2.1 The weight spaces

We choose once and for all $u=1+p$ as a generator of $\mathbb{Z}_{p}^{\times }$ . This choice determines an isomorphism $\mathbb{Z}_{p}^{\times }\cong (\mathbb{Z}/(p-1)\mathbb{Z})\times \mathbb{Z}_{p}$ . Let $g$ be a positive integer. Consider the Iwasawa algebra $\mathbb{Z}_{p}[[(\mathbb{Z}_{p}^{\times })^{g}]]$ . A construction by Berthelot [Reference de JongdeJ95, §7] attaches to the formal scheme $\operatorname{Spf}\mathbb{Z}_{p}[[(\mathbb{Z}_{p}^{\times })^{g}]]$ a rigid analytic space that we denote by ${\mathcal{W}}_{g}$ . If $A$ is a $\mathbb{Q}_{p}$ -algebra, the $A$ -points of ${\mathcal{W}}_{g}$ are the continuous characters $(\mathbb{Z}_{p}^{\times })^{g}\rightarrow A^{\times }$ . Denote by $(\mathbb{Z}/(p-1)\mathbb{Z})^{\wedge ,g}$ the group of characters of $(\mathbb{Z}/(p-1)\mathbb{Z})^{g}$ . The following map gives an isomorphism from ${\mathcal{W}}_{g}$ to a disjoint union of $g$ -dimensional open discs $B_{g}(0,1^{-})$ indexed by $(\mathbb{Z}/(p-1)\mathbb{Z})^{\wedge ,g}$ :

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D702}_{g}:{\mathcal{W}}_{g}\rightarrow (\mathbb{Z}/(p-1)\mathbb{Z})^{\wedge ,g}\times B_{g}(0,1^{-}), & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D705}\mapsto (\unicode[STIX]{x1D705}|_{(\mathbb{Z}/(p-1)\mathbb{Z})^{g}},(\unicode[STIX]{x1D705}(u,1,\ldots ,1)-1,\unicode[STIX]{x1D705}(1,u,1,\ldots ,1)-1,\ldots ,\unicode[STIX]{x1D705}(1,\ldots ,1,u)-1)). & \displaystyle \nonumber\end{eqnarray}$$

We denote by ${\mathcal{W}}_{g}^{\circ }$ the connected component of ${\mathcal{W}}_{g}$ that maps to $0\in (\mathbb{Z}/(p-1)\mathbb{Z})^{g}$ . We write $\unicode[STIX]{x1D6EC}_{g}$  for the algebra $\mathbb{Z}_{p}[[T_{1},T_{2},\ldots ,T_{g}]]$ of formal series in $g$ variables over $\mathbb{Z}_{p}$ . It is the ring of rigid analytic functions bounded by $1$ on a connected component of the weight space. The weight space ${\mathcal{W}}_{g}$ carries a universal character $\unicode[STIX]{x1D705}_{{\mathcal{W}}_{g}}:\mathbb{Z}_{p}^{\times }\rightarrow \mathbb{Z}_{p}[[(\mathbb{Z}_{p}^{\times })^{g}]]^{\times }$ .

We call arithmetic primes the primes of $\unicode[STIX]{x1D6EC}_{g}$ of the form $P_{\text{}\underline{k}}=(1+T_{1}-u^{k_{1}},1+T_{2}-u^{k_{2}},\ldots ,1+T_{g}-u^{k_{g}})$ for a $g$ -tuple of integers $\text{}\underline{k}=(k_{1},k_{2},\ldots ,k_{g})$ (in the usual definition an auxiliary character can appear, but we will never need it). We say that a $\mathbb{Q}_{p}$ -point $\unicode[STIX]{x1D705}:\mathbb{Z}_{p}^{\times }\rightarrow \mathbb{Q}_{p}^{\times }$ of ${\mathcal{W}}_{g}^{\circ }$ is classical if it is the specialization of the universal character of ${\mathcal{W}}_{g}$ at $P_{\text{}\underline{k}}$ , for some $\text{}\underline{k}\in \mathbb{Z}^{g}$ .

2.2 The abstract Hecke algebras

Let $\ell$ be a prime. Let $G$ be a $\mathbb{Z}$ -subgroup scheme of $\operatorname{GSp}_{2g}$ and let $K\subset G(\mathbb{Q}_{\ell })$ be a compact open subgroup. For $\unicode[STIX]{x1D6FE}\in G(\mathbb{Q}_{\ell })$ we denote by $\unicode[STIX]{x1D7D9}([K\unicode[STIX]{x1D6FE}K])$ the characteristic function of the double coset $[K\unicode[STIX]{x1D6FE}K]$ . Let ${\mathcal{H}}(G(\mathbb{Q}_{\ell }),K)$ be the $\mathbb{Q}$ -algebra generated by the functions $\unicode[STIX]{x1D7D9}([K\unicode[STIX]{x1D6FE}K])$ for $\unicode[STIX]{x1D6FE}\in G(\mathbb{Q}_{\ell })$ , equipped with the convolution product. We call spherical (or unramified) Hecke algebra of  $\operatorname{GSp}_{2g}$  at  $\ell$ the $\mathbb{Q}$ -algebra ${\mathcal{H}}(\operatorname{GSp}_{2g}(\mathbb{Q}_{\ell }),\operatorname{GSp}_{2g}(\mathbb{Z}_{\ell }))$ . It is generated by the elements $T_{\ell ,i}^{(g)}=\unicode[STIX]{x1D7D9}([\operatorname{GSp}_{2g}(\mathbb{Z}_{\ell })\operatorname{diag}(\unicode[STIX]{x1D7D9}_{i},\ell \unicode[STIX]{x1D7D9}_{2g-2i},\ell ^{2}\unicode[STIX]{x1D7D9}_{i})\operatorname{GSp}_{2g}(\mathbb{Z}_{\ell })])$ , for $i=0,1,\ldots g$ , and $(T_{\ell ,0}^{(g)})^{-1}$ .

