2.1 Modular curves

Given integers $M\geq 1$ , $N\geq 1$ , $m\geq 1$ and $n\geq 1$ with $M+N\geq 5$ , we denote by $Y(M(m),N(n))$ the affine modular curve over $\mathbb {Z}[1/MNmn]$ representing the functor taking a $\mathbb {Z}[1/MNmn]$ -scheme S to the set of isomorphism classes of 5-tuples $(E,P,Q,C,D)$ , where:

• • E is an elliptic curve over S,

• • P is an S-point of E of order M,

• • Q is an S-point of E of order N,

• • C is a cyclic order- $Mm$ subgroup of E defined over S and containing P,

• • D is a cyclic order- $Nn$ subgroup of E defined over S and containing Q,

and such that C and D have trivial intersection. If either $m=1$ or $n=1$ , we omit it from the notation, and we will often write $Y_1(N)$ for $Y(1,N)$ .

We will denote by

$$ \begin{align*}E(M(m),N(n))\rightarrow Y(M(m),N(n)) \end{align*} $$

the universal elliptic curve over $Y(M(m),N(n))$ .

Define the modular group

$$ \begin{align*}\Gamma(M(m),N(n))=\left\{\begin{pmatrix} a & b \\ c & d\end{pmatrix}\in \text{SL}_2(\mathbb{Z}): a\equiv 1\,(M), b\equiv 0\,(Mm), c\equiv 0\,(Nn),d\equiv 1\,(N)\right\}. \end{align*} $$

Then, letting $\mathcal {H}$ be the Poincaré upper half-plane, we have the complex uniformisation

(2.1) $$ \begin{align} Y(M(m),N(n))(\mathbb{C})\cong (\mathbb{Z}/M\mathbb{Z})^{\times}\times\Gamma(M(m),N(n))\backslash\mathcal{H}, \end{align} $$

with a pair $(a,\tau )$ on the right-hand side corresponding to the isomorphism class of the 5-tuple $(\mathbb {C}/\mathbb {Z}+\mathbb {Z}\tau , a\tau /M,1/N,\langle \tau /Mm\rangle , \langle 1/Nn\rangle )$ .

If $r\geq 1$ is an integer, there is an isomorphism of $\mathbb {Z}[1/MNmnr]$ -schemes

$$ \begin{align*}\varphi_r: Y(M(m),N(nr))\xrightarrow{\simeq} Y(M(mr),N(n)) \end{align*} $$

defined in terms of moduli by

$$ \begin{align*}(E,P,Q,C,D)\mapsto (E', P', Q', C',D'), \end{align*} $$

where $E'=E/NnD$ , $P'$ is the image of P in $E'$ , $Q'$ is the image of $r^{-1}(Q)\cap D$ in $E'$ , $C'$ is the image of $r^{-1}(C)$ in $E'$ and $D'$ is the image of D in $E'$ . Under the complex uniformisations (2.1), the isomorphism $\varphi _r$ sends $(a,\tau )\mapsto (a,r\cdot \tau )$ . If

$$ \begin{align*}\varphi_r^{\ast}(E(M(mr),N(n)))\rightarrow Y(M(m),N(nr)) \end{align*} $$

denotes the base change of $E(M(mr),N(n))\rightarrow Y(M(mr),N(n))$ under $\varphi _r$ , there is a natural degree-r isogeny

$$ \begin{align*}\lambda_r:E(M(m),N(nr))\rightarrow \varphi_r^{\ast}(E(M(mr),N(n))). \end{align*} $$

2.2 Degeneracy maps

With the same notations as above, we have natural degeneracy maps

$$ \begin{align*} & Y(M(m),Nr(n))\xrightarrow{\mu_r} Y(M(m),N(nr))\xrightarrow{\nu_r} Y(M(m),N(n)), \\ & Y(Mr(m),N(n))\xrightarrow{\check{\mu}_r} Y(M(mr),N(n))\xrightarrow{\check{\nu}_r} Y(M(m),N(n)), \end{align*} $$

forgetting the extra level structure, for example

$$ \begin{align*} & \mu_r(E,P,Q,C,D)=(E,P,r\cdot Q, C,D), \\ & \nu_r(E,P,Q,C,D)=(E,P,Q,C,rD). \end{align*} $$

We also define degeneracy maps

(2.2) $$ \begin{align} \begin{aligned} & \pi_1: Y(M(m),Nrs(nt))\rightarrow Y(M(m),N(ns)), \\ & \pi_2: Y(M(m),Nrs(nt))\rightarrow Y(M(m),N(ns)), \end{aligned} \end{align} $$

acting on the moduli space by

$$ \begin{align*} & \pi_1(E,P,Q,C,D)=(E,P,rs\cdot Q,C, rtD), \\ & \pi_2(E,P,Q,C,D)=(E',P',Q',C',D'), \end{align*} $$

where $E'=E/NnsD$ , $P'$ is the image of P in $E'$ , $Q'$ is the image of $t^{-1}(s\cdot Q)\cap D$ in $E'$ , $C'$ is the image of C in $E'$ and $D'$ is the image of D in $E'$ . Under the complex uniformisations in (2.1), the maps $\pi _1$ and $\pi _2$ correspond to the identity and to multiplication by $rt$ , respectively, on $\mathcal {H}$ . It is straightforward to check that the maps $\pi _1$ and $\pi _2$ are given by the compositions

$$ \begin{align*} & Y(M(m),Nrs(nt))\xrightarrow{\mu_{rs}} Y(M(m),N(nrst))\xrightarrow{\nu_{rt}} Y(M(m),N(ns)), \\ & Y(M(m),Nrs(nt))\kern-1pt\xrightarrow{\mu_{rs}}\kern-1pt Y(M(m),N(nrst))\kern-1pt\xrightarrow{\varphi_{rt}}\kern-1pt Y(M(mrt),N(ns))\kern-1pt\xrightarrow{\check{\nu}_{rt}}\kern-1pt Y(M(m),N(ns)), \end{align*} $$

respectively.

2.3 Relative Tate modules

Fix a prime p. Let S be a $\mathbb {Z}[1/MNmnp]$ -scheme, and let

$$ \begin{align*} v:E(M(m),N(n))_S\rightarrow Y(M(m),N(n))_S \end{align*} $$

be the structural morphism. For every ${\mathbb Z}[1/MNmnp]$ -scheme X, denote by $A=A_X$ either the locally constant constructible sheaf ${\mathbb Z}/p^t(j)$ or the locally constant p-adic sheaf ${\mathbb Z}_p(j)$ on $X_{\operatorname {\mathrm {et}}}$ , for fixed $t\geq 1$ and $j\in \mathbb {Z}$ . Set

$$ \begin{align*} \mathscr{T}_{M(m),N(n)}(A)=R^1v_{\ast} \mathbb{Z}_p(1)\otimes_{\mathbb{Z}_p} A\quad\text{and}\quad \mathscr{T}_{M(m),N(n)}^{\ast}(A)=\operatorname{\mathrm{Hom}}(\mathscr{T}_{M(m),N(n)}(A),A). \end{align*} $$

In particular, in the case $A=\mathbb {Z}_p$ , this gives the relative Tate module of the universal elliptic curve and its dual, respectively; in this case, we will often drop A from the notation.

From the proper base change theorem, both ${\mathscr T}_{M(m),N(n)}(A)$ and ${\mathscr T}_{M(m),N(n)}^*(A)$ are locally constant p-adic sheaves on $Y(M(m),N(n))_S$ of formation compatible with base changes along morphisms of $\mathbb {Z}[1/MNmnp]$ -schemes $S'\rightarrow S$ .

For every integer $r\geq 0$ , define

$$\begin{align*}{\mathscr L}_{M(m),N(n),r}(A) = \operatorname{\mathrm{Tsym}}_A^r {\mathscr T}_{M(m),N(n)}(A), \quad {\mathscr S}_{M(m),N(n),r}(A) = \operatorname{\mathrm{Symm}}_A^r {\mathscr T}_{M(m),N(n)}^*(A), \end{align*}$$

where, for any finite free module M over a profinite ${\mathbb Z}_p$ -algebra R, one denotes by $\operatorname {\mathrm {Tsym}}_R^r M$ the R-submodule of symmetric tensors in $M^{\otimes r}$ and by $\operatorname {\mathrm {Symm}}_R^r M$ the maximal symmetric quotient of $M^{\otimes r}$ .

When the level of the modular curve $Y(M(m),N(n))_S$ is clear, we may use the simplified notations

$$\begin{align*}{\mathscr L}_r(A) = {\mathscr L}_{M(m),N(n),r}(A), \quad {\mathscr L}_r = {\mathscr L}_r({\mathbb Z}_p), \quad {\mathscr S}_r(A) = {\mathscr S}_{M(m),N(n),r}(A), \quad {\mathscr S}_r = {\mathscr S}_r({\mathbb Z}_p). \end{align*}$$

2.4 Hecke operators

Let $\mathscr {F}_{M(m),N(n)}^r$ denote either $\mathscr {L}_{M(m),N(n),r}(A)$ or $\mathscr {S}_{M(m),N(n),r}(A)$ , and let q be a rational prime. Then there are natural isomorphisms of sheaves

(2.3) $$ \begin{align} \nu_q^{\ast}(\mathscr{F}_{M(m),N(n)}^r)\cong \mathscr{F}_{M(m),N(nq)}^r\quad\text{and}\quad \check{\nu}_q^{\ast}(\mathscr{F}_{M(m),N(n)}^r)\cong \mathscr{F}_{M(mq),N(n)}^r, \end{align} $$

and therefore pullback morphisms

$$ \begin{align*} & H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(n))_S,\mathscr{F}_{M(m),N(n)}^r)\xrightarrow{\nu_q^{\ast}} H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(nq))_S,\mathscr{F}_{M(m),N(nq)}^r), \\ & H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(n))_S,\mathscr{F}_{M(m),N(n)}^r)\xrightarrow{\check{\nu}_q^{\ast}} H^i_{\operatorname{\mathrm{et}}}(Y(M(mq),N(n))_S,\mathscr{F}_{M(mq),N(n)}^r), \end{align*} $$

and traces

(2.4) $$ \begin{align} \begin{aligned} & H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(nq))_S,\mathscr{F}_{M(m),N(nq)}^r)\xrightarrow{\nu_{q\ast}} H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(n))_S,\mathscr{F}_{M(m),N(n)}^r), \\ & H^i_{\operatorname{\mathrm{et}}}(Y(M(mq),N(n))_S,\mathscr{F}_{M(mq),N(n)}^r)\xrightarrow{\check{\nu}_{q\ast}} H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(n))_S,\mathscr{F}_{M(m),N(n)}^r). \end{aligned} \end{align} $$

