2.1 Notation

We begin by setting some notation that will remain fixed in the article. Let F be a totally real number field of degree g with ring of integers ${\mathcal O}_F$ and discriminant $\Delta _F$ . We fix a numbering $\{\sigma _1, \ldots , \sigma _g\}$ of real embeddings of F into $\mathbb C$ . We let $F^{\times +}$ (resp. ${\mathcal O}_F^{\times +})$ denote the totally positive elements in $F^{\times }$ (resp. ${\mathcal O}_F^{\times })$ . Let $\mathcal H$ denote the upper half plane and let $\mathcal H_F$ denote the set of elements of $F \otimes \mathbb C$ of totally positive imaginary part; note that $\mathcal H_F$ can be identified with the product of g copies of $\mathcal H$ . We let p be an odd prime that is unramified in F. We let $\mathbb A_f$ denote the finite adeles of $\mathbb Q$ , $\mathbb A_f^{(p)}$ the finite adeles away from p and $\mathbb A_{F, f}$ the finite adeles of F. We let G to be the algebraic group $\operatorname {\mathrm {Res}}_{F/\mathbb Q} \operatorname {GL}_2$ over $\mathbb Q$ .

2.2 Shimura varieties for G

If $K \subseteq G(\mathbb A_f)$ is an open compact subgroup, its corresponding Shimura variety $Y_G(K)$ is a quasi-projective variety with a canonical model over the reflex field $\mathbb Q$ whose complex points are given by

$$ \begin{align*}Y_G(K)(\mathbb C)= G(\mathbb Q)^+ \backslash [ \mathcal H_F \times G(\mathbb A_f)]/K. \end{align*} $$

The Shimura varieties $Y_G(K)$ are called Hilbert modular varieties.

Definition 2.1 We say than an open compact subgroup $K \subseteq G(\mathbb A_f)$ is sufficiently small if for every $h \in G(\mathbb A_f)$ the quotient

$$ \begin{align*}\frac{G(\mathbb Q)^+ \cap hKh^{-1}}{K \cap \left\{ \begin{pmatrix} u & 0 \\ 0 & u \end{pmatrix}: u \in {\mathcal O}_F^{\times} \right \} } \end{align*} $$

acts without fixed points on $\mathcal H_F$ .

If $K \subseteq G(\mathbb A_f)$ is sufficiently small, then $Y_G(K)$ is smooth. We also note that if ${K_1 \subseteq K_2}$ is an inclusion of open compact subgroups, $K_2$ is sufficiently small and $K_1$ is normal in $K_2$ , then the map $Y_G(K_1) \rightarrow Y_G(K_2)$ is a finite étale Galois cover.

2.3 The Hilbert modular variety $Y_G(U_n)$

Definition 2.2 For each $n \geq 1$ , we let

$$ \begin{align*}U_{n, p}:=\left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{GL}_2({\mathcal O}_F \otimes \mathbb Z_p): \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv \begin{pmatrix} x & * \\ 0 & x \end{pmatrix} \textrm{ mod } p^n \textrm{ for some } x \in {\mathcal O}_F \otimes \mathbb Z_p \right \}. \end{align*} $$

When $n=0$ , we let $U_{n, p}$ denote the Iwahori subgroup i.e. the subgroup of ${\operatorname {GL}_2({\mathcal O}_F \otimes \mathbb Z_p)}$ consisting of those matrices which are upper triangular mod p. Let $\mathfrak N$ be an ideal of ${\mathcal O}_F$ which does not divide 2, 3, or $\Delta _F$ . We fix a prime-to-p open compact subgroup

$$ \begin{align*}U^{(p)} =\left \{ g \in G(\mathbb A_f^{ (p)}): g \equiv \begin{pmatrix} * & * \\ 0 & 1 \end{pmatrix} \textrm{ mod } \mathfrak N \right \},\end{align*} $$

and let $U_n=U^{(p)} U_{n, p}$ . By [Reference Dimitrov5, Lemma 2.1], we have that $U_n$ is sufficiently small for all $n \geq 1$ and that the determinant map $\det : U_n \rightarrow ({\mathcal O}_F \otimes \widehat {\mathbb Z})^{\times }$ is surjective away from p.

The inclusion $U_{n+1} \hookrightarrow U_n$ induces a map of Shimura varieties $\phi _n: Y_G(U_{n+1}) \rightarrow Y_G(U_n).$

Proof Pick any element $[(x, g)] \in Y_G(U_n)(\mathbb C)=G(\mathbb Q)^+\backslash[ \mathcal H_F \times G(\mathbb A_f) ]/U_n$ and note that

$$ \begin{align*}\phi_n^{-1}( [(x, g)] ) = \{ [(x, g u_i)]: i \in I\}, \end{align*} $$

where $\{u_i\}_{i \in I}$ is a set of representatives of $U_n/U_{n+1}$ . It suffices to prove that $ [(x, gu_i)] \neq [(x, g u_j)] $ when $i \neq j$ . Suppose for contradiction that $[(x, gu_i)] = [(x, g u_j)] $ when $i \neq j$ . Then there exist $h \in G(\mathbb Q)^+$ and $k \in U_{n+1}$ such that $(hx, hgu_ik)=(x, gu_j)$ . In particular, we conclude that $h \in G^+(\mathbb Q) \cap g U_n g^{-1}$ . Since $hx=x$ and since $U_n$ is sufficiently small, we have by definition that

$$ \begin{align*}h \in U_n \cap \left\{ \begin{pmatrix} u & 0 \\ 0 & u \end{pmatrix}: u \in {\mathcal O}_F^{\times} \right\}= \left\{ \begin{pmatrix} u & 0 \\ 0 & u \end{pmatrix}: u \in {\mathcal O}_F^{\times} \text { and } u \equiv 1\quad \mod \mathfrak N \right\}. \end{align*} $$

Thus, $h \in U_{n+1}$ as well. Using the equality $hgu_ik= g u_j$ and the fact that h lies in the center of $G(\mathbb A_f)$ , we can now conclude that $u_i h k=u_j$ . Since $hk$ lies in $U_{n+1}$ , this contradicts the fact that $\{u_i\}_{i \in I}$ is a set of representatives for $U_n/U_{n+1}$ .

2.4 The number of components of $Y_G(U_n)$

Let $I(F)$ denote the group of fractional ideals of F, and let $\textrm {Cl}^{+}(F):= I(F)/\{ (\beta ): \beta \textrm {totally positive} \}$ denote the narrow class group of F. Let $h^+$ denote the narrow class number of F.

