(-1)^\vee, {\mathbb{Z}}_p)$ with $C_{\operatorname{cont}}(U_{K_{p,P}}(-1)^\vee, {\mathbb{Z}}_p)$ . Set $\mathscr{M}_{{U_{K_{p,P}}}} = \mathscr{M}_{{U_{K_{p,P}}}}^+[1/p]$ . We let

\[\Theta_{U_{K_{p,P}}} \colon \mathscr{M}_{U_{K_{p,P}}} \rightarrow \mathscr{M}_{U_{K_{p,P}}}\]

denote the map induced by the action of the identity function of $\mathbb{Q}_p$ in $C_{\operatorname{cont}}(U_{K_{p, P}}(-1)^\vee, {\mathbb{Z}}_p) \otimes \mathbb{Q}_p$ . Note that conjugation by $\mathrm{diag}(p^n,1)$ sends $\Theta_{U_{K_{p,P}}}$ to $p^n \Theta$ .

2.2 Relation to classical forms

To compare p-adic modular forms with classical modular forms (at this level of generality), we work over $\mathrm{Spa}(\mathbb{Q}_p^{\operatorname{cycl}}, {\mathbb{Z}}_p^{\operatorname{cycl}})$ . We fix an isomorphism ${\mathbb{Z}}_p \cong {\mathbb{Z}}_p(1)$ . Therefore, we can (and do) identify P’ and $\underline{P(\mathbb{Q}_p)}$ . We let $\mathcal{X}_{K^p, \mathbb{Q}_p^{\operatorname{cycl}}}$ be the perfectoid modular curve and we let $\mathcal{IG}_{K^p, \mathbb{Q}_p^{\operatorname{cycl}}} = \mathfrak{IG}_{K^p} \times \mathrm{Spa} (\mathbb{Q}_p^{\operatorname{cycl}}, {\mathbb{Z}}_p^{\operatorname{cycl}})$ . We have a natural $P(\mathbb{Q}_p)$ -equivariant map $\mathcal{IG}_{K^p, \mathbb{Q}_p^{\operatorname{cycl}}} \rightarrow \mathcal{X}_{K^p, \mathbb{Q}_p^{\operatorname{cycl}}}$ .

Let $K_p \subseteq G(\mathbb{Q}_p)$ be a compact open subgroup and let $g \in G(\mathbb{Q}_p)$ . The map

\[\mathcal{IG}_{K^p, \mathbb{Q}_p^{\operatorname{cycl}}} \xrightarrow{\cdot g} \mathcal{X}_{K^p, \mathbb{Q}_p^{\operatorname{cycl}}} \rightarrow \mathcal{X}_{K_p K^p, \mathbb{Q}_p^{\operatorname{cycl}}}\]

factors through an open immersion $\mathcal{IG}_{K_{p,P}K^p, \mathbb{Q}_p^{\operatorname{cycl}}} \rightarrow \mathcal{X}_{K_p K^p, \mathbb{Q}_p^{\operatorname{cycl}}}$ where $K_{p,P} = g K_{p} g^{-1} \cap P(\mathbb{Q}_p)$ . One can think of this open as a connected component of the ordinary locus in $\mathcal{X}_{K_p K^p, \mathbb{Q}_p^{\operatorname{cycl}}}$ .

Let $P_{\operatorname{dR}, K} \to X_{K}$ denote the right $\overline{P}$ -torsor parameterising frames of $\mathcal{H}_E$ respecting the Hodge filtration (see §A.3). We now consider the $M_{K_{p,P}}$ -torsor $\mathcal{IG}_{U_{K_{p,P}}K^p, \mathbb{Q}_p^{\operatorname{cycl}}} \rightarrow \mathcal{IG}_{K_{p,P}K^p, \mathbb{Q}_p^{\operatorname{cycl}}}$ . For ease of notation, set $K = K^pK_p$ . Then we have a commutative diagram

where $P^{\operatorname{an}}_{\operatorname{dR}, K, \mathbb{Q}_p^{\operatorname{cycl}}}$ denotes the analytification of $P_{\operatorname{dR}, K}$ . The top map is induced by the Hodge–Tate map and the unit root splitting. It is $M_{K_{p,P}}$ -equivariant via the natural map $M_{K_{p,P}} \rightarrow T^{\operatorname{an}} \subset \overline{P}^{\operatorname{an}}$ .

The following proposition describes the relation between nearly holomorphic modular forms and p-adic modular forms.

Proposition 2.8. We have a natural map

(2.1) \begin{equation} \mathrm{H}^0(P_{\operatorname{dR},K}, \mathcal{O}_{P_{\operatorname{dR},K}}) \otimes_{{\mathbb{Q}_p}} \mathbb{Q}_p^{\operatorname{cycl}} \rightarrow \mathscr{M}_{U_{K_{p,P}}} \hat{\otimes}_{\mathbb{Q}_p} \mathbb{Q}_p^{\operatorname{cycl}},\end{equation}

which is equivariant for the action of the prime-to-p Hecke algebra $\mathcal{H}_{K^p}$ , as well as for the action of $M_{K_{p,P}}$ via the map $M_{K_{p,P}} \rightarrow T \subset \overline{P}$ . Hence, for any $\kappa \in X^*(T)$ , we obtain a map

(2.2) \begin{equation}\mathrm{H}^0(X_K, \mathcal{H}_\kappa) \otimes_{{\mathbb{Q}_p}} \mathbb{Q}_p^{\operatorname{cycl}} \rightarrow \mathrm{Hom}_{M_{K_{p,P}}}( -w_0\kappa, \mathscr{M}_{U_{K_{p,P}}}) \hat{\otimes}_{\mathbb{Q}_p} \mathbb{Q}_p^{\operatorname{cycl}}\{w_0\kappa\},\end{equation}

where $\mathcal{H}_{\kappa}=\operatorname{Hom}_{T}(-w_0\kappa, \pi_*\mathcal{O}_{P_{\operatorname{dR}, K}} )$ (with $\pi \colon P_{\operatorname{dR}, K} \to X_K$ the structural map), and $\mathbb{Q}_p^{\operatorname{cycl}}\{w_0\kappa\}$ means that $T^+$ acts on $\mathbb{Q}_p^{\operatorname{cycl}}$ through the character $w_0\kappa$ .

Assume that $gK_pg^{-1}$ has an Iwahori decomposition

\[g K_p g^{-1} = U_{gK_p g^{-1}} \cdot T_{gK_pg^{-1}} \cdot \overline{U}_{gK_p g^{-1}}.\]

Let $\mathcal{H}_{K_p}^+$ be the subalgebra of the Hecke algebra $\mathcal{H}_{K_p}$ generated by $[K_p \cdot g^{-1} t g \cdot K_p]$ with $t \in T^+$ . Then the map (2.2) is equivariant for the morphism sending $[K_p \cdot g^{-1} t g \cdot K_p]$ to $t \in T^+$ .

Let $t \in T^+$ . Set $K' = (g^{-1}tg) K (g^{-1}tg)^{-1} \cap K$ and $K'' = (g^{-1}tg)^{-1}K' (g^{-1}tg)$ . set $U_{K_{p, P}} = g K_p g^{-1} \cap U_{P'}$ and $U_{K_{p, P}'} = g K_p' g^{-1} \cap U_{P'}$ . Then we have a correspondence

where $p_1$ is the natural map, and $p_2$ is the composition $X_{K'} \xrightarrow{\cdot (g^{-1}tg)} X_{K''} \to X_{K}$ . We also have a diagram of correspondences (over $\mathbb{Q}_p^{\operatorname{cycl}}$ )

where $q_1$ is the natural map induced from the inclusion $U_{K'_{p, P}} \subset U_{K_{p, P}}$ , $q_2$ is induced from right multiplication by t via the inclusion $t^{-1}U_{K'_{p, P}} t \subset U_{K_{p, P}}$ , and the map $\lambda$ is the action of t (via the torsor structure on $P_{\operatorname{dR}}$ ) composed with the morphism $p_1^{-1}P_{\operatorname{dR}, K} \to p_2^{-1}P_{\operatorname{dR}, K}$ arising from the $G(\mathbb{Q}_p)$ -equivariant structure on $P_{\operatorname{dR}}$ . The claim now follows from the fact that the left-hand square is Cartesian. Indeed, one has

\[[K : K'] = [gK_pg^{-1}: gK_p'g^{-1}] = [U_{gK_pg^{-1}} : tU_{gK_pg^{-1}}t^{-1} \cap U_{gK_pg^{-1}}] = [U_{K_{p, P}} : U_{K'_{p, P}}]\]

using the Iwahori decomposition for $gK_pg^{-1}$ .

By the results in §A.3, one has that $\mathcal{H}_{\kappa} = \operatorname{VB}_K^{\operatorname{can}}(M_{\kappa}^{\vee})$ is the quasi-coherent sheaf associated with the dual Verma module of lowest weight $w_0\kappa$ . In particular, $\mathcal{H}_{\kappa}$ is naturally a $\mathcal{D}_{X_K}$ -module on $X_K$ . By composing the induced connection $\mathcal{H}_{\kappa} \to \mathcal{H}_{\kappa} \otimes \Omega^1_{X_K}(\operatorname{log}D)$ with the map $\mathcal{H}_{\kappa} \otimes \Omega^1_{X_K}(\operatorname{log}D) \to \mathcal{H}_{\kappa+2\rho}$ induced from the Kodaira–Spencer isomorphism, we obtain a $\mathbb{Q}_p$ -linear derivation $\nabla$ on $\operatorname{H}^0(P_{\operatorname{dR}, K}, \mathcal{O}_{P_{\operatorname{dR}, K}}) = \bigoplus_{\kappa \in X^*(T)} \operatorname{H}^0(X_K, \mathcal{H}_{\kappa})$ (cf. Proposition A.14(2)).

Proposition 2.9. One has the following commutative diagram.

2.3 The main theorem

We can now introduce the space of nearly overconvergent modular forms and state the main theorem. Let $K_p \subset G(\mathbb{Q}_p)$ be a compact open subgroup, $g \in G(\mathbb{Q}_p)$ and $K_{p, P} = g K_p g^{-1} \cap P(\mathbb{Q}_p)$ as above. Recall that we have a morphism $\mathcal{IG}_{U_{K_{p, P}}K^p, \mathbb{Q}_p^{\operatorname{cycl}}} \to P_{\operatorname{dR}, K, \mathbb{Q}_p^{\operatorname{cycl}}}^{\operatorname{an}}$ .

