2.1 Formal models for Hilbert modular schemes

Following [Reference RapoportRap78], [Reference ChaiCha90] and [Reference DimitrovDim04], we introduce the Hilbert modular schemes with $\mu_{\mathfrak{n}}$ -level structure and their various compactifications. Let $T=\mathrm{Res}_{\mathfrak{o}/\mathbf{Z}}\mathbf{G}_m$ and denote by $\mathscr{T}$ (respectively, $\widehat{\mathbf{G}}_m$ ) the formal group obtained by completing T (respectively, $\mathbf{G}_m$ ) along the identity section. Let $\mathcal{O}$ be the ring of integers of a finite extension K of $\mathbf{Q}_p$ containing the values of $\phi_p$ and all the embedding of F into $\overline{\mathbf{Q}}_p$ . We let $\mathfrak{d}$ denote the different ideal of F.

Let $\mathfrak{n}$ be an integral ideal of $\mathfrak{o}$ relatively prime to p and divisible by a rational integer $\geqslant 4$ , and let $\mu_{\mathfrak{n}}$ be the Cartier dual of the constant group scheme $(\mathfrak{o}/\mathfrak{n})$ over $\mathbf{Z}_p$ . We set $\Delta_{\mathfrak{n}}=\mathfrak{o}_{+}^\times/E_{\mathfrak{n}}^2$ , where $\mathfrak{o}_{+}^\times$ denotes the group of totally positive units and $E_{\mathfrak{n}}=\{u\in \mathfrak{o}^\times\mid u\equiv 1 \pmod{\mathfrak{n}}\}$ . Given a fractional ideal $\mathfrak{c}$ of F, we consider the functor attaching to any $\mathbf{Z}_p$ -scheme S, the set of quadruplets $ (A, \iota, \alpha, \lambda)$ with:

• – $A \rightarrow S$ an abelian scheme of relative dimension d with real multiplication $\iota:\mathfrak{o} \hookrightarrow \mathrm{End}_S (A)$ ;

• – $\alpha:\mu_{\mathfrak{n}} \otimes \mathfrak{d}^{-1} \hookrightarrow A[\mathfrak{n}]$ an $\mathfrak{o}$ -linear embedding of group schemes étale locally over S;

• – $\lambda$ a $\mathfrak{c}$ -polarization, i.e., an $\mathfrak{o}$ -linear isomorphism $\lambda:A^\vee\xrightarrow{\sim} A\otimes \mathfrak{c}$ satisfying the Deligne–Pappas condition.

Under the above assumptions on $\mathfrak{n}$ , this functor is representable by a scheme $X_{\mathfrak{c}}$ of finite type over $\mathbf{Z}_p$ , endowed with a universal abelian scheme $\pi: A_{\mathfrak{c}} \rightarrow X_{\mathfrak{c}}$ . Let $\omega $ be the conormal $\mathfrak{o} \otimes \mathcal{O}_{X_{\mathfrak{c}}} $ -module $e^{*}(\Omega_{A_{\mathfrak{c}}/X_{\mathfrak{c}}})$ , where $e:X_{\mathfrak{c}}\rightarrow A_{\mathfrak{c}}$ is the identity section. Let $\overline{X}_{\mathfrak{c}}$ be a toroidal compactification of $X_{\mathfrak{c}}$ and let $\pi: G_{\mathfrak{c}} \rightarrow \overline{X}_{\mathfrak{c}}$ be the semi-abelian scheme extending $A_{\mathfrak{c}}$ to $\overline{X}_{\mathfrak{c}}$ . The conormal sheaf $\omega$ extends naturally to an $\mathfrak{o} \otimes \mathcal{O}_{\overline{X}_{\mathfrak{c}}}$ -module which is also denoted by $\omega$ and we denote by $\displaystyle X^*_{\mathfrak{c}}=\mathrm{Proj}(\bigoplus\nolimits_{k\geqslant 0} \mathrm{H}^0(\overline{X}_{\mathfrak{c}}, \det(\omega)^k ))$ the minimal compactification of $X_{\mathfrak{c}}$ . Let $\overline{\mathscr{X}_{\mathfrak{c}}} $ (respectively, $\mathscr{X}^*_{\mathfrak{c}}$ ) denote the formal completion of $\overline{X}_{\mathfrak{c}}$ (respectively, $X_{\mathfrak{c}}^*$ ) along its special fiber. The natural group action of $\epsilon\in \mathfrak{o}_{+}^\times$ on $X_{\mathfrak{c}}$ sending $(A, \iota, \alpha, \lambda)$ to $(A, \iota, \alpha, \epsilon \cdot \lambda)$ factors through the finite group $\Delta_{\mathfrak{n}}$ , because the multiplication by $\epsilon\in E_{\mathfrak{n}}$ induces an isomorphism between $(A, \iota, \alpha, \lambda)$ and $(A, \iota, \alpha, \epsilon^2 \cdot \lambda)$ . This action extends to both $X_{\mathfrak{c}}^*$ and $\overline{X}_{\mathfrak{c}}$ . All these constructions remain valid over $\mathbf{Z}[1/\mathrm{N}_{F/\mathbf{Q}}(\mathfrak{n})]$ ensuring the compatibility with the transcendental theory. Given $n\in\mathbf{Z}_{\geqslant 1}$ and $v \in [0,{1}/({4 p^{n-1}})]\cap \mathbf{Q}$ , we let $\overline{\mathcal{X}}_{\mathfrak{c}}(v)$ (respectively, $\mathcal{X}^{*}_{\mathfrak{c}}(v)$ ) denote the v-overconvergent strict neighborhood of the ordinary locus $\overline{\mathcal{X}}_{\mathfrak{c}}(0) \subset \overline{X}_{\mathfrak{c}}^{\mathrm{rig}}$ (respectively, $\mathcal{X}^{*}_{\mathfrak{c}}(0) \subset X^{*\mathrm{rig}}_{\mathfrak{c}}$ ), i.e., the locus where the troncated p-adic valuation of the total Hasse invariant is at most v. Under the assumption $p\geqslant 3$ , Fargues [Reference FarguesFar11, Theorem 7] constructed a canonical $\mathfrak{o}$ -stable subgroup of rank $p^{nd}$

$$ \mathscr{H}_n \hookrightarrow G_{\mathfrak{c}}[p^n]_{\mid \overline{\mathcal{X}}_{\mathfrak{c}}(v)}. $$

Consider the étale covering

$$\overline{\mathcal{X}}_{\mathfrak{c}}(\mu_{p^n},v)=\mathrm{Isom}^{\mathfrak{o}}_{\overline{\mathcal{X}}_{\mathfrak{c}}(v)}(\mathscr{H}_n,\mu_{p^n} \otimes \mathfrak{d}^{-1}) \to \overline{\mathcal{X}}_{\mathfrak{c}}(v), $$

with Galois group $T(\mathbf{Z}/p^{n}\mathbf{Z})$ . After enlarging K so that $p^{v}\in K$ , following [Reference Andreatta, Iovita and PilloniAIP16, § 3], we consider the normal formal model $\overline{\mathscr{X}_{\mathfrak{c}}}(v)$ (respectively, $\mathscr{X}^*_{\mathfrak{c}}(v)$ ) of $\overline{\mathcal{X}}_{\mathfrak{c}}(v)$ (respectively, $\mathcal{X}^{*}_{\mathfrak{c}}(v)$ ) obtained via normalization of a certain admissible blow-up of $\overline{\mathscr{X}_{\mathfrak{c}}}$ (respectively, $\mathscr{X}^*_{\mathfrak{c}}$ ) [Reference Andreatta, Iovita and PilloniAIP15, § 5.2], then we define the formal models $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)$ and $\mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)$ by taking normalizations in the generic fiber, and finally we define $\mathscr{X}_{\mathfrak{c}}(\mu_{p^n},v)$ as the largest open over which $G_{\mathfrak{c}}$ is an abelian scheme. To sum up, we have the following commutative diagram of formal schemes.

(2.1)

2.2 The Hodge–Tate map

As $\overline{\mathscr{X}_{\mathfrak{c}}}(v)$ is an admissible normal formal $\mathcal{O}$ -scheme, $\mathscr{H}_n$ extends to a finite flat $\mathfrak{o}$ -stable subgroup (see [Reference Andreatta, Iovita and PilloniAIP15, Proposition 4.1.3])

$$\mathscr{H}_n\hookrightarrow G_{\mathfrak{c}}[p^n]_{|\overline{\mathscr{X}_{\mathfrak{c}}}(v)} $$

whose generic fiber is isomorphic to $\mathfrak{o}/(p^n)$ for the étale topology (see [Reference Andreatta, Iovita and PilloniAIP16, Lemma 3.3]). We consider the conormal sheaf $\omega_{\mathscr{H}_n}=e^*(\Omega_{\mathscr{H}_n/\overline{\mathscr{X}_{\mathfrak{c}}}(v)})$ of $\mathscr{H}_n$ and we recall that the Hodge–Tate map is a morphism of abelian sheaves over $\overline{\mathscr{X}_{\mathfrak{c}}}(v)$ for the fppf-topology

$$ \mathrm{HT}:\mathscr{H}_n^D \to \omega_{\mathscr{H}_n}, $$

sending an S-point $x \in \mathscr{H}_n^D(S)=\mathrm{Hom}_{\mathfrak{o}}(\mathscr{H}_n, \mathbf{G}_m \otimes \mathfrak{d}^{-1})$ to $x^*(({dt}/{t}) \otimes 1) \in \omega_{\mathscr{H}_n}(S)$ , in view of the canonical isomorphism $\omega_{\mathbf{G}_m \otimes \mathfrak{d}^{-1} \mid S}=\mathfrak{o}\otimes \mathcal{O}_S\cdot ({dt}/{t})$ . As $n-\tfrac{1}{2} \leqslant n-v (({p^n-1})/({p-1}))$ , arguing exactly as in [Reference Andreatta, Iovita and PilloniAIP15, Proposition 4.2.1] yields a canonical isomorphism

\begin{align*}\omega\otimes_{\mathcal{O}_{\overline{\mathscr{X}_{\mathfrak{c}}}(v)}}\bigg(\frac{\mathcal{O}_{\overline{\mathscr{X}_{\mathfrak{c}}}(v)}}{(p^{n-({1}/{2})})}\bigg)&=\omega_{G_{\mathfrak{c}}[p^n]}\otimes_{\mathcal{O}_{\overline{\mathscr{X}_{\mathfrak{c}}}(v)}}\bigg(\frac{\mathcal{O}_{\overline{\mathscr{X}_{\mathfrak{c}}}(v)}}{(p^{n-({1}/{2})})}\bigg)\\ &\xrightarrow{\sim}\omega_{\mathscr{H}_n}\otimes_{\mathcal{O}_{\overline{\mathscr{X}_{\mathfrak{c}}}(v)}}\bigg(\frac{\mathcal{O}_{\overline{\mathscr{X}_{\mathfrak{c}}}(v)}}{(p^{n-({1}/{2})})}\bigg). \end{align*}

Moreover, by [Reference Andreatta, Iovita and StevensAIS14, Proposition 6.1], $p^{{v}/({p-1})}$ annihilates the cokernel of the linearized Hodge–Tate map

(2.2) \begin{align}\overline{\mathrm{HT}}: \mathscr{H}_n^D \otimes\frac{\mathcal{O}_{\overline{\mathscr{X}_{\mathfrak{c}}}(v)}}{(p^{n-({1}/{2})})}\to\omega\otimes_{\mathcal{O}_{\overline{\mathscr{X}_{\mathfrak{c}}}(v)}}\frac{\mathcal{O}_{\overline{\mathscr{X}_{\mathfrak{c}}}(v)}}{(p^{n-({1}/{2})})}.\end{align}

We let $\mathscr{F}_{n}$ be the subsheaf of $\omega$ over $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)$ constructed in [Reference Andreatta, Iovita and PilloniAIP16, Proposition 3.4]. It is locally free of rank 1 over $\mathfrak{o} \otimes \mathcal{O}_{ \overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)}$ and uniquely characterized by the facts that it contains $p^{{v}/({p-1})} \cdot \omega$ and that the induced Hodge–Tate map yields a $\mathfrak{o}$ -linear isomorphism

(2.3) \begin{align} \overline{\mathrm{HT}}: \mathscr{H}_n^D \otimes \frac{\mathcal{O}_{\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)}}{(p^{n-({1}/{2})})} \xrightarrow{\sim} \mathscr{F}_{n} /p^{n-({1}/{2})} \mathscr{F}_{n}.\end{align}

2.3 Andreatta–Iovita–Pilloni–Stevens overconvergent modular sheaves

The weight space

$$ \mathcal{W}=\mathrm{Hom}_{\mathrm{cont}}(T(\mathbf{Z}_p),\mathbf{G}_m) $$

is the rigid analytic generic fiber of the formal scheme $\mathrm{Spf}(\mathcal{O} [\![ T(\mathbf{Z}_p) ]\!])$ . The group of connected components of the weight space is isomorphic to the group of tamely ramified characters of the corresponding torus, while each connected component is isomorphic to a rigid open polydisc. Given an affinoid $\mathcal{V}$ in $\mathcal{W}$ , the universal character

(2.4) \begin{align}\kappa_{\mathcal{V}}: T(\mathbf{Z}_p) \longrightarrow \mathcal{O}_{\mathcal{V}}^\times\end{align}

is w-analytic for some $w \in \mathbf{Q}_{\gt0}$ (see [Reference Andreatta, Iovita and PilloniAIP15, § 2.2]), which after shrinking $\mathcal{V}$ can be assumed to be $n-\tfrac{1}{2}$ . Let $\mathscr{T}_{n}^\circ \subset \mathscr{T}$ denote the formal subgroup of units congruent to 1 modulo $p^{n-({1}/{2})}$ , and let $\mathscr{T}_{n} \subset \mathscr{T}$ denote the formal subgroup generated by $T(\mathbf{Z}_p)$ and $\mathscr{T}_{n}^\circ$ . By [Reference Andreatta, Iovita and PilloniAIP16, p. 9], $\kappa_{\mathcal{V}}$ extends uniquely to a character

(2.5) \begin{align} \kappa_{\mathcal{V}}:\mathscr{T}_{n} \times \mathscr{V} \to \widehat{\mathbf{G}}_m \times \mathscr{V},\end{align}

where $\mathscr{V}=\mathrm{Spf}(\mathcal{O}_{\mathcal{V}}^\circ)$ is a formal model of $\mathcal{V}$ . Henceforth, we fix $n \in \mathbf{Z}_{\geqslant 1}$ such that $\phi_p$ factors through $T(\mathbf{Z}/p^{n-1}\mathbf{Z})$ , while keeping the notation from previous subsections. Let $\gamma_{n}^{\circ}: \mathscr{I}_{n} \to \overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)$ be the formal $\mathscr{T}_{n}^\circ $ -torsor constructed in [Reference Andreatta, Iovita and PilloniAIP16, § 3.4]. A point $x\in \mathscr{I}_{n}(R)$ over a normal admissible $\mathcal{O}$ -algebra R corresponds to isomorphism classes of $(A,\iota,\alpha, \lambda, P, \Omega)$ , where:

• – P is an $\mathfrak{o}$ -generator of $\mathscr{H}^D_{n}(R)=\mathscr{H}^D_{n}(R\otimes_{\mathcal{O}} K) \simeq \mathfrak{o}/(p^n)$ such that $(A,\iota,\alpha,\lambda, P)$ corresponds to the point $\gamma_{n}^{\circ}(x) \in \overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)(R)$ ; and

• – $\Omega$ is an $\mathfrak{o} \otimes R$ -basis of $\mathscr{F}_{n}(R)$ such that $\overline{\mathrm{HT}}(P)=\Omega \mod p^{n-({1}/{2})}$ via (2.3).