Also, the isogeny $\lambda _q$ induces morphisms of sheaves

$$ \begin{align*}\lambda_{q\ast} : \mathscr{F}_{M(m),N(nq)}^r\rightarrow \varphi_q^{\ast}(\mathscr{F}_{M(mq),N(n)}^r) \quad\text{and}\quad \lambda_{q}^{\ast} : \varphi_q^{\ast}(\mathscr{F}_{M(mq),N(n)}^r)\rightarrow \mathscr{F}_{M(m),N(nq)}^r. \end{align*} $$

These morphisms allow us to define

$$ \begin{align*} & \Phi_{q\ast}:H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(nq))_S,\mathscr{F}_{M(m),N(nq)}^r)\rightarrow H^i_{\operatorname{\mathrm{et}}}(Y(M(mq),N(n))_S,\mathscr{F}_{M(mq),N(n)}^r), \\ & \Phi_{q}^{\ast}:H^i_{\operatorname{\mathrm{et}}}(Y(M(mq),N(n))_S,\mathscr{F}_{M(mq),N(n)}^r)\rightarrow H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(nq))_S,\mathscr{F}_{M(m),N(nq)}^r), \end{align*} $$

as the compositions

$$ \begin{align*}\Phi_{q\ast}=\varphi_{q\ast}\circ\lambda_{q\ast}\quad\text{and}\quad \Phi_q^{\ast}=\lambda_q^{\ast}\circ\varphi_q^{\ast}. \end{align*} $$

We define the Hecke operators $T_q$ and the adjoint Hecke operators $T_q'$ acting on the étale cohomology groups

$$ \begin{align*}H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(nq))_S,\mathscr{F}_{M(m),N(nq)}^r) \end{align*} $$

as the compositions

$$ \begin{align*}T_q=\check{\nu}_{q\ast}\circ\Phi_{q\ast}\circ\nu_q^{\ast}\quad\text{and}\quad T_q'=\nu_{q\ast}\circ\Phi_q^{\ast}\circ\check{\nu}_q^{\ast}. \end{align*} $$

If we define pullbacks

$$ \begin{align*} & H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(n))_S,\mathscr{F}_{M(m),N(n)}^r)\xrightarrow{\pi_1^{\ast}} H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(nq))_S,\mathscr{F}_{M(m),N(nq)}^r), \\ & H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(n))_S,\mathscr{F}_{M(m),N(n)}^r)\xrightarrow{\pi_2^{\ast}} H^i_{\operatorname{\mathrm{et}}}(Y(M(mq),N(n))_S,\mathscr{F}_{M(mq),N(n)}^r), \end{align*} $$

and pushforward

$$ \begin{align*} & H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(nq))_S,\mathscr{F}_{M(m),N(nq)}^r)\xrightarrow{\pi_{1\ast}} H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(n))_S,\mathscr{F}_{M(m),N(n)}^r), \\ & H^i_{\operatorname{\mathrm{et}}}(Y(M(mq),N(n))_S,\mathscr{F}_{M(mq),N(n)}^r)\xrightarrow{\pi_{2\ast}} H^i_{\operatorname{\mathrm{et}}}(Y(M(m),N(n))_S,\mathscr{F}_{M(m),N(n)}^r), \end{align*} $$

as

$$ \begin{align*}\pi_1^{\ast}=\nu_q^{\ast},\quad \pi_2^{\ast}=\Phi_q^{\ast}\circ\check{\nu}_q^{\ast},\quad \pi_{1\ast}=\nu_{q\ast}\quad\text{and}\quad \pi_{2\ast}=\check{\nu}_{q\ast}\circ\Phi_{q\ast}, \end{align*} $$

then we can write

$$ \begin{align*}T_q= \pi_{2\ast}\circ\pi_1^{\ast}\quad\text{and}\quad T_q'=\pi_{1\ast}\circ\pi_2^{\ast}. \end{align*} $$

Now we introduce diamond operators. For $d\in (\mathbb {Z}/MN\mathbb {Z})^{\times }$ , these are defined on the curves $Y(M(m),N(n))$ as the automorphisms $\langle d\rangle $ acting on the moduli space by

$$ \begin{align*}(E,P,Q,C,D)\mapsto (E,d^{-1}\cdot P,d\cdot Q, C, D). \end{align*} $$

We can also define the diamond operator $\langle d \rangle $ on the corresponding universal elliptic curve as the unique automorphism making the diagram

cartesian. This, in turn, induces automorphisms $\langle d\rangle = \langle d\rangle ^{\ast }$ and $\langle d\rangle '=\langle d\rangle _{\ast }$ on the group $H^i_{\operatorname {\mathrm {et}}}(Y(M(m),N(n))_S,\mathscr {F}_{M(m),N(n)}^r)$ which are inverses of each other. In general, we will be interested in modular curves of the form $Y(1(m),N(n))$ . In this case, the natural pairing $\mathscr {L}_r\otimes _{\mathbb {Z}_p}\mathscr {S}_r\rightarrow \mathbb {Z}_p$ together with cup-product yields a pairing

$$ \begin{align*}H^1_{\operatorname{\mathrm{et}}}(Y(1(m),N(n))_S,\mathscr{L}_r(1))\otimes_{\mathbb{Z}_p} H^1_{\operatorname{\mathrm{et}},c}(Y(1(m),N(n))_S,\mathscr{S}_r)\rightarrow \mathbb{Z}_p \end{align*} $$

which becomes perfect after inverting p. The operators $T_q$ , $T_q'$ , $\langle d\rangle $ , $\langle d\rangle '$ induce endomorphisms on compactly supported cohomology and

$$ \begin{align*}(T_q,T_q'),\quad (T_q',T_q),\quad (\langle d\rangle,\langle d\rangle'), \quad\text{and}\quad (\langle d\rangle',\langle d\rangle) \end{align*} $$

are adjoint pairs under this pairing.

2.5 Galois representations

Let $f\in S_k(N_f,\chi _f)$ be a newform of weight $k=r+2\geq 2$ , level $N_f$ , and character $\chi _f$ . Let p be a prime, and let E be a finite extension of $\mathbb {Q}_p$ with ring of integers $\mathcal {O}$ containing the Fourier coefficients of f. By the work of Eichler–Shimura and Deligne, there is a two-dimensional representation

$$ \begin{align*}\rho_f\,:\,G_{\mathbb{Q}}\longrightarrow \mathrm{GL}_2(E) \end{align*} $$

unramified outside $pN_f$ and characterised by the property that

$$ \begin{align*}\mathrm{trace}\,\rho_f(\mathrm{Fr}_{q})=a_{q}(f) \end{align*} $$

for all primes $q\nmid pN_f$ , where $\mathrm {Fr}_{q}$ denotes an arithmetic Frobenius element at q (in fact, this is the dual of the p-adic representation constructed by Deligne).

It will be convenient for our purposes to work with the following geometric realisation of $\rho _f$ . Let

$$\begin{align*}H^1_{\operatorname{\mathrm{et}}}(Y_{1}(N_f)_{\overline{\mathbb{Q}}},\mathscr{L}_{r}(1))\otimes_{\mathbb{Z}_p}E\twoheadrightarrow V_f \end{align*}$$

be the maximal quotient on which $T_q'$ and $\langle d\rangle '$ act as multiplication by $a_q(f)$ and $\chi _f(d)$ for all primes $q\nmid N_f$ and all $d\in (\mathbb {Z}/N_f\mathbb {Z})^{\times }$ . Then $V_f$ is a two-dimensional E-vector space affording the p-adic representation $\rho _f$ , and we let $T_f\subset V_f$ be the lattice defined by the image of

$$ \begin{align*} H^1_{\operatorname{\mathrm{et}}}(Y_{1}(N_f)_{\overline{\mathbb{Q}}},\mathscr{L}_{r}(1))\otimes_{\mathbb{Z}_p}\mathcal{O} \end{align*} $$

under the above quotient map.

5.1 Hida families

Let $\Lambda = {\mathbb Z}_p[[1+p{\mathbb Z}_p]]$ , and let

$$\begin{align*}\mathcal W=\mathrm{Spf} (\Lambda) \end{align*}$$

be the weight space. Then, for any extension E of $\mathbb {Q}_p$ , we have $\mathcal W(E)=\operatorname {\mathrm {Hom}}_{\text {cont}}(1+p\mathbb {Z}_p,E^{\times })$ . Points of the form $\nu _{r,\epsilon }(n)= \epsilon (n)n^r$ , where r is a nonnegative integer and $\epsilon $ is a finite order character, will be called arithmetic. We refer to $k=r+2$ as the weight of $\nu _{r,\epsilon }$ . Arithmetic points of the form $\nu _r=\nu _{r,1}$ will be called classical.

More generally, let $\mathcal {R}$ be a normal domain finite flat over $\Lambda $ , and let $\mathcal W_{\mathcal {R}}=\mathrm {Spf}(\mathcal {R})$ . Then, a point $x\in \mathcal W_{\mathcal {R}}(\overline {\mathbb {Q}}_p)$ will be called arithmetic if it lies above an arithmetic point $\nu _{r,\epsilon }$ of $\mathcal W(\overline {\mathbb {Q}}_p)$ , and classical if it lies above a classical point $\nu _r$ of $\mathcal W(\overline {\mathbb {Q}}_p)$ . Again, we refer to $k=r+2$ as the weight of x.