Definition 2.10. We define the space of nearly overconvergent modular forms to be

\[\mathscr{N}^{\dagger}_{U_{K_{p, P}}} := \operatorname{colim}_U \operatorname{H}^0(U, \mathcal{O}_U ),\]

where the colimit is over all open subsets of $P_{\operatorname{dR}, K, \mathbb{Q}_p^{\operatorname{cycl}}}^{\operatorname{an}}$ which contain the closure of $\mathcal{IG}_{U_{K_{p, P}}K^p, \mathbb{Q}_p^{\operatorname{cycl}}}$ (via the morphism above) with transition maps given by restriction. Similarly, we define the space of overconvergent modular forms to be

\[ \mathscr{M}^{\dagger}_{U_{K_{p, P}}} := \operatorname{colim}_V \operatorname{H}^0(V, \mathcal{O}_V), \]

where the colimit is over all open subsets of $M_{\operatorname{dR}, K, \mathbb{Q}_p^{\operatorname{cycl}}}^{\operatorname{an}} = P_{\operatorname{dR}, K, \mathbb{Q}_p^{\operatorname{cycl}}}^{\operatorname{an}} \times^{\overline{P}^{\operatorname{an}}} T^{\operatorname{an}}$ containing the closure of $\mathcal{IG}_{U_{K_{p, P}}K^p, \mathbb{Q}_p^{\operatorname{cycl}}}$ . These are $\operatorname{LB}$ -spaces of compact type, and one has a natural map $\mathscr{N}^{\dagger}_{U_{K_{p, P}}} \to \mathscr{M}_{U_{K_{p, P}}}$ induced from restriction.

Theorem 2.11. The space $\mathscr{N}^{\dagger}_{U_{K_{p, P}}}$ comes equipped with actions of $U_p$ , $\varphi$ , $S_p$ , $M_{K_{p, P}}$ and of locally analytic functions $C^{\operatorname{la}}(U_{K_{p, P}}(-1)^\vee, \mathbb{Q}_p) \subset C_{\operatorname{cont}}(U_{K_{p, P}}(-1)^\vee, \mathbb{Q}_p)$ which are compatible with the actions on p-adic modular forms via the map $\mathscr{N}^{\dagger}_{U_{K_{p, P}}} \to \mathscr{M}_{U_{K_{p, P}}}$ . Furthermore, it also carries an action of the lower triangular nilpotent Lie algebra $\overline{\mathfrak{n}} \subset \mathfrak{gl}_2$ obtained by differentiating the torsor structure on $P_{\operatorname{dR}, K, \mathbb{Q}_p^{\operatorname{cycl}}}^{\operatorname{an}}$ , providing an ascending filtration $\operatorname{Fil}_r \mathscr{N}^{\dagger}_{U_{K_{p, P}}} := \mathscr{N}^{\dagger}_{U_{K_{p, P}}}[\overline{\mathfrak{n}}^{r+1}]$ (the elements killed by $\overline{\mathfrak{n}}^{r+1}$ ).

The space $\mathscr{N}^{\dagger}_{U_{K_{p, P}}}$ satisfies the following additional properties.

1. (i) The filtration is stable under $U_p$ , $S_p$ , $\varphi$ , $M_{K_{p, P}}$ and $\mathrm{Fil}_0 \mathscr{N}^{\dagger}_{U_{K_{p, P}}}$ is the space of overconvergent modular forms. Furthermore, we have $U_p \circ \varphi = p S_p$ and $S_p$ commutes with both $U_p$ and $\varphi$ .

2. (ii) Let $R_0^+$ be an admissible $\mathbb{Z}_p$ -algebra, $R_0 = R_0^+[1/p]$ and set $\mathcal{W} = \operatorname{Spa}(R_0, R_0^+)$ . Suppose that $\kappa \colon M_{K_{p, P}} \rightarrow R_0^\times$ is a locally analytic character. Then for any $x \in \mathcal{W}(\overline{\mathbb{Q}}_p)$ and $h \in \mathbb{Q}$ , there exists a quasi-compact open affinoid neighbourhood $\Omega = \operatorname{Spa}(R, R^+) \subset \mathcal{W}$ containing x such that $\mathscr{N}^{\dagger}_{U_{K_{p, P}}, \kappa} := \mathrm{Hom}_{M_{K_{p, P}}}(-w_0\kappa, \mathscr{N}^{\dagger}_{U_{K_{p, P}}}\hat{\otimes} R)$ admits a slope $\leq h$ decomposition with respect to the operator $U_p$ . Furthermore, there exists an integer $r \geq 0$ such that

   \[\mathscr{N}^{\dagger, \leq h}_{U_{K_{p, P}}, \kappa} = (\operatorname{Fil}_r \mathscr{N}^{\dagger}_{U_{K_{p, P}}, \kappa} )^{\leq h} .\]

3. (iii) The analogous relations as in Theorem 2.5(3) hold for the actions of $U_p$ , $\varphi$ , $S_p$ , $M_{K_{p, P}}$ and $C^{\operatorname{la}}(U_{K_{p, P}}(-1)^\vee, \mathbb{Q}_p)$ on $\mathscr{N}^{\dagger}_{U_{K_{p, P}}}$ . In addition to this, there is a factorisation

which is compatible with filtrations. The operator $\nabla$ intertwines with the action of the identity function in $C^{\operatorname{la}}(U_{K_{p, P}}(-1)^\vee, \mathbb{Q}_p)$ .

We will prove this theorem when $K_p = \operatorname{GL}_2(\mathbb{Z}_p)$ and $g=1$ in §§5 and 6 using results from §§3 and 4. As explained in the following section, this is sufficient for proving the theorem for general $K_p$ and g.

2.4 Reduction to hyperspecial level

We now explain why it suffices to establish Theorem 2.11 when $K_p =\operatorname{GL}_2(\mathbb{Z}_p)$ and $g=1$ . Recall that we denoted $U_{P'} = ( \begin{smallmatrix} 1 & \ \mathbb{Q}_p(1) \\&\ 1\end{smallmatrix} )$ .

Proof. For the first part, let $U = U_{K_{p, P}} = U_{K'_{p, P}}$ . We have a commutative diagram

where f denotes the map induced by right multiplication by $g^{-1}g'$ , the diagonal map is induced from right translation by g’ and the horizontal map is induced from right translation by g. The map f is finite étale away from the cusps not lying in $\mathcal{IG}_{UK^p}$ , hence it induces a map $\mathscr{N}^{\dagger}_{U_{K'_{p, P}}} \to \mathscr{N}^{\dagger}_{U_{K_{p, P}}}$ .

Furthermore, we can find strict neighbourhoods $V_1$ (respectively, $V_2$ ) of $\mathcal{IG}_{U K^p}$ in $P^{\operatorname{an}}_{\operatorname{dR}, K_p K^p}$ (respectively, $P^{\operatorname{an}}_{\operatorname{dR}, K_p'K^p}$ ) such that $f \colon V_2 \to V_1$ is an isomorphism (cf. [Reference Kisin and LaiKL05, Proposition 2.2.1] – the key point is that we can find a strict neighbourhood V of $f^{-1}(\mathcal{IG}_{UK^p}) \subset P^{\operatorname{an}}_{\operatorname{dR}, K_p'K^p}$ such that the connected components of $f^{-1}(\mathcal{IG}_{UK^p})$ and V are in bijection with one another). This proves part (i).

Part (ii) follows immediately from the fact that the isomorphism $\mathcal{IG}_{U_{K'_{p, P}} K^p} \to \mathcal{IG}_{U_{K_{p, P}} K^p}$ , induced from right translation by t intertwines with the isomorphism $P_{\operatorname{dR}, K_p'K^p} \to P_{\operatorname{dR}, K_p K^p}$ in the statement of the lemma.

When working at hyperspecial level, we will omit any subscript which includes the level subgroup (e.g., $\mathscr{N}^{\dagger}$ , $\mathscr{M}^{\dagger}$ , $\mathscr{M}$ denote the spaces of (nearly) overconvergent and p-adic modular forms etc.). Furthermore, in this setting the morphism from the Igusa tower to $P_{\operatorname{dR}}$ is defined over $\mathbb{Q}_p$ , so it is not necessary to base-change to $\mathbb{Q}_p^{\operatorname{cycl}}$ . In fact, to consider families of nearly overconvergent modular forms, we will work over $\operatorname{Spa}(R, R^+)$ for some admissible $\mathbb{Z}_p$ -algebra $R^+$ .

3.1 The setting

Let $R^+$ be an admissible $\mathbb{Z}_p$ -algebra and set $R = R^+[1/p]$ . Fix a neat compact open subgroup $K^p \subset G(\mathbb{A}_f^p)$ and let $\mathcal{X}/\operatorname{Spa}(R, R^+)$ be the compact modular curve of level $\operatorname{GL}_2(\mathbb{Z}_p)K^p$ over $\operatorname{Spa}(R, R^+)$ . We have (see [Reference KatzKat73]) a system of neighbourhoods of the ordinary locus

\[\mathcal{X}_{\operatorname{ord}} \to \cdots \to \mathcal{X}_r \to \cdots \to \mathcal{X}_1 \to \mathcal{X,}\]

where $\mathcal{X}_r$ denotes the (unique) quasi-compact open whose rank-one points x satisfy $|h|_x \geq p^{-1/p^{r+1}}$ for any local lift h of the Hasse invariant.

For any integer $n \geq 1$ , let $\mathcal{IG}_n := \mathcal{IG}_{P'_n K^p}$ be the Igusa tower of level

\[P'_n = \left( \begin{matrix} 1 + p^n \mathbb{Z}_p & \mathbb{Z}_p(1) \\ & 1+p^n \mathbb{Z}_p \end{matrix} \right)\!,\]

which is a finite $T(\mathbb{Z}/p^n \mathbb{Z})$ -torsor over $\mathcal{X}_{\operatorname{ord}}$ . We also use the notation $\mathcal{IG}_{\infty} = \mathcal{IG}_{U_{P'}(\mathbb{Z}_p)K^p}$ . We define $\mathcal{H}_{\mathcal{E}}$ to be the canonical extension of the first relative de Rham homology of the universal elliptic curve over the good reduction locus in $\mathcal{X}$ . The formal model $\mathfrak{X}$ defines a $\mathcal{O}_{\mathcal{X}}^+$ -lattice $\mathcal{H}_{\mathcal{E}}^+ \subset \mathcal{H}_{\mathcal{E}}$ . We similarly define $\omega_{\mathcal{E}}^+ \subset \omega_{\mathcal{E}}$ and $\omega_{\mathcal{E}^D}^+ \subset \omega_{\mathcal{E}^D}$ for the Hodge bundle of the universal and dual universal generalised elliptic curve and the corresponding $\mathcal{O}_{\mathcal{X}}^+$ -lattices.