Proof Note that the map $\displaystyle { \mathbb A_{F, f}^{\times } \rightarrow \textrm {Cl}^+(F) }$ defined via $\displaystyle { (\alpha )_{\mathfrak p} \mapsto \prod _{\mathfrak p} \mathfrak p^{v_{\mathfrak p}(\alpha _{\mathfrak p}) }}$ has kernel $\prod _{\mathfrak p} {\mathcal O}_{F, \mathfrak p}^{\times } \cdot F^{\times +}$ . For each $x \in \textrm {Cl}^+(F)$ , choose a preimage $\alpha _x \in \mathbb A_{F, f}^{\times }$ . Hence, we have a decomposition $\displaystyle {\mathbb A_{F, f}^{\times }=\bigsqcup _{x \in \textrm {Cl}^+(F)} (\alpha _x \cdot \prod _{\mathfrak p} {\mathcal O}_{F, \mathfrak p}^{\times } \cdot F^{\times +})}$ as sets. By strong approximation, the connected components of $Y_G(U_n)$ are indexed by $F^{\times }_+\backslash \mathbb A_{F, f}^{\times }/\det (U_n)$ . This set is in bijection with

$$ \begin{align*} \bigsqcup_{x \in \textrm{Cl}^+(F)} F^{\times +}\backslash \prod_{\mathfrak p} (\alpha_x \cdot {\mathcal O}_{F, \mathfrak p}^{\times} \cdot F^{\times +} )/\det(U_n) &\cong \bigsqcup_{x \in \textrm{Cl}^+(F)} {\mathcal O}_F^{\times^+} \backslash (\alpha_x \cdot \prod_{\mathfrak p} {\mathcal O}_{F, \mathfrak p}^{\times} ) /\det(U_n) \\ & \cong \bigsqcup_{x \in \textrm{Cl}^+(F)} {\mathcal O}_F^{\times+} \backslash ({\mathcal O}_F \otimes \mathbb Z_p)^{\times} /\det(U_{n, p}), \end{align*} $$

where the last equality follows since the determinant map $\det : U_n \rightarrow ({\mathcal O}_F \otimes \widehat {\mathbb Z})^{\times }$ is surjective away from p.

2.5 Hecke action

For $K \subseteq G(\mathbb A_f)$ an open compact subgroup, we let $\mathbb T_K(G)=\mathbb Z[K\backslash G(\mathbb A_f)/K]$ be the Hecke-algebra of compactly supported bi-invariant functions on $G(\mathbb A_f)$ with multiplication given by convolution. Let $g \in G(\mathbb A_f)$ and let $K_g=K \cap g K g^{-1}$ ; we have a correspondence $[KgK]$

where the vertical maps are canonical projections and the upper-half horizontal map f on complex points is induced by multiplication by g. We obtain an action of $\mathbb T_K(G)$ on $H^i_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline {\mathbb Q}}, \mathbb F_p)$ and $H^i_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline {\mathbb Q}}, \mathbb Z_p)$ . Let $\Sigma $ be the set of places of F containing all primes dividing $\mathfrak N$ and all primes above p. Let

$$ \begin{align*}\mathbb T:=\bigotimes^{'}_{v \not \in \Sigma} \mathbb Z[ \operatorname{GL}_2({\mathcal O}_v) \backslash \operatorname{GL}_2(F_v)/ \operatorname{GL}_2({\mathcal O}_v) ] \end{align*} $$

denote the abstract spherical algebra away from $\Sigma $ . We note that $\mathbb T$ is a subalgebra of $\mathbb T_K(G)$ ; it is also commutative, generated by the following Hecke operators $\mathcal T_v$ and $\mathcal S_v^{\pm 1}$ for every finite place $v \not \in \Sigma $ ; for every such v we choose a uniformizer $\mathcal \varpi _v$ of ${\mathcal O}_v$ and define

• • $\mathcal T_v$ to be the double coset $\operatorname {GL}_2({\mathcal O}_v) \begin {pmatrix} \varpi _v & 0 \\ 0 & 1 \end {pmatrix} \operatorname {GL}_2({\mathcal O}_v).$

• • $\mathcal S_v$ to the double coset $\operatorname {GL}_2({\mathcal O}_v) \begin {pmatrix} \varpi _v & 0 \\ 0 & \varpi _v \end {pmatrix} \operatorname {GL}_2({\mathcal O}_v).$

2.6 Betti cohomology

Let $\Gamma $ be a subgroup of $G^+(\mathbb Q)$ and let $\overline {\Gamma }$ be its image in $G(\mathbb Q)/Z_G(\mathbb Q)$ . We assume that $\overline {\Gamma }$ has no nontrivial elements of finite order; hence, it acts freely and continuously on $\mathcal H_F$ . In this subsection, we closely follow [Reference Urban22] to establish a relation between the group cohomology of $\overline {\Gamma }$ and the Betti cohomology of the corresponding Hilbert modular variety.

By the work of Borel–Serre, there exists a canonical compactification $\overline {\Gamma } \backslash \overline {\mathcal H_F}$ , where $\overline {\mathcal H_F}$ is a contractible real manifold with corners. Since $\overline {\Gamma } \backslash \overline {\mathcal H_F}$ is compact, we may choose a finite triangulation of $\overline {\Gamma } \backslash \overline {\mathcal H_F}$ . We may pull it back to $\overline {\mathcal H_F}$ via the canonical projection $ \overline {\mathcal H_F} \rightarrow \overline {\Gamma } \backslash \overline {\mathcal H_F}$ . Let $C_q(\overline \Gamma )$ be the free $\mathbb Z$ -module over the set of q-dimensional simplices of the triangulation obtained by pull-back to $\overline {\mathcal H_F}$ . Since the action of $\overline {\Gamma }$ on $\overline {\mathcal H_F}$ is free, and since the triangulation of $\overline {\Gamma } \backslash \overline {\mathcal H_F}$ is finite, the $C_q(\overline \Gamma )$ ’s are free $\mathbb Z[\overline {\Gamma }]$ -modules of finite type. We also note that

$$ \begin{align*}0 \rightarrow C_d(\overline \Gamma) \xrightarrow{\partial_{d-1}} \cdots \xrightarrow{\partial_1} C_0(\overline \Gamma) \rightarrow 0\end{align*} $$

is a complex computing the homology of $\overline {\mathcal H_F}$ . Since $\overline {\mathcal H_F}$ is contractible, this complex is exact except in degree zero and $H_0(\overline {\mathcal H_F}, \mathbb Z)= C_0(\overline \Gamma )/\partial _0(C_1(\overline \Gamma ))=\mathbb Z$ . Thus, in summary, we have that

$$ \begin{align*}0 \rightarrow C_d(\overline \Gamma) \xrightarrow{\partial_{d-1}} \cdots \xrightarrow{\partial_1} C_0(\overline \Gamma) \rightarrow \mathbb Z \rightarrow 0\end{align*} $$

is an exact sequence of finite free $\mathbb Z[\overline {\Gamma }]$ -modules. If M is a $\Gamma $ -module, we let $\mathcal C^{\bullet }(\overline {\Gamma }, M)$ denote the complex

$$ \begin{align*}0 \rightarrow \textrm{Hom} (C_0(\overline{\Gamma}), M) \rightarrow \cdots \rightarrow \textrm{Hom}(C_d(\overline{\Gamma}), M). \end{align*} $$

Thus, $H^i(\overline {\Gamma }, M)$ is the ith cohomology group of the complex $\mathcal C^{\bullet }(\overline {\Gamma }, M)$ .