Let M be a positive integer coprime to p. A Hida family of tame level M and character $\chi :(\mathbb {Z}/Mp\mathbb {Z})^{\times }\rightarrow \overline {\mathbb {Q}}_p^{\times }$ is a formal q-expansion

$$\begin{align*}{\mathbf{f}} = \sum_{n\geq 1} a_n({\mathbf{f}}) q^n \in \Lambda_{{\mathbf{f}}}[[q]], \end{align*}$$

where $\Lambda _{\mathbf {f}}$ is a normal domain finite flat over $\Lambda $ , such that, for any arithmetic point $x\in \mathcal W_{\Lambda _{\mathbf {f}}}(\overline {\mathbb {Q}}_p)$ lying over some $\nu _{r,\epsilon }$ , the corresponding specialisation is a p-ordinary eigenform $f_x\in S_k(Mp^s,\chi \epsilon \omega ^{-r})$ . As above, we have denoted by k the weight of x, and we can take $s=\max \lbrace 1, {\mathrm {ord}}_p(\mathrm {cond}(\epsilon ))\rbrace $ . We say that a Hida family ${\mathbf {f}}$ is primitive if the specialisations $f_x$ at arithmetic points x are p-stabilised newforms. We say that it is normalised if $a_1({\mathbf {f}})=1$ .

Let ${\mathbf {f}}$ be a normalised primitive Hida family of tame level M. For each arithmetic point $x\in \mathcal W_{\Lambda _{\mathbf {f}}}(\overline {\mathbb {Q}}_p)$ , let ${\mathbf {f}}_x$ denote the specialisation of ${\mathbf {f}}$ at x, and let $f_x$ be the corresponding newform. There exists a locally free rank-two $\Lambda _{\mathbf {f}}$ -module $\mathbb {V}_{\mathbf {f}}$ equipped with a continuous action of $G_{\mathbb {Q}}$ , such that, for any arithmetic point $x\in \mathcal W_{\Lambda _{\mathbf {f}}}(\overline {\mathbb {Q}}_p)$ , the corresponding specialisation $\mathbb {V}_{\mathbf {f}}\otimes _{\Lambda _{\mathbf {f}},x}\overline {\mathbb {Q}}_p$ recovers the $G_{\mathbb {Q}}$ -representation $V_{f_x}$ attached to $f_x$ . In particular, the representation $\mathbb {V}_{\mathbf {f}}$ is unramified at any prime $q\nmid Mp$ and $\mathrm {Tr}(\mathrm {Fr}_q)=a_q({\mathbf {f}})$ . We refer to $\mathbb {V}_{\mathbf {f}}$ as the big Galois representation attached to ${\mathbf {f}}$ . If for some (equivalently all) arithmetic point $x_0\in \mathcal W_{\Lambda _{\mathbf {f}}}(\overline {\mathbb {Q}}_p)$ the $G_{\mathbb {Q}}$ -representation $T_{f_{x_0}}$ attached to $f_{x_0}$ is residually irreducible, then $\mathbb {V}_{\mathbf {f}}$ is a free $\Lambda _{\mathbf {f}}$ -module.

5.2 Continuous functions and distributions

Define the semigroups

$$\begin{align*}\Sigma_0(p)=\begin{pmatrix} \mathbb{Z}_p^{\times} & \mathbb{Z}_p \\ p\mathbb{Z}_p & \mathbb{Z}_p \end{pmatrix} \quad\text{and}\quad \Sigma_0'(p)=\begin{pmatrix} \mathbb{Z}_p & \mathbb{Z}_p \\ p\mathbb{Z}_p & \mathbb{Z}_p^{\times} \end{pmatrix}. \end{align*}$$

The sets $\mathsf {T}=\mathbb {Z}_p^{\times }\times \mathbb {Z}_p$ and $\mathsf {T}'=p\mathbb {Z}_p\times \mathbb {Z}_p^{\times }$ bear a right action of $\Sigma _0(p)$ and $\Sigma _0'(p)$ , respectively.

Let $\nu $ be a character of $\mathbb {Z}_p^{\times }$ taking values in a finite extension E of $\mathbb {Q}_p$ . Let $\mathcal {O}$ be the ring of integers of E and denote by $\mathfrak {m}$ its maximal ideal. Let $\text {Cont}(\mathbb {Z}_p,\mathcal {O})$ denote the module of continuous functions on $\mathbb {Z}_p$ with values in $\mathcal {O}$ . Define $\mathcal {O}$ -modules

$$\begin{align*}\mathcal{A}_{\nu}=\left\{ f:\mathsf{T}\rightarrow \mathcal{O}\; \vert\; f(1,z)\in \text{Cont}(\mathbb{Z}_p,\mathcal{O}) \text{ and } f(a\cdot t)=\nu(a)\cdot f(t) \text{ for all } a\in\mathbb{Z}_p^{\times},\, t\in \mathsf{T}\right\}, \end{align*}$$

$$\begin{align*}\mathcal{A}_{\nu}'=\left\{ f:\mathsf{T}'\rightarrow \mathcal{O}\; \vert\; f(pz,1)\in \text{Cont}(\mathbb{Z}_p,\mathcal{O}) \text{ and } f(a\cdot t)=\nu(a)\cdot f(t) \text{ for all } a\in\mathbb{Z}_p^{\times},\, t\in \mathsf{T}'\right\} \end{align*}$$

equipped with the $\mathfrak {m}$ -adic topology, and $\mathcal {O}$ -modules

$$\begin{align*}\mathcal{D}_{\nu}=\operatorname{\mathrm{Hom}}_{\text{cont},\mathcal{O}}(\mathcal{A}_{\nu},\mathcal{O}),\quad\mathcal{D}_{\nu}'=\operatorname{\mathrm{Hom}}_{\text{cont},\mathcal{O}}(\mathcal{A}_{\nu}',\mathcal{O}) \end{align*}$$

equipped with the weak- $\ast $ topology. The right $\Sigma _0^{\cdot }(p)$ -action on $\mathsf {T}^{\cdot }$ yields naturally a left $\Sigma _0^{\cdot }(p)$ -action on $\mathcal {A}_{\nu }^{\cdot }$ and a right $\Sigma _0^{\cdot }(p)$ -action on $\mathcal {D}_{\nu }^{\cdot }$ .

5.3 Group cohomology and étale cohomology

Let N and m be coprime positive integers which are also coprime to p, let $Y=Y(1,N(pm))$ , and let $\Gamma $ be the corresponding modular group. Let $\mathcal {E}\rightarrow Y$ be the universal elliptic curve over Y, and denote by $C_p$ the canonical cyclic p-subgroup. Let $\mathscr {T}$ be the relative p-adic Tate module of $\mathcal {E}$ over Y. Fix a geometric point $\eta :\text {Spec}(\overline {\mathbb {Q}})\rightarrow Y$ , and choose an isomorphism $\mathscr {T}_{\eta }\cong \mathbb {Z}_p\oplus \mathbb {Z}_p$ , such that the Weil pairing on $\mathscr {T}_{\eta }$ corresponds to the natural determinant map on the right and the reduction modulo p of the element (0,1) generates $C_{p,\eta }$ .

Let $\mathcal {G}=\pi _1^{\operatorname {\mathrm {et}}}(Y,\eta )$ . The action of $\mathcal {G}$ on $\mathscr {T}$ yields an action of $\mathcal {G}$ on $\mathbb {Z}_p\oplus \mathbb {Z}_p$ , and hence a continuous representation $\rho :\mathcal {G}\rightarrow \text {GL}_2(\mathbb {Z}_p)$ . More precisely, for any $g\in \mathcal {G}$ ,

$$ \begin{align*}g\cdot(a,b)=(a,b)\rho(g)^{-1}. \end{align*} $$

In fact, since the action of $\mathcal {G}$ preserves the canonical subgroup, we have a continuous representation $\rho :\mathcal {G}\rightarrow \Gamma _0(p\mathbb {Z}_p)$ , where

$$ \begin{align*}\Gamma_0(p\mathbb{Z}_p)=\left\{\begin{pmatrix} a & b \\ c & d\end{pmatrix}\in \text{GL}_2(\mathbb{Z}_p): p\mid c\right\}. \end{align*} $$

The anti-involution of $\text {GL}_2(\mathbb {Z}_p)$ given by $\gamma \mapsto \gamma ^{\iota }=\det (\gamma )\gamma ^{-1}$ restricts to $\Gamma _0(p\mathbb {Z}_p)$ and allows us to think of this group as acting on the right or left as convenient.

Taking the stalk at $\eta $ gives an equivalence of categories between the category $\mathbf {S}_f(Y_{\operatorname {\mathrm {et}}})$ of locally constant constructible sheaves with finite stalk of p-power order at $\eta $ and the category $\mathbf {M}_f(\mathcal {G})$ of finite $\mathcal {G}$ -sets of p-power order. For any topological group G, define $\mathbf {M}_f(G)$ as we did for $\mathcal {G}$ . Let $\mathbf {M}_{\text {cont}}(G)$ be the category of G-modules which are filtred unions $\cup _{i\in I} M_i$ with $M_i\in \mathbf {M}_f(G)$ , and let $\mathbf {M}(G)\subset \mathbf {M}_{\text {cont}}(G)^{\mathbb {N}}$ be the category of inverse systems of objects in $\mathbf {M}_{\text {cont}}(G)$ . Define $\mathbf {S}(Y_{\operatorname {\mathrm {et}}})$ similarly. Then, there is an equivalence of categories between $\mathbf {M}(\mathcal {G})$ and $\mathbf {S}(Y_{\operatorname {\mathrm {et}}})$ . Moreover, the representation $\rho $ defined above yields a functor $\mathbf {M}(\Gamma _0(p\mathbb {Z}_p))\rightarrow \mathbf {M}(\mathcal {G})$ . Regarding this functor, we adopt the following criterion: if an object $\mathcal {F}\in \mathbf {M}(\Gamma _0(p\mathbb {Z}_p))$ is given as a left $\Gamma _0(p\mathbb {Z}_p)$ -module, we define the left $\mathcal {G}$ -action via the map $\rho :\mathcal {G}\rightarrow \Gamma _0(p\mathbb {Z}_p)$ ; if it is given as a right $\Gamma _0(p\mathbb {Z}_p)$ -module, we define the left $\mathcal {G}$ -action via the map $g\mapsto \rho (g)^{-1}$ . Given an inverse system of sheaves $\boldsymbol {\mathcal {F}}=(\boldsymbol {\mathcal {F}}_i)_{i\in \mathbb {N}}\in \mathbf {S}(Y_{\operatorname {\mathrm {et}}})$ , we use the notation $H^j_{\operatorname {\mathrm {et}}}(Y,\boldsymbol {\mathcal {F}})$ for continuous étale cohomology as defined by Jannsen [Reference JannsenJan88], and write $\mathtt {H}^j_{\operatorname {\mathrm {et}}}(Y,\boldsymbol {\mathcal {F}})=\varprojlim _i H^j_{\operatorname {\mathrm {et}}}(Y,\boldsymbol {\mathcal {F}}_i)$ . There is a natural surjective morphism $H^j_{\operatorname {\mathrm {et}}}(Y,\boldsymbol {\mathcal {F}})\rightarrow \mathtt {H}^j_{\operatorname {\mathrm {et}}}(Y,\boldsymbol {\mathcal {F}})$ . The compactly supported cohomology groups $H^j_{\operatorname {\mathrm {et}},c}(Y,\boldsymbol {\mathcal {F}})$ and $\mathtt {H}^j_{\operatorname {\mathrm {et}},c}(Y,\boldsymbol {\mathcal {F}})$ are defined similarly.