The Hecke algebra ${\mathcal{H}}(T_{g}(\mathbb{Q}_{\ell }),T_{g}(\mathbb{Z}_{\ell }))$ carries a natural action of the Weyl group $W_{g}=\mathscr{S}_{g}\ltimes (\mathbb{Z}/2\mathbb{Z})^{g}$ of $\operatorname{GSp}_{2g}$ , where $\mathscr{S}_{g}$ is the group of permutations of $\{1,2,\ldots ,g\}$ : on an element $\operatorname{diag}(\unicode[STIX]{x1D708}t_{1},\ldots ,\unicode[STIX]{x1D708}t_{g},t_{g}^{-1},\ldots ,t_{1}^{-1})$ of the torus, $\mathscr{S}_{g}$ acts by permuting the $t_{i}$ and the non-trivial element in each $\mathbb{Z}/2\mathbb{Z}$ sends $t_{i}$ to $t_{i}^{-1}$ . We denote the action of $w\in W_{g}$ on $t\in T(\mathbb{Q}_{\ell })$ by $t\mapsto w.t$ . Via the twisted Satake transform, the algebra ${\mathcal{H}}(T_{g}(\mathbb{Q}_{\ell }),T_{g}(\mathbb{Z}_{\ell }))$ obtains a structure of Galois extension of ${\mathcal{H}}(\operatorname{GSp}_{2g}(\mathbb{Q}_{\ell }),\operatorname{GSp}_{2g}(\mathbb{Z}_{\ell }))$ . Its Galois group is $W_{g}$ .

For $i=0,1,\ldots ,g$ , let $t_{\ell ,i}^{(g)}=\unicode[STIX]{x1D7D9}([\operatorname{diag}(\unicode[STIX]{x1D7D9}_{i},\ell \unicode[STIX]{x1D7D9}_{2g-2i},\ell ^{2}\unicode[STIX]{x1D7D9}_{i})\mathbb{T}_{g}(\mathbb{Z}_{\ell })])$ . Note that $t_{\ell ,0}^{(g)}=S_{\operatorname{GSp}_{2g}}^{T_{g}}(T_{\ell ,0}^{g})$ . The set $(t_{\ell ,i}^{(g)})_{i=1,\ldots ,g}$ generates the extension ${\mathcal{H}}(T_{g}(\mathbb{Q}_{\ell }),T_{g}(\mathbb{Z}_{\ell }))$ over ${\mathcal{H}}(\operatorname{GSp}_{2g}(\mathbb{Q}_{\ell }),\operatorname{GSp}_{2g}(\mathbb{Z}_{\ell }))$ .

We call an element $\unicode[STIX]{x1D6FE}\in T_{g}(\mathbb{Z}_{\ell })$ dilating if $v_{p}(\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FE}))\leqslant 0$ for every positive root $\unicode[STIX]{x1D6FC}$ . Let $T_{g}(\mathbb{Z}_{\ell })^{-}$ be the subset of $T_{g}(\mathbb{Z}_{\ell })$ consisting of dilating elements and let ${\mathcal{H}}(T_{g}(\mathbb{Q}_{\ell }),T_{g}(\mathbb{Z}_{\ell }))^{-}$ be the $\mathbb{Q}$ -subalgebra of ${\mathcal{H}}(T_{g}(\mathbb{Q}_{\ell }),T_{g}(\mathbb{Z}_{\ell }))$ generated by the functions $\unicode[STIX]{x1D7D9}([\unicode[STIX]{x1D6FE}T_{g}(\mathbb{Z}_{\ell })])$ with $\unicode[STIX]{x1D6FE}\in T_{g}(\mathbb{Q}_{\ell })^{-}$ . The functions $\unicode[STIX]{x1D7D9}([\unicode[STIX]{x1D6FE}T_{g}(\mathbb{Z}_{\ell })])$ with $\unicode[STIX]{x1D6FE}\in T_{g}(\mathbb{Q}_{\ell })^{-}$ also form a basis of ${\mathcal{H}}(T_{g}(\mathbb{Q}_{\ell }),T_{g}(\mathbb{Z}_{\ell }))^{-}$ as a $\mathbb{Q}$ -vector space.

Let ${\mathcal{H}}(\operatorname{GSp}_{2g}(\mathbb{Q}_{\ell }),I_{g,\ell })^{-}$ be the subalgebra of ${\mathcal{H}}(\operatorname{GSp}_{2g}(\mathbb{Q}_{\ell }),I_{g,\ell })$ generated by the functions $\unicode[STIX]{x1D7D9}([I_{g,\ell }\unicode[STIX]{x1D6FE}I_{g,\ell }])$ with $\unicode[STIX]{x1D6FE}\in T(\mathbb{Z}_{\ell })^{-}$ . We call ${\mathcal{H}}(\operatorname{GSp}_{2g}(\mathbb{Q}_{\ell }),I_{g,\ell })^{-}$ the dilating Iwahori–Hecke algebra at $\ell$ . It is generated by the elements $U_{\ell ,i}^{(g)}=\unicode[STIX]{x1D7D9}([I_{g,\ell }\operatorname{diag}(\unicode[STIX]{x1D7D9}_{i},\ell \unicode[STIX]{x1D7D9}_{2g-2i},\ell ^{2}\unicode[STIX]{x1D7D9}_{i})I_{g,\ell }])$ , for $i=0,1,\ldots ,g$ , and $(U_{\ell ,0}^{(g)})^{-1}$ .

We define a morphism of $\mathbb{Q}$ -algebras $\unicode[STIX]{x1D704}_{I_{g,\ell }}^{T_{g}}:{\mathcal{H}}(\operatorname{GSp}_{2g}(\mathbb{Q}_{\ell }),I_{g,\ell })^{-}\rightarrow {\mathcal{H}}(T_{g}(\mathbb{Q}_{\ell }),T_{g}(\mathbb{Z}_{\ell }))^{-}$ by sending $\unicode[STIX]{x1D7D9}(I_{g,\ell }\unicode[STIX]{x1D6FE}I_{g,\ell })$ to $\unicode[STIX]{x1D7D9}(T_{g}(\mathbb{Z}_{\ell })\unicode[STIX]{x1D6FE}T_{g}(\mathbb{Z}_{\ell }))$ for every $\unicode[STIX]{x1D6FE}\in T(\mathbb{Z}_{\ell })^{-}$ . The map $\unicode[STIX]{x1D704}_{I_{g,\ell }}^{T_{g}}$ is an isomorphism; this can be proved as [Reference Bellaïche and ChenevierBC09, Proposition 6.4.1].