Let K be an open compact subgroup of $G(\mathbb A_f)$ which is sufficently small and let M be a left K-module acting via its projection to $K_p$ , the image of K in $G(\mathbb Q_p)$ . The corresponding Hilbert modular variety $Y_G(K)$ satisfies

$$ \begin{align*}Y_G(K) \cong \bigsqcup_{j \in J} \Gamma_j \backslash \mathcal H_F, \end{align*} $$

where J is a finite set and for each $j \in J$ , $\Gamma _j=g_j K g_j^{-1} \cap G^+(\mathbb Q)$ for some $g_j \in G(\mathbb A_f)$ . As before, we let $\overline {\Gamma _j}$ denote the image of $\Gamma _j$ in $G(\mathbb Q)/Z_G(\mathbb Q)$ . We set

(2.1) $$ \begin{align} C^{\bullet}(K, M):= \bigoplus_{j \in J} C^{\bullet}(\overline{\Gamma_j}, M). \end{align} $$

Let $\overline {Y_G}:=G(\mathbb Q)^+ \backslash G(\mathbb A_f) \times \overline {\mathcal H_F}$ . Then $\overline {Y_G}(K):=\overline {Y_G}/K$ is the Borel–Serre compactification of $Y_G(K)$ . Let $\pi : \overline {Y_G} \rightarrow \overline {Y_G}(K)$ denote the canonical projection. We choose a finite triangulation of $\overline {Y_G}(K)$ and pull it back via $\pi $ . Let $C^q(K)$ denote the corresponding chain complex equipped with a right action of K. Then

$$ \begin{align*}C^{\bullet}(K, M) = \textrm{Hom}_K(C^q(K), M). \end{align*} $$

Thus, $C^{\bullet }(K, M)$ also computes the cohomology of the local system M on $Y_G(K)$ and so we have an isomorphism

(2.2) $$ \begin{align} H^i(Y_G(K), M) \cong \bigoplus_{j \in J} H^i(\overline{\Gamma}_j, M). \end{align} $$

3.1 General notation

We consider the group scheme $\textrm {GL}_2$ over $\mathbb Z_p$ and we let $B_2$ , $N_2$ , and $T_2$ to be the subgroups of upper-triangular, unipotent, and diagonal matrices, respectively. Following [Reference Loeffler, Rockwood and Zerbes15], we set the following notation:

• • $Q= \textrm {Res}_{{\mathcal O}_F \otimes \mathbb Z_p/\mathbb Z_p} \hspace {1mm} B_2$ .

• • $N= \textrm {Res}_{{\mathcal O}_F \otimes \mathbb Z_p/\mathbb Z_p} \hspace {1mm} N_2$ .

• • $S= \textrm {Res}_{{\mathcal O}_F \otimes \mathbb Z_p/ \mathbb Z_p} \hspace {1mm} T_2$ .

• • $E/\mathbb Q_p$ is a finite extension with ring of integers ${\mathcal O}$ such that S splits over E.

• • $S^{0}= \textrm {Res}_{{\mathcal O}_F \otimes \mathbb Z_p / \mathbb Z_p} \hspace {1mm} \mathbb G_m$ , viewed as a subgroup of S via the diagonal embedding.

• • $Q^0$ denotes the preimage of $S^0$ under the projection $Q \rightarrow S$ i.e. $Q^0(\mathbb Z_p)= \left \{ \begin {pmatrix} a & b \\ 0 & a \end {pmatrix} \in \operatorname {GL}_2({\mathcal O}_F \otimes \mathbb Z_p)| a, b \in {\mathcal O}_F \otimes \mathbb Z_p \right \}$ .

• • ${\mathfrak S}=S(\mathbb Z_p)/S^0(\mathbb Z_p)$ identified with $({\mathcal O}_F \otimes \mathbb Z_p)^{\times }$ via the short exact sequence

  $$ \begin{align*}1 \rightarrow S^0(\mathbb Z_p) \rightarrow S(\mathbb Z_p) \rightarrow ({\mathcal O}_F \otimes \mathbb Z_p)^{\times} \rightarrow 1, \end{align*} $$
  where the map
  $$ \begin{align*}S(\mathbb Z_p) \rightarrow ({\mathcal O}_F \otimes \mathbb Z_p)^{\times} \hspace{3mm} \text{ is given by } \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \mapsto ab^{-1}. \end{align*} $$

• • .

• • $\tau =\begin {pmatrix} p & 0 \\ 0 & 1 \end {pmatrix} \in \operatorname {GL}_2(\mathbb Q_p \otimes {\mathcal O}_F)$ .

• • $N_r=\tau ^r N(\mathbb Z_p) \tau ^{-r}=\left \{ \begin {pmatrix} 1 & x \\ 0 & 1 \end {pmatrix} \in \operatorname {GL}_2({\mathcal O}_F \otimes \mathbb Z_p)| \hspace {1mm} x \equiv 0 \textrm { mod } p^r \right \} $ .

• • $\overline {N_r}=\tau ^{-r} \overline {N}(\mathbb Z_p) \tau ^{r}=\left \{ \begin {pmatrix} 1 & 0 \\ x & 1 \end {pmatrix} \in \operatorname {GL}_2({\mathcal O}_F \otimes \mathbb Z_p)| \hspace {1mm} x \equiv 0 \textrm { mod } p^r \right \} $ .

• • $L_r=\{\ell \kern1.6pt{\in}\kern1.6pt S(\mathbb Z_p): \ell \kern1.6pt{\in}\kern1.6pt S^0(\mathbb Z_p) \textrm { mod } p^r \}\kern1.5pt{=}\kern1.3pt\left \{ \begin {pmatrix} a &\!\! 0 \\ 0 &\!\! d \end {pmatrix} \kern1.6pt{\in}\kern1.6pt \operatorname {GL}_2({\mathcal O}_F \kern1.6pt{\otimes}\kern1.6pt \mathbb Z_p)|\kern1pt a \kern1.6pt{\equiv}\kern1.6pt d \textrm { mod } p^r \kern-1pt\right \}$ .

• • $V_r=\overline {N_r} L_r N_0$ .

We also set $V_0=\operatorname {GL}_2({\mathcal O}_F \otimes \mathbb Z_p)$ and $V_0^{(s)}:= \tau ^{-s}V_0\tau ^{s} \cap V_0$ for $s \geq 1$ ; in other words, $V_0^{(s)}$ is the group of upper-triangular matrices $\textrm {mod } p^s:$

$$ \begin{align*}V_0^{(s)=}\left \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{GL}_2({\mathcal O}_F \otimes \mathbb Z_p)| a, b, c, d \in {\mathcal O}_F \otimes \mathbb Z_p \textrm{ and } c \equiv 0 \textrm{ mod } p^s \right \}. \end{align*} $$

Similarly, for $s \geq 0$ and $n \geq 0$ , we set

$$ \begin{align*}U_{n, p}^{(s)}:= \tau^{-s}U_{n, p}\tau^{s} \cap U_{n, p}. \end{align*} $$

For our level groups $U_{n, p}$ , we define the Hecke operator $\mathcal U_p'$ to be the double coset $U_{n, p} \begin {pmatrix} 1 & 0 \\ 0 & p \end {pmatrix} U_{n, p}$ and we define the analogue of Hida’s anti-ordinary projection operator to be

$$ \begin{align*}e^{\prime}_{\operatorname{\mathrm{ord}}}= \lim_{n \to \infty} (\mathcal U_p')^{n!}. \end{align*} $$

The operator $e^{\prime }_{\operatorname {\mathrm {ord}}}$ acts on the cohomology of $Y_G(U_n)$ . Let M be a $\mathbb Z[U_{n}]$ -module (acting via projection to $U_{n, p}$ ) with a compatible action of $\mathcal U_p'$ (in the sense that for any $u \in U_{n, p}^{(s)}$ , the action of $\mathcal U_p'$ intertwines the action of u and $\tau ^{-s} u \tau ^s$ ). For $K \subseteq G(\mathbb A_f)$ an open compact subgroup, the operator $e^{\prime }_{\operatorname {\mathrm {ord}}}$ also acts on the complexes $C^{\bullet }(K, M)$ introduced in Section 2.6 by lifting the action on cohomology (see [Reference Loeffler, Rockwood and Zerbes15, p. 6]).