There is an isomorphism $\pi _1^{\operatorname {\mathrm {et}}}(Y_{\overline {\mathbb {Q}}},\eta )\cong \hat {\Gamma }$ . Thus, if $\mathcal {F}\in \mathbf {M}_f(\mathcal {G})$ is a discrete $\mathcal {G}$ -module and $\boldsymbol {\mathcal {F}}$ is the corresponding object in $\mathbf {S}_f(Y_{\operatorname {\mathrm {et}}})$ , there are natural isomorphisms

(5.1) $$ \begin{align} H^1_{\operatorname{\mathrm{et}}}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{F}})\cong H^1(\hat{\Gamma},\mathcal{F})\cong H^1(\Gamma,\mathcal{F}). \end{align} $$

Let $\mathcal {F}\in \mathbf {M}_f(\Gamma _0(p\mathbb {Z}_p))$ be a left $\Gamma _0(p\mathbb {Z}_p)$ -module, and assume that the $\Gamma _0(p\mathbb {Z}_p)$ -action on $\mathcal {F}$ extends to a left action of $\Sigma _0^{\cdot }(p)$ . Let $S=\Sigma _0^{\cdot }(p)\cap \text {GL}_2(\mathbb {Q})$ . The pair $(\Gamma ,S)$ is then a Hecke pair in the sense of [Reference Ash and StevensAS86a, Section 1.1], and there is a covariant (left) action of the Hecke algebra $D(\Gamma , S)$ on $H^1(\Gamma ,\mathcal {F})$ . For each $g\in S$ , let $T(g)=\Gamma g\Gamma $ . Following [Reference Greenberg and StevensGS93, Section 1], we define, for each positive integer n, the Hecke operators

$$ \begin{align*}T_n=T\left(\begin{pmatrix} 1 & \\ & n \end{pmatrix}\right), \quad T_n'=T\left(\begin{pmatrix} n & \\ & 1 \end{pmatrix}\right). \end{align*} $$

Also, for each positive integer a coprime to p, let

$$ \begin{align*}[a]_p = T\left(\begin{pmatrix} a & \\ & a \end{pmatrix}\right),\quad [a]_p'=T\left(\begin{pmatrix} a & \\ & a \end{pmatrix}\right). \end{align*} $$

Finally, for each positive integer a coprime to N, choose $\beta _a$ (respectively, $\beta _a'$ ) in $\Gamma _0(Npm)$ whose lower right entry is congruent to a (respectively, $a^{-1}$ ) modulo N, and let

$$ \begin{align*}[a]_N=T(\beta_a),\quad [a]_N'=T(\beta_a'). \end{align*} $$

The isomorphism (5.1) is compatible with Hecke actions in the following sense. To distinguish between different levels, we shall now write $\tilde {Y}(m)$ and $\tilde {\Gamma }(m)$ for the above Y and $\Gamma $ , respectively. Let s be a positive integer. Choose, as above, a geometric point $\eta :\text {Spec}(\overline {\mathbb {Q}})\rightarrow \tilde {Y}(m)$ , and let $\eta _s:\text {Spec}(\overline {\mathbb {Q}})\rightarrow \tilde {Y}(ms)$ be a geometric point lying above $\eta $ . Let $r=1+{\mathrm {ord}}_p(s)$ , and choose an isomorphism $\mathscr {T}_{\eta _s}\cong \mathbb {Z}_p\oplus \mathbb {Z}_p$ , such that the Weil pairing on $\mathscr {T}_{\eta _s}$ corresponds to the natural determinant map on the right, and the reduction modulo $p^r$ of the element (0,1) generates the canonical subgroup $C_{p^r,\eta _s}$ . Using these choices to define the corresponding isomorphisms between group cohomology and étale cohomology, there are commutative diagrams

Also, if $\left (\begin {smallmatrix}s & \\ & 1 \end {smallmatrix}\right )\in \Sigma _0^{\cdot }(p)$ , we have the commutative diagram

and, if $\left (\begin {smallmatrix} 1 & \\ & s \end {smallmatrix}\right )\in \Sigma _0^{\cdot }(p)$ , the commutative diagram

In the bottom lines of the previous two diagrams, $\varphi _s^{\ast }(\mathcal {F})$ is $\mathcal {F}$ with the action of $\Gamma _0(p^r\mathbb {Z}_p)$ conjugated by $\left (\begin {smallmatrix} s & \\ & 1\end {smallmatrix}\right )$ ; the map $\lambda _{s\ast }$ is induced by the map $\mathcal {F}\rightarrow \varphi _s^{\ast }(\mathcal {F})$ defined by $c\mapsto \left (\begin {smallmatrix} s & \\ & 1\end {smallmatrix}\right )c$ ; $\varphi _{s\ast }$ is induced by the pair of compatible maps $\Gamma (1(s),N(pm))\rightarrow \tilde {\Gamma }(ms)$ and $\varphi _s^{\ast }(\mathcal {F})\rightarrow \mathcal {F}$ defined by $\gamma \mapsto \left (\begin {smallmatrix} s^{-1} & \\ & 1\end {smallmatrix}\right ) \gamma \left (\begin {smallmatrix} s & \\ & 1\end {smallmatrix}\right )$ and $c\mapsto c$ , respectively; $\lambda _{s}^{\ast }$ is induced by the map $\varphi _s(\mathcal {F})\rightarrow \mathcal {F}$ defined by $c\mapsto \left (\begin {smallmatrix} 1 & \\ & s\end {smallmatrix}\right )c$ and $\varphi _{s}^{\ast }$ is induced by the pair of compatible maps $\tilde {\Gamma }(m)\rightarrow \Gamma (1(s),N(pm))$ and $\mathcal {F}\rightarrow \varphi _s^{\ast }(\mathcal {F})$ defined by $\gamma \mapsto \left (\begin {smallmatrix} 1 & \\ & s^{-1}\end {smallmatrix}\right ) \gamma \left (\begin {smallmatrix} 1 & \\ & s\end {smallmatrix}\right )$ and $c\mapsto c$ , respectively.

We shall denote by $\pi _{2\ast }$ and $\pi _2^{\ast }$ , respectively, the composition of the maps in the rows of the previous two diagrams, both in étale cohomology and in group cohomology. Similarly, we shall also use $\pi _{1\ast }$ and $\pi _1^{\ast }$ to denote the corresponding corestriction and restriction maps.

For any rational prime q, a simple calculation shows that the following identities hold in group cohomology whenever the maps involved are defined:

$$ \begin{align*}T_q=\pi_{1\ast}\circ\pi_2^{\ast},\quad T_q'=\pi_{2\ast}\circ\pi_1^{\ast}. \end{align*} $$

Therefore, under the isomorphism (5.1), the covariant action of the operators $T_q$ , $T_q'$ on étale cohomology corresponds to the covariant action of the operators $T_q$ , $T_q'$ on group cohomology, whenever defined. Similarly, the covariant action of the operators $\langle d\rangle $ , $\langle d\rangle '$ on étale cohomology corresponds to the covariant action of the operators $[d]_N$ , $[d]_N'$ on group cohomology.

The anti-involution $\iota $ extends to $\text {Mat}_{2\times 2}(\mathbb {Z}_p)$ in the obvious way, and turns a left (respectively, right) action of $\Sigma _0(p)$ into a right (respectively, left) action of $\Sigma _0'(p)$ . Thus, given an object $\mathcal {F}\in \mathbf {M}(\Gamma _0(p\mathbb {Z}_p))$ whose right $\Gamma _0(p\mathbb {Z}_p)$ -action extends to a right $\Sigma _0^{\cdot }(p)$ -action, there is an isomorphism $H^1_{\operatorname {\mathrm {et}}}(Y_{\overline {\mathbb {Q}}},\boldsymbol {\mathcal {F}})\cong H^1(\Gamma ,\mathcal {F})$ under which the contravariant action of the operators $T_q$ , $T_q'$ , $\langle d\rangle $ , $\langle d\rangle '$ on étale cohomology corresponds to the contravariant action of the operators $T_q$ , $T_q'$ , $[d]_N$ , $[d]_N'$ on group cohomology, whenever defined.