Let $p$ be a prime and $N$ be a positive integer such that $(N,p)=1$ . Set

(1) $$\begin{eqnarray}\displaystyle {\mathcal{H}}_{g}^{Np} & = & \displaystyle \bigotimes _{\mathbb{Q},\ell \nmid Np}{\mathcal{H}}(\operatorname{GSp}_{2g}(\mathbb{Q}_{\ell }),\operatorname{GSp}_{2g}(\mathbb{Z}_{\ell }))\end{eqnarray}$$

(2) $$\begin{eqnarray}\displaystyle {\mathcal{H}}_{g}^{N} & = & \displaystyle {\mathcal{H}}_{g}^{Np}\otimes _{\mathbb{Q}}{\mathcal{H}}(\operatorname{GSp}_{2g}(\mathbb{Q}_{p}),I_{g,p})^{-}.\end{eqnarray}$$

We call ${\mathcal{H}}_{g}^{N}$ the abstract Hecke algebra spherical outside  $N$ and Iwahoric dilating at  $p$ . With an abuse of notation, we will consider the elements of one of the local algebras as elements of ${\mathcal{H}}_{g}^{N}$ by tensoring with $1$ at all the other primes.

2.2.1 The Hecke polynomials

We record here some explicit formulas for the minimal polynomials $P_{\operatorname{min}}(t_{\ell ,i}^{(g)};X)$ of the elements $t_{\ell ,i}^{(g)}$ over ${\mathcal{H}}(\operatorname{GSp}_{2g}(\mathbb{Q}_{\ell }),\operatorname{GSp}_{2g}(\mathbb{Z}_{\ell }))$ when $g$ is $1$ or  $2$ .

For $g=1$ , the element $t_{\ell ,1}^{(1)}=\unicode[STIX]{x1D7D9}([\operatorname{diag}(1,\ell )T_{1}(\mathbb{Z}_{\ell })])$ generates the degree two extension ${\mathcal{H}}(T_{1}(\mathbb{Q}_{\ell }),T_{1}(\mathbb{Z}_{\ell }))$ of ${\mathcal{H}}(\operatorname{GL}_{2}(\mathbb{Q}_{\ell }),\operatorname{GL}_{2}(\mathbb{Z}_{\ell }))$ . Let $w$ be the only non-trivial element of the Weyl group of $\operatorname{GL}_{2}$ . The minimal polynomial of $t_{\ell ,1}^{(1)}$ is $P_{\operatorname{min}}(t_{\ell ,1}^{(1)})(X)=(X-t_{\ell ,1}^{(1)})(X-(t_{\ell ,1}^{(1)})^{w})$ . An explicit calculation gives

(3) $$\begin{eqnarray}P_{\operatorname{min}}(t_{\ell ,1}^{(1)};X)=(X-t_{\ell ,1}^{(1)})(X-(t_{\ell ,1}^{(1)})^{w})=X^{2}-T_{\ell }^{(1)}X+\ell T_{\ell ,0}^{(1)}.\end{eqnarray}$$

For $g=2$ , the degree eight extension ${\mathcal{H}}(T_{2}(\mathbb{Q}_{\ell }),T_{2}(\mathbb{Z}_{\ell }))$ over ${\mathcal{H}}(\operatorname{GSp}_{4}(\mathbb{Q}_{\ell }),\operatorname{GSp}_{4}(\mathbb{Z}_{\ell }))$ is generated by $t_{\ell ,1}^{(2)}=\unicode[STIX]{x1D7D9}([\operatorname{diag}(1,\ell ,\ell ,\ell ^{2})T_{2}(\mathbb{Z}_{\ell })])$ and $t_{\ell ,2}^{(2)}=\unicode[STIX]{x1D7D9}([\operatorname{diag}(1,1,\ell ,\ell )T_{2}(\mathbb{Z}_{\ell })])$ . Each of them has an orbit of order four under the action of the Weyl group. If $t=\operatorname{diag}(\unicode[STIX]{x1D708}t_{1},\unicode[STIX]{x1D708}t_{2},t_{1}^{-1},t_{2}^{-1})$ is an element of the torus we denote by $w_{0}$ , $w_{1}$ , $w_{2}$ the generators of the Weyl group satisfying $t^{w_{0}}=\operatorname{diag}(\unicode[STIX]{x1D708}t_{2},\unicode[STIX]{x1D708}t_{1},t_{2}^{-1},t_{1}^{-1})$ , $t^{w_{1}}=\operatorname{diag}(\unicode[STIX]{x1D708}t_{1}^{-1},\unicode[STIX]{x1D708}t_{2},t_{1},t_{2}^{-1})$ , $t^{w_{2}}=\operatorname{diag}(\unicode[STIX]{x1D708}t_{1},\unicode[STIX]{x1D708}t_{2}^{-1},t_{1}^{-1},t_{2})$ . Note that $t_{\ell ,2}^{(2)}$ is invariant under $w_{0}$ . The calculation in the proof of [Reference AndrianovAnd87, Lemma 3.3.35] gives

(4) $$\begin{eqnarray}\displaystyle P_{\operatorname{min}}(t_{\ell ,2}^{(2)};X) & = & \displaystyle (X-t_{\ell ,2}^{(2)})(X-(t_{\ell ,2}^{(2)})^{w_{1}})(X-(t_{\ell ,2}^{(2)})^{w_{2}})(X-(t_{\ell ,2}^{(2)})^{w_{1}w_{2}})\nonumber\\ \displaystyle & = & \displaystyle X^{4}-T_{\ell ,2}^{(2)}X^{3}+((T_{\ell ,2}^{(2)})^{2}-T_{\ell ,1}^{(2)}-\ell ^{2}T_{\ell ,0}^{(2)})X^{2}-\ell ^{3}T_{\ell ,2}^{(2)}T_{\ell ,0}^{(2)}X+\ell ^{6}(T_{\ell ,0}^{(2)})^{2}.\end{eqnarray}$$