Lemma 3.2 Let M be a $\mathbb {Z}[U_{n}]$ -module with a compatible action of $\mathcal U_p'$ . The following diagram commutes on cohomology

Proposition 3.3 The corestriction maps induce isomorphisms

$$ \begin{align*}e^{\prime}_{\operatorname{\mathrm{ord}}} H^i( Y_G(U^{(p)}U_{n, p}^{ (s)}), M) \cong e^{\prime}_{\operatorname{\mathrm{ord}}} H^i(Y_G(U_n), M). \end{align*} $$

3.2 Algebraic representations

Let $X^{\bullet }(S)$ denote the character lattice of S and $X^\bullet _+(S)$ be the set of dominant weights. For each $\lambda \in X^\bullet _+(S)$ , there is a unique isomorphism class of irreducible representations of $(\rho _\lambda , V_\lambda )$ of G (over E) of highest weight $\lambda $ . A representative of this isomorphism class can be constructed using the Borel–Weil–Bott theorem, as the space of all polynomials

$$ \begin{align*}\{ f \in E[G]: f(\overline{n} \ell g) = \lambda(\ell) f(g) \hspace{2mm} \forall \overline{n} \in \overline{N}, \ell \in S, g \in G \}, \end{align*} $$

with G acting by right translation. More concretely, each such $\lambda \in X^\bullet _+(S)$ can be identified with an integer tuple $(k_1,\ldots ,k_g, t_1, \ldots ,t_g)$ such that the associated $V_{\lambda }$ is given by the representation $\textrm {Sym}^{k_i} V \otimes \det ^{t_i}$ at the ith embedding, where V is the standard representation of G. All our Hecke operators defined above also act on cohomology with the $V_\lambda $ ’s as coefficient systems (see [Reference Loeffler, Rockwood and Zerbes15, Definition 2.5.1]).

3.3 Modules of measures

In this subsection, for brevity, we let U to be $U_{r, p}$ for some $r \geq 0$ . We also fix a character $\lambda = (k_1, \ldots ,k_g, t_1, \ldots t_g)$ such that $k_1+2t_1= \cdots = k_g+2t_g=0$ .

Proof This follows immediately from our normalization of the weights described in Section 3.2, namely,

$$ \begin{align*} \lambda \left ( \begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix} \right ) = \sigma_1(a)^{k_1+2t_1} \cdots \sigma_g^{k_g+2t_g}(a)=1.\\[-40pt] \end{align*} $$

Following [Reference Loeffler, Rockwood and Zerbes15], we set the following notation. We define U- modules of continuous functions

$$ \begin{align*}C_{\lambda, U}:=\{f: U \rightarrow {\mathcal O}: f \textrm{ continuous}, f( \ell n g)=\lambda^{-1}(\ell) f(g) \hspace{2mm} \forall ln \in (S \cap U) N_0, g \in U \} \end{align*} $$

and

$$ \begin{align*}C_{\textrm{univ}}:=\{f: U \rightarrow {\mathcal O}: f \textrm{ continuous}, f(\ell n g)= f(g) \hspace{2mm} \forall ln \in Q^0(\mathbb Z_p), g \in U \}, \end{align*} $$

with U acting by right translation. We endow these spaces with an action of $\tau $ given by

$$ \begin{align*}\tau \cdot f(n \ell \bar{n})= f(n \ell \tau^{-1} \bar{n} \tau). \end{align*} $$

We define modules of bounded distributions

$$ \begin{align*}D_{\lambda, U}:=\textrm{Hom}_{\textrm{cts}}(C_{\lambda, U}, {\mathcal O}), \end{align*} $$

and

$$ \begin{align*}D_{\textrm{univ}}:=\textrm{Hom}_{\textrm{cts}}(C_{\textrm{univ}}, {\mathcal O}), \end{align*} $$

which inherit actions of $V_0'$ and $\tau ^{-1}$ by duality. We let ${\mathfrak S}_U$ denote the image of U in ${\mathfrak S}$ (with respect to the Iwahori decomposition stated in Section 3.1); note that ${{\mathfrak S}_{U} = {\mathfrak S} \cong ({\mathcal O}_F \otimes \mathbb Z_p)^{\times }}$ when $U=U_{0, p}$ and $ {\mathfrak S}_{U}$ is the set of elements of $\mathcal ({\mathcal O}_F \otimes \mathbb Z_p)^{\times }$ which are congruent to 1 mod $p^n$ when $U=U_{r, p}$ and $r \geq 1$ . Similarly, we write $\bar {N_U}$ for $\bar {N_r}$ when $U=U_{r, p}$ for $r \geq 0$ . We let denote the Iwasawa algebra corresponding to ${\mathfrak S}_U$ . We have that $D_{\lambda , U}$ and $D_{\textrm {univ}}$ are modules over (with the action given by inverse translation) and this structure is given explicitly by the isomorphisms (see [Reference Loeffler, Rockwood and Zerbes15, p. 6])

and

where ${\mathcal O}[-\lambda ]$ denotes ${\mathcal O}$ regarded as an ${\mathfrak S}$ (and hence $\Lambda $ ) module via the inverse of $\lambda $ . In particular, we have that

as $\Lambda $ -modules. Since a power series ring in finitely many variables over a Noetherian ring is flat (see [Reference Bourbaki3, p. 146]), we have that $D_{\operatorname {\mathrm {univ}}}$ is flat as a $\Lambda $ -module.

Proposition 3.6 The anti-ordinary projector $ e^{\prime }_{\operatorname {\mathrm {ord}}}$ acts on $\mathcal C^{\bullet }(U, D_{\operatorname {\mathrm {univ}}})$ such that we have a decomposition

$$ \begin{align*}C^{\bullet}(U, D_{\operatorname{\mathrm{univ}}}) = e^{\prime}_{\operatorname{\mathrm{ord}}} C^{\bullet}(U, D_{\operatorname{\mathrm{univ}}}) \oplus (1-e^{\prime}_{\operatorname{\mathrm{ord}}}) C^{\bullet}(U, D_{\operatorname{\mathrm{univ}}}) \end{align*} $$

with $\mathcal U_p'$ acting invertibly on the first component and topologically nilpotently on the second. Moreover, the complex $e^{\prime }_{\operatorname {\mathrm {ord}}} C^{\bullet }(U, D_{\operatorname {\mathrm {univ}}})$ consists of flat $\Lambda $ -modules.

3.4 Proof of the Tor spectral sequence

In this subsection, we let $\lambda $ denote a weight that is self-dual in the sense of Section 3.3. We let s and n be integers with $s \geq n$ . Let $\Gamma _1=G(\mathbb Q)^+ \cap U^{(p)} V_0^{ (s) }$ and $\Gamma _2= G(\mathbb Q)^+ \cap ( U^{(p)} (U_{n, p} \cap V_0^{ (s) }))$ . Let $\overline {\Gamma _1}$ and $\overline {\Gamma _2}$ denote the images of $\Gamma _1$ and $\Gamma _2$ in $G/Z_G(\mathbb Q)$ . Write

$$ \begin{align*}Y_G( U^{ (p) } V_0^{(s)} )(\mathbb C) = \bigsqcup_{j \in J_1} \Gamma_1 \backslash \mathcal H_F, \end{align*} $$

and

$$ \begin{align*}Y_G( U^{ (p) } (V_0^{(s)} \cap U_{n, p}) )(\mathbb C) \cong \bigsqcup_{j \in J_2} \Gamma_2 \backslash \mathcal H_F. \end{align*} $$

Each $i \in J_1$ corresponds to a matrix $g_i \in G(\mathbb A_f)$ whose determinants form a set of representatives for $F^{\times +}\backslash \mathbb A_{F, f}^{\times }/\det ( U^{(p)} V_0^{(s)})$ . Similarly, each $i \in J_2$ corresponds to a matrix $g_i \in G(\mathbb A_f)$ whose determinants form a set of representatives for $F^{\times +}\backslash \mathbb A_{F, f}^{\times }/\det ( U^{(p)} (U_{n, p} \cap V_0^{(s)}))$ .