We set $\kappa_{\mathcal{V}}^{\circ}= \kappa_{\mathcal{V} \mid \mathscr{T}_{n}^\circ \times \mathscr{V} }$ . Andreatta, Iovita, Pilloni and Stevens [Reference Andreatta, Iovita and PilloniAIP16, Reference Andreatta, Iovita and StevensAIS14] defined the p-adic automorphic sheaf

(2.6) \begin{align}\omega^{\kappa_{\mathcal{V}}^{\circ}}=(\gamma_{n}^{\circ})_*(\mathcal{O}_{ \mathscr{I}_{n}}\widehat{\otimes} \mathcal{O}_{\mathcal{V}}^\circ)[-\kappa_{\mathcal{V}}^{\circ}],\end{align}

which is invertible on $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v) \times \mathscr{V}$ . In addition, as $\gamma_{n}^{\circ}$ is equivariant for the action of $t\in T(\mathbf{Z}_p)$ sending $(A,\iota,\alpha, \lambda, P, \Omega)$ to $(A,\iota,\alpha, \lambda, t\cdot P, t\cdot \Omega)$ in a compatible way with the $\mathscr{T}_{n}^\circ$ -torsor structure, we obtain a $\mathscr{T}_{n}$ -morphism

$$\gamma_{n}: \mathscr{I}_{n} \xrightarrow{\gamma_{n}^{\circ}} \overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v) \to \overline{\mathscr{X}_{\mathfrak{c}}}(v), $$

allowing us to define a coherent sheaf

(2.7) \begin{align}\omega^{\kappa_{\mathcal{V}}}=(\gamma_{n})_*(\mathcal{O}_{\mathscr{I}_{n}} \widehat{\otimes} \mathcal{O}_{\mathcal{V}}^\circ)[-\kappa_{\mathcal{V}}]\end{align}

on $\overline{\mathscr{X}_{\mathfrak{c}}}(v) \times \mathscr{V}$ which is invertible on the generic fiber. There is an inclusion of coherent sheaves $\omega^{\kappa_{\mathcal{V}}} \hookrightarrow \overline{h}_{n\ast}( \omega^{\kappa_{\mathcal{V}}^{\circ}})$ over $\overline{\mathscr{X}_{\mathfrak{c}}}(v) \times \mathscr{V}$ , where $\omega^{\kappa_{\mathcal{V}}}$ is exactly the sheaf of invariant sections of $\overline{h}_{n\ast}( \omega^{\kappa_{\mathcal{V}}^{\circ}})$ under the (twisted) action of $T(\mathbf{Z}/p^{n}\mathbf{Z})$ . This clarifies the relationship between these sheaves.

The sheaf $\omega^{\kappa_{\mathcal{V}}}$ can be used to give a geometric definition of Coleman families of overconvergent Hilbert modular forms. Namely, an element $f_{\mathcal{V}} \in \mathrm{H}^0(\overline{\mathcal{X}}_{\mathfrak{c}}(v) \times \mathcal{V}, \omega^{\kappa_{\mathcal{V}}})$ can be seen as a rule assigning to each normal admissible $\mathcal{O}_{\mathcal{V}}^{\circ}$ -algebra R and a tuple $(A,\iota,\alpha, \lambda, P, \Omega) \in \mathscr{I}_{n}(R)$ an element $f_R(A,\iota,\alpha, \lambda, P, \Omega) \in R[1/p]$ such that for all $ t \in T(\mathbf{Z}_p)$

(2.8) \begin{align}f_R(A,\iota,\alpha, \lambda, t^{-1} P, t^{-1} \Omega) =\kappa_{\mathcal{V}}(t) f_R(A,\iota,\alpha, \lambda, P, \Omega).\end{align}

2.4 Classicality for overconvergent modular forms

For any $k\in \mathbf{Z}[\Sigma]$ , we let

(2.9) \begin{align}\omega^{k}=\bigotimes_{\sigma \in \Sigma} \omega_\sigma^{k_\sigma },\end{align}

where $\omega=\bigoplus_{\sigma \in \Sigma} \omega_{\sigma}$ is the direct sum decomposition of the classical automorphic sheaf on any Hilbert modular variety over K. We let $X_{\mathfrak{c},1}(p^n)$ , respectively $X_{\mathfrak{c},0}(p^n)$ , be the Hilbert modular variety over K of level $\Gamma_1^1(\mathfrak{c},\mathfrak{n} p^n)$ , respectively $\Gamma_1^1(\mathfrak{c},\mathfrak{n} ) \cap \Gamma_0(\mathfrak{c},p^n)$ (see [Reference DimitrovDim04] for a precise definition). As the covering $\mathrm{pr}: X_{\mathfrak{c},1}(p^n) \to X_{\mathfrak{c},0}(p^n)$ is étale with Galois group $T(\mathbf{Z}/p^{n}\mathbf{Z})$ , there is a natural decomposition

(2.10) \begin{align}\mathrm{pr}_{*}( \mathrm{pr}^{*}(\omega^{k}))=\bigoplus_{\phi_p \,: \,T(\mathbf{Z}/p^{n}\mathbf{Z}) \to K^\times} \omega^{k}(\phi_p).\end{align}

It is unclear whether $\omega^{k}(\phi_p)$ extends to an invertible sheaf on a toroidal compactification $\overline{X}_{\mathfrak{c},0}(p^n)$ , as the morphism between the toroidal compactifications $\overline{X}_{\mathfrak{c},1}(p^n) \to \overline{X}_{\mathfrak{c},0}(p^n)$ may be ramified at the non-multiplicative cusps.

By (2.6), the specialization of $\omega^{\kappa_{\mathcal{V}}^{\circ}}$ at a point of $\mathcal{V}(K)$ corresponding to an algebraic character $k \in \mathbf{Z}[\Sigma]$ is the invertible sheaf $(\gamma_{n}^{\circ})_*(\mathcal{O}_{\mathscr{I}_{n}})[-k^{\circ}]$ on $\overline{\mathcal{X}}_{\mathfrak{c}}(\mu_{p^n},v)$ which is canonically isomorphic to the classical automorphic sheaf $\omega^{k}$ from (2.9). The étale covering of rigid spaces $\overline{\mathcal{X}}_{\mathfrak{c}}(\mu_{p^n},v)\to \overline{\mathcal{X}}_{\mathfrak{c}}(v)$ , defined using the canonical subgroup $\mathscr{H}_n$ , allows one to similarly decompose the pushforward of $\omega^{k}$ into a direct sum of invertible sheaves $\omega^{k}(\phi_p)$ over $\overline{\mathcal{X}}_{\mathfrak{c}}(v)$ . By construction, for any $\phi_p:T(\mathbf{Z}_p) \to T(\mathbf{Z}/p^{n}\mathbf{Z}) \to K^\times$ , $\omega^{k}(\phi_p)$ is the specialization of $\omega^{\kappa_{\mathcal{V}}}$ at the point of $\mathcal{V}(K)$ corresponding to the locally algebraic character

$$ T(\mathbf{Z}_p) \to \mathcal{O}_K^\times, \quad x \mapsto \phi_p(x) \cdot \prod_{\sigma \in \Sigma} \sigma(x)^{k_\sigma}. $$

The classicality results of [Reference Pilloni and StrohPS17, Reference BijakowskiBij16, Reference BijakowskiBij17] imply that the natural inclusion

(2.11) \begin{align}\mathrm{H}^0(\overline{X}_{\mathfrak{c},1}(p^n)^{\mathrm{an}}, \omega^{k}) \hookrightarrow \mathrm{H}^0(\overline{\mathcal{X}}_{\mathfrak{c}}(\mu_{p^n},v), \omega^{k})\end{align}

obtained from the restriction along the open immersion $\overline{\mathcal{X}}_{\mathfrak{c}}(\mu_{p^n},v) \hookrightarrow \overline{X}_{\mathfrak{c},1}(p^n)^{\mathrm{an}}$ becomes an isomorphism when restricting to the part where for all $\mathfrak{p}\in \Sigma_p$ , the Hecke operator $U_{\mathfrak{p}}$ has slope strictly less than $\min_{\sigma \in \Sigma_{\mathfrak{p}}}(k_{\sigma}) - d_{\mathfrak{p}}$ . One should be careful to use an mth power of $U_{\mathfrak{p}}$ so that $\mathfrak{c} \mathfrak{p}^m$ and $\mathfrak{c}$ have the same image in the narrow class group $\mathscr{C}\!\ell_F^+$ of F. Koecher’s principle and the rigid analytic GAGA yield isomorphisms

$$\mathrm{H}^0(X_{\mathfrak{c},0}(p^n),\omega^{k}(\phi_p)) \xrightarrow{\sim} \mathrm{H}^0(\overline{X}_{\mathfrak{c},1}(p^n),\omega^{k})[\phi_p] \xrightarrow{\sim}\mathrm{H}^0(\overline{X}_{\mathfrak{c},1}(p^n)^{\mathrm{an}}, \omega^{k}) [\phi_p]. $$

Hence, the inclusion $ \mathrm{H}^0(X_{\mathfrak{c},0}(p^n),\omega^{k}(\phi_p)) \hookrightarrow \mathrm{H}^0(\overline{\mathcal{X}}_{\mathfrak{c}}(v), \omega^{k}(\phi_p)) $ obtained by taking the $\phi_p$ -isotypic components in (2.11) restricted to the same small slope part for $U_{\mathfrak{p}}$ ( $\mathfrak{p}\in \Sigma_p$ ) is an isomorphism as well.

2.5 Twisted p-adic families of arithmetic Hilbert modular forms

In order to define Coleman families interpolating eigenforms for $G=\mathrm{Res}_{F/\mathbf{Q}}\mathrm{GL}(2)$ one needs to perform a descent from the fine moduli space to the Shimura variety for G using the action of $\Delta_{\mathfrak{n}}$ , while moving to a $(d+1)$ -dimensional weight space. Indeed, the algebraic automorphic forms for G have weights $(k, \mathsf{w})\in \mathbf{Z}[\Sigma]\times \mathbf{Z}$ , where for all $\sigma\in \Sigma$ we have $k_\sigma\geqslant 1$ and $k_\sigma\equiv \mathsf{w}\pmod{2}$ . Our normalization is such that $\mathsf{w}$ is the purity weight, i.e., the automorphic form has central character of the form $\phi \vert \cdot \vert_F^{\mathsf{w}}$ for some finite order Hecke character $\phi$ . As a consequence we only consider p-adic weights $\kappa$ which are in the image of

(2.12) \begin{align} \mathcal{W}\times \mathcal{W}^{\parallel} \longrightarrow \mathcal{W}, \quad (\nu,\mathsf{w})\mapsto \phi_p\cdot \nu^{2} \cdot \mathsf{w} \circ \mathrm{N}_{F/\mathbf{Q}},\end{align}

where $\mathcal{W}^{\parallel}=\mathrm{Hom}_{\mathrm{cont}}(\mathbf{Z}_p^\times,\mathbf{G}_m)$ is the rigid analytic generic fiber of the formal scheme $\mathrm{Spf}(\mathcal{O} [\![ \mathbf{Z}_p^\times ]\!])$ . As parallel weight satisfy $k_\sigma=2-\mathsf{w}$ for all $\sigma\in \Sigma$ (see, for example, [Reference Barrera, Dimitrov and JorzaBDJ22, I.1]), we consider the following embedding of the subspace of parallel weights

(2.13) \begin{align}\mathcal{W}^{\parallel} \to \mathcal{W}\times \mathcal{W}^{\parallel}, \quad \mathsf{w}\mapsto (\mathrm{N}_{F/\mathbf{Q}} \cdot\mathsf{w}\circ \mathrm{N}^{-1}_{F/\mathbf{Q}}, \mathsf{w}) .\end{align}

We let $\mathscr{G}_{n}^\circ \subset \widehat{\mathbf{G}}_m$ denote the formal subgroup of units congruent to 1 modulo $p^{n-({1}/{2})}$ and $\mathscr{G}_{n} \subset \widehat{\mathbf{G}}_m$ the formal subgroup generated by $\mathbf{Z}_p^\times$ and $\mathscr{G}_{n}^\circ$ . We recall that $n\in\mathbf{Z}_{\geqslant 1}$ is chosen so that $\phi_p$ factors through $T(\mathbf{Z}/p^{n-1}\mathbf{Z})$ , and we fix an affinoid $\mathcal{U}\subset \mathcal{W}\times \mathcal{W}^{\parallel}$ whose image by (2.12) is contained in an affinoid $\mathcal{V}\subset \mathcal{W}$ such that $\kappa_{\mathcal{V}}$ is $(n-\tfrac{1}{2})$ -analytic. By definition, the pushforward $\kappa_{\mathcal{U}}:\mathscr{T}_{n} \times \mathscr{U} \to \widehat{\mathbf{G}}_m \times \mathscr{U}$ of $\kappa_{\mathcal{V}}$ via the natural map $\mathcal{O}_{\mathcal{V}}^\circ \to \mathcal{O}_{\mathcal{U}}^\circ$ coming from (2.12) factors through

(2.14) \begin{align} (\nu_{\mathcal{U}}, \mathsf{w}_{\mathcal{U}}):\mathscr{T}_{n} \times \mathscr{G}_{n} \times\mathscr{U} \to \widehat{\mathbf{G}}_m \times \mathscr{U},\end{align}

i.e., $\kappa_{\mathcal{U}}=\phi_p\cdot \nu_{\mathcal{U}}^2\cdot \mathsf{w}_{\mathcal{U}}\circ\mathrm{N}_{F/\mathbf{Q}}$ . Henceforth, we assume that $\phi_{p}(\epsilon)=1$ for all $\epsilon\in E_{\mathfrak{n}}$ . It is worth noting that this condition is satisfied by the restriction to $T(\mathbf{Z}_p)$ of any totally odd finite order Hecke character $\phi:\mathbf{A}_F^\times/F^\times \to \overline{\mathbf{Q}}^\times$ having tame conductor $\mathfrak{n}$ . As $\mathrm{N}_{F/\mathbf{Q}}(\epsilon)=1$ for all $\epsilon\in E_{\mathfrak{n}}$ , applying (2.8) to a global section $f_{\mathcal{U}} \in \mathrm{H}^0(\overline{\mathcal{X}}_{\mathfrak{c}}(v) \times \mathcal{U}, \omega^{\kappa_{\mathcal{U}}})$ , where $v \leqslant {1}/({4p^{n-1}})$ , for all $\epsilon\in E_{\mathfrak{n}}$ the automorphism $A \xrightarrow{\epsilon\cdot } A$ yields

\begin{align*}f_{\mathcal{U}}(A,\iota,\alpha, \lambda, P, \Omega)&= f_{\mathcal{U}}(A,\iota,\alpha, \epsilon^2 \lambda, \epsilon P, \epsilon \Omega)=\kappa_{\mathcal{U}}(\epsilon^{-1}) f_{\mathcal{U}}(A,\iota,\alpha, \epsilon^2 \lambda, P, \Omega)\\ &= \phi_p(\epsilon^{-1})\mathsf{w}_{\mathcal{U}}(\mathrm{N}_{F/\mathbf{Q}}(\epsilon^{-1})) \cdot \nu_{\mathcal{U}} (\epsilon^{-2}) f_{\mathcal{U}}(A,\iota,\alpha, \epsilon^2 \lambda, P, \Omega)\\ &= \nu_{\mathcal{U}} (\epsilon^{-2}) f_{\mathcal{U}}(A,\iota,\alpha, \epsilon^2 \lambda, P, \Omega).\end{align*}

Hence, the following action of $\epsilon\in \mathfrak{o}_{+}^\times$ on $\mathrm{H}^0(\overline{\mathcal{X}}_{\mathfrak{c}}(v) \times \mathcal{U}, \omega^{\kappa_{\mathcal{U}}})$ factors through the finite group $\Delta_{\mathfrak{n}}$

(2.15) \begin{align}(\epsilon \cdot f_{\mathcal{U}})(A,\iota,\alpha, \lambda, P, \Omega)=\nu_{\mathcal{U}} (\epsilon^{-1}) \cdot f_{\mathcal{U}}(A,\iota,\alpha, \epsilon \lambda, P, \Omega).\end{align}

Let $\mathbf{M}^{\dagger}_{\mathcal{U}}(v)=\bigoplus_{\mathfrak{c} \in \mathscr{C}\!\ell_F^+} \mathrm{H}^0(\overline{\mathcal{X}}_{\mathfrak{c}}(v) \times \mathcal{U}, \omega^{\kappa_{\mathcal{U}}})^{\Delta_{\mathfrak{n}}}$ denote the Banach $\mathcal{O}(\mathcal{U})$ -module of $\Delta_{\mathfrak{n}}$ -invariants and let

(2.16) \begin{align}\mathbf{M}^{\dagger}_{\mathcal{U}}= \varinjlim_{v\gt0} \mathbf{M}^{\dagger}_{\mathcal{U}}(v)\end{align}

be the Fréchet $\mathcal{O}(\mathcal{U})$ -module of Coleman families. The corresponding submodule of cuspidal Coleman families $\mathbf{S}^{\dagger}_{\mathcal{U}}$ is defined by replacing the sheaf $\omega^{\kappa_{\mathcal{U}}}$ with $ \omega^{\kappa_{\mathcal{U}}}(-D)$ , where D is the divisor with normal crossings at infinity of $\overline{\mathscr{X}_{\mathfrak{c}}}(v)$ . For an affinoid $\mathcal{U}$ of $\mathcal{W}^\parallel$ , we let $\omega^{\mathsf{w}_{\mathcal{U}}}$ denote the sheaf of parallel weight $(\mathrm{N}_{F/\mathbf{Q}} \cdot\mathsf{w}_{\mathcal{U}}\circ \mathrm{N}^{-1}_{F/\mathbf{Q}}, \mathsf{w}_{\mathcal{U}})$ and we analogously define $\mathbf{M}^{\dagger}_{\mathcal{U}}$ and $\mathbf{S}^{\dagger}_{\mathcal{U}}$ .