If $\operatorname{Spa}(A, A^+) \subset \mathcal{X}$ is a quasi-compact open affinoid subspace, then we let $\operatorname{Spa}(A_r, A_r^+)$ (respectively, $\operatorname{Spa}(A_{\operatorname{ord}}, A_{\operatorname{ord}}^+)$ , $\operatorname{Spa}(A_{\operatorname{ord}, n}, A_{\operatorname{ord}, n}^+)$ ) denote the pullback to $\mathcal{X}_r$ (respectively, $\mathcal{X}_{\operatorname{ord}}$ , $\mathcal{IG}_n$ ). We warn the reader that we will often implicitly assume that $\operatorname{Spa}(A_{\operatorname{ord}}, A_{\operatorname{ord}}^+) \neq \varnothing$ , otherwise most of what we say will be vacuous. Note that if $\omega_{\mathcal{E}}^+$ is trivial over $\operatorname{Spa}(A, A^+)$ and $h \in A^+$ is a fixed local lift of the Hasse invariant, then $A_{\operatorname{ord}}^+ = A^+\langle 1/h \rangle$ . If $\{e, f\}$ is a fixed basis of $\mathcal{H}_{\mathcal{E}}^+$ over $\operatorname{Spa}(A, A^+)$ respecting the Hodge filtration (i.e., $e \in \omega_{\mathcal{E}}^+$ ), then we have

\[P_{\operatorname{dR}}^{\operatorname{an}} \times_{\mathcal{X}} \operatorname{Spa}(A, A^+) \cong \overline{P}^{\operatorname{an}}_{A} = \left( \begin{matrix} (\mathbb{G}_m^{\operatorname{an}})_{A} & \\ (\mathbb{G}_a^{\operatorname{an}})_{A} & (\mathbb{G}_m^{\operatorname{an}})_{A} \end{matrix} \right)\]

and the basis $\{e, f\}$ gives rise to coordinates $T_1, T_2, U \in \mathcal{O}^+(P_{\operatorname{dR}}^{\operatorname{an}} \times_{\mathcal{X}} \operatorname{Spa}(A, A^+))$ such that the universal trivialisation of $\mathcal{H}_{\mathcal{E}}^+$ over $P_{\operatorname{dR}}^{\operatorname{an}} \times_{\mathcal{X}} \operatorname{Spa}(A, A^+)$ takes e (respectively, f) to $(0, T_1)$ (respectively, $(T_2, U)$ ). Equivalently, if $\{ e_{\operatorname{univ}}, f_{\operatorname{univ}} \}$ denotes the universal basis of $\mathcal{H}^+_{\mathcal{E}}$ over $P_{\operatorname{dR}}^{\operatorname{an}} \times_{\mathcal{X}} \operatorname{Spa}(A, A^+)$ (with $e_{\operatorname{univ}} \in \omega_{\mathcal{E}}^+$ ) then we have $e = T_1 e_{\operatorname{univ}}$ and $f = U e_{\operatorname{univ}} + T_2 f_{\operatorname{univ}}$ .

For the rest of this section, we fix an open affinoid $\operatorname{Spa}(A, A^+) \subset \mathcal{X}$ over which $\omega_{\mathcal{E}}^+$ , $\mathcal{H}_{\mathcal{E}}^+$ and $\omega_{\mathcal{E}^D}^+$ are trivial, and fix a basis $\{ e, f \}$ of $\mathcal{H}_{\mathcal{E}}^+$ respecting the Hodge filtration

\[0 \to \omega_{\mathcal{E}^D}^+ \to \mathcal{H}_{\mathcal{E}}^+ \to (\omega_{\mathcal{E}}^+)^{\vee} \to 0 .\]

Recall that over $\mathcal{IG}_n$ , one has universal trivialisations

\[\psi_1 \colon (\mathcal{E}^D[p^{n}]^{\circ})^D \xrightarrow{\sim} \mathbb{Z}/p^n\mathbb{Z}, \quad \psi_2 \colon \mathcal{E}[p^{n}]^{\circ} \xrightarrow{\sim} \mu_{p^{n}},\]

which induce trivialisations

\[\phi_1 \colon \mathcal{O}^+_{\mathcal{IG}_n}/p^n \xrightarrow{\sim} (\omega_{\mathcal{E}^D}^+/p^n)|_{\mathcal{IG}_{n}}, \quad \phi_2 \colon \mathcal{O}^+_{\mathcal{IG}_n}/p^n \xrightarrow{\sim} (\omega_{\mathcal{E}}^+/p^n)^{\vee}|_{\mathcal{IG}_n}.\]

More precisely, $\phi_1(1)$ is the vector $\operatorname{dlog} \circ \; \psi_1^{-1}(1)$ and $\phi_2(1)$ is defined to be the vector dual to $\operatorname{dlog} \circ \; \psi_2^D(1)$ . We set $e_n := \phi_1(1)$ and define $f_n$ to be the image of $\phi_2(1)$ under the unit root splitting $(\omega_{\mathcal{E}}^+/p^n)^{\vee} \hookrightarrow \mathcal{H}_{\mathcal{E}}^+/p^n$ . Then $\{e_n, f_n \}$ defines a basis of $\mathcal{H}_{\mathcal{E}}^+/p^n$ over $\mathcal{IG}_n$ respecting the Hodge filtration.

Definition 3.2. Let $\alpha_n, \gamma_n \in (A_{\operatorname{ord}, n}^+)^{\times}$ and $\beta_n \in A_{\operatorname{ord}, n}^+$ be any elements such that

\[\begin{array}{l}e \equiv \alpha_n e_n, \\[3pt]f \equiv \beta_n e_n + \gamma_n f_n,\end{array} \quad \text{mod } p^n ,\]

and $\alpha_n^{-1} \beta_n \in A_{\operatorname{ord}}^+$ .

Remark 3.3. We can choose $\alpha_n, \beta_n, \gamma_n$ in Definition 3.2 such that the last condition is satisfied as follows. Let $f^{\operatorname{ur}} \in \mathcal{H}_{\mathcal{E}}^+|_{\operatorname{Spa}(A_{\operatorname{ord}}, A_{\operatorname{ord}}^+)}$ denote the image of f under the composition

\[\mathcal{H}_{\mathcal{E}}^+|_{\operatorname{Spa}(A_{\operatorname{ord}}, A_{\operatorname{ord}}^+)} \twoheadrightarrow (\omega_{\mathcal{E}}^+)^{\vee}|_{\operatorname{Spa}(A_{\operatorname{ord}}, A_{\operatorname{ord}}^+)} \hookrightarrow \mathcal{H}_{\mathcal{E}}^+|_{\operatorname{Spa}(A_{\operatorname{ord}}, A_{\operatorname{ord}}^+)},\]

where the first map arises from the Hodge filtration and the second map is the unit root splitting. There exists an element $\upsilon \in A_{\operatorname{ord}}^+$ such that $f^{\operatorname{ur}} = \upsilon e + f$ . We can then take $\alpha_n, \gamma_n \in (A_{\operatorname{ord}, n}^+)^{\times}$ such that $e \equiv \alpha_n e_n$ and $f^{\operatorname{ur}} \equiv \gamma_n f_n$ modulo $p^n$ , and set $\beta_n = -\upsilon \alpha_n$ .

Remark 3.5. The map $\mathcal{IG}_\infty \times_{\mathcal{X}} \operatorname{Spa}(A, A^+) = \operatorname{Spa}(A_{\operatorname{ord}, \infty}, A_{\operatorname{ord}, \infty}^+) \to P_{\operatorname{dR}}^{\operatorname{an}} \times_{\mathcal{X}} \operatorname{Spa}(A, A^+)$ induced from the universal trivialisations and the unit root splitting is described by the point

\[\left( \begin{matrix} \gamma_{\infty} & \\ \beta_{\infty} & \alpha_{\infty} \end{matrix} \right) \in \overline{P}^{\operatorname{an}}(A_{\operatorname{ord}, \infty}),\]

that is, the morphism sending $T_1 \mapsto \alpha_\infty$ , $U \mapsto \beta_{\infty}$ , and $T_2 \mapsto \gamma_{\infty}$ .

3.2 A preliminary lemma

We will first prove an elementary lemma that will be useful later for constructing strict neighbourhoods of the Igusa tower over the ordinary locus. Let $C^+$ be an admissible $\mathbb{Z}_p$ -algebra, $C = C^+[1/p]$ , and let $\mathcal{Y} = \mathbb{G}_m^{\operatorname{an}}$ over $\operatorname{Spa}(C, C^+)$ with coordinate T. Let $n \geq 1$ and suppose that we have a collection of elements $\{ c_{\lambda} \in (C^+)^{\times} : \lambda \in \left(\mathbb{Z}/p^n \mathbb{Z} \right)^{\times} \}$ with the property that $c_{\lambda} \equiv \lambda c_{1}$ modulo $p^nC^+$ for all $\lambda \in \left(\mathbb{Z}/p^n \mathbb{Z}\right)^{\times}$ . Define

\[ P(T) := \prod_{\lambda \in ({\mathbb{Z}} / p^n {\mathbb{Z}})^\times} (T - c_{\lambda}) \in C^+[T] \subset \mathcal{O}_{\mathcal{Y}}^+(\mathcal{Y}), \]

and, for every $0 \leq m \leq n$ , let

\begin{align*}\rho_{n, m} & := p^{-m p^{n - m}} \prod_{ \substack{x \in ({\mathbb{Z}} / p^n {\mathbb{Z}})^\times \\ x \not \equiv 1 \text{ (mod $p^m$)}}} |x - 1|_p, \\\nu_{n, m} & := - v_p(\rho_{n, m}),\end{align*}

where $v_p$ is the p-adic valuation satisfying $v_p(p) = 1$ . Note that $\nu_{n, m} \geq 0$ and $\rho_{n, m} = p^{-\nu_{n, m}} = |p^{\nu_{n, m}}|_p$ .

Proof. We have

\begin{align*} \nu_{n, m} &= m p^{n - m} + \sum_{k = 1}^{m - 1} k \cdot \, \# \{ 1+p^k a \, : \, 0 \leq a < p^{n-k}, (a, p) = 1\} \\&= m p^{n - m} + \sum_{k = 1}^{m - 1} k (p-1) p^{n - (k+1)}.\end{align*}

A simple computation shows that $\nu_{n, m+1} - \nu_{n, m} = p^{n - (m+1)}$ as required.

The following lemma relates estimates of the norm of P(T) to the norm of its linear factors.