Since $\det (U^{(p)}V_0^{(s)}))=({\mathcal O}_F \otimes \hat {\mathbb Z})^{\times }$ , we have that $|J_1|=h^+$ , and by Lemmma 2.4, $|J_2|=$ $h^+ \cdot |{\mathcal O}_F^{\times ^+}\backslash ({\mathcal O}_F \otimes \mathbb Z_p)^{\times } /\det (U_{n, p}) |$ (as written, Lemma 2.4 can only be applied when $s=n$ but the exact same proof goes through when $s \geq n$ since $\det (U_{n, p})=\det (U_{n, p} \cap V_0^{ (s) }) \textrm { in } ({\mathcal O}_F \otimes \mathbb Z_p)^{\times } )$ .

Proposition 3.7 Let $\varphi $ denote the natural map

$$ \begin{align*}\varphi: \Gamma_1/\Gamma_2 \rightarrow V_0^{ (s) }/(U_{n, p} \cap V_0^{ (s) }). \end{align*} $$

Then $\operatorname {\mathrm {im}}(\varphi )$ has index $ |{\mathcal O}_F^{\times +}\backslash ({\mathcal O}_F \otimes \mathbb Z_p)^{\times } /\det (U_{n, p}) | $ in $V_0^{ (s) }/(U_{n, p} \cap V_0^{ (s) })$ .

Proof By strong approximation for the semisimple group $\textrm {Res}_{F/\mathbb Q}\textrm {SL}_2$ , the diagram

is cartesian (see [Reference Hida10, Corollary 3.3] for a similar argument). On the other hand, note that $\det (U_{n, p})=({\mathcal O}_F^{\times })^2$ in $({\mathcal O}_F \otimes \mathbb Z_p)^{\times }$ by Hensel’s lemma; thus, we conclude that

$$ \begin{align*}\displaystyle{ |\textrm{coker}(\varphi)|= \frac{ [ ({\mathcal O}_F \otimes \mathbb Z_p)^{\times}: \det(U_{n, p} ) ]}{[{\mathcal O}_F^{\times +}: ({\mathcal O}_F^{\times})^2]}}\end{align*} $$

and that the natural map $({\mathcal O}_F \otimes \mathbb Z_p)^{\times }/\det (U_{n, p}) \rightarrow {\mathcal O}_F^{\times +}\backslash ({\mathcal O}_F \otimes \mathbb Z_p)^{\times } /\det (U_{n, p})$ has kernel ${\mathcal O}_F^{\times +}/ ({\mathcal O}_F^{\times })^2$ . Thus, $\displaystyle { \frac { [ ({\mathcal O}_F \otimes \mathbb Z_p)^{\times }: \det (U_{n, p} ) ]}{[{\mathcal O}_F^{\times +}: ({\mathcal O}_F^{\times })^2]}= |{\mathcal O}_F^{\times ^+}\backslash ({\mathcal O}_F \otimes \mathbb Z_p)^{\times } /\det (U_{n, p}) | }$ proving the claim.

Proposition 3.8 We have a Hecke-equivariant isomorphism

$$ \begin{align*}\bigoplus_{j \in J_1} H^i(\overline{\Gamma_1}, {\mathcal O}/(p^s)[-\lambda][V_0^{(s)}/ V_0^{(s)} \cap U_{n, p} ] ) \cong \bigoplus_{j \in J_2} H^i(\overline{\Gamma_1}, {\mathcal O}/(p^s)[-\lambda][\overline{\Gamma_1}/\overline{\Gamma_{2}}] ).\\[-22pt] \end{align*} $$

Theorem 3.9 We have an isomorphism of ${\mathcal O}$ -modules

$$ \begin{align*}e^{\prime}_{\operatorname{\mathrm{ord}}} H^i(Y_G(U_0), D_{\lambda, U_{n, p}}) \cong e^{\prime}_{\operatorname{\mathrm{ord}}} H^i(Y_G(U_n), V_{\lambda}). \end{align*} $$

Proof As explained in [Reference Loeffler, Rockwood and Zerbes15, Proposition 2.7.7], we have an isomorphism

$$ \begin{align*}e^{\prime}_{\textrm{ord}} H^i(Y_G(U_0), D_{\lambda, U_{n, p}}/p^s) \cong e^{\prime}_{\textrm{ord}} H^i(Y_G(U^{(p)} V_0^{ (s) }), {\mathcal O}/(p^s)[-\lambda] \otimes_{{\mathcal O}[{\mathfrak S}_{U_{n,p}}]} {\mathcal O}/(p^s)[{\mathfrak S}]). \end{align*} $$

Hence,

$$ \begin{align*} & H^i(Y_G( U^{ (p) } V_0^{(s)} ), {\mathcal O}/(p^s)[-\lambda] \otimes_{{\mathcal O}[{\mathfrak S}_{U_{n, p}}]} {\mathcal O}(p^n)[{\mathfrak S}]) \\ & \cong \bigoplus_{j \in J_1} H^i(\overline{\Gamma_1}, {\mathcal O}/(p^s)[-\lambda] \otimes_{{\mathcal O}[{\mathfrak S}_{U_{n,p}}]} {\mathcal O}/(p^s)[{\mathfrak S}]) \\ & \cong \bigoplus_{j \in J_1} H^i(\overline{\Gamma_1}, {\mathcal O}/(p^s)[-\lambda][{\mathfrak S}/{\mathfrak S}_{U_{n,p}}] )\\ & \cong \bigoplus_{j \in J_1} H^i(\overline{\Gamma_1}, {\mathcal O}/(p^s)[-\lambda][V_0^{(s)}/ V_0^{(s)} \cap U_{n, p} ] ) \\ & \cong \bigoplus_{j \in J_2} H^i(\overline{\Gamma_1}, {\mathcal O}/(p^s)[-\lambda][\overline{\Gamma_1}/\overline{\Gamma_{2}}] ) \\ & \cong \bigoplus_{j \in J_2} H^i(\overline{\Gamma_2}, {\mathcal O}/(p^s)[-\lambda] ) \\ & \cong H^i(Y_G(U^{(p)}( V_0^{ (s) } \cap U_{n, p})), {\mathcal O}/(p^s)[-\lambda] ). \end{align*} $$