By [Reference Andreatta, Iovita and PilloniAIP16, § 3.7] the Hecke correspondences $T_v$ , for $v \,\nmid\, \mathfrak{n} p$ , and $U_{v}$ , for $v \in \Sigma_p$ , act continuously on the Banach $\mathcal{O}(\mathcal{U})$ -module $\bigoplus_{\mathfrak{c} \in \mathscr{C}\!\ell_F^+} \mathrm{H}^0(\overline{\mathcal{X}}_{\mathfrak{c}}(v) \times \mathcal{U}, \omega^{\kappa_{\mathcal{U}}})$ . Moreover, they commute with the action of $\Delta_{\mathfrak{n}}$ , hence also act continuously on the Fréchet $\mathcal{O}(\mathcal{U})$ -modules $\mathbf{M}^{\dagger}_{\mathcal{U}}$ and $\mathbf{S}^{\dagger}_{\mathcal{U}}$ , generating a commutative algebra of endomorphisms. It is shown in [Reference Andreatta, Iovita and PilloniAIP16, Lemma 3.27] that $U_p=\prod_{v \in \Sigma_p} U_{v}^{e_{v}} $ is a compact operator.

2.6 Adic version of the formal function theorem

Recall that we have a proper morphism $\overline{\pi}_n: \overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v) \to \mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)$ of admissible formal $\mathcal{O}$ -schemes. Let C be a cusp and $\overline{C}=\overline{\pi}_n^{-1}(C)$ . Let:

• – $\widehat{\overline{\pi}}_n: \widehat{\overline{\mathscr{X}_{\mathfrak{c}}}}(\mu_{p^n},v)_{\mid \mathscr{V}} \longrightarrow \widehat{\mathscr{X}^*_{\mathfrak{c}}}(\mu_{p^n},v)_{\mid \mathscr{V}}$ be the formal completion of $\overline{\pi}_n$ along $ \overline{C} \times \mathscr{V} \to C \times \mathscr{V}$ ;

• – $\widehat{\mathrm{R}^n \overline{\pi}_{n \ast} }(\omega^{\kappa_{\mathcal{V}}^{\circ}})$ be the formal completion of the coherent $\mathcal{O}_{ \mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)_{\mid \mathscr{V}}}$ -module $\mathrm{R}^n \overline{\pi}_{n \ast} (\omega^{\kappa_{\mathcal{V}}^{\circ}})$ along $C \times \mathscr{V} $ and $\widehat{\omega^{\kappa_{\mathcal{V}}^{\circ}}}$ be the completion of $\omega^{\kappa_{\mathcal{V}}^{\circ}}$ along $ \overline{C} \times \mathscr{V}$ .

Proposition 2.1. The canonical homomorphism

(2.17) \begin{align} \widehat{\mathrm{R}^n \overline{\pi}_{n \ast} }(\omega^{\kappa_{\mathcal{V}}^{\circ}}) \longrightarrow \mathrm{R}^n \widehat{\overline{\pi}}_{n \ast} (\widehat{\omega^{\kappa_{\mathcal{V}}^{\circ}}})\end{align}

is an isomorphism.

1. (i) $\mathrm{Spf}(\widehat{A}) \subset \mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)_{\mid \mathscr{V}}$ is an affine open containing the closed formal subscheme $C \times \mathscr{V} $ , defined by an open ideal $\mathfrak{m}_C$ ;

2. (ii) $\pi: \mathscr{X} \longrightarrow \mathrm{Spf}(\widehat{A})$ is algebraizable by a proper morphism $\pi^{\mathrm{alg}}: X \to \mathrm{Spec}(A)$ of schemes of finite type over $\mathcal{O}$ ; and

3. (iii) $\overline{C} \times \mathscr{V} =\pi^{-1}(C \times \mathscr{V} )$ a divisor of $\mathscr{X} $ .

One has the following diagram of our schemes and their formal completions.

(2.18)

Here $\widehat{ \mathscr{X}}^{\overline{C}}$ is the formal completion of $\mathscr{X} $ along $\overline{C}\times \mathscr{V}$ and $\widehat{A}_{\mathfrak{m}_C}$ is the $\mathfrak{m}_C$ -adic completion of $\widehat{A}$ . Note that $\pi$ is also the p-adic completion of the proper morphism of noetherian schemes $\pi^{\mathrm{alg}}_{\mid \widehat{A}}$ which is obtained from $\pi^{\mathrm{alg}}$ by base change. By Grothendieck’s formal existence theorem [Reference GrothendieckGro60, Theorem 3], the sheaf $\omega^{\kappa_{\mathcal{V}}^{\circ}}$ is the p-adic completion of a coherent sheaf $\omega^{\kappa_{\mathcal{V}}^{\circ}}_{\mathrm{alg}}$ on the scheme $X_{\mid \widehat{A} }$ . Nevertheless, since the completion is transitive with a finer topology, $\widehat{\omega^{\kappa_{\mathcal{V}}^{\circ}}}$ is the completion $\omega^{\kappa_{\mathcal{V}}^{\circ}}_{\mathrm{alg}}$ with respect to $\overline{C} \times \mathrm{Spec}(\mathcal{O}( \mathscr{V})/(p))$ . Applying the theorem on formal functions to $\pi^{\mathrm{alg}}_{\mid \widehat{A}}:X_{\mid \widehat{A} } \to \mathrm{Spec}(\widehat{A} )$ with respect to the ideal $\mathfrak{m}_C \cdot( \widehat{A} )$ yields

$$ \widehat{\mathrm{R}^n (\pi^{\mathrm{alg}}}_{\mid \widehat{A} })_{\ast}(\omega^{\kappa_{\mathcal{V}}^{\circ}}_{\mathrm{alg}}) \simeq \mathrm{R}^n (\widehat{\pi} )_{\ast} (\widehat{\omega^{\kappa_{\mathcal{V}}^{\circ}}}). $$

The desired assertion then follows from the isomorphism

$$ \mathrm{R}^n (\pi^{\mathrm{alg}}_{\mid \widehat{A} })_{\ast}(\omega^{\kappa_{\mathcal{V}}^{\circ}}_{\mathrm{alg}}) \simeq \mathrm{R}^n (\pi)_{\ast}(\omega^{\kappa_{\mathcal{V}}^{\circ}}) $$

resulting again from the theorem on formal functions, this time applied to $\pi^{\mathrm{alg}}_{\mid \widehat{A}}: X_{\mid \widehat{A} } \to \mathrm{Spec}(\widehat{A} )$ and the ideal (p).

2.7 Toroidal compactifications and descent to the minimal compactification

Throughout this subsection, we only consider parallel weights. We will recall the geometric q-expansions of Coleman families at various cusps. For any $U\subset F\otimes_{\mathbf{Q}} \mathbf{R}$ , we let $U_+$ denote the subset consisting of totally positive elements. By [Reference DimitrovDim04, Definition 3.2], any cusp C of $\mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)$ gives rise to:

1. (i) two fractional ideals $\mathfrak{a}$ and $\mathfrak{b}$ of F and a positive polarization $\beta:\mathfrak{b}^{-1}\mathfrak{a} \simeq \mathfrak{c}$ ;

2. (ii) a decomposition $\mathfrak{n}=\mathfrak{n}' \mathfrak{n}''$ with isomorphisms $\mathfrak{n}'' \mathfrak{b}/\mathfrak{b} \simeq \mu_{\mathfrak{n}''} \otimes \mathfrak{d}^{-1}$ and $\mathfrak{a}/p^n\mathfrak{n}' \mathfrak{a} \simeq \mathfrak{o}/p^n \mathfrak{n}' $ (the cusp is unramified at $\mathfrak{n}'$ ).

We denote by $\mathscr{P}(\mathfrak{c},p^n)$ the set of cusps on $\mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)$ and denote by $\mathscr{P}(\mathfrak{c})=\mathscr{P}(\mathfrak{c},1)$ the set of cusps on $\mathscr{X}_{\mathfrak{c}}^*(v)$ , which can be described in the same manner as above by replacing $n\in \mathbf{Z}_{ \geqslant 1}$ by 0. Set $M=\mathfrak{a}\mathfrak{b}\mathfrak{n}''{}^{-1}$ and $M^* = \mathrm{Hom}(\mathfrak{a}\mathfrak{b}\mathfrak{n}''{}^{-1},\mathbf{R})$ . The construction of the toroidal compactification $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)$ of $\mathscr{X}_{\mathfrak{c}}(\mu_{p^n},v)$ at the cusp $C \in \mathscr{P}(\mathfrak{c},p^n)$ involves the following data (see [Reference RapoportRap78]).

1. (C1) A smooth projective rational polyhedral cone decomposition $\Sigma^{C}$ of $M_{+}^*$ invariant under $E_{\mathfrak{n} p^n}$ with finitely many orbits. We recall that $\epsilon \in E_{\mathfrak{n} p^n}$ acts by multiplication by $\epsilon^2$ .

2. (C2) A toric $\mathcal{O}$ -scheme $S_{\Sigma^C}$ covered by $S_{\sigma}=\mathrm{Spec}(\mathcal{O}[q^\xi; \xi \in M, \sigma(\xi) \geqslant 0 ])$ for $\sigma \in \Sigma^C$ , and endowed with a natural toric embedding $S^0_{\Sigma^C}=\mathrm{Spec}(\mathcal{O}[q^\xi]_{\xi \in M}) \hookrightarrow S_{\sigma}$ .

3. (C3) The formal completion of $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)$ along the divisor $\overline{\pi}_n^{-1}(C)$ is isomorphic to $\widehat{S}_{\Sigma^{C}}/E_{\mathfrak{n} p^n}$ , where $\widehat{S}_{\Sigma^{C}}$ is the formal completion of $S_{\Sigma^{C}}$ along the special fiber of the divisor $S_{\Sigma^C} \setminus S^0_{\Sigma^C}$ .

4. (C4) The completed local ring $\widehat{\mathcal{O}}_{\mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v),C}$ at the point $C:\mathrm{Spf}(\mathcal{O}) \to \mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)$ is isomorphic to the $E_{\mathfrak{n} p^n}$ -invariant global sections $\mathrm{H}^0(\widehat{S}_{\Sigma^{C}},\mathcal{O}_{\widehat{S}_{\Sigma^{C}}})^{E_{\mathfrak{n} p^n}}=\mathcal{O}[\![ q^\xi ]\!]_{\xi \in M_{+} \cup \{0\}}^{E_{\mathfrak{n} p^n}}$ .

5. (C5) A Tate object $(G_{\mathfrak{c}})_{\mid\widehat{S}_{\Sigma^{C}}}$ uniformized by the 1-motive $[\mathfrak{b} \xrightarrow{q} \mathbb{G}_m \otimes \mathfrak{a}^{-1}\mathfrak{d}^{-1}]$ . In particular, one has $\omega_{\mid \widehat{S}_{\Sigma^{C}}} \simeq \omega_{\mathbb{G}_m \otimes \mathfrak{a}^{-1}\mathfrak{d}^{-1} }\simeq \mathfrak{a} \otimes \mathcal{O}_{\widehat{S}_{\Sigma^{C}}}\! \cdot ({dt}/{t})$ , where ${dt}/{t}$ is the canonical invariant differential of $\mathbb{G}_m$ .

6. (C6) The action of $\epsilon \in E_{\mathfrak{n} p^n}$ induces at the level of the conormal sheaves

   $$\omega_{\mid \widehat{S}_{\epsilon^2 \cdot \sigma}} \longrightarrow \omega_{\mid \widehat{S}_{ \sigma}}, \quad \frac{dt}{t} \longmapsto \epsilon \cdot \frac{dt}{t}. $$

Recall the canonical proper adic morphism $\overline{\pi}_n: \overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v) \to \mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)$ from (2.1). Choose an affinoid $\mathcal{U}$ in $\mathcal{W}^{\parallel}$ such that the universal character $\mathsf{w}_{\mathcal{U}}$ is $(n-({1}/{2}))$ -analytic and let $\mathscr{U}=\mathrm{Spf}(\mathcal{O}(\mathcal{U})^\circ)$ .

The canonical subgroup $\mathscr{H}_{n \mid \widehat{S}_{\Sigma^{C}}} \hookrightarrow (G_{\mathfrak{c}})_{\mid \widehat{S}_{\Sigma^{C}}}$ is the multiplicative group given by an injection

$$\mu_{p^n} \otimes \mathfrak{d}^{-1} \mathfrak{a}^{-1} \hookrightarrow \mathbf{G}_m \otimes \mathfrak{d}^{-1}\mathfrak{a}^{-1}. $$

It follows from [Reference Andreatta, Iovita and StevensAIS14, Lemma 2.9, § 4.1] that we have the following $\mathfrak{o}$ -linear maps.