2.3 The cuspidal $\operatorname{GSp}_{2g}$ -eigenvariety

Let $g$ be a positive integer. Let $p$ be an odd prime and $N$ a positive integer such that $(N,p)=1$ . Let ${\mathcal{H}}_{g}^{N}$ be the abstract Hecke algebra for $\operatorname{GSp}_{2g}$ , spherical outside $N$ and Iwahoric dilating at $p$ . Let ${\mathcal{W}}_{g}$ be the $g$ -dimensional weight space. For every affinoid $A=\operatorname{Spm}R\subset {\mathcal{W}}_{g}$ and every sufficiently large rational number $w$ , Andreatta, Iovita and Pilloni [Reference Andreatta, Iovita and PilloniAIP15, §8.2] defined a Banach $R$ -module $M_{g}(A,w)$ of $w$ -overconvergent cuspidal $\operatorname{GSp}_{2g}$ -modular forms of weight $\unicode[STIX]{x1D705}_{A}$ and tame level $\unicode[STIX]{x1D6E4}_{1}(N)$ . For each pair $(A,w)$ there is an action $\unicode[STIX]{x1D719}_{A,w}^{g}:{\mathcal{H}}_{g}^{N}\rightarrow \operatorname{End}_{R,\operatorname{cont}}M_{g}(A,w)$ . Set $U_{p}^{(g)}=\prod _{i=1}^{g}U_{p,g}^{(g)}$ . It is shown in [Reference Andreatta, Iovita and PilloniAIP15, §8.1] that $({\mathcal{W}}_{g},{\mathcal{H}}_{g}^{N},(M_{g}(A,w))_{A,w},(\unicode[STIX]{x1D719}^{g})_{A,w},U_{p}^{(g)})$ is an eigenvariety datum in the sense of [Reference BuzzardBuz07, §5]. Buzzard’s ‘eigenvariety machine’ [Reference BuzzardBuz07, Construction 5.7] produces from this datum a rigid analytic variety over $\mathbb{Q}_{p}$ , equidimensional of dimension $g$ . We call it the $\operatorname{GSp}_{2g}$ -eigenvariety of tame level $N$ and we denote it by ${\mathcal{D}}_{g}^{N}$ . It is equipped with a weight morphism $w_{g}:{\mathcal{D}}_{g}^{N}\rightarrow {\mathcal{W}}_{g}$ and a homomorphism $\unicode[STIX]{x1D713}_{g}:{\mathcal{H}}_{g}^{N}\rightarrow {\mathcal{O}}({\mathcal{D}}_{g}^{N})$ , that interpolates the normalized systems of Hecke eigenvalues of classical cuspidal $\operatorname{GSp}_{2g}$ -eigenforms of tame level $\unicode[STIX]{x1D6E4}_{1}(N)$ . The images of the elements $T_{i,\ell }^{(g)}$ and $U_{i,p}^{(g)}$ , $1\leqslant i\leqslant g$ , belong to ${\mathcal{O}}({\mathcal{D}}_{g}^{N})^{\circ }$ .

When $g=1$ we call ${\mathcal{D}}_{1}^{N}$ the eigencurve. It was constructed by Coleman and Mazur in [Reference Coleman and MazurCM98] for $N=1$ and $p>2$ , building on earlier ideas of Coleman. Their construction was extended to all $N$ and $p$ by Buzzard in [Reference BuzzardBuz07].

We call a point $x\in {\mathcal{D}}_{g}^{N}(\mathbb{C}_{p})$ classical if the evaluation of $\unicode[STIX]{x1D713}_{g}$ at $x$ is the Hecke eigensystem a classical $\operatorname{GSp}_{2g}$ -eigenform $f$ of level $\unicode[STIX]{x1D6E4}_{1}(N)\cap \unicode[STIX]{x1D6E4}_{0}(p)$ and weight $w_{g}(x)$ . In this case, $w_{g}(x)$ is clearly a classical weight.

There is a slope function $\operatorname{sl}:{\mathcal{D}}_{g}^{N}(\mathbb{C}_{p})\rightarrow \mathbb{R}^{+}$ defined as the $p$ -adic valuation of $\unicode[STIX]{x1D713}_{g}(U_{p,g}^{(g)})$ . Let $x$ be a $\overline{\mathbb{Q}}_{p}$ -point of ${\mathcal{D}}_{g}^{N}$ of weight $\text{}\underline{k}=(k_{1},k_{2},\ldots ,k_{g})\in \mathbb{Z}^{g}$ , so that $k_{1}\geqslant k_{2}\geqslant \cdots \geqslant k_{g}$ . We recall the following result.

2.4 The Galois pseudocharacters on the eigenvarieties

In this section, $p$ is a fixed prime, $M$ is a positive integer prime to $p$ and $g$ is $1$ or $2$ . For a point $x\in {\mathcal{D}}_{g}^{M}(\mathbb{C}_{p})$ , we denote by $\operatorname{ev}_{x}:{\mathcal{O}}({\mathcal{D}}_{g}^{M})\rightarrow \mathbb{C}_{p}$ both the evaluation at $x$ and the map $\operatorname{GSp}_{2g}({\mathcal{O}}({\mathcal{D}}_{g}^{M}))\rightarrow \operatorname{GSp}_{2g}(\mathbb{C}_{p})$ induced by $\operatorname{ev}_{x}$ . Recall that the $\operatorname{GSp}_{2g}$ -eigenvariety ${\mathcal{D}}_{g}^{M}$ is endowed with a morphism $\unicode[STIX]{x1D713}_{g}:{\mathcal{H}}_{g}^{M}\rightarrow {\mathcal{O}}({\mathcal{D}}_{g}^{M})$ that interpolates the normalized systems of Hecke eigenvalues associated with the cuspidal $\operatorname{GSp}_{2g}$ -eigenforms of level $\unicode[STIX]{x1D6E4}_{1}(N)\cap \unicode[STIX]{x1D6E4}_{0}(p)$ . Also recall that the images of $T_{i,\ell }^{(g)}$ and $U_{i,p}^{(g)}$ , $1\leqslant i\leqslant g$ , are elements of ${\mathcal{O}}({\mathcal{D}}_{g}^{M})^{\circ }$ . Let $S^{\operatorname{cl}}$ denote the set of classical $\overline{\mathbb{Q}}_{p}$ -points of ${\mathcal{D}}_{g}^{M}$ . For $x\in S^{\operatorname{cl}}$ let $\unicode[STIX]{x1D713}_{x}=\operatorname{ev}_{x}\circ \,\unicode[STIX]{x1D713}_{g}$ . Let $f_{x}$ be the classical $\operatorname{GSp}_{2g}$ -eigenform having system of Hecke eigenvalues $\unicode[STIX]{x1D713}_{x}$ . Let $\unicode[STIX]{x1D70C}_{x}:G_{\mathbb{Q}}\rightarrow \operatorname{GSp}_{2g}(\overline{\mathbb{Q}}_{p})$ be the $p$ -adic Galois representation attached to $f_{x}$ and let $T_{x}$ be the pseudocharacter defined as the trace of $\unicode[STIX]{x1D70C}_{x}$ . When $x$ varies, the traces $T_{x}$ can be interpolated by a pseudocharacter with values in ${\mathcal{O}}({\mathcal{D}}_{g}^{M})^{\circ }$ . This is stated precisely in the following proposition.