Here, the first isomorphism follows from Equation (2.2), the second is a general property of tensor products, the third follows the fact that ${\mathfrak S}/{\mathfrak S}_{U_{n, p}} \cong V_0^{(s)}/ V_0^{(s)} \cap U_{n, p}$ , the fourth follows from Proposition 3.8, the fifth follows from Shapiro’s Lemma, and the sixth follows from Equation (2.2) again. As explained in [Reference Loeffler, Rockwood and Zerbes15, Proposition 2.7.7], we can use Proposition 3.3 to conclude that $e^{\prime }_{\textrm {ord}} H^i(Y_G(U^{(p)}( V_0^{ (s) } \cap U_{n, p})), {\mathcal O}/(p^s)[-\lambda ] )$ is in turn isomorphic to $e^{\prime }_{\textrm {ord}} H^i(Y_G(U_n), V_\lambda /p^s)$ . Thus, combining our isomorphims, we conclude that

$$ \begin{align*} e^{\prime}_{\textrm{ord}} H^i(Y_G(U_0), D_{\lambda, U_{n, p}}) &= \varprojlim_{s} e^{\prime}_{\textrm{ord}} H^i(Y_G(U_0), D_{\lambda, U_{n, p}}/p^s) \\ &\cong \varprojlim_{s} e^{\prime}_{\textrm{ord}} H^i(Y_G(U_n), V_\lambda/p^s) \\ &=e^{\prime}_{\textrm{ord}} H^i(Y_G(U_n) , V_{\lambda}).\\[-37pt] \end{align*} $$

We note that Theorem 3.9 is compatible with the comparision isomorphism between Betti and étale cohomology; for the rest of the article, we work with étale cohomology of Hilbert modular varieties.

Corollary 3.11 We have an isomorphism

$$ \begin{align*}e^{\prime}_{\operatorname{\mathrm{ord}}} H^i_{\acute{\rm e}\textrm{t}}(Y_G(U_0), D_{\operatorname{\mathrm{univ}}}) \cong e^{\prime}_{\operatorname{\mathrm{ord}}} \varprojlim_{n} H^i_{\acute{\rm e}\textrm{t}} (Y_G(U_n), {\mathcal O}). \end{align*} $$

Proof Setting $\lambda $ to be the trivial character, we deduce from the previous theorem that

$$ \begin{align*} e^{\prime}_{\textrm{ord}} H^i_{\acute{\rm e}\textrm{t}}(Y_G(U_0), D_{\textrm{univ}}) & \cong \varprojlim_{Q^0 \subseteq U, n} e^{\prime}_{\textrm{ord}} H^i_{\acute{\rm e}\textrm{t}}(Y_G( U_0), D_{\lambda, U}/(p^n)) \\ & \cong \varprojlim_{n} e^{\prime}_{\textrm{ord}} H^i_{\acute{\rm e}\textrm{t}}(Y_G(U_n), {\mathcal O}/(p^n)) \\ & \cong e^{\prime}_{\textrm{ord}} \varprojlim_{n} H^i_{\acute{\rm e}\textrm{t}} (Y_G(U_n), {\mathcal O}). \end{align*} $$

Here the last isomorphism follows from the facts that our inverse system satisfies the Mittag–Leffler property because the cohomology groups in the system are finitely generated, and that $e^{\prime }_{\operatorname {\mathrm {ord}}}$ commutes with the maps in the inverse limit.

Define $M^{\bullet }$ to be the image of $e^{\prime }_{\textrm {ord}} C^{\bullet }(U_0, D_{\textrm {univ}})$ in the subcategory $D^{\textrm {flat}}(R)$ of the derived category of R-modules generated by flat objects. We also set for all $n \geq 1$ .

Theorem 3.12 For all $n \geq 1$ , we have a quasi-isomorphism

$$ \begin{align*}M^{\bullet} \otimes^{\mathbb L}_ {\Lambda_n} {\mathcal O}[-\lambda] \cong e^{\prime}_{\operatorname{\mathrm{ord}}} C^{\bullet}(U_n, V_\lambda). \end{align*} $$

Proof Since $M^{\bullet }$ is represented by the flat complex $e^{\prime }_{\textrm {ord}} \mathcal C^\bullet (U_0, D_{\textrm {univ}})$ , we can compute the derived tensor product as

$$ \begin{align*}e^{\prime}_{\textrm{ord}} \mathcal C^\bullet (U_0, D_{\textrm{univ}}) \otimes_{\Lambda_n} {\mathcal O}[-\lambda] = e^{\prime}_{\textrm{ord}} \mathcal C^\bullet(U_0, D_{\textrm{univ}} \otimes_{\Lambda_n} {\mathcal O}[-\lambda]). \end{align*} $$

By Theorem 3.9, this complex is isomorphic to $\ e^{\prime }_{\textrm {ord}} C^{\bullet }(U_n, V_\lambda )$ .

Corollary 3.13 For all $n \geq 1$ , there is a spectral sequence

$$ \begin{align*}E_2^{i, j}: {\operatorname{Tor}}_{-i}^{\Lambda_n}(e^{\prime}_{\operatorname{\mathrm{ord}}} \varprojlim_{s} H^j_{\acute{\rm e}\textrm{t}} (Y_G(U_s), {\mathcal O}) _{\overline{\mathbb Q}}, {\mathcal O}[-\lambda]) \implies e^{\prime}_{\operatorname{\mathrm{ord}}} H^{i+j}_{\acute{\rm e}\textrm{t}} (Y_G(U_n)_{\overline{\mathbb Q}}, V_\lambda). \end{align*} $$

An analogous argument to the one given in this section allows us to obtain a similar result at the Iwahori level $U_0$ .

Corollary 3.14 There is a spectral sequence

$$ \begin{align*}E_2^{i, j}: {\operatorname{Tor}}_{-i}^{\Lambda}(e^{\prime}_{\operatorname{\mathrm{ord}}} \varprojlim_{s} H^j_{\acute{\rm e}\textrm{t}} (Y_G(U_s), {\mathcal O})_{\overline{\mathbb Q}}, {\mathcal O}[-\lambda]) \implies e^{\prime}_{\operatorname{\mathrm{ord}}} H_{\acute{\rm e}\textrm{t}} ^{i+j}(Y_G(U_0)_{\overline{\mathbb Q}}, V_\lambda). \end{align*} $$

4.1 The results of Caraiani–Tamiozzo

Let $K \subseteq G(\mathbb A_f)$ be a neat compact open subgroup and take a maximal ideal ${\mathfrak m} \subseteq \mathbb T$ in the support of $H^i(Y_G(K)_{\overline {\mathbb Q}}, \mathbb F_p)$ . By the work of Scholze [Reference Scholze20], we have a unique continuous semisimple Galois representation

$$ \begin{align*}\overline{\rho_{{\mathfrak m}}}: {\operatorname{Gal}} (\overline{F}/F) \rightarrow \operatorname{GL}_2(\overline{\mathbb F_p}), \end{align*} $$

such that $\overline {\rho _{{\mathfrak m}}}$ is unramified for all v not in a finite set of suitable places of F and such that the characteristic polynomial of $\overline {\rho _{{\mathfrak m}}}(\mathrm {Frob}_v)$ equals $X^2-\mathcal T_vX+\mathcal S_v N(v) \textrm { mod } {\mathfrak m}$ .