Consider the modules $\mathcal {A}_{\nu }^{\cdot }$ and $\mathcal {D}_{\nu }^{\cdot }$ defined earlier in this section. The action of $\Gamma _0(p\mathbb {Z}_p)$ on $\mathsf {T}'$ is transitive and the stabiliser of the element $(0,1)\in \mathsf {T}'$ is the subgroup

$$\begin{align*}P(\mathbb{Z}_p)=\left\{\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}\in \text{GL}_2(\mathbb{Z}_p)\right\}, \end{align*}$$

so we can identify $\mathsf {T}'$ with $P(\mathbb {Z}_p)\backslash \Gamma _0(p\mathbb {Z}_p)$ . Similarly, the action of $\Gamma _0(p\mathbb {Z}_p)$ on $\mathsf {T}$ is transitive and the stabiliser of the element $(1,0)\in \mathsf {T}$ is the subgroup

$$\begin{align*}P(\mathbb{Z}_p)^w=\left\{\begin{pmatrix} 1 & 0 \\ pc & d \end{pmatrix}\in \text{GL}_2(\mathbb{Z}_p)\right\}, \end{align*}$$

so we can identify $\mathsf {T}$ with $P(\mathbb {Z}_p)^w\backslash \Gamma _0(p\mathbb {Z}_p)$ . For any positive integer j, let

$$\begin{align*}\Gamma_1(p^j\mathbb{Z}_p)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}\in\text{GL}_2(\mathbb{Z}_p): c\equiv 0\text{ (mod }p^j),\, d\equiv 1\text{ (mod }p^j)\right\}, \end{align*}$$

$$\begin{align*}\Gamma_1(p^j\mathbb{Z}_p)^w=\left\{\begin{pmatrix}a & b \\ pc & d\end{pmatrix}\in\text{GL}_2(\mathbb{Z}_p): a\equiv 1\text{ (mod }p^j),\, b\equiv 0\text{ (mod }p^{j-1})\right\}. \end{align*}$$

Then, for any positive integers $i,j$ , we can define

$$ \begin{align*} \mathcal{A}_{\nu,i,j}'=\big\{ f:\Gamma_1(p^j\mathbb{Z}_p)\backslash \Gamma_0(p\mathbb{Z}_p)\rightarrow \mathcal{O}/\mathfrak{m}^i\; \vert\; f(a\cdot \gamma)=\nu(a)\cdot f(\gamma) \\ \text{ for all } a\in\mathbb{Z}_p^{\times},\, \gamma\in \Gamma_1(p^j\mathbb{Z}_p)\backslash \Gamma_0(p\mathbb{Z}_p)\big\}, \end{align*} $$

$$ \begin{align*} \mathcal{A}_{\nu,i,j}=\big\{ f:\Gamma_1(p^j\mathbb{Z}_p)^w\backslash \Gamma_0(p\mathbb{Z}_p)\rightarrow \mathcal{O}/\mathfrak{m}^i\; \vert\; f(a\cdot \gamma)=\nu(a)\cdot f(\gamma) \\ \text{ for all } a\in\mathbb{Z}_p^{\times},\, \gamma\in \Gamma_1(p^j\mathbb{Z}_p)^w\backslash \Gamma_0(p\mathbb{Z}_p)\big\}. \end{align*} $$

The objects $\mathcal {A}_{\nu ,i,j}^{\cdot }$ can be regarded as left $\mathcal {O}[\Sigma ^{\cdot }_0(p)]$ -modules. Let $\mathcal {A}_{\nu ,i}^{\cdot }=\varinjlim _j \mathcal {A}_{\nu ,i,j}^{\cdot }$ . Then $\mathcal {A}_{\nu }^{\cdot }\cong \varprojlim _i \mathcal {A}_{\nu ,i}^{\cdot }$ . We denote by $\boldsymbol {\mathcal {A}}_{\nu }^{\cdot }$ the object in $\mathbf {S}(Y_{\operatorname {\mathrm {et}}})$ corresponding to $\lbrace \mathcal {A}_{\nu ,i}^{\cdot }\rbrace _i\in \mathbf {M}(\Gamma _0(p\mathbb {Z}_p))$ . We also define $\mathcal {D}_{\nu ,i}^{\cdot }=\operatorname {\mathrm {Hom}}_{\mathcal {O}}(\mathcal {A}_{\nu ,i,i}^{\cdot },\mathcal {O}/\mathfrak {m}^i)$ . These objects can be regarded as right $\mathcal {O}[\Sigma ^{\cdot }_0(p)]$ -modules, and we have $\mathcal {D}_{\nu }^{\cdot }\cong \varprojlim _i \mathcal {D}_{\nu ,i}^{\cdot }$ . We denote by $\boldsymbol {\mathcal {D}}_{\nu }^{\cdot }$ the object in $\mathbf {S}(Y_{\operatorname {\mathrm {et}}})$ corresponding to $\lbrace \mathcal {D}_{\nu ,i}^{\cdot }\rbrace _i\in \mathbf {M}(\Gamma _0(p\mathbb {Z}_p))$ . There are natural morphisms of $\mathcal {O}$ -modules

$$ \begin{align*} H^1_{\operatorname{\mathrm{et}}}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{A}}_{\nu}^{\cdot})\rightarrow \mathtt{H}^1_{\operatorname{\mathrm{et}}}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{A}}_{\nu}^{\cdot})\cong H^1(\Gamma,\mathcal{A}_{\nu}^{\cdot}) \end{align*} $$

and

$$ \begin{align*} H^1_{\operatorname{\mathrm{et}}}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{D}}_{\nu}^{\cdot})\cong \mathtt{H}^1_{\operatorname{\mathrm{et}}}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{D}}_{\nu}^{\cdot})\cong H^1(\Gamma,\mathcal{D}_{\nu}^{\cdot}) \end{align*} $$

compatible with the action of Hecke operators. We also have Hecke-equivariant isomorphisms

$$ \begin{align*} H^1_{\operatorname{\mathrm{et}},c}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{D}}_{\nu}^{\cdot})\cong \mathtt{H}^1_{\operatorname{\mathrm{et}},c}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{D}}_{\nu}^{\cdot})\cong H^1_c(\Gamma,\mathcal{D}_{\nu}^{\cdot}), \end{align*} $$

where $H^j_c(\Gamma ,-)=H^{j-1}(\Gamma ,\operatorname {\mathrm {Hom}}_{\mathbb {Z}}(\text {Div}^0(\mathbb {P}^1(\mathbb {Q})),-))$ . These isomorphisms allow us to define continuous $G_{\mathbb {Q}}$ -actions on the groups $H^1(\Gamma ,\mathcal {A}_{\nu }^{\cdot })$ , $H^1(\Gamma ,\mathcal {D}_{\nu }^{\cdot })$ and $H^1_c(\Gamma ,\mathcal {D}_{\nu }^{\cdot })$ .

Given a character $\chi :\mathbb {Z}_p^{\times }\rightarrow \mathcal {O}^{\times }$ , let $\mathcal {O}(\chi )$ be the module $\mathcal {O}$ with $\Gamma _0(p\mathbb {Z}_p)$ acting via $\chi \circ \det $ , where $\det :\Gamma _0(p\mathbb {Z}_p)\rightarrow \mathbb {Z}_p^{\times }$ is the determinant map.

The natural $\mathcal {G}$ -equivariant evaluation map $\mathcal {A}_{\nu }^{\cdot }\otimes _{\mathcal {O}}\mathcal {D}_{\nu }^{\cdot }\rightarrow \mathcal {O}$ yields a $G_{\mathbb {Q}}$ -equivariant cup-product pairing

(5.2) $$ \begin{align} H^1(\Gamma,\mathcal{A}_{\nu}^{\cdot})\otimes_{\mathcal{O}} H^1_c(\Gamma,\mathcal{D}_{\nu}^{\cdot})\longrightarrow \mathcal{O}(-1) ,\end{align} $$

under which the Hecke operators $T_q$ , $T_q'$ , $[d]_N$ , $[d]_N'$ acting covariantly on the left, whenever defined, are adjoint to these same operators acting contravariantly on the right.

Let $\det :\mathsf {T}'\times \mathsf {T}\rightarrow \mathbb {Z}_p^{\times }$ be the function defined by $\det ((x_1,x_2),(y_1,y_2))=x_1y_2-x_2y_1$ , and let $\det _{\nu }$ be the composition of this function with $\nu :\mathbb {Z}_p^{\times }\rightarrow \mathcal {O}$ . Evaluation at this function defines a $\mathcal {G}$ -equivariant map $\mathcal {D}_{\nu }'\otimes _{\mathcal {O}} \mathcal {D}_{\nu }\rightarrow \mathcal {O}(-\nu )$ which yields a $G_{\mathbb {Q}}$ -equivariant cup-product pairing

(5.3) $$ \begin{align} H^1(\Gamma,\mathcal{D}_{\nu}')\otimes_{\mathcal{O}} H^1_c(\Gamma,\mathcal{D}_{\nu})\longrightarrow \mathcal{O}(\boldsymbol{\nu})(-1), \end{align} $$

where $\boldsymbol {\nu }=\nu \circ \epsilon _{\text {cyc}}:G_{\mathbb {Q}}\rightarrow \mathcal {O}^{\times }$ . Under this pairing, the Hecke operators $T_q$ , $T_q'$ , $[d]_N$ , $[d]_N'$ acting contravariantly on the left, whenever defined, are adjoint to the Hecke operators $T_q'$ , $T_q$ , $[d]_N'$ , $[d]_N$ acting contravariantly on the right. We obtain a similar pairing interchanging the roles of $\mathcal {D}_{\nu }$ and $\mathcal {D}_{\nu }'$ .

5.4 Ordinary cohomology

For any $\mathbb {Z}_p$ -algebra B, let $S_r(B)$ be the set of two-variable homogeneous polynomials of degree r in $B[x_1,x_2]$ . It is a left $B[\Sigma _0^{\cdot }(p)]$ -module with the action of $\Sigma _0^{\cdot }(p)$ defined by

$$\begin{align*}gP(x_1,x_2)=P((x_1,x_2)\cdot g) \end{align*}$$

for all $g\in \Sigma _0^{\cdot }(p)$ and $P(x_1,x_2)\in S_r(B)$ . To the p-adic $\Gamma _0(p\mathbb {Z}_p)$ -representation $S_r=S_r(\mathbb {Z}_p)$ , there corresponds the locally constant p-adic sheaf $\mathscr {S}_{r}$ on $Y_{\operatorname {\mathrm {et}}}$ defined in Section 2.3. Therefore, we have an isomorphism

$$\begin{align*}H^1_{\operatorname{\mathrm{et}}}(Y_{\overline{\mathbb{Q}}},\mathscr{S}_r)\cong H^1(\Gamma,S_r) \end{align*}$$

which is Hecke-equivariant when we consider the covariant action of Hecke operators on both sides, and we use this isomorphism to define an action of $G_{\mathbb {Q}}$ on $H^1(\Gamma ,S_r)$ .