(2.19)

Here the horizontal Hodge–Tate map sends the generator $(\xi\mod{p^n})$ of $\mathfrak{a}/p^n \mathfrak{a}$ to ${dt^\xi}/{t^\xi} \pmod{p^n}$ . According to the universal property of $\mathscr{I}_{n}$ , the data $((G_{\mathfrak{c}})_{\mid \widehat{S}_{\Sigma^{C}}},\xi \in \mathscr{H}_{n}^D(\widehat{S}_{\Sigma^{C}}), {dt^{\xi}}/{t^{\xi}})$ gives a section $\widehat{S}_{\Sigma^{C}} \to \mathscr{I}_{n \mid \widehat{S}_{\Sigma^{C}}} $ . This implies that $\mathscr{I}_{n \mid \widehat{S}_{\Sigma^{C}}} \simeq \widehat{S}_{\Sigma^{C}} \times \mathscr{T}_{n}^\circ$ and that $\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}}$ can be trivialized on $\widehat{S}_{\Sigma^{C}} \times \mathscr{U}$ , namely,

(2.20) \begin{align}\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}}_{\mid \widehat{S}_{\Sigma^{C}} \times \mathscr{U}} = \mathcal{O}_{\widehat{S}_{\Sigma^{C}}\times \mathscr{U}}\cdot \mathsf{w}_{\mathcal{U}}^{\circ},\end{align}

where $\mathsf{w}_{\mathcal{U}}^{\circ}: \mathscr{T}_{n}^\circ \times\mathscr{U} \to \widehat{\mathbf{G}}_m \times \mathscr{U}$ is defined as the pullback of the $(n-\tfrac{1}{2})$ -analytic character $\kappa_{\mathcal{V}}^{\circ}$ from § 2.3 by the composed map of (2.12) and (2.13). As $\mathsf{w}_{\mathcal{U}}^{\circ}(\epsilon)=1$ for all $\epsilon \in E_{\mathfrak{n} p^n}$ (because $\mathsf{w}_{\mathcal{U}}^{\circ}$ factors through $\mathrm{N}_{F/\mathbf{Q}}$ ), (C6) yields that the trivialization (2.20) descends to a trivialization on $\widehat{S}_{\Sigma^{C}}/E_{\mathfrak{n} p^n} \times \mathscr{U}$ , namely,

(2.21) \begin{align}\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}}_{\mid \widehat{S}_{\Sigma^{C}/E_{\mathfrak{n} p^n}}\times \mathscr{U}} =\mathcal{O}_{\widehat{S}_{\Sigma^{C}/E_{\mathfrak{n} p^n}}\times \mathscr{U}}\cdot \mathsf{w}_{\mathcal{U}}^{\circ}.\end{align}

Thus, (C3) and Proposition 2.1 yield that the completed stalk of $(\overline{\pi}_n)_* (\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}})$ is free of rank 1 at any cusp C and, hence, $(\overline{\pi}_n)_* (\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}})$ is invertible on $\mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)\times \mathscr{U}$ .

2.8 The specialization map for overconvergent p-adic modular forms

The geometric results of the previous subsection have the following important consequence.

Proof. Consider the adjunction homomorphism for n such that $\phi_p$ factors through $T(\mathbf{Z}/p^{n-1}\mathbf{Z})$ :

(2.22) \begin{equation}\nu_n: (\overline{\pi}_n)_* (\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}})\otimes_{\mathcal{O}_{\mathscr{U}},\mathsf{w}^{\circ}}\mathcal{O} \longrightarrow(\overline{\pi}_n)_* (\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}}\otimes_{\mathcal{O}_{ \mathscr{U}},\mathsf{w}^{\circ}}\mathcal{O})=(\overline{\pi}_n)_*(\omega^{\mathsf{w}^{\circ}}).\end{equation}

Proposition 2.2 asserts that $(\overline{\pi}_n)_* (\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}})$ is an invertible sheaf on $\mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v) \times \mathscr{U}$ , hence its specialization $(\overline{\pi}_n)_* (\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}})\otimes_{\mathcal{O}_{ \mathscr{U}},\mathsf{w}^{\circ}} \mathcal{O}$ is an invertible sheaf on $\mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)$ . Similarly, $(\overline{\pi}_n)_*(\omega^{\mathsf{w}^{\circ}})$ is an invertible sheaf on $\mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)$ . As $\mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)$ is normal and $\nu_n$ is an isomorphism over the open $\mathscr{X}_{\mathfrak{c}}(\mu_{p^n},v)$ whose complement has codimension at least 2, it follows that $\nu_n$ is an isomorphism of invertible sheaves. Passing to the generic fiber $\mathcal{X}^*_{\mathfrak{c}}(\mu_{p^n},v)$ and using the finiteness of $h_n$ from (2.1), we obtain a canonical isomorphism $\nu: \overline{\pi}_{*} (\omega^{\mathsf{w}_{\mathcal{U}}})\otimes_{\mathcal{O}(\mathcal{U}),\mathsf{w}} K \xrightarrow{\sim}\overline{\pi}_{*}(\omega^{\mathsf{w}})$ of coherent sheaves on $\mathcal{X}^*_{\mathfrak{c}} (v)$ . As $\mathcal{X}^*_{\mathfrak{c}}(v)$ is an affinoid, the resulting specialization map is surjective:

$$ \mathrm{H}^0(\mathcal{X}^{*}_{\mathfrak{c}} (v)\times \mathcal{U}, \overline{\pi}_{*}(\omega^{\mathsf{w}_{\mathcal{U}}}) ) \twoheadrightarrow \mathrm{H}^0(\mathcal{X}^{*}_{\mathfrak{c}} (v), \overline{\pi}_{*}( \omega^{\mathsf{w}}) ).$$

2.9 Geometric q-expansion, the residue map and the fundamental exact sequence

Keeping the notation from the previous subsections, we obtain from (2.21) and (C3) a geometric q-expansion at a cusp $C \in \mathscr{P}(\mathfrak{c},p^n)$

(2.23) \begin{equation}\mathrm{Ev}_{C,v}: \mathrm{H}^0(\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)\times \mathscr{U},\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}}) \longrightarrow \mathcal{O}(\mathscr{U}) [\![ q^\xi ]\!]_{\xi \in M_{+} \cup \{0\}}^{E_{\mathfrak{n} p^n}},\end{equation}

and we denote its geometric constant term by $a_{C}(0,\cdot)$ . Since all the cusps of $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)$ belong to the ordinary locus $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},0) \subset \overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)$ , one has the following commutative diagram.

(2.24)

Here the horizontal map corresponds to the restriction along the open immersion $\overline{\mathcal{X}_{\mathfrak{c}}}(\mu_{p^n},0) \hookrightarrow \overline{\mathcal{X}_{\mathfrak{c}}}(\mu_{p^n},v)$ at the level of the generic fiber. By the irreducibility of the Igusa tower [Reference HidaHid09, Reference RibetRib75], the special fiber of $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},0) $ is geometrically irreducible and the map (2.23) is injective for $v=0$ ; it then follows from the diagram (2.24) that injectivity also holds for $v\gt0$ after inverting p. Combining these observations with (2.7), we obtain an injective map

$$\mathrm{Ev}_{C}: \mathrm{H}^0(\overline{\mathcal{X}}_{\mathfrak{c}}(v) \times \mathcal{U} ,\omega^{\mathsf{w}_{\mathcal{U}}}) \hookrightarrow \mathrm{H}^0(\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)\times \mathscr{U},\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}}) \otimes_{\mathbf{Z}_p} \mathbf{Q}_p \hookrightarrow \mathcal{O}(\mathscr{U}) [\![ q^\xi ]\!]_{\xi \in M_{+} \cup \{0\}}^{E_{\mathfrak{n} p^n}}\otimes_{\mathbf{Z}_p} \mathbf{Q}_p. $$

Note that these Fourier coefficients are unnormalized, while we use the normalized Fourier coefficient in next subsection. We refer the reader to [Reference Dasgupta, Darmon and PollackDDP11, § 2] or [Reference ShihShi20, § 2] for their relationship.

Recall that for $n\in \mathbf{Z}_{ \geqslant 1}$ , we denote by $\mathscr{P}(\mathfrak{c},p^n)$ the set of cusps of the formal scheme $\mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v)$ . Set $\mathscr{P}_1(\mathfrak{c},p^n)=\mathscr{P}(\mathfrak{c},p^n)/\Delta_{\mathfrak{n}}$ and $\mathcal{O}(\mathcal{U})[\mathscr{P}_1(p^n)]=\bigoplus_{\mathfrak{c}\in \mathscr{C}\!\ell_F^+} \mathcal{O}(\mathcal{U})[\mathscr{P}_1(\mathfrak{c},p^n)]$ . We have $\mathcal{O}(\mathcal{U})[\mathscr{P}_1(\mathfrak{c},p^n)]=\mathcal{O}(\mathcal{U})[\mathscr{P}(\mathfrak{c},p^n)]^{\Delta_{\mathfrak{n}}}$ . By the definition of the space $\mathbf{M}^{\dagger}_{\mathcal{U}}$ and the above discussion, we obtain a morphism

(2.25) \begin{align}\mathrm{res}: \mathbf{M}^{\dagger}_{\mathcal{U}} \to \mathcal{O}(\mathcal{U})[\mathscr{P}_1(p^n)]^{(\phi_p)}, \quad \mathcal{F} \mapsto \sum_{C\in \mathscr{P}_1(p^n)} a_{C}(0,\mathcal{F}) \cdot [C],\end{align}

where $\mathcal{O}(\mathcal{U})[\mathscr{P}_1(p^n)]^{(\phi_p)}$ is the subgroup of $\mathcal{O}(\mathcal{U})[\mathscr{P}_1(p^n)]$ on which $T(\mathbf{Z}/p^{n}\mathbf{Z})$ acts via $\phi_p$ .

Theorem 2.5. One has a Hecke equivariant exact sequence of $\mathcal{O}(\mathcal{U})$ -modules

(2.26) \begin{align}0\to \mathbf{S}^{\dagger}_{\mathcal{U}} \to \mathbf{M}^{\dagger}_{\mathcal{U}} \xrightarrow{\mathrm{res}} \mathcal{O}(\mathcal{U})[\mathscr{P}_1(p^n)]^{(\phi_p)} \to 0.\end{align}

Proof. Using the notation of diagram (2.1), it has been shown in [Reference Andreatta, Iovita and PilloniAIP16, Corollary 3.20] that

$$ \mathrm{R}^1 (h_n\circ \overline{\pi}_n)_{*}(\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}}(-D)) =0. $$

As the rigid analytic space $\mathcal{X}^*_{\mathfrak{c}}(v)$ is an affinoid, it follows from [Reference BoschBos14, § 6.3], that

(2.27) \begin{equation}\mathrm{H}^1(\overline{\mathcal{X}}_{\mathfrak{c}}(\mu_{p^n},v)\times \mathcal{U},\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}}(-D))=\mathrm{R}^1 (h_n\circ \overline{\pi}_n)_{*}(\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}}(-D))(\mathcal{X}^*_{\mathfrak{c}}(v)\times \mathcal{U})=0.\end{equation}

We have an exact sequence

(2.28) \begin{equation}0 \to \mathcal{O}_{\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)\times \mathscr{U}} (-D) \to \mathcal{O}_{\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)\times \mathscr{U}} \to (\mathbf{j} \times \mathbf{1})_{*}(\mathcal{O}_{D \times \mathscr{U}}) \to 0,\end{equation}

where $\mathbf{j}: D \hookrightarrow \overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v) $ is the closed immersion of the divisor with normal crossings at infinity. The following exact sequence is obtained by taking the tensor product of (2.28) with the invertible sheaf $\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}}$ and the projection formula:

(2.29) \begin{equation} 0 \to \omega^{\mathsf{w}_{\mathcal{U}}^{\circ}} (-D) \to \omega^{\mathsf{w}_{\mathcal{U}}^{\circ}} \to (\mathbf{j} \times \mathbf{1})_{*}( (\mathbf{j} \times \mathbf{1})^{*}(\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}} ) ) \to 0. \end{equation}

On the other hand, it follows from (2.21) that

(2.30) \begin{equation}(\mathbf{j} \times \mathbf{1})^{*}(\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}} ) \simeq \mathcal{O}_{D \times \mathscr{U}}.\end{equation}

Finally, (2.27), (2.29) and (2.30) yield the following short exact sequence (we take the generic fiber)

(2.31) \begin{align}0 \to \mathrm{H}^0(\overline{\mathcal{X}}_{\mathfrak{c}}(\mu_{p^n},v) \times \mathcal{U},\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}}(-D))&\to \mathrm{H}^0(\overline{\mathcal{X}}_{\mathfrak{c}}(\mu_{p^n},v) \times \mathcal{U},\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}})\nonumber\\&\to \mathrm{H}^0(\overline{\mathcal{X}}_{\mathfrak{c}}(\mu_{p^n},v) \times \mathcal{U}, (\mathbf{j} \times \mathbf{1})_{*}(\mathcal{O}_{D \times \mathcal{U} }) ) \to 0.\end{align}

Since D is proper over $\mathrm{Spf}(\mathcal{O})$ and $\mathbf{j}$ is affine, it follows that

(2.32) \begin{align}\mathrm{H}^0(\overline{\mathcal{X}}_{\mathfrak{c}}(\mu_{p^n},v) \times \mathcal{U}, (\mathbf{j} \times \mathbf{1})_{*}(\mathcal{O}_{D \times \mathcal{U} }) )&=\mathrm{H}^0(D \times \mathcal{U}, \mathcal{O}_{D \times \mathcal{U}})\nonumber\\&= \mathrm{H}^0(D, \mathcal{O}_{D}) \otimes \mathcal{O}(\mathcal{U}) = \mathcal{O}(\mathcal{U})[\mathscr{P}(\mathfrak{c},p^n)].\end{align}

The desired short exact sequence (2.26) is obtained from (2.31) and (2.32) by taking a direct sum over $\mathscr{C}\!\ell_F^+$ , then a direct limit on v and, finally, $\Delta_{\mathfrak{n}}$ -invariants.

3.1 Eisenstein families

In this section, we recall the definitions of weight 1 Eisenstein series and Eisenstein families with coefficients in the Iwasawa algebra $\varLambda_{\mathcal{O}}=\mathcal{O} [\![ X ]\!]$ . We fix a topological generator $\gamma$ of the Galois group $\mathrm{Gal}(F_{\infty}/F)$ , where $F_{\infty}$ is the cyclotomic $\mathbf{Z}_p$ -extension of F, and we let $u\in 1+p\mathbf{Z}_p$ be such that $\gamma\cdot \zeta=\zeta^u$ for all $\zeta\in \mu_{p^{\infty}}$ . For any integer $k\in \mathbf{Z}_{\geqslant 1}$ , the weight k specialization $\varLambda_{\mathcal{O}}\to \mathcal{O}$ map sends X to $(u^{k-1}-1)$ .