Lemma 3.7. For any point $x \in \mathcal{Y}$ with associated valuation $| \cdot |_x$ , one has

\[|P(T)|_x \leq |p^{\nu_{n, m}}|_x \Longleftrightarrow \exists \lambda\in ({\mathbb{Z}} / p^n {\mathbb{Z}})^\times \quad \text{such that } |T - c_{\lambda} |_x \leq |p^m|_x.\]

Suppose that there exists an element $\mu \in \left(\mathbb{Z}/p^n\mathbb{Z} \right)^{\times}$ such that $|p^{i+1}|_x < |T - c_{\mu}|_x< |p^{i}|_x$ for some integer $i \geq 0$ . Then $|T - c_{\mu}|_x$ is never equal to $|c_{\mu} - c_{\lambda}|_x = |\mu - \lambda|_x$ for any $\lambda \neq \mu$ . Let $J_{\mu, i+1} \subset \left(\mathbb{Z}/p^n \mathbb{Z} \right)^{\times}$ be the subset of all elements $\lambda$ such that $\lambda \equiv \mu$ modulo $p^{i+1}$ , and let $J_{\mu, i+1}^c$ denote its complement. Then we have

\begin{align*}|P(T)|_x &= \prod_{\lambda \in J_{\mu, i+1}} |T-c_{\mu}|_x \cdot\prod_{\lambda \in J_{\mu, i+1}^c} | \lambda - \mu |_x\\&= \left\{\begin{array}{l@{\quad}l} |T- c_{\mu}|_x^{p^{n-(i+1)}}|p^{-(i+1)p^{n-(i+1)}}|_x |p^{\nu_{n, i+1}}|_x & \text{if } 0\leq i <n, \\[3pt] |T-c_{\mu}|_x |p^{-n}|_x |p^{\nu_{n, n}}|_x & \text{if } i \geq n. \end{array} \right.\end{align*}

We now return to the proof of the lemma. Suppose that there exists $\lambda \in \left(\mathbb{Z}/p^n \mathbb{Z} \right)^{\times}$ such that $|T - c_{\lambda}|_x \leq |p^{m}|_x$ . Then we must have that

\[|P(T)|_x \leq \prod_{\lambda' \in J_{\lambda, m}} |p^{m}|_x \cdot \prod_{\lambda' \in J_{\lambda, m}^c} |c_{\lambda'} - c_{\lambda}|_x = |p^{\nu_{n, m}}|_x .\]

Conversely, suppose that $|P(T)|_x \leq |p^{\nu_{n, m}}|_x$ . Suppose there exists $\mu \in \left(\mathbb{Z}/p^n \mathbb{Z} \right)^{\times}$ such that $|p^{i+1}|_x < |T - c_{\mu}|_x < |p^{i}|_x$ for some integer $i \geq 0$ . We may assume $i \leq m-1$ otherwise there is nothing to prove. Then by the above calculation, we find that

\[|P(T)|_x = |T - c_{\mu}|_x^{p^{n-(i+1)}} |p^{-(i+1)p^{n-(i+1)}}|_x |p^{\nu_{n, i+1}}|_x \leq |p^{\nu_{n, m}}|_x.\]

But since $|T - c_{\mu}|_x > |p^{i+1}|_x$ , this implies that $\nu_{n, i+1} > \nu_{n, m}$ for some $i \leq m-1$ . This contradicts Lemma 3.6.

Finally, we suppose that for every $\mu$ , there exists an integer $i \geq 0$ such that $|T - c_{\mu}|_x = |p^{i}|_x$ , and assume that $|P(T)|_x \leq |p^{\nu_{n, m}}|_x$ . Let $j = \operatorname{max}\{ i : |T - c_{\lambda}|_x = |p^{i}|_x \text{ for some } \lambda \}$ , and we assume $0 \leq j \leq m-1$ , otherwise there is nothing to prove. Let $\mu$ be such that $|T - c_{\mu}|_x = |p^{j}|_x$ . Then we see that

\[|P(T)|_x = \prod_{\lambda \in J_{\mu, j}} |T - c_{\lambda}|_x \cdot \prod_{\lambda \in J_{\mu, j}^c} |c_{\lambda} - c_{\mu}|_x \geq |p^{jp^{n-j}}|_x \prod_{\substack{y \in \left(\mathbb{Z}/p^n \mathbb{Z}\right)^{\times} \\ y \not\equiv 1 \pmod{{p^j}}}} | y - 1 |_x = |p^{\nu_{n, j}}|_x .\]

This implies that $\nu_{n, j} \geq \nu_{n, m}$ , which contradicts Lemma 3.6 because $j \leq m-1$ .

We note the following corollary.

Corollary 3.8. Let $1 \leq m \leq n$ and suppose that $\{ d_{\mu} \in (C^+)^{\times} : \mu \in \left(\mathbb{Z}/p^m \mathbb{Z} \right)^{\times} \}$ is a collection of elements such that $d_{\mu} \equiv c_{\lambda}$ modulo $p^m C^+$ whenever $\mu \equiv \lambda$ modulo $p^m$ . Set $Q(T) = \prod_{\mu \in \left(\mathbb{Z}/p^m \mathbb{Z}\right)^{\times}} (T - d_{\mu})$ . Then for any point $x \in \mathcal{Y}$ , one has

\[|P(T)|_x \leq |p^{\nu_{n, m}}|_x \; \Longleftrightarrow \; |Q(T)|_x \leq |p^{\nu_{m, m}}|_x.\]

Proof. By Lemma 3.7, it is enough to show that

\[\exists \lambda \in \left(\mathbb{Z}/p^n \mathbb{Z} \right)^{\times} \quad \text{such that } |T - c_{\lambda} |_x \leq |p^{m}|_x \Longleftrightarrow \exists \mu \in \left(\mathbb{Z}/p^m \mathbb{Z} \right)^{\times} \quad \text{such that } |T - d_{\mu} |_x \leq |p^{m}|_x .\]

But this holds by construction.

3.3 Ordinary neighbourhoods

Fix a quasi-compact open affinoid $\operatorname{Spa}(A, A^+) \subset \mathcal{X}$ as above. We now define the strict (local) neighbourhoods of the Igusa tower over the ordinary locus. For any integer $n \geq 1$ , we define:

• – $P_{\alpha_n}(T_1) = \prod_{\lambda = (\lambda_1, 1) \in T(\mathbb{Z}/p^n \mathbb{Z})} (T_1 - \sigma_{\lambda}(\alpha_n)) \; \in \; A_{\operatorname{ord}}^+[T_1]$ ;

• – $P_{\gamma_n}(T_2) = \prod_{\lambda = (1, \lambda_2) \in T(\mathbb{Z}/p^n \mathbb{Z})} (T_2 - \sigma_{\lambda}(\gamma_n)) \; \in \; A_{\operatorname{ord}}^+[T_2]$ ;

• – $Q_n(T_1, U) = U - \alpha_n^{-1}\beta_n T_1 \; \in \; A_{\operatorname{ord}}^+[T_1, U]$ .

We view all of these polynomials as global sections of $\mathcal{O}^+_{P_{\operatorname{dR}, A_{\operatorname{ord}}}^{\operatorname{an}}}$ via the coordinates $T_1, T_2, U$ arising from the fixed basis $\{ e, f \}$ as above.

Definition 3.9. For $1 \leq m \leq n$ , let $\mathcal{U}_{\operatorname{HT}, n, m} \subset P_{\operatorname{dR}, A_{\operatorname{ord}}}^{\operatorname{an}}$ be the quasi-compact open affinoid whose points $| \cdot |_x$ satisfy the inequalities

\[|P_{\alpha_n}(T_1)|_x \leq |p^{\nu_{n, m}}|_x, \quad |P_{\gamma_n}(T_2)|_x \leq |p^{\nu_{n, m}}|_x, \quad |Q_n(T_1, U) |_x \leq |p^{m}|_x.\]

For any element $x \in\overline{P}^{\operatorname{an}}(A_{\operatorname{ord}, n})$ and integer $m \leq n$ , we let $\mathcal{B}_n(x, p^{-m}) \subset\overline{P}^{\operatorname{an}}_{A_{\operatorname{ord}, n}} \cong P_{\operatorname{dR}, A_{\operatorname{ord}, n}}^{\operatorname{an}}$ denote the rigid ball of radius $p^{-m}$ , that is, if we have $x =\big( \begin{smallmatrix} x_2 & \\ y & \ x_1 \end{smallmatrix}\big)$ , then it is the quasi-compact open affinoid subspace defined by the conditions $|T_1 - x_1| \leq |p^{m}|$ , $|U - y| \leq |p^{m}|$ , and $|T_2 - x_2| \leq |p^{m}|$ .

Lemma 3.10. Let $g_n = \big( \begin{smallmatrix} \gamma_n & \\ \beta_n &\\alpha_n \end{smallmatrix} \big)$ . Then

\[ \mathcal{U}_{\operatorname{HT}, n, m} \times_{\mathcal{X}_{\operatorname{ord}}} \mathcal{IG}_n = \bigsqcup_{\lambda \in T({\mathbb{Z}} / p^m {\mathbb{Z}})} \mathcal{B}_n(\sigma_\lambda(g_n), p^{-m}) = \bigsqcup_{\lambda \in T({\mathbb{Z}} / p^m {\mathbb{Z}})} \sigma_\lambda(g_n) \cdot \mathcal{B}_n(1, p^{-m}), \]

where, by a slight abuse of notation, $\sigma_\lambda$ denotes any lift of $\sigma_\lambda \in \mathscr{G}_m$ to an element of $\mathscr{G}_n$ . In particular, we have $\mathcal{U}_{\operatorname{HT}, n} := \mathcal{U}_{\operatorname{HT}, n, n} = \mathcal{U}_{\operatorname{HT}, n, m}$ for any $1 \leq m \leq n$ .

Proof. It suffices to prove this after base-change along the map $\operatorname{Spa}(A_{\operatorname{ord}, \infty}, A_{\operatorname{ord}, \infty}^+) \to \operatorname{Spa}(A_{\operatorname{ord}}, A_{\operatorname{ord}}^+)$ , because this morphism is profinite proétale and hence closed. But

\[\mathcal{U}_{\operatorname{HT}, n} \times_{\operatorname{Spa}(A_{\operatorname{ord}}, A_{\operatorname{ord}}^+)} \operatorname{Spa}(A_{\operatorname{ord}, \infty}, A_{\operatorname{ord}, \infty}^+) = \bigsqcup_{\lambda \in T(\mathbb{Z}/p^n\mathbb{Z})} \mathcal{B}_{\infty}(\sigma_{\lambda}(g_{\infty}), p^{-n}),\]

where $\sigma_{\lambda}$ denotes a lift to $\mathscr{G}_{\infty}$ (here we are using the fact that $g_n \equiv g_{\infty}$ modulo $p^n$ ). These are clearly cofinal.

3.4 Overconvergent neighbourhoods

Fix a quasi-compact open affinoid $\operatorname{Spa}(A, A^+) \subset \mathcal{X}$ over which $\mathcal{H}_{\mathcal{E}}^+$ , $\omega_{\mathcal{E}}^+$ and $\omega_{\mathcal{E}^D}^+$ trivialise. We now define a cofinal system of strict neighbourhoods of $\mathcal{IG}_{\infty, A}$ inside $P_{\operatorname{dR}, A}^{\operatorname{an}}$ .