We also define $L_r(B)=\operatorname {\mathrm {Hom}}_{B}(S_r(B),B)$ , which we regard as a right $B[\Sigma _0^{\cdot }(p)]$ -module defining the $\Sigma _0^{\cdot }(p)$ -action by

$$\begin{align*}(\mu\cdot g)(P(x_1,x_2))=\mu(g P(x_1,x_2)) \end{align*}$$

for all $g\in \Sigma _0^{\cdot }(p)$ , $\mu \in L_r(B)$ and $P(x_1,x_2)\in S_r(B)$ . To the p-adic $\Gamma _0(p\mathbb {Z}_p)$ -representation $L_r=L_r(\mathbb {Z}_p)$ , there corresponds the locally constant p-adic sheaf $\mathscr {L}_{r}$ on $Y_{\operatorname {\mathrm {et}}}$ . Therefore, we have an isomorphism

$$\begin{align*}H^1_{\operatorname{\mathrm{et}}}(Y_{\overline{\mathbb{Q}}},\mathscr{L}_r)\cong H^1(\Gamma,L_r) \end{align*}$$

which is Hecke-equivariant when we consider the contravariant action of Hecke operators on both sides, and we use this isomorphism to define an action of $G_{\mathbb {Q}}$ on $H^1(\Gamma ,L_r)$ .

The natural $\Gamma _0(p\mathbb {Z}_p)$ -equivariant evaluation map $S_r\otimes _{\mathbb {Z}_p} L_r\rightarrow \mathbb {Z}_p$ yields a $G_{\mathbb {Q}}$ -equivariant cup-product pairing

(5.4) $$ \begin{align} H^1(\Gamma,S_r)\otimes_{\mathbb{Z}_p} H^1_c(\Gamma,L_r)\longrightarrow \mathbb{Z}_p(-1) ,\end{align} $$

under which the Hecke operators $T_q$ , $T_q'$ , $[d]_N$ , $[d]_N'$ acting covariantly on the left, whenever defined, are adjoint to these same operators acting contravariantly on the right. This pairing becomes perfect after inverting p.

Let $\nu _r: \mathbb {Z}_p^{\times } \rightarrow \mathbb {Z}_p^{\times }$ be the character defined by $\nu _r(z)=z^r$ . Evaluation at the polynomial $(x_1y_2-x_2y_1)^r\in S_r\otimes _{\mathbb {Z}_p}S_r$ defines a $\Gamma _0(p\mathbb {Z}_p)$ -equivariant map $L_r\otimes _{\mathbb {Z}_p}L_r\rightarrow \mathbb {Z}_p(-\nu _r)$ and thus yields a $G_{\mathbb {Q}}$ -equivariant cup-product pairing

(5.5) $$ \begin{align} H^1(\Gamma,L_r)\otimes_{\mathbb{Z}_p} H^1_c(\Gamma,L_r)\longrightarrow \mathbb{Z}_p(r-1) ,\end{align} $$

under which the Hecke operators $T_q$ , $T_q'$ , $[d]_N$ , $[d]_N'$ acting contravariantly on the left, whenever defined, are adjoint to the Hecke operators $T_q'$ , $T_q$ , $[d]_N'$ , $[d]_N$ acting contravariantly on the right. This pairing becomes perfect after inverting p.

Combining these two pairings, we can define a morphism

$$\begin{align*}\mathtt{s}_{r\ast}: H^1(\Gamma,S_r(\mathbb{Q}_p))\longrightarrow H^1(\Gamma,L_r(\mathbb{Q}_p))(-r). \end{align*}$$

This map is $G_{\mathbb {Q}}$ -equivariant and intertwines the covariant action of the operators $T_q$ , $[d]_N$ , $[a]_p$ on the source with the contravariant action of the operators $T_q'$ , $[d]_N'$ , $[a]_p'$ on the target. We can also define $\mathtt {s}_{r\ast }$ directly via the isomorphism $S_r(\mathbb {Q}_p)\cong L_r(\mathbb {Q}_p)(\nu _r)$ arising from the perfect pairing $L_r(\mathbb {Q}_p)\otimes _{\mathbb {Q}_p}L_r(\mathbb {Q}_p)\rightarrow \mathbb {Q}_p(-\nu _r)$ defined by evaluation at $(x_1y_2-x_2y_1)^r$ . Therefore, the denominators introduced by this map are bounded by $r!$ , that is, an element in

$$\begin{align*}\operatorname{\mathrm{im}}\left(H^1(\Gamma,S_r)\rightarrow H^1(\Gamma,S_r(\mathbb{Q}_p))\right) \end{align*}$$

is mapped to an element in

$$\begin{align*}\frac{1}{r!}\operatorname{\mathrm{im}}\left(H^1(\Gamma,L_r)\rightarrow H^1(\Gamma,L_r(\mathbb{Q}_p))\right), \end{align*}$$

as follows from [Reference Bertolini, Seveso and VenerucciBSV22, Remark 3.3].

To slightly simplify the notation, we will write $\mathcal {A}_r^{\cdot }$ and $\mathcal {D}_r^{\cdot }$ for $\mathcal {A}_{\nu _r}^{\cdot }$ and $\mathcal {D}_{\nu _r}^{\cdot }$ , respectively. Regarding two-variable polynomials as functions on $\mathsf {T}^{\cdot }$ , we obtain a natural morphism of left $\mathbb {Z}_p[\Sigma _0^{\cdot }(p)]$ -modules $S_r\rightarrow \mathcal {A}_{r}^{\cdot }$ . Also, dualizing this map, we obtain a morphism of right $\mathbb {Z}_p[\Sigma _0^{\cdot }(p)]$ -modules $\mathcal {D}_{r}^{\cdot }\rightarrow L_r$ . Thus, we have $G_{\mathbb {Q}}$ -equivariant and Hecke-equivariant morphisms

$$\begin{align*}H^1(\Gamma,S_r)\rightarrow H^1(\Gamma,\mathcal{A}_{r}^{\cdot}) \quad\text{and}\quad H^1(\Gamma,\mathcal{D}_r^{\cdot})\rightarrow H^1(\Gamma,L_r). \end{align*}$$

Applying Hida’s (anti-)ordinary projector $e_{\text {ord}}^{\cdot }:= \lim _{n\to \infty } (T_p^{\cdot })^{n!}$ , the previous morphisms become isomorphisms

$$\begin{align*}e_{\text{ord}}^{\cdot} H^1(\Gamma,S_r)\cong e_{\text{ord}}^{\cdot} H^1(\Gamma,\mathcal{A}_r^{\cdot}),\quad e_{\text{ord}}^{\cdot} H^1(\Gamma,\mathcal{D}_r^{\cdot})\cong e_{\text{ord}}^{\cdot} H^1(\Gamma,L_r). \end{align*}$$

Under these isomorphisms, the pairings (5.4) and (5.5) correspond to the pairings (5.2) and (5.3), respectively, after applying the corresponding (anti-)ordinary projector to every term involved.

5.5 $\Lambda $ -adic Poincaré pairing

It will be convenient to write $\langle a; b\rangle $ , with $a\in (\mathbb {Z}/N\mathbb {Z})^{\times }$ and $b\in (\mathbb {Z}/p^r\mathbb {Z})^{\times }$ , for the diamond operator $\langle d\rangle $ , where $d\in (\mathbb {Z}/Np^r)^{\times }$ is congruent to a modulo N and to b modulo $p^r$ . We also write $\epsilon _N:G_{\mathbb {Q}}\rightarrow (\mathbb {Z}/N\mathbb {Z})^{\times }$ for the mod N cyclotomic character.

For any positive integer r, let

$$ \begin{align*}G_r=1+p(\mathbb{Z}/p^r\mathbb{Z}), \quad \tilde{G}_r=(\mathbb{Z}/p^r\mathbb{Z})^{\times}, \end{align*} $$

and define

$$ \begin{align*}\Lambda_r=\mathbb{Z}_p[G_r],\quad \tilde{\Lambda}_r=\mathbb{Z}_p[\tilde{G}_r], \quad \Lambda=\varprojlim_r\Lambda_r=\mathbb{Z}_p[[1+p\mathbb{Z}_p]],\quad \tilde{\Lambda}=\varprojlim_r \tilde{\Lambda}_r=\mathbb{Z}_p[[\mathbb{Z}_p^{\times}]]. \end{align*} $$

We have natural factorisations $(\mathbb {Z}/p^r\mathbb {Z})^{\times }=\mu _{p-1}\times (1+p\mathbb {Z}/p^r\mathbb {Z})$ and $\mathbb {Z}_p^{\times }=\mu _{p-1}\times (1+p\mathbb {Z}_p)$ which give natural embeddings and . We define idempotents

$$ \begin{align*}e_i=\frac{1}{p-1}\sum_{\zeta\in\mu_{p-1}}\zeta^{-i}[\zeta] \end{align*} $$

for any integer i modulo $p-1$ . Let $\kappa _i:\mathbb {Z}_p^{\times }\rightarrow \Lambda ^{\times }$ be the character defined by $z\mapsto \omega ^i(z)[\langle z\rangle ]$ , and let $\boldsymbol {\kappa }_i=\kappa _i\circ \epsilon _{\text {cyc}}:G_{\mathbb {Q}}\rightarrow \Lambda ^{\times }$ .

We will shorten notation by writing

(5.6) $$ \begin{align} X_r(m)=X(1,Np^r(m)), \quad H^1_{\operatorname{\mathrm{et}}}(X_{\infty}(m)_{\overline{\mathbb{Q}}},\mathbb{Z}_p)=\varprojlim_r H^1_{\operatorname{\mathrm{et}}}(X_r(m)_{\overline{\mathbb{Q}}},\mathbb{Z}_p). \end{align} $$

We have a natural action of $\tilde {\Lambda }_r$ and $\tilde {\Lambda }$ on the previous groups defined by letting group-like elements $[u]$ act like the diamond operators $\langle 1; u\rangle '$ .