Let $\phi_1$ and $\phi_2$ be two primitive finite order Hecke characters of F having conductors $\mathfrak{n}_1$ and $\mathfrak{n}_2$ , respectively, such that $\phi=\phi_1^{-1}\phi_2$ is totally odd and $\phi_1$ has conductor relatively prime to p. By enlarging the field K, we may assume that its ring of integers $\mathcal{O}$ contains all the values of $\phi_1$ and $\phi_2$ . We assume, in addition, that $\phi\not\equiv\omega_p^{-1}\pmod{\varpi}$ , with $\varpi$ a uniformizer of $\mathcal{O}$ , which is automatically satisfied if $\Sigma_p^{\mathrm{irr}}\ne\varnothing$ . The weight 1 Eisenstein series $E_1(\phi_1,\phi_2)$ has level $\mathfrak{n}_1\mathfrak{n}_2$ and central character $\phi_1\phi_2$ which we will see as a (possibly imprimitive) character of modulus $\mathfrak{n}_1\mathfrak{n}_2$ . Let $\delta(\phi_i)=1$ , if $\mathfrak{n}_i=\mathfrak{o}$ , and 0 otherwise ( $i=1,2$ ). Following [Reference Dasgupta, Darmon and PollackDDP11, Proposition 2.1], the adelic Fourier coefficients of $E_1(\phi_1,\phi_2)$ are given by

(3.1) \begin{equation}\begin{aligned}C(\mathfrak{b},E_1(\phi_1,\phi_2)) & =\sum_{\mathfrak{a}|\mathfrak{b}} \phi_1\left(\frac{\mathfrak{b}}{\mathfrak{a}}\right)\phi_2(\mathfrak{a}),\quad\text{for } (0)\ne \mathfrak{b}\subset \mathfrak{o}, \\C((0),E_1(\phi_1,\phi_2)) & =2^{-d}\sum_{[\mathfrak{c}]\in\mathscr{C}\!\ell_F^+}(\delta(\phi_1)\phi_1^{-1}(\mathfrak{c}\mathfrak{d})L(0,\phi)+\delta(\phi_2)\phi_2^{-1}(\mathfrak{c}\mathfrak{d})L(0,\phi^{-1}))[\mathfrak{c}].\end{aligned}\end{equation}

As $\mathfrak{n}_1$ is relatively prime to p, the p-stabilization $f_{\phi_1}$ of $E_1(\phi_1,\phi_2)$ , defined recursively for all $v\in \Sigma_p$ relatively prime to $\mathfrak{n}_2$ as

$$ E_1(\phi_1,\phi_2)-\phi_2(v) (\begin{smallmatrix} \varpi_v & 0 \\ 0 & 1 \end{smallmatrix})\cdot E_1(\phi_1,\phi_2), $$

is a p-ordinary modular form. Following [Reference WilesWil88, § 1.3] there exists a $\varLambda_{\mathcal{O}}$ -adic Eisenstein series $\mathcal{E}_{\phi_1,\phi_2}$ having adelic Fourier coefficients given by

(3.2) \begin{align}C(\mathfrak{b},\mathcal{E}_{\phi_1,\phi_2}) &=\sum_{\mathfrak{a}|\mathfrak{b}, (p)+\mathfrak{a}=\mathfrak{o}}\phi_1\left(\frac{\mathfrak{b}}{\mathfrak{a}}\right)\phi_2(\mathfrak{a})(1+X)^{{\log_p(\mathrm{N}_{F/\mathbf{Q}}(\mathfrak{a}))}/{\log_p(u)}},\quad\text{for } (0)\ne \mathfrak{b}\subset \mathfrak{o}, \end{align}

(3.3) \begin{align}C((0),\mathcal{E}_{\phi_1,\phi_2}) & =2^{-d} \zeta_{\phi_1,\phi_2}(X)\sum_{[\mathfrak{c}]\in\mathscr{C}\!\ell_F^+} \delta(\phi_1) \phi_1^{-1}(\mathfrak{c}\mathfrak{d}) [\mathfrak{c}]. \end{align}

The constant term of $\mathcal{E}_{\phi_1,\phi_2}$ is related to the imprimitive p-adic zeta function

(3.4) \begin{equation}\zeta_{\phi_1,\phi_2}(X)=\zeta_{\phi}(X)\prod_{\mathfrak{q}|\mathfrak{n}, \mathfrak{q}\,\nmid\, \mathrm{cond}(\phi)}(1-\phi(\mathfrak{q})^{-1}(1+X)^{-({{\log_p(\mathrm{N}_{F/\mathbf{Q}}(\mathfrak{q}))}/{\log_p(u)}})}\mathrm{N}_{F/\mathbf{Q}}(\mathfrak{q})^{-1})\in \varLambda_{\mathcal{O}},\end{equation}

where we recall that $\zeta_{\phi}(X)$ is the p-adic inverse Mellin transform of the Deligne–Ribet p-adic L-function $L_p(s,\phi\omega_p)$ satisfying the interpolation property that, for all positive integers k, one has

(3.5) \begin{equation} \zeta_{\phi}(u^{k-1}-1)=L_p(1-k,\phi\omega_p)=L(1-k,\phi\omega_p^{1-k})\cdot \prod_{ \mathfrak{q}\in\Sigma_p} (1-\phi\omega_p^{1-k}(\mathfrak{q}) \mathrm{N}_{F/\mathbf{Q}}(\mathfrak{q})^{k-1}).\end{equation}

One sees that the weight 1 specialization of $\mathcal{E}_{\phi_1,\phi_2}$ is $f_{\phi_1}$ . As the Eisenstein family $\mathcal{E}_{\phi_1,\phi_2}$ is p-ordinary, the argument of [Reference PilloniPil13, § 6] shows that for any open admissible affinoid $\mathcal{U}$ of $\mathcal{W}^{\parallel}$ , the push-forward of the Eisenstein family $\mathcal{E}_{\phi_1,\phi_2}$ along the natural inclusion $\Lambda_{\mathcal{O}} \hookrightarrow \mathcal{O}(\mathcal{U})$ belongs to the module $\mathbf{M}^{\dagger}_{\mathcal{U}}$ introduced in (2.16), for $\mathfrak{n}$ the prime-to-p part of $\mathfrak{n}_1\mathfrak{n}_2$ and n such that $\mathfrak{n}_1\mathfrak{n}_2$ divides $\mathfrak{n} p^n$ .

3.2 Eigenvarieties

We recall that a Banach $\mathcal{O}(\mathcal{U})$ -module P satisfies the property $(\mathbf{Pr})$ if there is a Banach $\mathcal{O}(\mathcal{U})$ -module Q such that $P \oplus Q$ is potentially orthonormalizable [Reference BuzzardBuz07, § 2].

Proof. We recall that $(\overline{\pi}_n)_* (\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}})$ is a line bundle on $\mathscr{X}^*_{\mathfrak{c}}(\mu_{p^n},v) \times \mathscr{U}$ by Proposition 2.2. We choose a finite affinoid cover $\{ \mathcal{B}_i \}_{1\leqslant i \leqslant r}$ of $\mathcal{X}^*_{\mathfrak{c}}(\mu_{p^n},v)$ such that $(\overline{\pi}_n)_* (\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}})_{\mid \mathcal{B}_i\times\mathcal{U} }$ is a free sheaf of rank 1 for every i, and we consider the augmented Čech complex associated to that line bundle:

$$0 \to M_0=\mathrm{H}^0(\mathcal{X}^*_{\mathfrak{c}}(\mu_{p^n},v) \times \mathcal{U}, (\overline{\pi}_n)_* (\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}})) \to M_1 \to \cdots \to M_r \to 0.$$

This complex is exact because $h_n:\mathcal{X}^*_{\mathfrak{c}}(\mu_{p^n},v) \to \mathcal{X}^*_{\mathfrak{c}}(v)$ is finite and $\mathcal{X}^*_{\mathfrak{c}}(v)$ is an affinoid (so it has no cohomology in degree $\geqslant\!1$ by [Reference BoschBos14, § 6]), hence $M_0$ satisfies $(\mathbf{Pr})$ by [Reference PilloniPil13, Lemma 5.2]. We deduce that both $\mathbf{M}^{\dagger}_{\mathcal{U}}(v)$ and $\mathbf{S}^{\dagger}_{\mathcal{U}}(v)$ satisfy $(\mathbf{Pr})$ as direct $\mathcal{O}(\mathcal{U})$ -factors in $\bigoplus_{\mathfrak{c} \in \mathscr{C}\!\ell_F^+} \mathrm{H}^0(\mathcal{X}^*_{\mathfrak{c}}(\mu_{p^n},v) \times \mathcal{U}, (\overline{\pi}_n)_* (\omega^{\mathsf{w}_{\mathcal{U}}^{\circ}}))$ .

Buzzard’s eigenvariety machine [Reference BuzzardBuz07, § 5] then allows us to associate an eigencurve $\mathcal{C}$ to the data

$$(\mathcal{O}_{\mathcal{U}}, \mathbf{M}^{\dagger}_{\mathcal{U}}, \mathbf{Z}[T_{\mathfrak{l}}, U_{\mathfrak{p}}: \mathfrak{l} \,\nmid\, \mathfrak{n} p, \mathfrak{p}\in \Sigma_p]),$$

where the affinoids $\mathcal{U}$ form an admissible cover of $\mathcal{W}^{\parallel}$ such that for v sufficiently small the $\mathcal{O}(\mathcal{U})$ -module $\mathbf{M}^{\dagger}_{\mathcal{U}}(v)$ admits a slope decomposition with respect to the compact operator $U_p$ . Let $\mathbf{M}^{ \leqslant s}_{\mathcal{U}}(v)$ denote the projective finite type $\mathcal{O}(\mathcal{U})$ -module of elements of slope at most $s\in \mathbf{Q}_{ \geqslant 0}$ and consider the finite and flat affinoid $\mathcal{O}(\mathcal{U})$ -algebra $\mathcal{T}^{ \leqslant s}_{\mathcal{U}}$ defined as the image of $\mathcal{O}(\mathcal{U})[T_{\mathfrak{l}}, U_{\mathfrak{p}}: \mathfrak{l} \,\nmid\, \mathfrak{n} p, \mathfrak{p} \in \Sigma_p]$ in $\mathrm{End}_{\mathcal{O}(\mathcal{U})}(\mathbf{M}^{ \leqslant s}_{\mathcal{U}}(v))$ . The Hilbert eigencurve $\mathcal{C}$ , obtained by glueing together $\mathrm{Spm}(\mathcal{T}^{ \leqslant s}_{\mathcal{U}})$ above the spectral curve for $U_p$ , is reduced and endowed with a locally finite flat morphism $\mathsf{w}: \mathcal{C} \rightarrow \mathcal{W}^{\parallel}$ called the weight map.

Similarly, we obtain a cuspidal Hilbert eigencurve $\mathsf{w}: \mathcal{C}_{\mathrm{cusp}} \rightarrow \mathcal{W}^{\parallel}$ which is locally finite and flat and a Hilbert cuspidal eigenvariety $\kappa:\mathcal{E}\to \mathcal{W}\times\mathcal{W}^{\parallel}$ of dimension $d+1$ which is locally finite and torsion-free. The ordinary locus (i.e., the locus where the $U_p$ -slope is zero) of $\mathcal{C}$ (respectively, $\mathcal{C}_{\mathrm{cusp}}$ and $\mathcal{E}$ ) has a formal model given by the ordinary Hida Hecke algebra $\mathfrak{h}$ (see [Reference HidaHid89]) (respectively, cuspidal ordinary Hecke algebra $\mathfrak{h}^{\mathrm{cusp}}$ and the nearly-ordinary Hecke algebra $\mathfrak{h}^{\mathrm{no}}$ ) of level $\mathfrak{n} p^\infty$ constructed using the Hecke operators $\{T_{\mathfrak{l}}, U_{\mathfrak{p}}: \mathfrak{l} \,\nmid\, \mathfrak{n} p, \mathfrak{p}\in \Sigma_p\}$ . By construction of $\mathcal{C}$ , there exist bounded global sections $\{ T_{\mathfrak{l}},U_{\mathfrak{p}}\}_{\mathfrak{l} \,\nmid\,\mathfrak{n} p, \mathfrak{p}\in \Sigma_p}$ . A fundamental tool in the study of the geometry of $\mathcal{C}$ is the universal two-dimensional pseudo-character

(3.6) \begin{equation}\tau_{\mathcal{C}}:\mathrm{G}_{F} \rightarrow \mathcal{O}_{\mathcal{C}}(\mathcal{C}),\end{equation}

which is unramified at all primes $\mathfrak{l}\,\nmid\, p\mathfrak{n}$ and maps an arithmetic Frobenius $\mathrm{Frob}_{\mathfrak{l}}$ to $T_{\mathfrak{l}}$ (see [Reference ChenevierChe04, Proposition 7.1.1]). While $\tau_{\mathcal{C}}$ interpolates p-adically the traces of semi-simple p-adic Galois representations attached to the classical points of $\mathcal{C}$ which are De Rham at p, the semi-simple p-adic Galois representation attached to an arbitrary specialization of $\tau_{\mathcal{C}}$ is only trianguline at $v\in\Sigma_p$ and $U_{v}$ interpolates the crystalline period (see, for example, [Reference Barrera, Dimitrov and JorzaBDJ22, § 5.3]). The above discussion also holds if one replaces $\mathcal{C}$ by $\mathcal{E}$ . Let $\mathcal{C}^{\mathrm{full}}$ (respectively, $\mathcal{E}^{\mathrm{full}}$ ) be the p-adic eigencurve (respectively, eigenvariety) over $\mathcal{W}^{\parallel}$ (respectively, $\mathcal{W}\times\mathcal{W}^{\parallel}$ ) of tame level $\mathfrak{n}$ constructed by the data

$$(\mathcal{O}_{\mathcal{U}}, \mathbf{M}^{\dagger}_{\mathcal{U}}, \mathbf{Z}[T_{\mathfrak{l}}, U_{\mathfrak{p}}, U_{\mathfrak{q}} : \mathfrak{l} \,\nmid\, \mathfrak{n} p, \mathfrak{p}\in \Sigma_p, \mathfrak{q} \mid \mathfrak{n} ]), $$

It is endowed with a locally finite surjective morphism $\mathcal{C}^{\mathrm{full}}\to \mathcal{C}$ (respectively, $\mathcal{E}^{\mathrm{full}}\to \mathcal{E}$ ). We recall that $\mathfrak{n}_1$ is relatively prime to p. The following proposition asserts that these surjective morphisms are indeed isomorphisms locally at (not necessarily classical) intersection points of $\mathcal{E}_{\phi_1,\phi_2}$ with $\mathcal{C}_{\mathrm{cusp}}$ .

For $\mathcal{P}$ as in the above proposition, we denote by $\varLambda_{\mathsf{w}(\mathcal{P})}$ (respectively $\mathcal{T}_{\mathcal{P}}$ and $\mathcal{T}_{\mathcal{P}}^{\mathrm{cusp}}$ ) the complete localization of $\varLambda_{\mathcal{O}}[1/p]$ (respectively $\mathcal{C}^{\mathrm{full}}$ and $\mathcal{C}^{\mathrm{cusp},\mathrm{full}}$ ) at $\mathsf{w}(\mathcal{P})$ (respectively at $\mathcal{P}$ ). Set $\mathbf{M}^{\mathrm{ord}}_{\mathcal{U}}=\mathbf{M}_{\mathcal{U}}^{\dagger, \leqslant 0}$ and $\mathbf{S}^{\mathrm{ord}}_{\mathcal{U}}=\mathbf{S}_{\mathcal{U}}^{\dagger, \leqslant 0}$ , and let $\mathbf{S}^{\mathrm{ord}}_{\mathcal{P}}=\mathbf{S}^{\mathrm{ord}}_{\mathcal{U}} \widehat{\otimes}_{\mathcal{T}^{\mathrm{cusp},\leqslant 0}_{\mathcal{U}}} \mathcal{T}_{\mathcal{P}}^{\mathrm{cusp}} $ and $\mathbf{M}^{\mathrm{ord}}_{\mathcal{P}}=\mathbf{M}^{\mathrm{ord}}_{\mathcal{U}} \widehat{\otimes}_{\mathcal{T}^{\leqslant 0}_{\mathcal{U}}} \mathcal{T}_{\mathcal{P}}$ . By [Reference HidaHid93, Theorem 5] (or [Reference HidaHid04, Theorem 4.30]), we have the following perfect pairing

(3.7) \begin{equation}\mathcal{T}_{\mathcal{P}}^{\mathrm{cusp}} \times \mathbf{S}^{\mathrm{ord}}_{\mathcal{P}}\to \varLambda_{\mathsf{w}(\mathcal{P})} \quad(h,\mathcal{F})\mapsto C(\mathfrak{o},h\cdot \mathcal{F}).\end{equation}

3.3 Constant terms of Eisenstein families

Applying Hida’s ordinary idempotent $e=\displaystyle\lim_{n\to\infty} U_p^{n!}$ to (2.26) yields a short exact sequence

(3.8) \begin{equation}0\to \mathbf{S}^{\mathrm{ord}}_{\mathcal{U}} \to \mathbf{M}^{\mathrm{ord}}_{\mathcal{U}} \xrightarrow{\mathrm{res}} e\cdot \mathcal{O}(\mathcal{U})[\mathscr{P}_1(p^n)]^{(\phi_p)} \to 0.\end{equation}

We let $(e_1,e_2)$ be the canonical basis of $F^2$ . The group $\mathrm{GL}_2(F)$ acts on the left on $F^2$ by

$$g \ast (x,y)= (x,y) \cdot g^{-1} \quad \text{for all }g \in \mathrm{GL}_2(F). $$

This action extends to an action of $\mathrm{GL}_2(\mathbf{A}_{F,f})$ on $\mathbf{A}_{F,f}^2$ . As we showed in § 2.7, a cusp of level $\Gamma_1(\mathfrak{n} p^n)$ corresponds to two fractional ideals $\mathfrak{a}$ and $\mathfrak{b}$ with extra data (see [Reference DimitrovDim04, Definition 3.1] for more details). In fact, we can associate to a cusp an $\mathfrak{o}$ -lattice L of $F^2$ (see [Reference DimitrovDim04, § 3]) and $\mathrm{GL}_2(\mathbf{A}_{F,f})$ acts on such lattices (hence acts on the cusps) as follows

$$g \ast L = F^2 \cap (L \cdot g^{-1}) \quad \text{for all } g \in \mathrm{GL}_2(\mathbf{A}_{F,f}). $$

In particular, $\mathfrak{b}= g \ast L \cap (F\cdot e_1)$ and $ \mathfrak{a}^{-1}\mathfrak{d}^{-1}= g \ast L \cap (F\cdot e_2)$ . For $[\mathfrak{c}]\in \mathscr{C}\!\ell_F^+$ , we let $\infty(\mathfrak{c})=( \mathfrak{o} \oplus \mathfrak{d}^{-1}\mathfrak{c}^{-1}, \alpha_{\mathfrak{n} p^n}: \mathfrak{c} /\mathfrak{n} p^n \mathfrak{c} \simeq \mathfrak{o}/\mathfrak{n} p^n )$ be the unramified cusp in the sense of [Reference DimitrovDim04, Definition 3.2] associated to the pair of ideals $(\mathfrak{a},\mathfrak{b}) = (\mathfrak{c}, \mathfrak{o})$ . As $\mathrm{GL}_2(\mathbf{A}_{F,f}) $ acts transitively, it follows that the set of cusps of level $\Gamma_1(\mathfrak{n} p^n)$ can be identified with the orbit of $\infty=\infty(\mathfrak{d}^{-1})$ given, up to isomorphisms, by

(3.9) \begin{equation}B(F)^+ \backslash \mathrm{GL}_2(\mathbf{A}_{F,f}) / \Gamma_1(\mathfrak{n}p^n),\end{equation}

where $B(F)^+$ is the group of upper triangular matrices having a totally positive determinant.