For every ring $R$ , we implicitly extend a character of the Hecke algebra ${\mathcal{H}}_{g}^{M}\rightarrow R^{\times }$ to a morphism of polynomial algebras ${\mathcal{H}}_{g}^{M}[X]\rightarrow R[X]$ by applying it to the coefficients. Recall that we fixed an embedding $G_{\mathbb{Q}_{\ell }}{\hookrightarrow}G_{\mathbb{Q}}$ for every prime $\ell$ , hence an embedding of the inertia subgroup $I_{\ell }$ in $G_{\mathbb{Q}}$ . As usual $\operatorname{Frob}_{\ell }$ denotes a lift of the Frobenius at $\ell$ to $G_{\mathbb{Q}_{\ell }}$ .

3.1 Trianguline parameters of overconvergent $\operatorname{GSp}_{4}$ -eigenforms

We refer to [Reference BergerBer02, Reference ColmezCol08] for the definitions and results that we need from the theory of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules and trianguline representations. Let $F$ be as in the beginning of the section. Part (i) of the following theorem is a classical result of Faltings [Reference FaltingsFal89]. Part (ii) is a combination of [Reference KisinKis03, Theorem 6.3] and [Reference ColmezCol08, Proposition 4.3], as Berger observed in [Reference BergerBer11, §4.3].

If $g=2$ , an analogue of Theorem 3.1(ii) for $\unicode[STIX]{x1D70C}_{F,p}$ can be deduced from the work of Kedlaya, Pottharst and Xiao [Reference Kedlaya, Pottharst and XiaoKPX14]. Moreover, the results of [Reference Kedlaya, Pottharst and XiaoKPX14] allow us to write the parameters of the triangulation of $\unicode[STIX]{x1D70C}_{F,p}$ in terms of a Hecke polynomial, as for classical points. Recall that we are working on an eigenvariety ${\mathcal{D}}_{2}^{M}$ with a fixed residual Galois representation $\overline{\unicode[STIX]{x1D70C}}$ . Suppose that $\overline{\unicode[STIX]{x1D70C}}$ is irreducible. By lifting pseudocharacters to representations and considering the associated $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules we can define a family of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules over ${\mathcal{D}}_{2}^{M}$ in the sense of [Reference Kedlaya, Pottharst and XiaoKPX14, §2.1]. For $x\in {\mathcal{D}}_{2}^{M}(\mathbb{C}_{p})$ , let $\unicode[STIX]{x1D70C}_{x}:G_{\mathbb{Q}}\rightarrow \operatorname{GL}_{4}(\overline{\mathbb{Q}}_{p})$ and $\unicode[STIX]{x1D713}_{x}:{\mathcal{H}}_{2}^{N}\rightarrow \overline{\mathbb{Q}}_{p}$ be the Galois representation and the system of Hecke eigenvalues, respectively, attached to $x$ . Let $M_{x}$ be the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -module over $\overline{\mathbb{Q}}_{p}$ attached to $\unicode[STIX]{x1D70C}_{x}$ . Denote by $\operatorname{ev}_{x}$ the evaluation of rigid analytic functions on ${\mathcal{D}}_{2}^{M}$ at $x$ . We identify the weight of $x$ with a character $(\unicode[STIX]{x1D705}_{1}(x),\unicode[STIX]{x1D705}_{2}(x)):(\mathbb{Z}_{p}^{\times })^{2}\rightarrow \mathbb{C}_{p}^{\times }$ . Let $\operatorname{id}:\mathbb{Z}_{p}^{\times }\rightarrow \mathbb{Z}_{p}^{\times }$ be the identity. Let $\unicode[STIX]{x1D6FF}_{i}$ , $1\leqslant i\leqslant 4$ be the characters $\mathbb{Z}_{p}^{\times }\rightarrow {\mathcal{O}}({\mathcal{D}}_{2}^{M})^{\times }$ defined by

$$\begin{eqnarray}\begin{array}{@{}c@{}}\unicode[STIX]{x1D6FF}_{1}|_{\mathbb{Z}_{p}^{\times }}=1,\quad \unicode[STIX]{x1D6FF}_{1}(p)=\unicode[STIX]{x1D713}_{x}(U_{p,2}^{(2)});\quad \unicode[STIX]{x1D6FF}_{2}|_{\mathbb{Z}_{p}^{\times }}=\unicode[STIX]{x1D705}_{1}/\!\operatorname{id},\quad \unicode[STIX]{x1D6FF}_{2}(p)=\unicode[STIX]{x1D713}_{x}((U_{p,2}^{(2)})^{w_{1}});\\ \unicode[STIX]{x1D6FF}_{3}|_{\mathbb{Z}_{p}^{\times }}=\unicode[STIX]{x1D705}_{2}/\!\operatorname{id}^{2},\quad \unicode[STIX]{x1D6FF}_{3}(p)=\unicode[STIX]{x1D713}_{x}((U_{p,2}^{(2)})^{w_{2}});\quad \unicode[STIX]{x1D6FF}_{4}|_{\mathbb{Z}_{p}^{\times }}=\unicode[STIX]{x1D705}_{1}\unicode[STIX]{x1D705}_{2}(p)/\!\operatorname{id}^{3},\quad \unicode[STIX]{x1D6FF}_{4}(p)=\unicode[STIX]{x1D713}_{x}((U_{p,2}^{(2)})^{w_{1}w_{2}}).\end{array}\end{eqnarray}$$

For $x\in {\mathcal{D}}_{2}^{M}(\mathbb{C}_{p})$ , let $\unicode[STIX]{x1D6FF}_{i,x}=\operatorname{ev}_{x}\circ \,\unicode[STIX]{x1D6FF}_{i}:\mathbb{Q}_{p}^{\times }\rightarrow \overline{\mathbb{Q}}_{p}$ .

Now let $N$ be a positive integer prime to $p$ and let $M=N^{3}$ . Let $F$ be an overconvergent $\operatorname{GSp}_{4}$ -eigenform corresponding to a point of $\widetilde{{\mathcal{D}}}_{2}^{M}$ . Suppose that there is a $\operatorname{GL}_{2}$ -eigenform $f$ of level $N$ such that $\unicode[STIX]{x1D70C}_{F,p}\cong \operatorname{Sym}^{3}\unicode[STIX]{x1D70C}_{f,p}$ , with the usual notation. Let $\unicode[STIX]{x1D712}_{F}:{\mathcal{H}}_{2}^{N}\rightarrow \overline{\mathbb{Q}}_{p}$ and $\unicode[STIX]{x1D712}_{f}:{\mathcal{H}}_{1}^{N}\rightarrow \overline{\mathbb{Q}}_{p}$ be the systems of Hecke eigenvalues of the two forms. For $1\leqslant i\leqslant 8$ , define $\unicode[STIX]{x1D712}_{2}^{\operatorname{st},i}$ as in Proposition 9.7 by replacing $\unicode[STIX]{x1D712}_{1}^{\operatorname{st}}$ by $\unicode[STIX]{x1D712}_{f}$ .