Fix compatible primitive p-power roots of unity $\zeta _{p^r}$ and a primitive N-th root of unity $\zeta _N$ . Then one can define Atkin–Lehner automorphisms $w_r$ and w for the curve $X_r(m)$ similarly as in [Reference Darmon and RotgerDR17, Section 1.2]. More precisely, $X_r(m)$ parameterises quadruples $(E,P,Q,C)$ , where E is an elliptic curve, P is a point of order N, Q is a point of order $p^r$ and C is a cyclic subgroup of E of order $Nm$ containing P. Then, we define

$$ \begin{align*}w_r(E,P,Q,C)=(E/C_Q,P+C_Q,Q'+C_Q,C+C_Q/C_Q), \end{align*} $$

where $C_Q\subseteq E$ is the subgroup generated by Q, and $Q'\in E[p^r]$ satisfies $\langle Q,Q'\rangle = \zeta _{p^r}$ . Similarly, we define

$$ \begin{align*}w(E,P,Q,C)=(E/C,P'+C,Q+C,E[Nm]/C), \end{align*} $$

where $P'\in E[N]$ satisfies $\langle P,P'\rangle = \zeta _N$ . These Atkin–Lehner automorphisms satisfy, for any $\sigma \in G_{\mathbb {Q}}$ ,

$$ \begin{align*}w_r^{\sigma} =\langle 1;\epsilon_{\text{cyc}}(\sigma)\rangle w_r,\quad w^{\sigma} =\langle \epsilon_{N}(\sigma); 1 \rangle w. \end{align*} $$

We let w and $w_r$ act on cohomology via pullback.

Define $G_{\mathbb {Q}}$ -equivariant pairings

$$ \begin{align*}\langle\,,\rangle_{G_r}: e_i H^1_{\operatorname{\mathrm{et}}}(X_{r}(m)_{\overline{\mathbb{Q}}},\mathbb{Z}_p)\times e_{-i} H^1_{\operatorname{\mathrm{et}}}(X_{r}(m)_{\overline{\mathbb{Q}}},\mathbb{Z}_p)\longrightarrow \Lambda_r(-1) \end{align*} $$

by the formula

$$ \begin{align*}\langle a,b\rangle_{G_r}=\sum_{\sigma\in G_r}\langle a^{\sigma},b\rangle_r\cdot \sigma^{-1}, \end{align*} $$

where $\langle \,,\rangle _r$ stands for the natural Poincaré pairing. These pairings are $\Lambda _r$ -linear and antilinear in the first and second argument, respectively. Then we get $G_{\mathbb {Q}}$ -equivariant $\Lambda _r$ -pairings

$$ \begin{align*}[\,,]_{G_r}: e_i H^1_{\operatorname{\mathrm{et}}}(X_{r}(m)_{\overline{\mathbb{Q}}},\mathbb{Z}_p)\times e_i H^1_{\operatorname{\mathrm{et}}}(X_{r}(m)_{\overline{\mathbb{Q}}},\mathbb{Z}_p)(\langle\epsilon_N^{-1};1\rangle')\longrightarrow \Lambda_r(\boldsymbol{\kappa}_i)(-1) \end{align*} $$

via the following modification of the previous pairing:

$$ \begin{align*}[a,b]_{G_r}=\langle a,ww_r\cdot (T_p')^r\cdot b\rangle_{G_r}. \end{align*} $$

These pairings are compatible in the sense that the diagram

commutes, which can be proved as in [Reference Darmon and RotgerDR17, Lemma 1.1]. This yields a $\Lambda $ -adic perfect $G_{\mathbb {Q}}$ -equivariant pairing

(5.7) $$ \begin{align} e_i H^1_{\operatorname{\mathrm{et}}}(X_{\infty}(m)_{\overline{\mathbb{Q}}},\mathbb{Z}_p)^{\text{ord}}\times e_i H^1_{\operatorname{\mathrm{et}}}(X_{\infty}(m)_{\overline{\mathbb{Q}}},\mathbb{Z}_p)^{\text{ord}}(\langle\epsilon_N^{-1};1\rangle') \longrightarrow \Lambda(\boldsymbol{\kappa}_i)(-1), \end{align} $$

where $H^1_{\operatorname {\mathrm {et}}}(X_{\infty }(m)_{\overline {\mathbb {Q}}},\mathbb {Z}_p)^{\text {ord}}=e_{\text {ord}}' H^1_{\operatorname {\mathrm {et}}}(X_{\infty }(m)_{\overline {\mathbb {Q}}},\mathbb {Z}_p)$ . All Hecke operators are self-adjoint for this pairing.

5.6 Big Galois representations

Let $\mathfrak {m}_{\Lambda }$ be the maximal ideal of $\Lambda $ , let $\text {Cont}(\mathbb {Z}_p,\Lambda )$ be the $\Lambda $ -module of continuous functions on $\mathbb {Z}_p$ with values in $\Lambda $ and let $\kappa $ be any of the $\kappa _i$ above. Define the $\Lambda $ -module

$$ \begin{align*} \mathcal{A}_{\kappa}'=\big\{ f:\mathsf{T}'\rightarrow \Lambda \; \vert\; f(pz,1)\in \text{Cont}(\mathbb{Z}_p,\Lambda) \text{ and } f(a\cdot \gamma)=\kappa(a)\cdot f(\gamma) \text{ for all } a\in\mathbb{Z}_p^{\times},\, \gamma\in \mathsf{T}'\big\}, \end{align*} $$

equipped with the $\mathfrak {m}_{\Lambda }$ -adic topology, and the $\Lambda $ -module

$$\begin{align*}\mathcal{D}_{\kappa}'=\operatorname{\mathrm{Hom}}_{\text{cont},\Lambda}(\mathcal{A}_{\kappa}',\Lambda) ,\end{align*}$$

equipped with the weak- $\ast $ topology. As in Section 5.2, we can regard $\mathcal {A}_{\kappa }'$ (respectively, $\mathcal {D}_{\kappa }'$ ) as a left (respectively, right) $\Lambda [\Sigma ^{\prime }_0(p)]$ -module.

Similarly to what we did in Section 5.3, define, for any positive integers $j,r$ ,

$$ \begin{align*} \mathcal{A}_{\kappa,j,r}'=\big\{ f:\Gamma_1(p^r\mathbb{Z}_p)\backslash \Gamma_0(p\mathbb{Z}_p)\rightarrow \Lambda/\mathfrak{m}_{\Lambda}^j\; \vert\; f(a\cdot \gamma)=\kappa(a)\cdot f(\gamma) \\ \text{ for all } a\in\mathbb{Z}_p^{\times},\, \gamma\in \Gamma_1(p^r\mathbb{Z}_p)\backslash \Gamma_0(p\mathbb{Z}_p)\big\} \end{align*} $$

and $\mathcal {A}_{\kappa ,j}'=\varinjlim _r \mathcal {A}_{\kappa ,j,r}'$ . Then $\mathcal {A}_{\kappa }^{\prime }=\varprojlim _j \mathcal {A}_{\kappa ,j}'$ . We denote by $\boldsymbol {\mathcal {A}}_{\kappa }'$ the object in $\mathbf {S}(Y_{\operatorname {\mathrm {et}}})$ corresponding to $\lbrace \mathcal {A}_{\kappa ,j}'\rbrace _j\in \mathbf {M}(\Gamma _0(p\mathbb {Z}_p))$ . We also define $\mathcal {D}_{\kappa ,j}'=\operatorname {\mathrm {Hom}}_{\Lambda }(\mathcal {A}_{\kappa ,j,j}',\Lambda /\mathfrak {m}_{\Lambda }^j)$ , so that $\mathcal {D}_{\kappa }'=\varprojlim _j \mathcal {D}_{\kappa ,j}'$ , and denote by $\boldsymbol {\mathcal {D}}_{\kappa }'$ the object in $\mathbf {S}(Y_{\operatorname {\mathrm {et}}})$ corresponding to $\lbrace \mathcal {D}_{\kappa ,j}'\rbrace _j\in \mathbf {M}(\Gamma _0(p\mathbb {Z}_p))$ . There are natural Hecke-equivariant morphisms of $\Lambda $ -modules

$$ \begin{align*} H^1_{\operatorname{\mathrm{et}}}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{A}}_{\kappa}')\rightarrow \mathtt{H}^1_{\operatorname{\mathrm{et}}}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{A}}_{\kappa}')\cong H^1(\Gamma,\mathcal{A}_{\kappa}'), \end{align*} $$

$$ \begin{align*} H^1_{\operatorname{\mathrm{et}}}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{D}}_{\kappa}')\cong \mathtt{H}^1_{\operatorname{\mathrm{et}}}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{D}}_{\kappa}')\cong H^1(\Gamma,\mathcal{D}_{\kappa}'), \end{align*} $$

$$ \begin{align*} H^1_{\operatorname{\mathrm{et}},c}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{D}}_{\kappa}')\cong \mathtt{H}^1_{\operatorname{\mathrm{et}},c}(Y_{\overline{\mathbb{Q}}},\boldsymbol{\mathcal{D}}_{\kappa}')\cong H^1_c(\Gamma,\mathcal{D}_{\kappa}'), \end{align*} $$

which allow us to define continuous $G_{\mathbb {Q}}$ -actions on the groups $H^1(\Gamma ,\mathcal {A}_{\kappa }')$ , $H^1(\Gamma ,\mathcal {D}_{\kappa }')$ and $H^1_c(\Gamma ,\mathcal {D}_{\kappa }')$ .

The evaluation map $\mathcal {A}_{\kappa }'\otimes _{\Lambda }\mathcal {D}_{\kappa }'\rightarrow \Lambda $ yields a $G_{\mathbb {Q}}$ -equivariant cup-product pairing

(5.8) $$ \begin{align} H^1(\Gamma,\mathcal{A}_{\kappa}')\otimes_{\Lambda} H^1_c(\Gamma,\mathcal{D}_{\kappa}')\longrightarrow \Lambda(-1) ,\end{align} $$

under which the Hecke operators $T_q$ , $T_q'$ , $[d]_N$ , $[d]_N'$ acting covariantly on the left, whenever defined, are adjoint to these same operators acting contravariantly on the right.