Recall that $\mathfrak{n}_i=\mathrm{cond}(\phi_i)$ for $i=1,2$ . As before, we denote by $\mathfrak{n}$ the prime-to-p part of $\mathfrak{n}_1\mathfrak{n}_2$ and let n be the smallest positive integer such that $\mathfrak{n}_1\mathfrak{n}_2$ divides $\mathfrak{n} p^n$ . Denote by $\mathfrak{n}'_1$ the prime-to- $\mathfrak{n}_2$ part of $\mathfrak{n}_1$ and denote by $\mathfrak{n}'_2$ the prime-to- $\mathfrak{n}_1$ part of $\mathfrak{n}_2$ . For any $\alpha\in \mathbf{A}_{F,f}$ and for $i=1,2$ , we denote by $\alpha_{\mathfrak{n}_i}$ the $\mathfrak{n}_i$ -part of $\alpha$ . We next recall the constant terms of $E_k(\phi_1,\phi_2)$ for all $k\geqslant 2$ computed in [Reference ShihShi20, § 3] (see also [Reference Dasgupta and KakdeDK21]). In [Reference ShihShi20], the Eisenstein series $E_k(\phi_1,\phi_2)$ is a member of a family of adelic Eisenstein series evaluated at $s=({1-k})/{2}$ (see also [Reference BumpBum97, § 3.7]). This family is defined via explicit choices of local sections $f_{\phi_1,\phi_2,s,v}$ in the normalized induced representations $\mathrm{Ind}_{B(F_v)}^{\mathrm{GL}_2(F_v)} (\phi_2|\cdot|^s_v,\phi_1|\cdot|_v^{-s})$ for v a place of F (see [Reference ShihShi20, § 3.4]). We consider a specific value of the local intertwining operator (see [Reference BumpBum97, (7.22)])

$$M_{\phi_1,\phi_2,v,s}=\int_{F_v} f_{\phi_1,\phi_2,s,v}\bigg(\bigg(\begin{matrix}0 & -1 \\ 1 & x \end{matrix}\bigg)\bigg(\begin{matrix}1 & 0 \\ \varpi_v^{\mathrm{val}_v(\mathfrak{n}_2)} & 1 \end{matrix}\bigg)\bigg)\, dx. $$

1. (i) $\mathrm{val}_v(c)\geqslant \mathrm{val}_v(\mathfrak{n}_2)$ for all $v\mid \mathfrak{n}'_2$ ;

2. (ii) $\mathrm{val}_v(c)=0$ for all $v\mid \mathfrak{n}'_1$ ;

3. (iii) $\mathrm{val}_v(c)=\mathrm{val}_v(\mathfrak{n}_2)$ for all v dividing both $\mathfrak{n}_1$ and $\mathfrak{n}_2$ ;

in which case, letting $\varepsilon_v(s,\phi_{1,v},\psi_v)$ denote the local epsilon factor, it is given by

(3.10) \begin{align}&\phi_1(t_1)\phi_2(t_2) \mathrm{N}_{F/\mathbf{Q}}\bigg(\frac{t_1}{t_2}\bigg)^{-k/2}\phi_2^{-1}(\mathfrak{n}_1')\mathrm{N}_{F/\mathbf{Q}}(\mathfrak{n}_1')^{-1}\phi_{2}(d_{\mathfrak{n}_2}\hat{\mathfrak{o}})\phi_1^{-1}\bigg(\frac{-c_{\mathfrak{n}_1}}{\mathfrak{n}_2'}\bigg) \nonumber\\&\quad \times L^{(\mathfrak{n})}(1-k,\phi)\cdot \prod_{v|\mathfrak{n}_1,v|\mathfrak{n}_2} M_{\phi_1,\phi_2,v,(1-k)/2}\prod_{v|\mathfrak{n}_1}\varepsilon_v(2-k,\phi_{1,v},\psi_v)^{-1}.\end{align}

3.4 Hecke duality and Proof of Theorem B

The Eisenstein ideal $\mathcal{J}_{\phi_1,\phi_2}\subset \mathfrak{h}^{\mathrm{cusp}}$ associated to $\mathcal{E}_{\phi_1,\phi_2}$ is the image of the kernel of the $\mathcal{E}_{\phi_1,\phi_2}$ -projection $\mathfrak{h} \to \varLambda_{\mathcal{O}}$ via the natural surjection $\mathfrak{h} \twoheadrightarrow \mathfrak{h}^{\mathrm{cusp}}$ of the Hida Hecke algebra introduced in § 3.2. The congruence ideal $\mathcal{I}_{\phi_1,\phi_2}\subset \varLambda_{\mathcal{O}}$ of the Eisenstein family $\mathcal{E}_{\phi_1,\phi_2}$ is uniquely characterized by the following commutative diagram of $ \varLambda_{\mathcal{O}}$ -algebras.

(3.11)

We next establish a precise relation between the congruence ideal and the constant term of $\mathcal{E}_{\phi_1,\phi_2}$ . Previously, it was only known by the work of Wiles [Reference WilesWil90, Theorem 4.1] that, after inverting p and the height one primes corresponding to trivial zeros, $(\zeta_{\phi_1,\phi_2})$ divides $ \mathcal{I}_{\phi_1,\phi_2}$ .

Proof. As $\varLambda_{\mathcal{O}}[1/p]$ is a Dedekind domain, it suffices to show equality after localization at an arbitrary maximal ideal $\mathcal{P}$ . Localizing the isomorphism in (3.11) at $\mathcal{P}$ , we obtain

(3.12) \begin{equation} {\mathcal{T}^{\mathrm{cusp}}_{\mathcal{P}}}/{\mathcal{J}_{\phi_1,\phi_2,\mathcal{P}}} \simeq {\varLambda_{\mathsf{w}(\mathcal{P})}}/{\mathcal{I}_{\phi_1,\phi_2,\mathcal{P}}}.\end{equation}

Choose a sufficiently small open admissible affinoid $\mathcal{U}$ of $\mathcal{W}^{\parallel}$ containing $\mathcal{P}$ . The image of $\mathcal{E}_{\phi_1,\phi_2}$ under the map res from (3.8) is contained in a free of rank 1 direct summand in $ \mathcal{O}(\mathcal{U})[\mathscr{P}_1(p^n)]^{(\phi_p)}$ having $\mathcal{O}(\mathcal{U})$ -basis $c_{\phi_1,\phi_2}$ . The module $\mathbf{M}_{\phi_1,\phi_2}=\mathrm{res}^{-1}(\mathcal{O}(\mathcal{U})\cdot c_{\phi_1,\phi_2})$ is Hecke stable and fits in a short exact sequence of free $\mathcal{O}(\mathcal{U})$ -modules

$$0\to \mathbf{S}^{\mathrm{ord}}_{\mathcal{U}} \to \mathbf{M}_{\phi_1,\phi_2} \xrightarrow{\mathrm{res}} \mathcal{O}(\mathcal{U})\cdot c_{\phi_1,\phi_2} \to 0. $$

Recall that $\mathrm{Spec}(\varLambda_{\mathcal{O}})$ is the irreducible component of $\mathrm{Spec}(\mathfrak{h})$ corresponding to $\mathcal{E}_{\phi_1,\phi_2}$ . Taking the complete localization of the above short exact sequence at $\mathcal{P}$ viewed as a maximal ideal of $\mathcal{T}^{\leqslant 0}_{\mathcal{U}}$ , we obtain a short exact sequence of free $\varLambda_{\mathsf{w}(\mathcal{P})}$ -modules

(3.13) \begin{equation}0\to \mathbf{S}^{\mathrm{ord}}_{\mathcal{U},\mathcal{P}} \to \mathbf{M}_{\phi_1,\phi_2,\mathcal{P}} \xrightarrow{\mathrm{res}} \varLambda_{\mathsf{w}(\mathcal{P})}\cdot c_{\phi_1,\phi_2} \to 0.\end{equation}

If $\zeta_{\phi_1,\phi_2}\notin \mathcal{P}$ , then $ (\zeta_{\phi_1,\phi_2}) =\varLambda_{\mathsf{w}(\mathcal{P})}\supset\mathcal{I}_{\phi_1,\phi_2,\mathcal{P}}$ . Otherwise, as $\mathrm{res}(\mathcal{E}_{\phi_1,\phi_2})\in \zeta_{\phi_1,\phi_2} \cdot \varLambda_{\mathsf{w}(\mathcal{P})}^\times \cdot c_{\phi_1,\phi_2}$ by Proposition 3.7, there exists a $\varLambda_{\mathsf{w}(\mathcal{P})}$ -adic cuspform $\mathcal{H}\in \mathbf{S}^{\mathrm{ord}}_{\mathcal{U},\mathcal{P}}$ congruent to $\mathcal{E}_{\phi_1,\phi_2}$ modulo $\zeta_{\phi_1,\phi_2}$ . Using (3.7), one obtains a surjective $\varLambda_{\mathsf{w}(\mathcal{P})}$ -algebra homomorphism

(3.14) \begin{equation}{\mathcal{T}^{\mathrm{cusp}}_{\mathcal{P}}}/{\mathcal{J}_{\phi_1,\phi_2,\mathcal{P}}}\twoheadrightarrow{\varLambda_{\mathsf{w}(\mathcal{P})}}/{(\zeta_{\phi_1,\phi_2})}, \quad h\mapsto C(\mathfrak{o},h\cdot \mathcal{H}) \mod \zeta_{\phi_1,\phi_2}.\end{equation}

In view of (3.12), this precisely shows the inclusion of principal ideals $(\zeta_{\phi_1,\phi_2}) \supset \mathcal{I}_{\phi_1,\phi_2,\mathcal{P}} $ in the discrete valuation ring $ \varLambda_{\mathsf{w}(\mathcal{P})}$ . It remains to show the opposite inclusion.

Observe that the freeness of $\mathcal{T}^{\mathrm{cusp}}_{\mathcal{P}} $ as $\varLambda_{\mathsf{w}(\mathcal{P})}$ -module allows us to lift the isomorphism (3.12) to a morphism $\mathcal{T}^{\mathrm{cusp}}_{\mathcal{P}} \to \varLambda_{\mathsf{w}(\mathcal{P})}$ of $\varLambda_{\mathsf{w}(\mathcal{P})}$ -modules (though not necessarily to an algebra homomorphism). Using (3.7), this lift gives rise to a $\varLambda_{\mathsf{w}(\mathcal{P})}$ -adic cuspform $\mathcal{G}$ (not necessarily an eigenform) with Fourier coefficients in $\varLambda_{\mathsf{w}(\mathcal{P})}$ , such that $C(\mathfrak{m},\mathcal{G}) \equiv C(\mathfrak{m},\mathcal{E}_{\phi_1,\phi_2}) \mod \mathcal{I}_{\phi_1,\phi_2,\mathcal{P}}$ for all non-zero ideals $\mathfrak{m} \subset \mathfrak{o}$ . As this does not provide in general a congruence between the constant terms, we are led to split the remainder of the argument in two cases. As $\mathcal{I}_{\phi_1,\phi_2,\mathcal{P}} \subset (\zeta_{\phi_1,\phi_2})$ , one can define a $\varLambda_{\mathsf{w}(\mathcal{P})}$ -adic family

(3.15) \begin{equation} \mathcal{F}_0=\frac{\mathcal{E}_{\phi_1,\phi_2}- \mathcal{G}}{2^{-d} \zeta_{\phi_1,\phi_2}} \in \mathbf{M}_{\phi_1,\phi_2,\mathcal{P}}, \end{equation}

which provides a section of the map res in (3.13), i.e., $\mathbf{M}_{\phi_1,\phi_2,\mathcal{P}}=\mathbf{S}^{\mathrm{ord}}_{\mathcal{U},\mathcal{P}} \oplus \varLambda_{\mathsf{w}(\mathcal{P})}\cdot \mathcal{F}_0$ as $ \varLambda_{\mathsf{w}(\mathcal{P})} $ -module.

Assume first that $\mathfrak{n}_1\ne\mathfrak{o}$ . In this case (3.3) yields $C((0),\mathcal{E}_{\phi_1,\phi_2}) =0$ , hence $\mathcal{E}_{\phi_1,\phi_2} \equiv \mathcal{G} \mod \mathcal{I}_{\phi_1,\phi_2,\mathcal{P}}$ by the q-expansion principle. Consequently, $\mathrm{res}(\mathcal{E}_{\phi_1,\phi_2})\in \mathcal{I}_{\phi_1,\phi_2,\mathcal{P}}\cdot c_{\phi_1,\phi_2}$ which combined with $\mathrm{res}(\mathcal{E}_{\phi_1,\phi_2})\in \zeta_{\phi_1,\phi_2} \cdot \varLambda_{\mathsf{w}(\mathcal{P})}^\times \cdot c_{\phi_1,\phi_2}$ from Proposition 3.7, implies that $(\zeta_{\phi_1,\phi_2})\subset \mathcal{I}_{\phi_1,\phi_2,\mathcal{P}}$ .