Proof. Let $n \geq 1$ . Then $\mathcal{U}_{\operatorname{HT}, n+1} = \mathcal{U}_{\operatorname{HT}, n+1, n+1}$ (respectively, $\mathcal{U}_{\operatorname{HT}, n} = \mathcal{U}_{\operatorname{HT}, n+1, n}$ ) is described by the inequalities

\begin{gather*}|P_{\alpha_{n+1}}(T_1)| \leq |p^{\nu_{n+1, n+1}}|, \quad |P_{\gamma_{n+1}}(T_2)| \leq |p^{\nu_{n+1, n+1}}|, \quad |Q_{n+1}(T_1, U)| \leq |p^{n+1}| \\\text{(respectively, } |P_{\alpha_{n+1}}(T_1)| \leq |p^{\nu_{n+1, n}}|, \quad |P_{\gamma_{n+1}}(T_2)| \leq |p^{\nu_{n+1, n}}|, \quad |Q_{n+1}(T_1, U)| \leq |p^{n}| ).\end{gather*}

Since $A_{\operatorname{ord}}^+ = A^+\langle 1/h \rangle$ , there exists a sufficiently large integer $N \geq 0$ , and elements $P_{\alpha_{n+1}}'(T_1) \in A^+[T_1]$ , $P_{\gamma_{n+1}}'(T_2) \in A^+[T_2]$ , $Q_{n+1}'(T_1, U) \in A^+[T_1, U]$ such that

\[P'_{\alpha_{n+1}}(T_1) \equiv h^N P_{\alpha_{n+1}}(T_1), \quad P'_{\gamma_{n+1}}(T_2) \equiv h^N P_{\gamma_{n+1}}(T_2), \quad Q_{n+1}'(T_1, U) \equiv h^N Q_{n+1}(T_1, U),\]

modulo $p^{t}A_{\operatorname{ord}}^+$ , for some fixed integer t such that $p^{-t} \leq \rho_{n+1, n+1}$ .

Define $\mathcal{V}$ (respectively, $\mathcal{U}$ ) to be the rational subset of $P_{\operatorname{dR}, A}^{\operatorname{an}}$ defined by the inequalities

\begin{gather*} |P'_{\alpha_{n+1}}(T_1)| \leq |p^{\nu_{n+1, n+1}}|, \quad |P'_{\gamma_{n+1}}(T_2)| \leq |p^{\nu_{n+1, n+1}}|, \quad |Q'_{n+1}(T_1, U)| \leq |p^{n+1}| \\ \text{(respectively, } |P'_{\alpha_{n+1}}(T_1)| \leq |p^{\nu_{n+1, n}}|, \quad |P'_{\gamma_{n+1}}(T_2)| \leq |p^{\nu_{n+1, n}}|, \quad |Q'_{n+1}(T_1, U)| \leq |p^{n}| ).\end{gather*}

Then $\mathcal{IG}_{\infty, A} \subset \mathcal{V} \subset \mathcal{U}$ . Since $|p^{\nu_{n+1, n+1}}| < |p^{\nu_{n+1, n}}|$ (Lemma 3.6) and $|p^{n+1}| < |p^{n}|$ , we see that $\overline{\mathcal{V}} \subset \mathcal{U}$ , where the closure is in $P_{\operatorname{dR}, A}^{\operatorname{an}}$ . This implies that $\mathcal{U}$ is an overconvergent extension of $\mathcal{U}_{\operatorname{HT}, n}$ . The rest of the proposition follows from a standard compactness argument, and the fact that $\{\mathcal{U}_{\operatorname{HT}, n}: n \geq 1\}$ are cofinal over the ordinary locus.

Let $\mathcal{U}$ be an overconvergent extension of $\mathcal{U}_{\operatorname{HT}, n}$ for some integer $n \geq 1$ . Then we have a chain of quasi-compact open affinoid neighbourhoods

\[\mathcal{U}_1 \supset \mathcal{U}_2 \supset \cdots \supset\mathcal{U}_r \supset \cdots \supset \mathcal{U}_{\infty} := \mathcal{U}_{\operatorname{HT}, n},\]

such that $\mathcal{U}_{\infty}$ is the locus in $\mathcal{U}_r$ ( $:= \mathcal{U} \cap P_{\operatorname{dR}, A_r}^{\operatorname{an}}$ ) where the Hasse invariant $h \in \mathcal{O}^+(\mathcal{U}_r)$ is invertible, for any $r \geq 1$ . On sections, this chain of inclusions is induced by maps

\[B_1^+ \to B_2^+ \to \cdots \to B_{\infty}^+,\]

where $B_r^+ = \mathcal{O}^+(\mathcal{U}_r)$ . Each $B_r := \mathcal{O}(\mathcal{U}_r)$ is a Banach space – denote the norm for which $B_r^+$ is the unit ball by $|\!| \cdot |\!|_r$ . The key result of this section is to show that these morphisms are very close to being isometries, with the failure to be an isometry measured by powers of the Hasse invariant (whose valuation can be made arbitrarily small). This will allow us to overconverge the Gauss–Manin connection to p-adic powers.

Proposition 3.14. Let $r \geq 1$ . With notation as above, there exists an integer $M \geq 0$ (depending on r) such that for any $k \geq 1$ , the kernel of the map

\[B^+_{r}/p^k \to B^+_{\infty}/p^k\]

is killed by $h^{kM}$ .

This proves the proposition in the case $k=1$ . The general case now follows from a simple induction argument using the fact that $B_{\infty}^+$ is p-torsion-free.

The following corollary is a generalisation of [Reference Andreatta and IovitaAI21, Proposition 4.10] and will be key for the proof of our main result.

Corollary 3.15. With notation as above, for any real number $0 < \delta < 1$ there exists an integer $s = s(\delta) \geq r$ such that the following assertion holds: for all $k \geq 1$ , $x \in B_{r}$ and $c \in \mathbb{Q}$ , one has

\[|\!| x |\!|_{\infty} \leq p^{c-k} \quad\text{ and }\quad |\!| x |\!|_{r} \leq p^c \Rightarrow |\!| x |\!|_{s} \leq p^{c-\delta k} .\]

Proof. By raising x to an integral power, it is enough to prove the statement for $c \in \mathbb{Z}$ , and by rescaling it is enough to prove the statement for $c=0$ . Therefore, we have an element $x \in B_{r}^+$ whose image is in $p^k B_{\infty}^+$ . By Proposition 3.14, there exists an $M \geq 0$ such that $|\!| (p^{-1} h^M)^k x | \!|_s \leq |\!| (p^{-1} h^M)^k x | \!|_r \leq 1$ for any $s \geq r$ and any $k \geq 1$ . Since $|\!|h^{-1}|\!|_s \to 1^+$ as $s \to +\infty$ , taking s large enough such that $|\!|h^{-1}|\!|_s \leq p^{{(1 - \delta)}/{M}}$ , we obtain

\[|\!|x|\!|_s \leq |\!| (p h^{-M})^k |\!|_s \leq p^{-\delta k}\]

as required.

4.1 Function spaces

Let $R^+$ be an admissible ${{\mathbb{Z}}_p}$ -algebra and set $R = R^+[1/p]$ . Let $C_{\operatorname{cont}}({{\mathbb{Z}}_p}, R)$ , $C^{h\operatorname{-an}}({{\mathbb{Z}}_p}, R)$ and $C^{\operatorname{la}}({{\mathbb{Z}}_p}, R)$ denote the R-algebras of continuous, analytic of radius $p^{-h}$ ( $h \in {\mathbb{N}}$ ), and locally analytic functions from ${{\mathbb{Z}}_p}$ to R, respectively. We also set $C_{\operatorname{cont}}({{\mathbb{Z}}_p}, R^+)$ to be the $R^+$ -algebra of continuous functions from $\mathbb{Z}_p$ to $R^+$ . We recall the following classical result.

For any $\varepsilon > 0$ we will denote by $C_\varepsilon(\mathbb{Z}_p, R)$ the subspace of functions $f = \sum_{k \in {\mathbb{N}}} a_k \binom{x}{k}$ with $p^{k \varepsilon} |a_k| \to 0$ as $k \to +\infty$ . Observe that $C^{\operatorname{la}}({{\mathbb{Z}}_p}, R) = \varinjlim_{h \to +\infty} C^{h\operatorname{-an}}({{\mathbb{Z}}_p}, R) = \varinjlim_{\varepsilon > 0} C_\varepsilon({{\mathbb{Z}}_p}, R)$ .

4.2 Continuous and analytic actions

We begin with some elementary calculations. Let $R^+$ be an admissible ${\mathbb{Z}}_p$ -algebra, $R = R^+[1/p]$ , and let V be a ${\mathbb{Q}_p}$ -Banach vector space equipped with a topological R-module structure.

Definition 4.3. Let $C = C_{\operatorname{cont}}({{\mathbb{Z}}_p}, R)$ or $C_\varepsilon({{\mathbb{Z}}_p}, R)$ or $C^{\operatorname{la}}({{\mathbb{Z}}_p}, R)$ and let W be a topological R-module. If $T \in \mathrm{End}_R(W)$ is a continuous operator, then we say that T extends to a continuous (respectively, $\varepsilon$ -analytic, locally analytic) action if there exists a continuous R-algebra action of $C_{\operatorname{cont}}({{\mathbb{Z}}_p}, R)$ (respectively, $C_\varepsilon(\mathbb{Z}_p, R)$ , $C^{\operatorname{la}}({{\mathbb{Z}}_p}, R)$ ), that is, a continuous R-bilinear map

\[ C \times W \to W \]

(where the left-hand side is equipped with the product topology) that respects the algebra structure on C given by multiplication of functions, such that the structural map ${{\mathbb{Z}}_p} \to R$ acts as T.

We will also need the following uniqueness property for $\varepsilon$ -analytic/locally analytic actions on $\operatorname{LB}$ -spaces with injective transition maps.

Proof. In this proof we will continually use the fact that if X is a $\mathbb{Q}_p$ -Banach space and $Y = \varinjlim_{i \in I}Y_i$ is a countable filtered colimit of $\mathbb{Q}_p$ -Banach spaces (with the direct limit topology) where all the transition maps are injective, then any continuous $\mathbb{Q}_p$ -linear map

\[U \colon X \to \varinjlim_{i \in I} Y_i = Y,\]

factors through $Y_i$ for some $i \in I$ (see [Reference SchneiderSch02, Corollary 8.9]). Note that $C_{\varepsilon}(\mathbb{Z}_p, R)$ is a $\mathbb{Q}_p$ -Banach space, and $C^{\operatorname{la}}(\mathbb{Z}_p, R) = \varinjlim_{\varepsilon > 0} C_{\varepsilon}(\mathbb{Z}_p, R)$ with the direct limit topology.

For part (i), let $f \in C_{\varepsilon}(\mathbb{Z}_p, R)$ and $v \in V_j$ . Then f can be written as the limit of $\{f_k\}_{k \geq 1}$ , where $f_k \in C(\mathbb{Z}_p, R)^{\operatorname{pol}}$ . The map

\[C_{\varepsilon}(\mathbb{Z}_p, R) \to V, \quad g \mapsto g \cdot v,\]

is continuous (by assumption), so by the above, factors though some $V_i$ . Since $V_i$ is Hausdorff, we must have that $f \cdot v$ is the (unique) limit of $f_k \cdot v$ , which are already determined.