Proof This is explained in [Reference Caraiani and Tamiozzo4, Corollary 7.1.2]; we give the proof below for the convenience of the reader. We consider the short exact sequence

$$ \begin{align*}0 \rightarrow \mathbb Z_p \xrightarrow{\cdot p} \mathbb Z_p \rightarrow \mathbb F_p \rightarrow 0, \end{align*} $$

localize the corresponding long exact sequence at ${\mathfrak m}$ and employ Theorem 4.1 to obtain a surjective map

$$ \begin{align*}H^i_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline{\mathbb Q}}, \mathbb Z_p)_{{\mathfrak m}} \xrightarrow{\cdot p} H^i_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline{\mathbb Q}}, \mathbb Z_p)_{{\mathfrak m}}, \end{align*} $$

for all $i \neq g$ . Since $H^i_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline {\mathbb Q}}, \mathbb Z_p)_{{\mathfrak m}}$ is a finitely generated $\mathbb Z_p$ -module, we can apply Nakayama’s lemma to conclude that $H^i_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline {\mathbb Q}}, \mathbb Z_p)_{{\mathfrak m}}=0$ for all $i \neq g$ . When $i=g$ , we get that

$$ \begin{align*}\frac{H^g_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline{\mathbb Q}}, \mathbb F_p)_{{\mathfrak m}}}{\textrm{im}(H^g_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline{\mathbb Q}}, \mathbb Z_p)_{{\mathfrak m}} \rightarrow H^g_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline{\mathbb Q}}, \mathbb F_p)_{{\mathfrak m}}}=0. \end{align*} $$

Since $H^g_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline {\mathbb Q}}, \mathbb F_p)_{{\mathfrak m}} \neq 0$ , it follows that $H^g_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline {\mathbb Q}}, \mathbb Z_p)_{{\mathfrak m}} \neq 0$ as well. Finally, note that the long-exact sequence also yields an injection $H^g_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline {\mathbb Q}}, \mathbb Z_p)_{\mathfrak m} \hookrightarrow {\cdot p} H^g_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline {\mathbb Q}}, \mathbb Z_p)_{\mathfrak m}$ . Hence, $H^g_{\acute{\rm e}\textrm{t}}(Y_G(K)_{\overline {\mathbb Q}}, \mathbb Z_p)_{\mathfrak m}$ has no p-torsion, hence is torsion-free and hence free (using the fact that torsion-free modules over a PID are free).

4.2 Proof of the control theorem

We begin by recalling some notation and assumptions. We let $H^g_{\textrm {Iw}}(Y_G(U_\infty )_{\overline {\mathbb Q}}, {\mathcal O})= \varprojlim _{n} H^g_{\acute{\rm e}\textrm{t}} (Y_G(U_n)_{\overline {\mathbb Q}}, {\mathcal O})$ , and . We recall that $\Lambda _n$ is just the Iwasawa algebra corresponding to elements of $\mathcal ({\mathcal O}_F \otimes \mathbb Z_p)^{\times }$ which are congruent to 1 mod $p^n$ . Let ${\mathfrak m}$ be a maximal ideal in the support of $H^g_{\acute{\rm e}\textrm{t}}(Y_G(U_n)_{\overline {\mathbb Q}} , \mathbb F_p)$ such that the image of the associated Galois representation $\overline {\rho }_{\mathfrak m}$ is not solvable. Let $\lambda $ denote a weight of G that is self-dual in the sense of Section 3.3. Let ${\mathcal O}[-\lambda ]$ denote ${\mathcal O}$ with $\Lambda $ acting via the inverse of the character $\lambda $ . The first step in the proof of the control theorem is to analyze the $E_2^{i, j}$ term in Corollary 3.13. To do so, we first recall the following result from commutative algebra.

Proposition 4.3 Let R be a commutative ring, let $x_1, \ldots , x_N$ be elements of R, and let $R^N$ be the free R-module of rank N with basis $\{e_i : 1 \leq i \leq N \}$ . Consider the Koszul complex $K_{\bullet }(x_1, \ldots , x_N)$ associated with $x_1, \ldots , x_N$ given by

$$ \begin{align*}0 \rightarrow \bigwedge^{N} R^{N} \xrightarrow{d} \bigwedge^{N-1} R^{N} \xrightarrow{d} \cdots \xrightarrow{d} \bigwedge^{1} R^N \xrightarrow{d} R \rightarrow 0, \end{align*} $$

where

$$ \begin{align*}d(e_{j_1} \wedge \cdots \wedge e_{j_i} )= \sum_{s=1}^{i} (-1)^{s-1} x_{j_s} e_{j_1} \wedge \cdots \wedge \widehat{e_{j_s}} \wedge \cdots \wedge e_{j_i}. \end{align*} $$

If $x_1, \ldots ,x_N$ form a regular sequence in R, then $K_{\bullet }(x_1, \ldots , x_N)$ is a free-resolution of $R/(x_1, \ldots , x_N)$ as an R-module.

Proof Since $E_2^{i, j}= {\operatorname {Tor}}_{-i}^{\Lambda _n}(e^{\prime }_{\operatorname {\mathrm {ord}}} \varprojlim _{s} H^j_{\acute{\rm e}\textrm{t}} (Y_G(U_s)_{\overline {\mathbb Q}}, {\mathcal O}), {\mathcal O}[-\lambda ])$ , it suffices to construct a free resolution of ${\mathcal O}[-\lambda ]$ as a $\Lambda _n$ -module of length g. On the other hand, the Iwasawa algebra $\Lambda _n$ can be identified with products of copies of ${\mathcal O}[[T_1, \ldots , T_g]]$ ; it thus suffices to construct a free resolution of ${\mathcal O}[-\lambda ]$ as an ${\mathcal O}[[T_1, \ldots , T_g]]$ module of length g. To do this, we note that the sequence $T_1-\lambda ^{-1}(T_1), \ldots , T_g-\lambda ^{-1}(T_g)$ is a regular sequence in $ O[[T_1, \ldots , T_g]]$ and that

$$ \begin{align*}O[[T_1, \ldots, T_g]]/(T_1-\lambda^{-1}(T_1), \ldots, T_g-\lambda^{-1}(T_g)) \cong {\mathcal O}[-\lambda]\end{align*} $$

as ${\mathcal O}[[T_1, \ldots , T_g]]$ -modules. We can thus apply Proposition 4.3 to deduce that the Koszul complex associated with $T_1-\lambda ^{-1}(T_1), \ldots , T_g-\lambda ^{-1}(T_g)$ provides a free resolution of ${\mathcal O}[-\lambda ]$ as an ${\mathcal O}[[T_1, \ldots , T_g]]$ module of length g.

Proposition 4.6 The following hold:

(a) For all $n \geq 1$ , we have that

$$ \begin{align*}{\operatorname{Tor}}^{\Lambda_n}_{-i}(e^{\prime}_{\operatorname{\mathrm{ord}}}H^g_{\operatorname{\mathrm{Iw}}}(Y_G(U_\infty)_{\overline{\mathbb Q}}, {\mathcal O} )_{\mathfrak m}, {\mathcal O}[-\lambda])= e^{\prime}_{\operatorname{\mathrm{ord}}} H^{i+g}_{\acute{\rm e}\textrm{t}} (Y_G(U_n)_{\overline{\mathbb Q}}, V_\lambda)_{{\mathfrak m}}. \end{align*} $$

(b) When $n=0$ , we have that

$$ \begin{align*}{\operatorname{Tor}}^{\Lambda}_{-i}(e^{\prime}_{\operatorname{\mathrm{ord}}}H^g_{\operatorname{\mathrm{Iw}}}(Y_G(U_\infty)_{\overline{\mathbb Q}}, {\mathcal O} )_{\mathfrak m}, {\mathcal O}[-\lambda])= e^{\prime}_{\operatorname{\mathrm{ord}}} H^{i+g}_{\acute{\rm e}\textrm{t}} (Y_G(U_0)_{\overline{\mathbb Q}}, V_\lambda)_{{\mathfrak m}}.\end{align*} $$