Let $F$ be a finite slope overconvergent $\operatorname{GSp}_{4}$ -eigenform of tame level $M$ .

Now let $f$ be a finite slope overconvergent $\operatorname{GL}_{2}$ -eigenform of tame level $N$ .

3.2 Representations with symmetric cube of automorphic origin

Let $V$ be a two-dimensional representation of $G_{\mathbb{Q}}$ over a $p$ -adic field. We recall a special case of a result of Di Matteo.

When ‘de Rham’ is replaced by ‘trianguline’, the techniques of [Reference Di MatteoDiM13b] can be modified to prove the following result.

We refer to [Reference ContiCon16, Proposition 3.10.25] for the proof. We only remark here that to obtain triangulinity, rather than potential triangulinity, we need a result of Berger and Chenevier [Reference Berger and ChenevierBC10, Théorème A].

Given a two-dimensional modulo $p$ representation $\overline{\unicode[STIX]{x1D70F}}$ of $G_{\mathbb{Q}}$ , we list here for future reference some assumptions that we need in applying the results of [Reference EmertonEme14]. Here $\unicode[STIX]{x1D712}$ denotes the modulo $p$ cyclotomic character:

For convenience we also restate the symmetric cube of the above assumptions, for a four-dimensional modulo $p$ representation $\overline{\unicode[STIX]{x1D70F}}$ of $G_{\mathbb{Q}}$ :

We will always replace the subscript $\overline{\unicode[STIX]{x1D70F}}$ with the representation we are making the above assumptions on.

Let $\unicode[STIX]{x1D70C}_{1}:G_{\mathbb{Q}}\rightarrow \operatorname{GL}_{2}(\overline{\mathbb{Q}}_{p})$ and $\unicode[STIX]{x1D70C}_{2}:G_{\mathbb{Q}}\rightarrow \operatorname{GSp}_{4}(\overline{\mathbb{Q}}_{p})$ be two continuous representations. We will deduce the following theorem from the two previous propositions.

3.3 A big image result for classical $\operatorname{GSp}_{4}$ -eigenforms

In the following definitions, let $E$ be a finite extension of $\mathbb{Q}_{p}$ . Let $R$ be a local ring with maximal ideal $\mathfrak{m}_{R}$ and residue field $\mathbb{F}$ . Let $\unicode[STIX]{x1D70F}:G_{E}\rightarrow \operatorname{GSp}_{4}(R)$ be a representation. Let $\operatorname{PGSp}_{4}(R)=\operatorname{GSp}_{4}(R)/R^{\times }$ , where $R^{\times }$ is identified with the subgroup of scalar matrices. We denote by $\overline{\unicode[STIX]{x1D70F}}:G_{E}\rightarrow \operatorname{GSp}_{4}(\mathbb{F})$ the reduction of $\unicode[STIX]{x1D70F}$ modulo $\mathfrak{m}_{R}$ . Recall that $T_{2}$ is the torus consisting of diagonal matrices in $\operatorname{GSp}_{4}$ . We give a notion of $\mathbb{Z}_{p}$ -regularity of $\unicode[STIX]{x1D70F}$ , analogous to that in [Reference Hida, Tilouine, Dieulefait, Faltings, Heath-Brown, Manin, Moroz and WintenbergerHT15, Lemma 4.5(2)].

From now on we focus on representations that are either ‘residually full’ or ‘residually of symmetric cube type’, in the sense of the following definition. Note that these two types of representations appear in [Reference PilloniPil12, §5.8] as examples of those for which Pilloni can construct a sequence of Taylor–Wiles primes.

We write $\mathfrak{s}\mathfrak{p}_{4}(K)$ for the Lie algebra of $\operatorname{Sp}_{4}(K)$ and $\operatorname{Ad}:\operatorname{GSp}_{4}(K)\rightarrow \operatorname{End}(\mathfrak{s}\mathfrak{p}_{4}(K))$ for the adjoint representation. Let $F$ and $\unicode[STIX]{x1D70C}_{F,p}:G_{\mathbb{Q}}\rightarrow \operatorname{GSp}_{4}({\mathcal{O}}_{K})$ be as in the beginning of the section. Let $E$ be the subfield of $K$ generated over $\mathbb{Q}_{p}$ by the set $\{\operatorname{Tr}(\operatorname{Ad}(\unicode[STIX]{x1D70C}(g)))\}\text{}_{g\in G_{\mathbb{Q}}}$ . Let ${\mathcal{O}}_{E}$ be the ring of integers of $E$ . For a $\operatorname{GL}_{2}$ -eigenform $f$ , we denote by $\unicode[STIX]{x1D70C}_{f,p}$ the associated $p$ -adic Galois representation. We will prove the following result.

For use in the proof of Theorem 3.12 we state a result of Pink.

We also need the following lemma, that results from an application of Theorem 5.12. We refer to [Reference ContiCon16, Lemma 3.11.5] for a detailed proof.

The rest of the section is devoted to the proof of Theorem 3.12. Let $(\operatorname{Im}\unicode[STIX]{x1D70C}_{F,p})^{\prime }$ be the derived subgroup of $\operatorname{Im}\unicode[STIX]{x1D70C}_{F,p}$ and let $G=(\operatorname{Im}\unicode[STIX]{x1D70C}_{F,p})\cap \operatorname{Sp}_{4}(K)$ . We denote by $\overline{G}$ the Zariski-closure of $G$ in $\operatorname{Sp}_{4}(K)$ . As in [Reference Hida, Tilouine, Dieulefait, Faltings, Heath-Brown, Manin, Moroz and WintenbergerHT15, §3], we will show first that under the hypotheses of Theorem 3.12 we have $\overline{G}=\operatorname{Sp}_{4}(K)$ , and second that $G$ is $p$ -adically open in $\overline{G}$ . We will replace the ordinarity assumption in [Reference Hida, Tilouine, Dieulefait, Faltings, Heath-Brown, Manin, Moroz and WintenbergerHT15, §3] by that of $\mathbb{Z}_{p}$ -regularity. Let $\overline{G}^{\circ }$ denote the connected component of the identity in $\overline{G}$ .