Part (ii) is similar, using the fact that the action of polynomial functions are already determined by T (because it is an algebra action). Part (ii) just follows from part (ii).

The following useful result explains how one can perturb a locally analytic action to produce an $\varepsilon$ -analytic action for some appropriately chosen $\varepsilon$ .

Proof. Let $N \geq 0$ and denote $T = T_1 + p^N T_2$ . Let $f_k(X)$ denote the polynomial

\[ f_k(X) = \frac{X (X-1) \cdots (X - k + 1)}{k!}. \]

Let $\Lambda$ denote the set of all ordered tuples $I = (-1 = t_0 \leq t_1 < t_2 < \cdots < t_{2r-1} \leq t_{2r} = k-1)$ with $r \geq 1$ . For such an $I \in \Lambda$ and any $1 \leq i \leq r$ , let $k_i = t_{2i-1} - t_{2i-2}$ and $\ell_i = t_{2i} - t_{2i - 1}$ , and set

\[ z_I = f_{k_1}(T_1 - (t_0 + 1)) \cdot T_2^{\ell_1} \cdot f_{k_2}(T_1 - (t_2 + 1)) \cdot T_2^{\ell_2} \cdot \ldots \cdot f_{k_r}(T_1 - (t_{2r} + 1)) \cdot T_2^{\ell_r} .\]

Then we can write

(4.1) \begin{equation} f_k(T) = f_k(T_1) + \sum_{I \in \Lambda, b \neq 0} \frac{p^{N b}}{b!}\bigg(\begin{array}{c}{k}\\[3pt]{a}\end{array}\bigg)^{\!{-}1} \bigg(\begin{array}{c}{a}\\[3pt]{k_1, \ldots, k_r}\end{array}\bigg)^{\!{-}1} z_I,\end{equation}

where we have denoted $a = \sum_{i=1}^r k_i$ and $b = \sum_{i=1}^r \ell_i$ so that $a + b = k$ .

We now give a bound for the operators $z_I$ . Note that for any integer $t \in {\mathbb{Z}}$ , $f_k(X - t)$ is also a polynomial in X and the Mahler expansion of $f_k(X - t)$ is of the form

\[ f_k(X - t) = \sum_{i = 0}^k a_i f_i(X) \]

for $a_i \in {\mathbb{Z}}_p$ because $f_k(x - t) \in {{\mathbb{Z}}_p}$ for any $x \in {{\mathbb{Z}}_p}$ and the coefficients are just computed using the discrete difference operator. This implies that, if $\varepsilon > 0$ and $C \in \mathbb{R}_{> 0}$ is such that

\[ p^{-k \varepsilon / 2} |\!| f_k(T_1) |\!| \leq C \]

for all $k \geq 0$ , then, for any $t \in {\mathbb{Z}}$ and $k \geq 0$ , we have

\[p^{-k \varepsilon / 2} |\!|f_k(T_1 - t)|\!| \leq C.\]

Hence, by the fact that $|\!|T_2|\!| \leq 1$ , we deduce

(4.2) \begin{equation} p^{- a \varepsilon / 2} |\!|z_I|\!| \leq C^{r}.\end{equation}

Using that for any multinomial coefficient we have $v_p\big( \binom{n}{a_1, \ldots, a_s}^{-1} \big) \geq - \log_p(n)$ , we get from (4.1) and (4.2) that

\[ p^{-k \varepsilon} |\!|f_k(T) |\!| \leq \mathrm{max}(C, p^{-k \varepsilon} p^{-Nb + b/(p-1)} p^{2 \log_p(k)} p^{(k - b) \varepsilon / 2} C^r ). \]

Since $b \geq r - 1$ , taking N such that $p^{-N + 1/(p-1)} \leq C^{-1}$ we deduce that $p^{-Nb + b/(p-1)} C^r \leq C$ and the right-hand side is bounded above by

\[ p^{-k \varepsilon / 2 + 2 \log_p(k)} C, \]

which is uniformly bounded in k as $\varepsilon > 0$ . This concludes the proof.

4.3 Nilpotent operators

Let $R^+ \to S^+$ be a morphism of p-adically complete and separated, p-torsion-free ${{\mathbb{Z}}_p}$ -algebras with $R^+$ admissible, and denote by $R = R^+[1/p], S = S^+[1/p]$ their associated ${\mathbb{Q}_p}$ -Banach algebras.

We equip $V_0$ with the norm induced by the lattice $L_0$ , or equivalently the norm given by $|\!| \sum_{a,b, c} s_{a, b, c} X^a Y^b Z^c |\!| = \max_{a,b, c} |\!|s_{a,b, c}|\!|$ , where the norm on the right-hand side is the ${\mathbb{Q}_p}$ -Banach norm of S. This allows us to view $V_0$ as a Banach algebra with unit ball $L_0$ . Analogously, for $m \geq 1$ , we can view $V_m$ as a Banach algebra with the norm induced by the lattice $L_m$ , or equivalently defined by $|\!| \sum_{a,b, c} s_{a, b, c} (X-1)^a Y^b (Z-1)^c |\!|_m = \max_{a,b,c} p^{-m(a + b + c)} |\!|s_{a,b, c}|\!|$ .

Let $\theta: S^+ \to S^+$ be an $R^+$ -linear derivation, which can naturally be viewed as an $S^+$ -linear functional $\Omega^1_{S^+ / R^+} \to S^+$ . We define $\nabla_\theta: V_m \to V_m$ to be the unique R-linear derivation such that it acts as $\theta$ on S and satisfies

\[ \nabla_\theta(X) = Y, \quad \nabla_\theta(Y) = \nabla_\theta(Z) = 0. \]

In particular, we have a commutative diagram

where the vertical maps are induced by sending $X \mapsto 1$ , $Y \mapsto 0$ , $Z \mapsto 1$ .

We will show next that, whenever $\theta : S^+ \to S^+$ extends to a continuous action, the action of $C^{\operatorname{la}}({{\mathbb{Z}}_p}, R) \subseteq C_{\operatorname{cont}}({{\mathbb{Z}}_p}, R)$ on S extends to $V_m$ for any $m \geq 0$ . We begin with some elementary lemmas for which we introduce some notation. For any $k \in {\mathbb{N}}_{>0}$ and $0 \leq r \leq k$ , let $\Sigma_{k, r}$ be the set of subsets of $\{0, \ldots, k-1\}$ of size r. If $I \in \Sigma_{k, r}$ , let $k_1, \ldots, k_\ell$ (with $\ell$ and the $k_i$ depending on I) be the lengths of the largest blocks of consecutive integers in I, so that $\sum_{i=1}^\ell k_i = r$ . More precisely, I can be written as

\[I = \bigcup_{1 \leq j \leq \ell} [i_j, i_j + k_j - 1] = \bigcup_{1 \leq j \leq \ell} I_j,\]

with no adjacent intervals, that is, $i_j > i_{j-1} + k_{j-1}$ for all $2 \leq j \leq \ell$ . For any $I \in \Sigma_{k, r}$ , we denote

\[g_I(X) = \prod_{i \in I} (X - i), \quad f_I(X) = \prod_{1 \leq j \leq \ell} \frac{1}{k_j!} \prod_{i \in I_j} (X - i),\]

with the convention that $g_{\emptyset} = f_{\emptyset} = 1$ . The following easy lemma will be very useful.

Lemma 4.8. We have

\[ f_k(\nabla_\theta)(s X^a Y^b Z^c) = \sum_{r = 0}^{\min(k, a)} \sum_{I \in \Sigma_{k, k-r}} \bigg(\begin{array}{c}{k - r}\\[3pt]{k_1, \ldots, k_\ell}\end{array}{\kern-6pt}\bigg)^{\!{-}1} \bigg({\kern-5pt}\begin{array}{c}{k}\\[3pt]{r}\end{array}{\kern-6pt}\bigg)^{\!{-}1} \bigg({\kern-5pt}\begin{array}{c}{a}\\[3pt]{r}\end{array}{\kern-6pt}\bigg) f_I(\theta)(s) X^{a-r} Y^{b+r} Z^c. \]

Proof. First observe that, since everything is linear with respect to the variable Z, we may assume that $c = 0$ . We have

\[ \bigg({\kern-5pt}\begin{array}{c}{k - r}\\[3pt]{k_1, \ldots, k_\ell}\end{array}{\kern-6pt}\bigg)^{\!{-}1} \bigg({\kern-5pt}\begin{array}{c}{k}\\[3pt]{r}\end{array}{\kern-6pt}\bigg)^{\!{-}1} \bigg({\kern-5pt}\begin{array}{c}{a}\\[3pt]{r}\end{array}{\kern-6pt}\bigg) f_I(\theta)(s) = \frac{a (a - 1) \cdots (a - r + 1)}{k!} g_I(\theta)(s), \]

so that we need to prove the formula

\[f_k(\nabla_\theta)(s X^a Y^b) = \sum_{r = 0}^{\min(k, a)} \sum_{I \in \Sigma_{k, k-r}} \frac{a (a - 1) \cdots (a - r + 1)}{k!} g_I(\theta)(s) X^{a-r} Y^{b+r}. \]

We check the formula by induction. For $k=1$ we need to prove that

\[ f_k(\nabla_\theta)(s X^a Y^b) = \theta(s) X^a Y^b + a s X^{a-1} Y^{b + 1}, \]

which follows from Leibniz rule. Assume the result holds for k. We calculate

\begin{eqnarray*}f_{k+1}(\nabla_\theta)(s X^a Y^b) &=& \frac{(\nabla_\theta - k)}{k+1} f_k(\nabla_\theta)(s X^a Y^b) \\&=& (\nabla_\theta - k) \sum_{r = 0}^{\min(k, a)} \sum_{I \in \Sigma_{k, k-r}} \frac{a (a - 1) \cdots (a - r + 1)}{(k+1)!} g_I(\theta)(s) X^{a-r} Y^{b+r}.\end{eqnarray*}

We have

\[ (\nabla_\theta - k) g_I(\theta)(s) X^{a - r} Y^{b + r} = (\theta - k) g_I(\theta)(s) X^{a - r} Y^{b + r} + g_I(\theta)(s) (a - r) X^{a - (r + 1)} Y^{b + (r + 1)}, \]

and observe that $(\theta - k) g_I(X) = g_{I \cup \{ k \}}(X)$ . The result follows by decomposing $\Sigma_{k+1, k-r}$ as those subsets containing k and those not containing k.

The following corollary implies that, if the extension of $\nabla_\theta$ to $V_m$ extends to an $\varepsilon$ -analytic action, then the same holds for any $m' \geq m$ .