Suppose now that $\mathfrak{n}_1=\mathfrak{o}$ . Given $[\mathfrak{c}]\in\mathscr{C}\!\ell_F^+$ , it follows from (3.3) that the value of $\mathcal{E}_{\phi_1,\phi_2}$ at the ‘infinity’ cusp $\infty(\mathfrak{c})$ of $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)$ equals $2^{-d} \zeta_{\phi_1,\phi_2}(X) \phi_1^{-1}(\mathfrak{c}\mathfrak{d})\ne 0$ (see § 3.3). Then $ \mathcal{F}_0$ has constant term 1 at $\infty=\infty(\mathfrak{d}^{-1})$ and its specialization $F_{\mathcal{P}}$ in weight $\mathsf{w}(\mathcal{P})$ lies the generalized eigenspace $\mathbf{M}^{\dagger}_{\mathsf{w}(\mathcal{P})}[\![ E_{\mathcal{P}} ]\!]$ attached to the specialization $E_{\mathcal{P}}$ of $\mathcal{E}_{\phi_1,\phi_2}$ in weight $\mathsf{w}(\mathcal{P})$ . In particular, for any prime $\mathfrak{q}\,\nmid\, \mathfrak{n} p$ the operator $S_{\mathfrak{q}}$ acts on $F_{\mathcal{P}}\!\!\mod\! \varpi$ as $(\phi_1\phi_2 \omega_p)(\mathfrak{q})$ . The operator $S_{\mathfrak{q}}$ defined in [Reference Andreatta, Iovita and PilloniAIP16, p. 24] is renormalized by the factor $\mathrm{N}_{F/\mathbb{Q}}(\mathfrak{q})^{-2}$ to match the determinant formula of the p-adic Galois representations described in [Reference Andreatta, Iovita and PilloniAIP16, Theorem 5.1]. Throughout this work (as in [Reference De MariaDeM20, p. 44]) we do not renormalize $S_{\mathfrak{q}}$ .

If the inclusion of principal ideals $\mathcal{I}_{\phi_1,\phi_2,\mathcal{P}} \subset (\zeta_{\phi_1,\phi_2}) $ of $\varLambda_{\mathsf{w}(\mathcal{P})}$ was strict, then:

1. (i) $C(\mathfrak{m},F_{\mathcal{P}}) =0$ for all non-zero ideals $\mathfrak{m} \subset \mathfrak{o}$ ; and

2. (ii) for $[\mathfrak{c}]\in\mathscr{C}\!\ell_F^+$ , the value of $F_{\mathcal{P}}$ at the cusp $\infty(\mathfrak{c})$ of $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^n},v)$ equals $\phi_1^{-1}(\mathfrak{c}\mathfrak{d})$ .

Inspired by the work of Emerton [Reference EmertonEme99, Proposition 1] when $F=\mathbf{Q}$ , further extended by Lafferty [Reference LaffertyLaf19, p. 766], we now provide a contradiction using a geometrically flavored argument based on the irreducibility of the Igusa tower $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^\infty},0)$ . As the special fiber $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^\infty},0)_1$ is geometrically irreducible by a theorem of Ribet [Reference RibetRib75] (see also [Reference HidaHid09]), the q-expansion principle shows that $F_{\mathcal{P}}$ coincides with its constant term $C((0),F_{\mathcal{P}})$ seen as a modular form of weight 0 (see [Reference Andreatta, Iovita and StevensAIS14, Proposition 4.2] for more details). Hence, by the definition of the operator $S_{\mathfrak{q}}$ , the value of $S_{\mathfrak{q}}(C((0),F_{\mathcal{P}}))$ on $\overline{\mathscr{X}_{\mathfrak{c}}}(\mu_{p^\infty},0)$ is exactly the value of $C((0),F_{\mathcal{P}})$ on $\overline{\mathscr{X}_{\mathfrak{c}\mathfrak{q}^{-2}}}(\mu_{p^\infty},0)$ . Combining this with the fact that $S_{\mathfrak{q}}$ acts on $F_{\mathcal{P}} \mod \varpi$ as $(\phi_1\phi_2 \omega_p)(\mathfrak{q})$ , we find

$$ (\phi_1\phi_2 \omega_p)(\mathfrak{q})\cdot \phi_1^{-1}(\mathfrak{c}\mathfrak{d})\equiv \phi_1^{-1}(\mathfrak{c}\mathfrak{q}^{-2}\mathfrak{d}) \pmod{\varpi}, $$

that is, $\phi=\phi_2 \phi_1^{-1}\equiv \omega_p^{-1}\pmod{\varpi}$ . This contradicts our running assumption on $\phi$ . Hence, we conclude that $\mathcal{I}_{\phi_1,\phi_2,\mathcal{P}} = (\zeta_{\phi_1,\phi_2}) $ .

The techniques employed in the above proof, namely the construction of the $\varLambda_{\mathsf{w}(\mathcal{P})}$ -adic form $\mathcal{F}_0$ in (3.15), have the following important consequence.

Proposition 3.9. Let $\mathcal{P}$ be a maximal ideal of the ordinary Hecke algebra $\mathfrak{h}[1/p]$ corresponding to an Eisenstein intersection point of $\mathcal{E}_{\phi_1,\phi_2}$ and $\mathcal{C}_{\mathrm{cusp}}$ , and let $\mathcal{T}_{\phi_1,\phi_2,\mathcal{P}}$ be the quotient of $\mathcal{T}_{\mathcal{P}}$ acting faithfully on the $\varLambda_{\mathsf{w}(\mathcal{P})}$ -module $\mathbf{M}_{\phi_1,\phi_2,\mathcal{P}} $ in (3.13). Then, one has an isomorphism of $\varLambda_{\mathsf{w}(\mathcal{P})}$ -modules

(3.16) \begin{equation}\mathcal{T}_{\phi_1,\phi_2,\mathcal{P}} \simeq \mathcal{T}_{\mathcal{P}}^{\mathrm{cusp}} \times_{\varLambda_{\mathsf{w}(\mathcal{P})}/(\zeta_{\phi_1,\phi_2})} \varLambda_{\mathsf{w}(\mathcal{P})},\end{equation}

and a perfect pairing of $\Lambda_{\mathsf{w}(\mathcal{P})}$ -modules

(3.17) \begin{equation}\mathcal{T}_{\phi_1,\phi_2,\mathcal{P}}\times \mathbf{M}_{\phi_1,\phi_2,\mathcal{P}} \to \varLambda_{\mathsf{w}(\mathcal{P})}, \quad (h,\mathcal{G})\mapsto C(\mathfrak{o},h\cdot \mathcal{G}).\end{equation}

In particular, there exists an element $h_0 \in \mathcal{T}_{\phi_1,\phi_2,\mathcal{P}}$ such that for any $\mathcal{G} \in \mathbf{M}_{\phi_1,\phi_2,\mathcal{P}}$ , we have

$$C(\mathfrak{o},h_0 \cdot \mathcal{G}) \cdot \sum_{[\mathfrak{c}]\in\mathscr{C}\!\ell_F^+} \phi_1^{-1}(\mathfrak{c}\mathfrak{d}) [\mathfrak{c}] = C((0), \mathcal{G}).$$

We assume first that $\mathfrak{n}_1 \ne \mathfrak{o}$ . By Nakayama’s lemma, to show the perfectness of the pairing (3.17) of free $\varLambda_{\mathsf{w}(\mathcal{P})}$ -modules, it suffices to establish that it is right injective residually and left injective. As $C((0),\mathcal{E}_{\phi_1,\phi_2}) = 0$ by (3.3) we deduce that $C((0),\mathcal{G}) = 0$ for all $\mathcal{G} \in \mathbf{M}_{\phi_1,\phi_2,\mathcal{P}}$ . The left injectivity in (3.17) then follows from q-expansion principle and from the formula $C(\mathfrak{o} , h\cdot (T_{\mathfrak{q}} \cdot \mathcal{G})) = C(\mathfrak{q}, h\cdot \mathcal{G})$ , where $\mathfrak{q}$ is any non-zero integral ideal of $\mathfrak{o}$ (see [Reference HidaHid04, (4.67)]), since by definition $\mathcal{T}_{\phi_1,\phi_2,\mathcal{P}}$ acts faithfully on $\mathbf{M}_{\phi_1,\phi_2,\mathcal{P}}$ . For right injectivity of the reduction of (3.17) modulo the maximal ideal of $\varLambda_{\mathsf{w}(\mathcal{P})}$ , we use again the vanishing of the constant term, the q-expansion principle and the formula $C(\mathfrak{o} , T_{\mathfrak{q}} \cdot \overline{\mathcal{G}}) = C(\mathfrak{q}, \overline{\mathcal{G}})$ for $\overline{\mathcal{G}}$ in the generalized eigenspace $\mathbf{M}_{\phi_1,\phi_2,\mathcal{P}}\otimes_{\varLambda_{\mathrm{\mathsf{w}}(\mathcal{P})}}\overline{\mathbf{Q}}_p$ .

If $\mathfrak{n}_1=\mathfrak{o}$ , we consider a $\varLambda_{\mathsf{w}(\mathcal{P})}$ -basis of $\mathbf{M}_{\phi_1,\phi_2,\mathcal{P}}$ adapted to the direct sum decomposition $\mathbf{S}^{\mathrm{ord}}_{\mathcal{U},\mathcal{P}} \oplus \varLambda_{\mathsf{w}(\mathcal{P})}\cdot \mathcal{F}_0$ and a $\varLambda_{\mathsf{w}(\mathcal{P})}$ -basis of $\mathcal{T}_{\phi_1,\phi_2,\mathcal{P}}$ obtained by completing a lift of a basis of $\mathcal{T}_{\mathcal{P}}^{\mathrm{cusp}}$ by the element $h_0=(0,2^{-d} \zeta_{\phi_1,\phi_2})$ which belongs to $\mathcal{T}_{\phi_1,\phi_2,\mathcal{P}}$ by (3.16). The pairing (3.17) is represented by a block upper triangular matrix with coefficients in $\varLambda_{\mathsf{w}(\mathcal{P})}$ , because $h_0$ annihilates $\mathbf{S}^{\mathrm{ord}}_{\mathcal{U},\mathcal{P}}$ . Moreover, by (3.7) the matrix has an invertible block of co-dimension 1. The perfectness of the pairing (3.17) is then equivalent to the last diagonal coefficient being in $\varLambda_{\mathsf{w}(\mathcal{P})}^\times$ . By construction, this coefficient is given by

$$C(\mathfrak{o},h_0\cdot \mathcal{F}_0)=C\left(\mathfrak{o},h_0\cdot \frac{\mathcal{E}_{\phi_1,\phi_2}}{2^{-d} \zeta_{\phi_1,\phi_2}}\right)= 1. $$

4.1 Non-vanishing of an $\mathscr{L}$ -invariant

Let $w_0\mid \mathfrak{p}$ be the prime of $H=\overline{\mathbf{Q}}^{\ker(\phi)}$ singled out by $\iota_p$ . As $\mathfrak{p}$ splits completely in H, i.e., $F_{\mathfrak{p}}=H_{w_0}$ , the group $\mathrm{Gal}(H/F)$ acts freely and transitively on the set of primes of H above $\mathfrak{p}$ and we have a canonical identification $ \Sigma_{\mathfrak{p}} \simeq \mathrm{Hom}_{\mathbf{Q}_p-\text{alg}}(H_{w_0},\overline{\mathbf{Q}}_p)$ via $\iota_p$ . Consider the homomorphisms

(4.1) \begin{equation}\begin{aligned}\mathrm{ord}_{\mathfrak{p}} &:\mathcal{O}_{H}\bigg[\frac{1}{\mathfrak{p}}\bigg]^\times\otimes \overline{\mathbf{Q}}_p \longrightarrow \overline{\mathbf{Q}}_p, \quad u\otimes 1\mapsto \mathrm{ord}_{w_0}(u), \\ \log_{\mathfrak{p}} &:\mathcal{O}_{H}\bigg[\frac{1}{\mathfrak{p}}\bigg]^\times\otimes \overline{\mathbf{Q}}_p \longrightarrow \overline{\mathbf{Q}}_p,\quad u\otimes 1\mapsto \log_p( \mathrm{N}_{F_{\mathfrak{p}}/\mathbf{Q}_p}(\iota_p(u)))=\sum_{\sigma\in \Sigma_{\mathfrak{p}}} \log_p(\sigma(\iota_p(u))),\end{aligned}\end{equation}

where $\mathcal{O}_{H}[{1}/{\mathfrak{p}}]^\times$ denotes the group of $\mathfrak{p}$ -units and we recall that $\iota_p$ embeds H into $H_{w_0}$ . As in [Reference Dasgupta, Darmon and PollackDDP11, (7)], the p-adic number

(4.2) \begin{equation}\mathscr{L}^{\mathfrak{p}}(\phi)=-\frac{\log_{\mathfrak{p}}(u_{\phi})}{\mathrm{ord}_{\mathfrak{p}}(u_{\phi})}\in \overline{\mathbf{Q}}_p,\end{equation}

depends only on $\phi$ and $\mathfrak{p}$ , and not on the particular choice of a basis $u_{\phi}$ of the line $(\mathcal{O}_{H}[{1}/{\mathfrak{p}}]^\times\otimes \overline{\mathbf{Q}}_p)[\phi]$ . If $\mathfrak{p}\in\Sigma_p$ is the unique irregular prime for $\phi$ , then $\mathscr{L}^{\mathfrak{p}}(\phi)$ coincides with Gross’ $\mathscr{L}$ -invariant $\mathscr{L}(\phi)$ from [Reference GrossGro81]. We next describe an explicit basis of $\mathrm{H}^1(F,\phi)$ . One has an exact sequence

(4.3) \begin{equation}0\to \mathrm{Hom}(\mathrm{G}_H,\overline{\mathbf{Q}}_p)\to \mathrm{Hom}((\mathcal{O}_H\otimes \mathbf{Z}_p)^\times, \overline{\mathbf{Q}}_p)\to \mathrm{Hom}(\mathcal{O}_H^\times,\overline{\mathbf{Q}}_p),\end{equation}

of left $\overline{\mathbf{Q}}_p[\mathrm{Gal}(H/F)]$ -modules. Consider the coset decomposition as in [Reference Betina, Deo and FitéBDF21, (3)]

(4.4) \begin{equation} \mathrm{Hom}_{\mathbf{Q}-\text{alg}}(H,\overline{\mathbf{Q}})= \coprod_{\sigma\in \Sigma} \tilde{\sigma}\cdot\mathrm{Gal}({H}/{F}),\end{equation}

where:

• – for $\sigma\in\Sigma_{\mathfrak{p}}$ , $\tilde{\sigma}$ denotes the unique extension to H determining via $\iota_p$ the place $w_0$ ;

• – for $\sigma\in \Sigma\setminus\Sigma_{\mathfrak{p}}$ , $\tilde{\sigma}$ denotes an arbitrary extension of $\sigma$ to H.

As in [Reference BetinaBet18, § 4], using the natural identification

\begin{align*}\mathrm{Hom}_{\mathbf{Q}-\text{alg}}(H,\overline{\mathbf{Q}}_p)=\coprod_{w|p}\mathrm{Hom}_{\mathbf{Q}_p-\text{alg}} (H_w,\overline{\mathbf{Q}}_p),\end{align*}

we consider the basis $(\log_p(\iota_p \circ \tilde\sigma\circ g^{-1}(\cdot)))_{\sigma\in \Sigma, g\in \mathrm{Gal}(H/F) }$ of $\mathrm{Hom}((\mathcal{O}_H\otimes \mathbf{Z}_p)^\times,\overline{\mathbf{Q}}_p)$ . As $(\mathcal{O}_H^\times \otimes \overline{\mathbf{Q}}_p)[\phi]$ vanishes, it follows from (4.3) that

\begin{align*}\mathrm{Hom}(\mathrm{G}_H,\overline{\mathbf{Q}}_p)[\phi^{-1}]\simeq\mathrm{Hom}((\mathcal{O}_H\otimes\mathbf{Z}_p)^\times,\overline{\mathbf{Q}}_p)[\phi^{-1}],\end{align*}

yielding the following.