Proof (a) Since localization is exact and commutes with the Tor functor, we deduce from Corollary 3.13 the following spectral sequence:

$$ \begin{align*}F_2^{i, j}:=(E_2^{i,j})_{\mathfrak m} ={\operatorname{Tor}}^{\Lambda_n}_{-i}(e^{\prime}_{\textrm{ord}}H^j_{\textrm{Iw}}(Y_G(U_\infty)_{\overline {\mathbb Q}}, {\mathcal O})_{\mathfrak m}, {\mathcal O}[-\lambda]) \implies e^{\prime}_{\textrm{ord}} H^{i+j}_{\acute{\rm e}\textrm{t}} (Y_G(U_n), V_\lambda)_{{\mathfrak m}}. \end{align*} $$

By Proposition 4.4 and Corollary 4.2, we deduce that $F_2^{i,j}=0$ unless $(i, j) \in \{ (0,g), \ldots (-g, g)\}$ . It follows that the spectral sequence degenerates on the second page, and thus, the desired isomorphism follows directly from applying the filtration theorem.

(b) Using the spectral sequence in Corollary 3.14 and Remark 4.5, this follows by using a similar argument as in part (a).

Theorem 4.7 With the notation as above, the following hold:

(a) For all $n \geq 1$ , we have that $e^{\prime }_{\textrm {ord}} H^g_{\operatorname {\mathrm {Iw}}}(Y_G(U_\infty )_{\overline {\mathbb Q}}, {\mathcal O})_{{\mathfrak m}}$ is free as a $\Lambda _n$ -module.

(b) For all $n \geq 1$ , we have an isomorphism of ${\mathcal O}$ -modules

$$ \begin{align*}e^{\prime}_{\textrm{ord}} H^g_{\operatorname{\mathrm{Iw}}}(Y_G(U_\infty)_{\overline {\mathbb Q}}, {\mathcal O})_{{\mathfrak m}} \otimes_{\Lambda_n} {\mathcal O}[-\lambda] \cong e^{\prime}_{\textrm{ord}}H^g_{\acute{\rm e}\textrm{t}}(Y_G(U_n)_{\overline{\mathbb Q}}, V_\lambda)_{{\mathfrak m}}. \end{align*} $$

that is compatible with the action of $G_{\mathbb Q, \Sigma }$ and $\mathbb T$ .

(c) When $n=0$ , we have a similar isomorphism

$$ \begin{align*}e^{\prime}_{\operatorname{\mathrm{ord}}} H^g_{\operatorname{\mathrm{Iw}}}(Y_G(U_\infty)_{\overline {\mathbb Q}}, {\mathcal O})_{{\mathfrak m}} \otimes_{\Lambda} {\mathcal O}[-\lambda] \cong e^{\prime}_{\operatorname{\mathrm{ord}}}H^g_{\acute{\rm e}\textrm{t}}(Y_G(U_0)_{\overline{\mathbb Q}}, V_\lambda)_{{\mathfrak m}}. \end{align*} $$

Proof We first deduce parts (b) and (c) from the results above.

(b) This follows by setting $i=0$ in Proposition 4.6(a).

(c) This follows by setting $i=0$ in Proposition 4.6(b).

We now prove part (a).

(a) Let $\pi $ and k denote the uniformizer and residue field of ${\mathcal O}$ , respectively. Consider the short-exact sequence

$$ \begin{align*}0 \rightarrow {\mathcal O} \xrightarrow{\cdot \pi} {\mathcal O} \twoheadrightarrow k \rightarrow 0.\end{align*} $$

This gives rise to a long-exact sequence of Tor groups:

$$ \begin{align*} & \cdots \rightarrow {\operatorname{Tor}}^{\Lambda_n}_{1}(e^{\prime}_{\textrm{ord}} H^g_{\textrm{Iw}}(Y_G(U_\infty)_{\overline {\mathbb Q}}, {\mathcal O})_{{\mathfrak m}}, {\mathcal O}) \xrightarrow{\cdot \pi} {\operatorname{Tor}}^{\Lambda_n}_{1}(e^{\prime}_{\textrm{ord}} H^g_{\textrm{Iw}}(Y_G(U_\infty)_{\overline {\mathbb Q}}, {\mathcal O})_{{\mathfrak m}}, {\mathcal O}) \rightarrow \\ &\quad {\operatorname{Tor}}^{\Lambda}_{1}(e^{\prime}_{\textrm{ord}}H^g_{\acute{\rm e}\textrm{t}}(Y_G(U_\infty)_{\overline {\mathbb Q}}, {\mathcal O})_{\mathfrak m}, k ) \rightarrow e^{\prime}_{\textrm{ord}} H^g_{\textrm{Iw}}(Y_G(U_\infty)_{\overline {\mathbb Q}}, {\mathcal O})_{{\mathfrak m}} \otimes_{\Lambda_n} {\mathcal O} \\&\quad\xrightarrow{\cdot \pi} H^g_{\textrm{Iw}}(Y_G(U_\infty)_{\overline {\mathbb Q}}, {\mathcal O})_{{\mathfrak m}} \otimes_{\Lambda_n} {\mathcal O} \rightarrow \cdots \end{align*} $$

By setting $i=-1$ and $\lambda $ to be the trivial character in Proposition 4.6, and then applying Corollary 4.2, we get that

$$ \begin{align*}{\operatorname{Tor}}^{\Lambda_n}_{1}(e^{\prime}_{\textrm{ord}}H^g_{\textrm{Iw}}(Y_G(U_\infty)_{\overline{\mathbb Q}}, {\mathcal O})_{\mathfrak m}, {\mathcal O})=0. \end{align*} $$

On the other hand, Corollary 4.2 implies that $e^{\prime }_{\textrm {ord}} H^g_{\textrm {Iw}}(Y_G(U_\infty )_{\overline {\mathbb Q}}, {\mathcal O})_{{\mathfrak m}} \otimes _{\Lambda _n} {\mathcal O}$ has no $\pi $ -torsion, so the long exact sequence gives us that ${\operatorname {Tor}}^{\Lambda _n}_{1}(e^{\prime }_{\textrm {ord}}H^g_{\acute{\rm e}\textrm{t}}(Y_G(U_\infty )_{\overline {\mathbb Q}}, {\mathcal O})_{\mathfrak m}, k )=0$ . Finally, we note that part (b) and the topological Nakayama’s lemma (see [Reference Balister and Howson1, p. 226]) imply that $e^{\prime }_{\textrm {ord}} H^g_{\textrm {Iw}}(Y_G(U_\infty )_{\overline {\mathbb Q}}, {\mathcal O})_{{\mathfrak m}}$ is finitely generated as a $\Lambda _n$ -module; we can now use the local criterion of flatness to conclude that $e^{\prime }_{\textrm {ord}} H^g_{\textrm {Iw}}(Y_G(U_\infty )_{\overline {\mathbb Q}}, {\mathcal O})_{{\mathfrak m}} $ is flat as a $\Lambda _n$ -module. Since a finitely generated flat module over a noetherian local ring is free, we have that $e^{\prime }_{\textrm {ord}} H^g_{\textrm {Iw}}(Y_G(U_\infty )_{\overline {\mathbb Q}}, {\mathcal O})_{{\mathfrak m}}$ is free as a $\Lambda _n$ -module.