Corollary 4.9. Let $m' \geq m$ , $s \in S$ , $a,b \in {\mathbb{N}}$ and $k \geq 0$ . Then

\[ \bigg|\! \bigg| f_k(\nabla_\theta) \bigg(s \left(\frac{X -1}{p^m}\right)^{\!a} \left( \frac{Y}{p^m} \right)^{\!b} \left( \frac{Z -1}{p^m}\right)^{\!c} \bigg)\! \bigg| \! \bigg|_{m} = \bigg| \!\bigg| f_k(\nabla_\theta) \bigg(s \left(\frac{X -1}{p^{m'}}\right)^{\!a} \left( \frac{Y}{p^{m'}} \right)^{\!b} \left(\frac{Z - 1}{p^{m'}}\right)^{\!c} \bigg)\! \bigg|\! \bigg|_{m'}. \]

In particular,

\[ \sup_{x \in L_{m}} |\! | f_k(\nabla_\theta) (x)) |\!|_m = \sup_{x \in L_{m'}} |\!| f_k(\nabla_\theta) (x))|\!|_{m'},\]

for all $m' \geq m$ .

Proof. The first assertion follows immediately from Lemma 4.8. Indeed, calling

\[ \alpha_r(s) = \sum_{I \in \Sigma_{k, k-r}} \bigg({\kern-5pt}\begin{array}{c}{k - r}\\[3pt]{k_1, \ldots, k_\ell}\end{array}{\kern-6pt}\bigg)^{\!{-}1} \bigg({\kern-5pt}\begin{array}{c}{k}\\[3pt]{r}\end{array}{\kern-6pt}\bigg)^{\!{-}1} \bigg({\kern-5pt}\begin{array}{c}{a}\\[3pt]{r}\end{array}{\kern-6pt}\bigg) f_I(\theta)(s), \]

which is independent of m or m’ we have, for all $m' \geq m$ ,

\begin{align*}\bigg|\! \bigg| f_k(\nabla_\theta) \bigg(\!s \left(\!\frac{X -1}{p^{m'}}\right)^{\!a} \!\left( \frac{Y}{p^{m'}} \right)^{\!b}\!\left( \!\frac{Z - 1}{p^{m'}}\right)^{\!c} \bigg) \bigg| \!\bigg|_{m'} &= \bigg|\!\bigg| \sum_{r = 0}^{\min(k, a)}\!\alpha_{r}(s) \!\left(\frac{X - 1}{p^{m'}}\right)^{\!a-r}\!\! \left(\frac{Y}{p^{m'}} \right)^{\!b+r} \!\!\left( \frac{Z -1}{p^{m'}}\!\right)^{\!c} \!\bigg|\! \bigg|_{m'} \\&= \sup_{r} |\!|\alpha_r(s)|\!|,\end{align*}

which is independent of m’. The last assertion follows from the first one. This finishes the proof.

Proof. By Corollary 4.9 and a change of coordinates, we can assume $m = 0$ . By Lemma 4.4, it suffices to show that, for any $\varepsilon > 0$ , there exists a constant $C_\varepsilon \in {\mathbb{R}}_{>0}$ such that for all $k \geq 0$ we have

\[ p^{-k \varepsilon} |\!|f_k(\nabla_\theta) |\!| \leq C_\varepsilon, \]

that is, for all $F \in V_0$ , we have

\[ p^{-k \varepsilon} |\!|f_k(\nabla_\theta) (F) |\!| \leq C_\varepsilon |\!|F|\!|. \]

It suffices to prove this for $F = s X^a Y^b Z^c$ with $s \in S^+$ . By Lemma 4.8, it suffices to show that, for each $0 \leq r \leq \min(k, a)$ and any $I \in \Sigma_{k, k-r}$ , the value

\[ p^{-k \varepsilon} \bigg|\! \bigg| \bigg({\kern-5pt}\begin{array}{c}{k - r}\\[3pt]{k_1, \ldots, k_\ell}\end{array}{\kern-6pt}\bigg)^{\!{-}1} \bigg({\kern-5pt}\begin{array}{c}{k}\\[3pt]{r}\end{array}{\kern-6pt}\bigg)^{\!{-}1} \bigg({\kern-5pt}\begin{array}{c}{a}\\[3pt]{r}\end{array}{\kern-6pt}\bigg) f_I(\theta)(s) \bigg| \! \bigg| \]

is uniformly bounded. But, since $|\!|f_I(\theta)|\!| \leq 1$ (because $\theta$ extends to a continuous action on $S^+$ ) this value is bounded above by $p^{-k \varepsilon + 2\log_p(k)}$ , which is uniformly bounded in k as $-k \varepsilon + 2 \log_p(k) \to - \infty$ as $k \to + \infty$ . This finishes the proof.

4.4 Overconvergence

We finish this section with the abstract results that will allow us to extend the locally analytic action from the ordinary locus to overconvergent neighbourhoods. The setting is as follows. Let

\[ V_0 \subseteq \cdots \subseteq V_r \subseteq \cdots \subseteq V_\infty\]

be a sequence of topological R-modules which are also ${\mathbb{Q}_p}$ -Banach spaces and let $\nabla : V_\infty \to V_\infty$ be a continuous R-linear operator stabilising each $V_i$ . Let $| \! | \cdot | \! |_i$ denote the Banach norm on $V_i$ and suppose that $| \! | x | \! |_s \leq | \! | x | \! |_r$ for all $x \in V_r$ and $r \leq s \leq \infty$ .

Proposition 4.11. Assume that the following property holds: for any $0 < \delta < 1$ and $r \in {\mathbb{N}}$ , there exists $s = s(\delta) \geq r$ such that, for all $c \in \mathbb{Q}$ , $h \in {\mathbb{N}}$ and $x \in V_r$ , we have

(4.3) \begin{equation} |\!|x|\!|_r \leq p^c \text{ and } |\!|x|\!|_{\infty} \leq p^{c - h} \implies |\!|x|\!|_s \leq p^{c -\delta h}.\end{equation}

Assume that, for some $\varepsilon > 0$ , the operator $\nabla$ extends to an $\varepsilon$ -analytic action on $V_\infty$ . Then, for any $r \in \mathbb{N}$ and $\gamma > \varepsilon$ , there exists $s \in {\mathbb{N}}$ (depending only on $\varepsilon$ , $\gamma$ and $|\!|\nabla|\!|_r$ ) such that, for any $x\in V_r$ ,

\[ p^{-k \gamma} |\!|f_k(\nabla)(x)|\!|_s \to 0 \quad \text{as } k \to +\infty. \]

In particular, the operator $\nabla$ extends to a continuous R-linear action $C_{\gamma}(\mathbb{Z}_p, R) \times V_r \to V_s$ .

Proof. Let $x \in V_r$ be such that $|\!|x|\!|_r \leq 1$ (and hence $|\!|x|\!|_\infty \leq 1$ ), and let $C_r \in \mathbb{Q}_{\geq 0}$ be a constant such that $|\!|\nabla|\!|_r \leq p^{C_r}$ . From the inequality $|\!|\nabla|\!|_r \leq p^{C_r}$ we deduce that

\[ |\!|k! f_k(\nabla)(x)|\!|_r \leq p^{k C_r}. \]

On the other hand, since $\nabla$ extends to an $\varepsilon$ -analytic action on $V_\infty$ , there exists $C_\varepsilon > 0$ such that

\[ |\!| k! f_k(\nabla)(x)|\!|_\infty \leq p^{-v_p(k!) + C_\varepsilon + k \varepsilon} = p^{kC_r - (k C_r - C_\varepsilon - k \varepsilon + v_p(k!))} \]

for all $k \in {\mathbb{N}}$ . Let k be large enough such that $kC_r - C_\varepsilon - k \varepsilon + v_p(k!) > 0$ , which is always possible as soon as $C_r \geq \varepsilon$ . Applying (4.3) to the element $k! f_k(\nabla)(x)$ with $c = kC_r$ and $h = k C_r - C_\varepsilon - k \varepsilon + v_p(k!) > 0$ (which we may assume to be an integer) we deduce that, for any $\delta < 1$ , there exists $s \in {\mathbb{N}}$ such that

\[ |\!| k! f_k(\nabla)(x)|\!|_s \leq p^{k C_r - \delta(k C_r -C_\varepsilon - k \varepsilon + v_p(k!))}.\]

Now let $\gamma > \varepsilon$ . We obtain

(4.4) \begin{align} p^{-k \gamma} |\!|f_k(\nabla)(x)|\!|_s &\leq p^{k C_r - \delta k C_r + \delta C_\varepsilon + k \delta \varepsilon - \delta v_p(k!) - k \gamma + v_p(k!)} \notag\\& \leq p^{-k \left( (\delta - 1) C_r + (\gamma - \delta \varepsilon) + (\delta - 1) \right) + \delta C_{\varepsilon},}\end{align}

where the last inequality follows from $v_p(k!) \leq k$ . One can easily show that one can choose $\delta$ (which will depend on $\varepsilon$ , $\gamma$ and $C_r$ ) in such a way that $ \left( (\delta - 1) C_r + (\gamma - \delta \varepsilon) + (\delta - 1) \right) > 0$ , which implies that (4.4) goes to 0 as $k \to + \infty$ . This finishes the proof.

5.1 The Gauss–Manin connection

Recall from Appendix A that $\pi_* \mathcal{O}_{P_{\operatorname{dR}}} = \operatorname{VB}^{\operatorname{can}}(\mathcal{O}_{\overline{P}})$ is naturally a $\mathcal{D}_X$ -module, where $\pi \colon P_{\operatorname{dR}} \to X$ denotes the structural map and $\mathcal{O}_{\overline{P}}$ denotes sections of the lower-triangular Borel $\overline{P} \subset \operatorname{GL}_2$ . The $(\mathfrak{g}, \overline{P})$ -module $\mathcal{O}_{\overline{P}}$ carries some additional structure, namely $\overline{P}$ acts through algebra automorphisms of $\mathcal{O}_{\overline{P}}$ and $\mathfrak{g}$ acts through derivations $\mathcal{O}_{\overline{P}} \to \mathcal{O}_{\overline{P}}$ . Therefore, we obtain a derivation $\nabla \colon \pi_*\mathcal{O}_{P_{\operatorname{dR}}} \to \pi_*\mathcal{O}_{P_{\operatorname{dR}}}$ as the composition

(5.1) \begin{equation} \pi_*\mathcal{O}_{P_{\operatorname{dR}}} \to \pi_*\mathcal{O}_{P_{\operatorname{dR}}} \otimes_{\mathcal{O}_X} \Omega^1_{X/R}(\operatorname{log}D) \to \pi_*\mathcal{O}_{P_{\operatorname{dR}}},\end{equation}

where the first map is induced from the $\mathcal{D}_X$ -module structure, and the second is induced from the Kodaira–Spencer isomorphism $\Omega^1_{X/R}(\operatorname{log}D) \cong \omega_{\mathcal{E}} \otimes \omega_{\mathcal{E}^D}$ , the adjoint map to the universal trivialisation $\pi^*(\omega_{\mathcal{E}} \otimes \omega_{\mathcal{E}^D}) \xrightarrow{\sim} \mathcal{O}_{P_{\operatorname{dR}}}$ , and the multiplication structure on $\pi_*\mathcal{O}_{P_{\operatorname{dR}}}$ .