Let $H$ be any connected, Zariski-closed subgroup of $\operatorname{Sp}_{4}$ , defined over $K$ . As in [Reference Hida, Tilouine, Dieulefait, Faltings, Heath-Brown, Manin, Moroz and WintenbergerHT15, §3.4], we have six possibilities for the isomorphism class of $H$ over $K$ :

1. (i) $H\cong \operatorname{Sp}_{4}$ ;

2. (ii) $H\cong \operatorname{SL}_{2}\times \operatorname{SL}_{2}$ ;

3. (iii) $H\cong \operatorname{SL}_{2}$ embedded in a Klingen parabolic subgroup;

4. (iv) $H\cong \operatorname{SL}_{2}$ embedded in a Siegel parabolic subgroup;

5. (v) $H\cong \operatorname{SL}_{2}$ embedded via the symmetric cube representation $\operatorname{SL}_{2}\rightarrow \operatorname{Sp}_{4}$ (in this case we write $H\cong \operatorname{Sym}^{3}\operatorname{SL}_{2}$ );

6. (vi) $H\cong \{1\}$ .

We show that only choice (i) is possible for $H=\overline{G}^{\circ }$ .

We equip all groups with their $p$ -adic topology. The proof of Theorem 3.12 is completed by the following proposition.

Theorem 3.12 states that, when $\overline{\unicode[STIX]{x1D70C}}_{F,p}$ is either full or of symmetric cube type, the image of $\unicode[STIX]{x1D70C}_{F,p}$ is large if and only if $F$ is not a lift of an eigenform from a smaller group, the only possible such lift under these assumptions being associated with the symmetric cube representation of $\operatorname{GL}_{2}$ . We think that a similar result should hold under more general assumptions on the residual representation, and that it would follow from Pink’s theorem together with an analogue of Corollary 3.9 for the other possible Langlands lifts to $\operatorname{GSp}_{4}$ .

4.1 Families defined over $\mathbb{Z}_{p}$

For our purpose of studying the images of Galois representations, we will need to have our finite slope families defined over $\mathbb{Z}_{p}$ . For this reason we specialize to families over weight discs for which we can construct a $\mathbb{Z}_{p}$ -model. For simplicity we only work on the connected component ${\mathcal{W}}_{g}^{\circ }$ . Recall that we defined coordinates $T_{1},T_{2},\ldots ,T_{g}$ on ${\mathcal{W}}_{g}^{\circ }$ . Let $\unicode[STIX]{x1D705}$ be a point of ${\mathcal{W}}_{g}^{\circ }$ with coordinates $(\unicode[STIX]{x1D705}_{1},\unicode[STIX]{x1D705}_{2},\ldots ,\unicode[STIX]{x1D705}_{g})$ in $\mathbb{Z}_{p}^{g}$ ; for instance, we can take as $\unicode[STIX]{x1D705}$ the arithmetic prime $P_{\text{}\underline{k}}$ for some $\text{}\underline{k}\in \mathbb{Z}^{g}$ . Let $r_{h,\unicode[STIX]{x1D705}}$ be the largest radius in $p^{\mathbb{Q}}$ such that the map $w_{\unicode[STIX]{x1D705},B_{g}(\unicode[STIX]{x1D705},r_{h,\unicode[STIX]{x1D705}}^{-})}:{\mathcal{D}}_{g,B_{g}(\unicode[STIX]{x1D705},r_{h,\unicode[STIX]{x1D705}}^{-})}^{N,h}\rightarrow B_{g}(\unicode[STIX]{x1D705},r_{h,\unicode[STIX]{x1D705}}^{-})$ is finite. Such a radius is non-zero thanks to Remark 4.2(i). Let $s_{h}$ be a rational number satisfying $r_{h}=p^{s_{h}}$ . We define a model for $B_{g}(\unicode[STIX]{x1D705},r_{h}^{-})$ over $\mathbb{Q}_{p}$ by adapting Berthelot’s construction for the wide open unit disc (see [Reference de JongdeJ95, §7]). Write $s_{h}=b/a$ for some $a,b\in \mathbb{N}$ . For $i\geqslant 1$ , let $s_{i}=s_{h}+1/2^{i}$ and $r_{i}=p^{-s_{i}}$ . Set

$$\begin{eqnarray}A_{r_{i}}^{\circ }=\mathbb{Z}_{p}\langle t_{1},t_{2},\ldots ,t_{g},X_{i}\rangle /(t_{j}^{2^{i}a}-p^{a+2^{i}b}X_{i})_{j=1,2,\ldots ,g}\end{eqnarray}$$

and $A_{r_{i}}=A_{r_{i}}^{\circ }[p^{-1}]$ . Set $B_{i}=\operatorname{Spm}A_{r_{i}}$ . Then $B_{i}$ is a $\mathbb{Q}_{p}$ -model of the disc of centre $\unicode[STIX]{x1D705}$ and radius  $r_{i}$ . We define morphisms $A_{r_{i+1}}^{\circ }\rightarrow A_{r_{i}}^{\circ }$ by

$$\begin{eqnarray}\displaystyle & X_{i+1}\mapsto p^{a}X_{i}^{2}, & \displaystyle \nonumber\\ \displaystyle & t_{j}\mapsto t_{j}\quad \text{for }j=1,2,\ldots ,g. & \displaystyle \nonumber\end{eqnarray}$$

They induce compact maps $A_{r_{i+1}}\rightarrow A_{r_{i}}$ which give open immersions $B_{i}{\hookrightarrow}B_{i+1}$ . We define $B_{g,h}=\mathop{\varinjlim }\nolimits_{i}B_{i}$ where the limit is taken with respect to the above immersions. Let $\unicode[STIX]{x1D6EC}_{g,h}={\mathcal{O}}(B_{g,h})^{\circ }$ . Then $\unicode[STIX]{x1D6EC}_{g,h}=\mathop{\varprojlim }\nolimits_{i}{\mathcal{O}}(\operatorname{Spm}B_{i})^{\circ }=\mathop{\varprojlim }\nolimits_{i}A_{r_{i}}^{\circ }$ . We call $\unicode[STIX]{x1D6EC}_{g,h}$ the genus  $g$ , $h$ -adapted Iwasawa algebra; we leave its dependence on $\unicode[STIX]{x1D705}$ implicit. We define $t_{1},t_{2},\ldots ,t_{g}\in \unicode[STIX]{x1D6EC}_{g,h}$ as the projective limits of the variables $t_{1},t_{2},\ldots ,t_{g}$ , respectively, of $A_{r_{i}}^{\circ }$ .