Lemma 4.1. The space $\mathrm{H}^1(F,\phi)\simeq \mathrm{Hom}(\mathrm{G}_H,\overline{\mathbf{Q}}_p)[\phi^{-1}]$ is d-dimensional with basis $([\eta_{\phi}^{(\tilde{\sigma})}])_{\sigma\in \Sigma}$ , where $\eta_{\phi}^{(\tilde{\sigma})}$ is the cocycle whose image in $\mathrm{Hom}((\mathcal{O}_H\otimes \mathbf{Z}_p)^\times,\overline{\mathbf{Q}}_p)$ equals

$$ (u \otimes v) \mapsto \displaystyle\sum_{g\in \mathrm{Gal}(H/F)} \phi(g) \log_p ( \iota_p \circ \tilde\sigma\circ g^{-1} (u) \cdot v). $$

Moreover $[\eta_{\phi}^{(\tilde{\sigma})}]$ is only ramified at the prime of F determined by $\sigma$ via the decomposition $\Sigma=\coprod_{\mathfrak{q}\in\Sigma_p} \Sigma_{\mathfrak{q}}$ .

As $(\phi-\mathbf{1})$ is a basis of the coboundaries and $\phi(\tau)=-1\ne 1$ , where $\tau \in \mathrm{G}_F$ denotes a complex conjugation, each $[\eta]\in \mathrm{H}^1(F,\phi)$ is represented by a unique cocycle $\eta$ such that $\eta(\tau)=0$ . We will fix $\tau$ and $\eta$ throughout this article and make no further comment. The image of $\eta_{\mathbf{1}}=\log_p\circ \epsilon_{\mathrm{cyc}} \in \mathrm{Hom}(\mathrm{G}_\mathbf{Q},\overline{\mathbf{Q}}_p)$ , where $\epsilon_{\mathrm{cyc}}$ denotes the p-adic cyclotomic character, under the natural restrictions

corresponds to $\log_p\circ \mathrm{N}_{H/\mathbf{Q}}$ . The following cohomological interpretation of $\mathscr{L}^{\mathfrak{p}}(\phi)$ will be used in § 5.4.

1. (i) There exists a unique $[\eta_{\phi}^{(\mathfrak{p})}]\in \mathrm{H}^1(F,\phi)$ unramified outside $\mathfrak{p}$ and such that

   $$ \mathrm{loc}_{\mathfrak{p}}([\eta_{\phi}^{(\mathfrak{p})}-\eta_{\mathbf{1}}]) \in \ker ( \mathrm{Hom}(\mathrm{G}_{\mathfrak{p}},\overline{\mathbf{Q}}_p) \to \mathrm{Hom}(\mathrm{I}_{\mathfrak{p}},\overline{\mathbf{Q}}_p) ). $$

2. (ii) Under the local Artin map, one has the following equality in $\mathrm{Hom}(F_{\mathfrak{p}}^\times,\overline{\mathbf{Q}}_p)$ :

   $$ \mathrm{loc}_{\mathfrak{p}}([\eta_{\phi}^{(\mathfrak{p})}-\eta_{\mathbf{1}}] ) = \mathscr{L}^{\mathfrak{p}}(\phi) \mathrm{ord}_{\mathfrak{p}}. $$

3. (iii) For all $\sigma\in \Sigma\setminus \Sigma_{\mathfrak{p}}$ , $\mathrm{loc}_{\mathfrak{p}}([\eta_{\phi}^{(\tilde{\sigma})}]) \in \overline{\mathbf{Q}}_p^\times \cdot \mathrm{ord}_{\mathfrak{p}}$ .

4. (iv) One has $\mathscr{L}^{\mathfrak{p}}(\phi) \ne0$ .

It follows from part (i) and the local–global compatibility in class field theory that

where we have used that $\log_p( \mathrm{N}_{H/\mathbf{Q}}(u_0))=0$ as $\mathrm{N}_{H/\mathbf{Q}}(u_0) \in \pm p^{\mathbf{Z}}$ .

1. (iii) For each $\sigma\in \Sigma\setminus \Sigma_{\mathfrak{p}}$ , we have seen in Lemma 4.1 that $\eta_{\phi}^{(\tilde{\sigma})}$ is unramified at $\mathfrak{p}$ . A similar computation to the one above shows that

   $$\eta_{\phi}^{(\tilde{\sigma})}(\mathrm{Frob}_{\mathfrak{p}})=-\sum_{g\in \mathrm{Gal}(H/F)} \phi(g)\log_p( \iota_p \circ \tilde\sigma\circ g^{-1}(u_0)),$$
   which is non-zero by the Baker–Brumer theorem as the elements $( g^{-1}(u_0))_{g \in \mathrm{Gal}(H/F)}$ are $\mathbf{Z}$ -linearly independent (this follows from examining the valuations at the $[H:F]$ places of H above $\mathfrak{p}$ ).

2. (iv) The proof mixes ingredients all present in Gross [Reference GrossGro81]. For w a place of H above $\mathfrak{p}$ and $u\in H_w^\times$ , we let $|u|^\circ_{w}=q_{w}^{-\mathrm{ord}_{w}(u)} \mathrm{N}_{H_w/\mathbf{Q}_p}(u)$ denote its local absolute value, where $q_w$ is the order of the residue field (see [Reference GrossGro81, (1.8)] where it is denoted by $||u||_{w,p}$ ). Exactly as in [Reference GrossGro81, Proposition 1.16], one checks using the Baker–Brumer theorem, the injectivity of the $\overline{\mathbf{Q}}$ -linear map:

   \begin{align*} \lambda_{\mathfrak{p}}: \mathcal{O}_H\left[\frac{1}{\mathfrak{p}} \right]^{\times,\tau=-1} \otimes \overline{\mathbf{Q}} & \hookrightarrow \overline{\mathbf{Q}}_p[\mathrm{Gal}({H}/{F})] \\ u \otimes 1 & \mapsto \sum_{g \in \mathrm{Gal}(H/F)} \log_p(|\iota_p\circ g^{-1}(u)|^\circ_{w_0})[g]= \sum_{g \in \mathrm{Gal}(H/F)} \log_{\mathfrak{p}}(g^{-1}(u))[g], \end{align*}
   where $\mathcal{O}_H [{1}/{\mathfrak{p}} ]^{\times,\tau=-1} $ is the submodule on which the complex conjugation $\tau$ acts by $-1$ . As $\lambda_{\mathfrak{p}}$ is $\mathrm{Gal}(H/F)$ -equivariant, taking the $\phi$ -isotypic component (note that $\phi$ is totally odd), yields
   $$ 0\ne \lambda_{\mathfrak{p}}(u_{\phi})=\mathrm{ord}_{w_0}(u_0) \mathscr{L}^{\mathfrak{p}}(\phi)\sum_{g\in \mathrm{Gal}(H/F)} \phi(g^{-1})[g].$$

For any $\sigma \in \Sigma_{\mathfrak{p}}=\Sigma\simeq\mathrm{Gal}(F/\mathbf{Q})$ and $g \in \mathrm{Gal}(H/F)$ , we let $g_{\sigma}= \tilde\sigma g \tilde\sigma^{-1}$ . Using the notation from the proof of Proposition 4.2(ii), we have

\begin{align*}-\mathrm{ord}_{w_0}(u_0) \cdot \mathscr{L}(\phi)& = \sum_{\sigma \in \Sigma_{\mathfrak{p}}} \sum_{g\in\mathrm{Gal}(H/F)} \phi(g) \log_p( \iota_p \circ \tilde\sigma g^{-1} (u_0))\\ &= \sum_{\sigma \in \Sigma} \sum_{g\in \mathrm{Gal}(H/F)} \phi(g) \log_p(\iota_p \circ g_{\sigma}^{-1} \tilde\sigma (u_0)) \\&= \sum_{\sigma \in \Sigma} \sum_{g\in \mathrm{Gal}(H/F)} \phi(g)\log_p(\iota_p \circ g_{\sigma}^{-1} (\tilde\sigma (u_0)/u_0))\\ &\quad + \sum_{\sigma \in \Sigma_{\mathfrak{p}}} \sum_{g\in\mathrm{Gal}(H/F)} \phi(g) \log_p( \iota_p \circ g_{\sigma}^{-1} (u_0 )) \\ &= \sum_{\sigma \in \Sigma_{\mathfrak{p}}} \sum_{g\in\mathrm{Gal}(H/F)} \phi(g) \log_p(\iota_p \circ g_{\sigma}^{-1} (u_0)) \\ &= \sum_{\sigma \in \Sigma} \sum_{g\in \mathrm{Gal}(H/F)} \phi(\tilde\sigma^{-1}g\tilde\sigma) \log_p(\iota_p \circ g^{-1}(u_0)),\end{align*}

the penultimate equality following from $ {\tilde{\sigma}(u_0)}/{u_0} \in \mathcal{O}_H^{\times}$ and the vanishing of

\begin{align*}(\mathcal{O}_H^{\times} \otimes_{\mathbf{Z}} \overline{\mathbf{Q}})[\phi(\tilde\sigma^{-1} \text{$\cdot$ }\tilde\sigma) ].\end{align*}

We proceed by contradiction assuming that $\mathscr{L}(\phi) + \mathscr{L}(\phi^{-1})=0$ , i.e.,

(4.7) \begin{equation}\sum_{g\in \mathrm{Gal}(H/F)}\log_p(\iota_p \circ g^{-1}(u_0)) \sum_{\sigma \in \Sigma}( \phi(\tilde\sigma^{-1}g\tilde\sigma) +\phi(\tilde\sigma^{-1}g^{-1}\tilde\sigma)) =0.\end{equation}

Since the kernel of $\log_p$ on $\overline{\mathbf{Q}}^\times$ consists of elements of the form $p^a \cdot u$ , where $a \in \mathbf{Q}$ and u is a root of unity, one can check using the valuations at places w of H dividing p that the only possible non-trivial $\mathbf{Q}$ -linear relation between these logarithms is

$$ \sum_{g\in \mathrm{Gal}(H/F)}\log_p(\iota_p \circ g^{-1}(u_0))=0. $$

The Baker–Brumer theorem then implies that this is the only possible non-trivial relation over $\overline{\mathbf{Q}}$ . As the $\overline{\mathbf{Q}}$ -valued function $g\mapsto \sum_{\sigma \in \Sigma} ( \phi(\tilde\sigma^{-1}g\tilde\sigma) +\phi(\tilde\sigma^{-1}g^{-1}\tilde\sigma))$ is non-constant (e.g., the parity of $\phi$ implies that the values at $\mathbf{1}$ and $\tau$ differ), this contradicts (4.7).

4.2 Nearly ordinary cuspidal deformation functor

Let $\phi_1$ and $\phi_2$ be finite order Hecke characters such that $\phi=\phi_2\phi_1^{-1}$ is totally odd and irregular at a unique prime $\mathfrak{p}$ above p, i.e., $ \Sigma_p^{\mathrm{irr}}=\{\mathfrak{p} \}$ . The goal of this subsection is to introduce a nearly-ordinary deformation functor and compute its tangent space. This is used in §§ 5.2 and 5.4 to prove Theorem A(i) and (iv), respectively. Let $\mathcal{A}$ be the category of Artinian local $\overline{\mathbf{Q}}_p$ -algebras A with maximal ideal $\mathfrak{m}_A$ and residue field $\overline{\mathbf{Q}}_p$ , where the morphisms are local homomorphisms of $\overline{\mathbf{Q}}_p$ -algebras inducing identity on the residue field. Let $[\eta]$ be a non-trivial cocycle of the $d_{\mathfrak{p}}$ -dimensional space $\ker (\mathrm{H}^1(F,\phi^{-1}) \to \prod_{\mathfrak{q} \in \Sigma_p\setminus\{\mathfrak{p}\}} \mathrm{H}^1(F_{\mathfrak{q}},\phi^{-1}) )$ (see Lemma 4.4) and fix a representative $\eta$ such that $\eta(\tau)=0$ . As the centralizer of the image of $\rho=(\begin{smallmatrix} \phi_1 & \phi_2 \eta \\ 0 & \phi_2 \end{smallmatrix})$ consists only of scalar matrices, the functor $\mathcal{D}_{\eta}$ assigning to $A\in \mathcal{A}$ the set of lifts $\rho_A:\mathrm{G}_F\to \mathrm{GL}_2(A)$ of $\rho$ modulo strict equivalence is pro-representable by a complete Noetherian local $\overline{\mathbf{Q}}_p$ -algebra $\mathcal{R}^{\mathrm{u}}_\eta$ (see [Reference MazurMaz89]).

1. (i) The functor $\mathcal{D}^{\mathrm{no}}_{\eta}$ is pro-representable by a complete Noetherian local $\overline{\mathbf{Q}}_p$ -algebra $\mathcal{R}^{\mathrm{no}}_{\rho_{\mathrm{u}}}$ , which is the quotient of $\mathcal{R}^{\mathrm{u}}_\eta[\![ Y]\!]$ by the ideal $\mathscr{J}^{\mathrm{no}}_{( \rho_{\mathrm{u}},Y)}$ defined in (4.12). The universal deformation is given by the pushforward $\rho^{\mathrm{no}}$ of $ \rho_{\mathrm{u}}$ along $\mathcal{R}^{\mathrm{u}}_\eta \hookrightarrow \mathcal{R}^{\mathrm{u}}_\eta[\![ Y]\!] \twoheadrightarrow \mathcal{R}^{\mathrm{no}}_{\rho_{\mathrm{u}}}$ together with the universal nearly-ordinary line $\mathcal{R}^{\mathrm{no}}_{\rho_{\mathrm{u}}}\cdot (e_1+ Y e_2)$ at $\mathfrak{p}$ . Changing the representative $\rho_{\mathrm{u}}$ with $\rho_{\mathrm{u}}'=M \rho_{\mathrm{u}} M^{-1}$ for $ M=( \begin{smallmatrix}\alpha & \beta \\ \gamma & \delta \end{smallmatrix} ) \in \ker (\mathrm{GL}_2(\mathcal{R}^{\mathrm{u}}_{\eta}) \to \mathrm{GL}_2(\overline{\mathbf{Q}}_p) )$ induces an isomorphism $\mathcal{R}^{\mathrm{no}}_{\rho_{\mathrm{u}}} \simeq \mathcal{R}^{\mathrm{no}}_{\rho'_{\mathrm{u}}} $ of $\mathcal{R}^{\mathrm{u}}_\eta$ -algebras sending Y to $M\cdot Y=({\gamma + \delta Y})/({\alpha + \beta Y})$ .

2. (ii) The functor $\mathcal{D}^{\mathrm{ord}}_{\eta}$ is pro-representable by a complete Noetherian local $\overline{\mathbf{Q}}_p$ -algebra $\mathcal{R}^{\mathrm{ord}}_\eta$ which is the quotient of $\mathcal{R}^{\mathrm{u}}_\eta$ by the ideal $\mathscr{J}^{\mathrm{ord}}$ defined in (4.11), and which is independent from the particular choice of $\rho_{\mathrm{u}}$ .

3. (iii) We have an injective map for the tangent space $t^{\mathrm{no}}_{\eta}$ of $\mathcal{R}^{\mathrm{no}}_{\rho_{\mathrm{u}}}$ given by

   $$\mathrm{Hom}_{\overline{\mathbf{Q}}_p-\mathrm{alg}}(\mathcal{R}^{\mathrm{no}}_{\rho_{\mathrm{u}}},\overline{\mathbf{Q}}_p[\varepsilon]) \hookrightarrow \mathrm{H}^1(F,\mathrm{ad}\rho) \oplus \overline{\mathbf{Q}}_p,\quad \varphi \mapsto \left(\left[\frac{\varphi(\rho^{\mathrm{no}}) \rho^{-1} -\mathbf{1}_2}{\varepsilon} \right], \frac{\varphi(Y)}{\varepsilon}\right),$$
   where by an abuse of notation ${\lambda \varepsilon}/{\varepsilon}$ simply means $\lambda \in \overline{\mathbf{Q}}_p$ .