Recall that in this section, we have set $\Gamma =\Gamma (1,N(pm))$ , and let $S=\Sigma _0'(p)\cap \text {GL}_2(\mathbb {Q})$ . For any positive integer r, define

$$\begin{align*}\Sigma_1'(p^r)=\begin{pmatrix} \mathbb{Z}_p & \mathbb{Z}_p \\ p^r\mathbb{Z}_p & 1+p^r\mathbb{Z}_p \end{pmatrix},\quad S_r=\Sigma_1'(p^r)\cap\text{GL}_2(\mathbb{Q}),\quad \Gamma_r=\Gamma(1,Np^r(m)). \end{align*}$$

We define compatibility of Hecke pairs as in [Reference Ash and StevensAS86a, Definition 1.1.2] but changing left-right conventions. More precisely, we say that the Hecke pair $(\Gamma _{\alpha },S_{\alpha })$ is compatible to the Hecke pair $(\Gamma _{\beta },S_{\beta })$ if $(\Gamma _{\alpha },S_{\alpha })\subseteq (\Gamma _{\beta },S_{\beta })$ , $S_{\alpha }\Gamma _{\beta }=S_{\beta }$ and $\Gamma _{\beta }\cap S_{\alpha }^{-1}S_{\alpha }=\Gamma _{\alpha }$ . With this definition, the Hecke pair $(\Gamma _r, S_r)$ is compatible to the Hecke pair $(\Gamma _t,S_t)$ , if $r\geq t$ , and to the Hecke pair $(\Gamma ,S)$ .

Suppose that the Hecke pair $(\Gamma _{\alpha },S_{\alpha })$ is compatible to $(\Gamma _{\beta }, S_{\beta })$ , and that $\Gamma _{\alpha }$ has finite index in $\Gamma _{\beta }$ . For any right $S_{\alpha }$ -module E, we define

$$\begin{align*}\operatorname{\mathrm{Ind}}_{\Gamma_{\alpha}}^{\Gamma_{\beta}} E = \big\{ \varphi :\Gamma_{\beta}\rightarrow E \; \vert \; \varphi (xy)=\varphi (y)x^{-1} \text{ for all } x\in \Gamma_{\alpha},\, y\in \Gamma_{\beta}\big\}. \end{align*}$$

This module is equipped with a right action of $S_{\beta }$ : given $\varphi \in \operatorname {\mathrm {Ind}}_{\Gamma _{\alpha }}^{\Gamma _{\beta }} E$ and $g\in S_{\beta }$

$$\begin{align*}(\varphi g)(x)=\sum \varphi (\gamma)\gamma g x^{-1}, \end{align*}$$

where the sum is over representatives $\gamma $ for the cosets in $\Gamma _{\alpha }\backslash \Gamma \cap S_{\alpha } x g^{-1}$ .

Now define

$$ \begin{align*} A_{\kappa,r}'=\big\{ f:\Gamma_1(p^r\mathbb{Z}_p)\backslash \Gamma_0(p\mathbb{Z}_p)\rightarrow \Lambda_{r}\; \vert\; f(a\cdot \gamma)=\kappa(a)\cdot f(\gamma) \\ \text{ for all } a\in\mathbb{Z}_p^{\times},\, \gamma\in \Gamma_1(p^r\mathbb{Z}_p)\backslash \Gamma_0(p\mathbb{Z}_p)\big\} ,\end{align*} $$

and let $D_{\kappa ,r}'=\operatorname {\mathrm {Hom}}_{\Lambda _{r}}(A_{\kappa ,r}',\Lambda _r)$ . With these definitions, $\mathcal {D}_{\kappa }'=\varprojlim _r D_{\kappa , r}'$ . Let $S_r$ act trivially on $\mathbb {Z}_p$ and consider the right $\mathbb {Z}_p[S_1]$ -module $\text {Ind}_{\Gamma _r}^{\Gamma _1}\,\mathbb {Z}_p$ . Let R be a set of representatives for the cosets in $\Gamma _r\backslash \Gamma _1$ . The map $\operatorname {\mathrm {Ind}}_{\Gamma _r}^{\Gamma _1}\mathbb {Z}_p\rightarrow D_{\kappa ,r}'$ defined by

$$\begin{align*}\varphi \mapsto \big[ f\mapsto \sum_{r\in R} \varphi(r) f(r)\big] \end{align*}$$

is an isomorphism of right $\mathbb {Z}_p[S_1]$ -modules. Therefore, there are natural isomorphisms

$$\begin{align*}H^1(\Gamma_1,\mathcal{D}_{\kappa}')\cong \varprojlim_r H^1(\Gamma_1,D_{\kappa,r}')\cong \varprojlim_r H^1(\Gamma_r,\mathbb{Z}_p). \end{align*}$$

According to [Reference Ash and StevensAS86a, Lemma 1.1.3] and [Reference Ash and StevensAS86a, Lemma 1.1.4], both corestriction and the Shapiro isomorphism commute with the action of $D(\Gamma ,S)$ via restriction of Hecke algebras, so the previous isomorphisms are Hecke-equivariant.

Similarly to (5.6), but omitting m from the notation, we let $Y_r=Y(1,Np^r(m))$ and put

$$\begin{align*}H^1_{\operatorname{\mathrm{et}}}(Y_{\infty,{\overline{\mathbb{Q}}}},\mathbb{Z}_p):=\varprojlim_r H^1_{\operatorname{\mathrm{et}}}(Y_{r,{\overline{\mathbb{Q}}}},\mathbb{Z}_p), \end{align*}$$

where the inverse limit is with respect to the maps $\pi _{1\ast }$ . Then

$$\begin{align*}H^1(\Gamma_1,\mathcal{D}_{\kappa}')\cong \varprojlim_r H^1(\Gamma_r,\mathbb{Z}_p) \cong H^1_{\operatorname{\mathrm{et}}}(Y_{\infty,{\overline{\mathbb{Q}}}},\mathbb{Z}_p), \end{align*}$$

where the last isomorphism is defined by choosing a compatible system of geometric points for the curves $Y_r$ and suitable compatible bases for the corresponding Tate modules. Under the isomorphisms above, the contravariant operators $T_q'$ , $[d]_N'$ , $[a]_p'$ on the first term correspond to the contravariant operators $T_q'$ , $\langle d; 1\rangle '$ , $\langle 1; a\rangle '$ defined on the last term via the compatibility of these operators with the pushforward maps $\pi _{1\ast }$ .

Also, according to [Reference Ash and StevensAS86a, Lemma 1.1.5], the restriction map yields a Hecke-equivariant isomorphism

$$\begin{align*}H^1(\Gamma, \mathcal{D}_{\kappa}')\cong e_i H^1(\Gamma_1,\mathcal{D}_{\kappa}') \end{align*}$$

(recall that we have set $\kappa =\kappa _i$ ). Combining this isomorphism with the previous ones, we obtain a Hecke-equivariant isomorphism

$$\begin{align*}H^1(\Gamma, \mathcal{D}_{\kappa}')\cong e_i H^1_{\operatorname{\mathrm{et}}}(Y_{\infty,{\overline{\mathbb{Q}}}},\mathbb{Z}_p). \end{align*}$$

Similarly, using [Reference Ash and StevensAS86b, Proposition 4.2], one proves that there is a Hecke-equivariant isomorphism

(5.9) $$ \begin{align} H^1_c(\Gamma,\mathcal{D}_{\kappa}') \cong e_i H^1_{\operatorname{\mathrm{et}},c}(Y_{\infty,{\overline{\mathbb{Q}}}},\mathbb{Z}_p). \end{align} $$

7.1 Triple product p-adic L-function

Fix a triple $({\mathbf {f}},g,h)$ consisting of a primitive Hida family ${\mathbf {f}}$ of tame level $N_{{\mathbf {f}}}$ and character $\chi _{{\mathbf {f}}}$ and two p-ordinary newforms $g, h$ of weights $l,m\geq 2$ , levels $N_g,N_h$ prime-to-p and nebentypus $\chi _g,\chi _h$ . Assume that ${\mathbf {f}}$ has coefficients in a ring $\Lambda _{{\mathbf {f}}}$ as in Section 5.1. Assume that $\chi _{{\mathbf {f}}}\chi _g\chi _h=\omega ^{r_1}$ for some even integer $r_1$ , and put

$$\begin{align*}N=\mathrm{lcm}(N_{{\mathbf{f}}},N_g,N_h). \end{align*}$$

Let ${\mathbf {g}}$ and ${\mathbf h}$ be primitive Hida families with coefficients in $\Lambda _{\mathbf {g}}$ and $\Lambda _{\mathbf h}$ passing through g and h, respectively. More precisely, there exist arithmetic points $y_0\in \mathcal W_{\Lambda _{{\mathbf {g}}}}(\overline {\mathbb {Q}}_p)$ and $z_0\in \mathcal W_{\Lambda _{{\mathbf h}}}(\overline {\mathbb {Q}}_p)$ , such that $g_{y_0}$ and $h_{z_0}$ are the ordinary p-stabilisations of g and h, respectively. The rings $\Lambda _{\mathbf {g}}$ and $\Lambda _{\mathbf h}$ need not be regular. However, for our purposes, we can consider the $\Lambda $ -adic families, denoted again ${\mathbf {g}}$ and ${\mathbf h}$ , that result from embedding $\Lambda _{\mathbf {g}}$ and $\Lambda _{\mathbf h}$ in the rings of functions of suitable wide open connected subsets $U_{\mathbf {g}}$ and $U_{\mathbf h}$ of $\mathcal W(\overline {\mathbb {Q}}_p)=\mathrm {Spf}(\Lambda )(\overline {\mathbb {Q}}_p)$ defined over some finite extension E of $\mathbb {Q}_p$ and containing the points $y_0$ and $z_0$ , respectively. From now on, it is these rings of functions that we will denote by $\Lambda _{\mathbf {g}}$ and $\Lambda _{\mathbf h}$ . These rings are now noncanonically isomorphic to ${\mathcal O}[[T]]$ , where $\mathcal {O}$ is the ring of integers of E; in particular, they are regular. Let $\boldsymbol {l}-l$ and $\boldsymbol {m}-m$ be generators in $\Lambda _{\mathbf {g}}$ and $\Lambda _{\mathbf h}$ of the prime ideals corresponding to the points $y_0$ and $z_0$ , respectively.