Since $\pi$ is affine, the derivation $\nabla$ is equivalent to a derivation $\nabla \colon \mathcal{O}_{P_{\operatorname{dR}}} \to \mathcal{O}_{P_{\operatorname{dR}}}$ . We also let $\nabla \colon \mathcal{O}_{P_{\operatorname{dR}}^{\operatorname{an}}} \to \mathcal{O}_{P_{\operatorname{dR}}^{\operatorname{an}}}$ denote the corresponding derivation on the associated adic space. This immediately induces a derivation

\[\nabla \colon \mathcal{N}^{\dagger} \to \mathcal{N}^{\dagger},\]

where $\mathcal{N}^{\dagger} = \operatorname{colim}_U \pi_* \mathcal{O}_U$ is the colimit over all quasi-compact strict open neighbourhoods of $\mathcal{IG}_{\infty}$ in $P_{\operatorname{dR}}^{\operatorname{an}}$ (with structural map $\pi \colon U \to \mathcal{X}$ ). This is compatible with the Atkin–Serre operator via the map to p-adic modular forms.

5.1.1 Local description.

We note the following local description of $\nabla \colon\mathcal{O}_{P_{\operatorname{dR}}^{\operatorname{an}}} \to \mathcal{O}_{P_{\operatorname{dR}}^{\operatorname{an}}}$ , which follows from the way the $\mathcal{D}_X$ -module structure is constructed in the proof of Lemma A.3. Let $\operatorname{Spa}(A, A^+) \subset \mathcal{X}$ be a quasi-compact open affinoid subspace as above, over which $\mathcal{H}_{\mathcal{E}}$ , $\omega_{\mathcal{E}}$ , $\omega_{\mathcal{E}^D}$ trivialise. Let $\{ e, f \}$ denote a basis of $\mathcal{H}_{\mathcal{E}}$ respecting the Hodge filtration, and let $\bar{f} \in \omega_{\mathcal{E}}^{\vee}$ denote the image of f under the projection $\mathcal{H}_{\mathcal{E}} \twoheadrightarrow \omega_{\mathcal{E}}^{\vee}$ (we also denote this projection by $\overline{(-)}$ ). Via the Kodaira–Spencer isomorphism, there exists a unique differential $\kappa \in \Omega^1_{A/R}(\operatorname{log}D)$ such that

\[\overline{\nabla(e)} = \bar{f} \otimes \kappa \in \omega_{\mathcal{E}}^{\vee} \otimes \Omega^1_{A/R}(\operatorname{log}D) .\]

Let $D \colon A \to A$ denote the derivation dual to $\kappa$ . Then we can write

\[\nabla_D( (f \; e) ) = (f \; e) \cdot \delta,\]

for some $\delta \in \mathfrak{g}_A$ . Let $F \in A[U, T_1, T_2,T_1^{-1}, T_2^{-1}] \subset\mathcal{O}_{P_{\operatorname{dR}}^{\operatorname{an}}}(P_{\operatorname{dR},A}^{\operatorname{an}})$ be a polynomial, which we view as an element of $\mathcal{O}_{\overline{P}} \otimes_{\mathbb{Q}_p} A = A[U, T_1,T_2, T_1^{-1}, T_2^{-1}]$ – here we write a general element in $\overline{P}$ as $\big( \begin{smallmatrix} T_2 & \\ U &\,\, T_1\end{smallmatrix} \big)$ . Then the action of $\nabla$ is described as

\[\nabla(F)(U, T_1, T_2) = \left( D \cdot F(U, T_1, T_2) + \delta \star_l F(U, T_1, T_2) \right) \cdot T_1 T_2^{-1},\]

where $D \cdot F(U, T_1, T_2)$ is the application of the derivation D on the coefficients, and $\star_l$ denotes the $\mathfrak{g}$ -action on $\mathcal{O}_{\overline{P}}$ . The extra factor $T_1T_2^{-1}$ arises from the second map in (5.1).

Example 5.1. Suppose that $\delta = \big( \begin{smallmatrix} 0 &\,\, 1 \\ 0 &\,\, 0 \end{smallmatrix} \big)$ . Then the action $\delta \star_l -$ is given by the differential $U T_1 T_2^{-1} \partial_{T_1} - U \partial_{T_2}$ .

5.2 The local construction

Fix a quasi-compact open affinoid $\operatorname{Spa}(A, A^+) \subset \mathcal{X}$ as in §5.1.1. Let $n \geq 1$ be an integer. In this subsection, we will prove the local version of Theorem 2.11 using results from §§3 and 4. The first step is to understand the derivation

\[\nabla \colon \mathcal{O}_{\mathcal{U}_{\operatorname{HT}, n}} \to \mathcal{O}_{\mathcal{U}_{\operatorname{HT}, n}},\]

where $\mathcal{U}_{\operatorname{HT}, n}$ is as in Definition 3.12. Note that, since $\mathcal{IG}_{\infty} \to \mathcal{X}_{\operatorname{ord}}$ is a proétale $T(\mathbb{Z}_p)$ -torsor, this derivation extends uniquely to a derivation $\nabla \colon \mathcal{O}_{\mathcal{U}_{\operatorname{HT}, n, A_{\operatorname{ord}, \infty}}} \to \mathcal{O}_{\mathcal{U}_{\operatorname{HT}, n, A_{\operatorname{ord}, \infty}}}$ , and via the decomposition in Lemma 3.10 and the local description in §5.1.1, this decomposes as

\[\nabla = \oplus_{\sigma_{\lambda} \in \mathscr{G}_n} \nabla^{\lambda} \colon \bigoplus_{\lambda \in T(\mathbb{Z}/p^n \mathbb{Z})} \mathcal{O}_{\mathcal{B}_{\infty}(\sigma_{\lambda}(g_\infty), p^{-n})} \to \bigoplus_{\lambda \in T(\mathbb{Z}/p^n \mathbb{Z})} \mathcal{O}_{\mathcal{B}_{\infty}(\sigma_{\lambda}(g_\infty), p^{-n})} .\]

Proof. First recall that for $\lambda = (\lambda_1, \lambda_2)$ we have $\sigma_{\lambda}(\alpha_{\infty}) = \lambda_1 \alpha_{\infty}$ , $\sigma_{\lambda}(\beta_{\infty}) = \lambda_1 \beta_{\infty}$ and $\sigma_{\lambda}(\gamma_{\infty}) = \lambda_2 \gamma_{\infty}$ . Via the local description in §5.1.1, we can calculate the Gauss–Manin connection using the basis $\{\lambda_1^{-1} e_{\infty}, \lambda_2^{-1} f_{\infty} \}$ of $\mathcal{H}_{\mathcal{E}}$ over $A_{\operatorname{ord}, \infty}$ . Let $\kappa \in \Omega^1_{A_{\operatorname{ord}, \infty}/R}(\operatorname{log}D)$ denote the unique differential satisfying

\[ \overline{\nabla(\lambda_1^{-1}e_{\infty})} = \lambda_2^{-1}\bar{f}_{\infty} \otimes \kappa,\]

and let $D \colon A_{\operatorname{ord}, \infty} \to A_{\operatorname{ord}, \infty}$ denote the derivation dual to $\kappa$ . It is well known (see, for example, [Reference HoweHow20]) that $D = s_{\lambda} \theta$ for some $s_{\lambda} \in \mathbb{Z}_p^{\times}$ (this factor arises from comparing the universal trivialisations over $\mathfrak{IG}_{\infty}$ via the polarisation on $\mathcal{E}$ ). Let $\delta \in \mathfrak{g}_{A_{\operatorname{ord}, \infty}}$ be the element such that

\[ \nabla_{D}(\lambda_2^{-1}f_{\infty} \; \lambda_1^{-1}e_{\infty} ) = (\lambda_2^{-1}f_{\infty} \; \lambda_1^{-1}e_{\infty}) \cdot \delta.\]

We have that $\delta = \big( \begin{smallmatrix} 0 &\,\, 1 \\ 0 &\,\, 0\end{smallmatrix} \big)$ . The coordinates corresponding to this basis are given by X, Y, Z’ satisfying the identity

\[\bigg( \begin{matrix} T_2 & \\ U & T_1 \end{matrix} \bigg) = \bigg(\begin{matrix} Z' & \\ Y & X \end{matrix} \bigg)\sigma_{\lambda}(g_{\infty}), \quad g_{\infty} = \bigg(\begin{matrix} \gamma_{\infty} & \\ \beta_{\infty} & \alpha_{\infty}\end{matrix} \bigg),\]

and we find that the action of $\delta \star_l -$ on $\mathcal{O}_{\overline{P}} \otimes_{\mathbb{Q}_p} A_{\operatorname{ord}, \infty}$ is given by $Y X (Z')^{-1} \partial_{X} - Y \partial_{Z'}$ (see Example 5.1). Set $Z = Z' X$ . Then we calculate that:

• – $\nabla^{\lambda}(g) = (Dg) \cdot X (Z')^{-1} = s_{\lambda} \cdot \theta(g) X^2 Z^{-1}$ for any $g \in A_{\operatorname{ord}, \infty}$ ;

• – $\nabla^{\lambda}(X) = YX (Z')^{-1} \cdot X (Z')^{-1} = Y X^4 Z^{-2}$ ;

• – $\nabla^{\lambda}(Y) = 0$ ;

• – $\nabla^{\lambda}(Z) = \nabla^{\lambda}(Z' X) = (-Y X (Z')^{-1})X + Z' Y X^4 Z^{-2} = - Y X^3 Z^{-1} + Y X^3 Z^{-1} = 0$ .

This completes the proof of part (i) of the proposition since $X \equiv Z \equiv 1$ (mod $p^n$ ). Part (ii) follows from the fact that $\mathfrak{IG}_{\infty} \to \mathfrak{X}_{\operatorname{ord}}$ is a proétale $T(\mathbb{Z}_p)$ -torsor, so it is enough to check integrality of $\nabla$ after base-change along this torsor.

We are now in a position to apply the general results in §4 as we have an integral derivation $\nabla$ which is congruent mod $p^n$ to a nilpotent derivation that extends the Atkin–Serre operator. Recall that the Atkin–Serre operator $\theta \colon A^+_{\infty} \to A^+_{\infty}$ extends to a $R^+$ -algebra action of $C_{\operatorname{cont}}(\mathbb{Z}_p, R^+)$ . Set $\mathscr{N}_{\mathcal{U}_{\operatorname{HT}, n}} = \mathcal{O}_{\mathcal{U}_{\operatorname{HT}, n}}(\mathcal{U}_{\operatorname{HT}, n})$ . We have the following proposition.