There is a map of $\mathbb{Z}_{p}$ -algebras $\unicode[STIX]{x1D704}_{g,h}^{\ast }:\unicode[STIX]{x1D6EC}_{g}\rightarrow \unicode[STIX]{x1D6EC}_{g,h}$ defined by $T_{j}\mapsto t_{j}+\unicode[STIX]{x1D705}_{j}$ for $j=1,2,\ldots ,g$ . The inclusion $\unicode[STIX]{x1D704}_{g,h}:B_{g,h}{\hookrightarrow}{\mathcal{W}}_{g}^{\circ }$ induced by $\unicode[STIX]{x1D704}_{g,h}^{\ast }$ makes $B_{g,h}$ into a $\mathbb{Q}_{p}$ -model of $B_{g}(\unicode[STIX]{x1D705},r_{h}^{-})$ , endowed with the integral structure defined by $\unicode[STIX]{x1D6EC}_{g,h}$ .

Let $\unicode[STIX]{x1D702}_{h}$ be an element of $\overline{\mathbb{Q}}_{p}$ satisfying $v_{p}(\unicode[STIX]{x1D702}_{h})=s_{h}$ . Let $K_{h}=\mathbb{Q}_{p}(\unicode[STIX]{x1D702}_{h})$ and let ${\mathcal{O}}_{h}$ be the ring of integers of $K_{h}$ . The algebra $\unicode[STIX]{x1D6EC}_{g,h}$ is not a ring of formal series over $\mathbb{Z}_{p}$ , but there is an isomorphism $\unicode[STIX]{x1D6EC}_{g,h}\otimes _{\mathbb{Z}_{p}}{\mathcal{O}}_{h}\cong {\mathcal{O}}_{h}[[t_{1},t_{2},\ldots ,t_{g}]]$ .

We say that a prime of $\unicode[STIX]{x1D6EC}_{g,h}$ is arithmetic if it lies over an arithmetic prime of $\unicode[STIX]{x1D6EC}_{g}$ . By an abuse of notation we will write again $P_{\text{}\underline{k}}$ for an arithmetic prime of $\unicode[STIX]{x1D6EC}_{g,h}$ lying over the arithmetic prime $P_{\text{}\underline{k}}$ of $\unicode[STIX]{x1D6EC}_{g}$ .

Let ${\mathcal{D}}_{g,h}^{N}$ be the rigid analytic space fitting in the following Cartesian diagram.

(5)

The rigid analytic space ${\mathcal{D}}_{g,h}^{N}$ is a model of ${\mathcal{D}}_{g,B_{g}(\unicode[STIX]{x1D705},r_{h,\unicode[STIX]{x1D705}}^{-})}^{N,h}$ over a $p$ -adic field, but it is not necessarily defined over $\mathbb{Q}_{p}$ since the map $\unicode[STIX]{x1D704}_{g,h}$ may not be. We say that a $\mathbb{C}_{p}$ -point of ${\mathcal{D}}_{g,h}^{N}$ is classical if it is a classical point of ${\mathcal{D}}_{g,B_{g}(\unicode[STIX]{x1D705},r_{h,\unicode[STIX]{x1D705}}^{-})}^{N,h}$ .

Let $\mathbb{T}_{g,h}={\mathcal{O}}({\mathcal{D}}_{g,h}^{N})^{\circ }$ . We call $\mathbb{T}_{g,h}$ the genus  $g$ , $h$ -adapted Hecke algebra; we leave its dependence on $\unicode[STIX]{x1D705}$ implicit again. The weight map induces a morphism $w_{g,h}:{\mathcal{D}}_{g,h}^{N}\rightarrow B_{g,h}$ , hence a morphism $w_{g,h}^{\ast }:\unicode[STIX]{x1D6EC}_{g,h}\rightarrow \mathbb{T}_{g,h}$ . Thanks to our choice of $r_{h}$ , $w_{g,h}^{\ast }$ gives $\mathbb{T}_{g,h}$ a structure of finite $\unicode[STIX]{x1D6EC}_{g,h}$ -algebra. The ${\mathcal{D}}_{g,h}^{N}\rightarrow {\mathcal{D}}_{g}^{N,h}$ appearing in the diagram induces a map ${\mathcal{O}}({\mathcal{D}}_{g}^{N,h})^{\circ }\rightarrow \mathbb{T}_{g,h}$ , that we compose with $\unicode[STIX]{x1D713}_{g}:{\mathcal{H}}_{g}^{N}\rightarrow {\mathcal{O}}({\mathcal{D}}_{g}^{N,h})^{\circ }$ to obtain a morphism $\unicode[STIX]{x1D713}_{g,h}:{\mathcal{H}}_{g}^{N}\rightarrow \mathbb{T}_{g,h}$ .

For a prime $\mathfrak{P}$ of $\mathbb{T}_{g,h}$ , we denote by $\operatorname{ev}_{\mathfrak{P}}:\mathbb{T}_{g,h}\rightarrow \overline{\mathbb{Z}}_{p}$ the evaluation at $\mathfrak{P}$ . We say that $\mathfrak{P}$ is a classical point of $\operatorname{Spec}\mathbb{T}_{g,h}$ if $\operatorname{ev}_{\mathfrak{P}}\circ \,\unicode[STIX]{x1D713}_{g,h}:{\mathcal{H}}_{g}^{N}\rightarrow \overline{\mathbb{Z}}_{p}$ is the system of Hecke eigenvalues attached to a classical $\operatorname{GSp}_{2g}$ -eigenform. These systems of eigenvalues also appear at classical points of ${\mathcal{D}}_{g,h}^{N}$ .

We will usually refer to an $I$ as in Definition 4.4 simply as a finite slope family. Let $\mathbb{I}^{\circ }={\mathcal{O}}(I)$ . Then $\mathbb{I}^{\circ }$ is a finite $\unicode[STIX]{x1D6EC}_{g,h}$ -algebra that is also profinite and local. The component $I$ is determined by the surjective morphism $\unicode[STIX]{x1D703}:\mathbb{T}_{g,h}{\twoheadrightarrow}\mathbb{I}^{\circ }$ . We sometimes refer to $\unicode[STIX]{x1D703}$ as a finite slope family. The family $I$ is equipped with maps $w_{\unicode[STIX]{x1D703}}:I\rightarrow B_{g,h}$ and $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D703}}:{\mathcal{H}}_{g}^{N}\rightarrow \mathbb{I}^{\circ }$ induced by $w_{g,h}$ and $\unicode[STIX]{x1D713}_{g,h}$ , respectively. Here $^{\circ }$ denotes the fact that we are working with integral objects. When introducing relative Sen theory in §7, we will need to invert $p$ and we will drop the $^{\circ }$ from all rings.

Proposition 2.2 implies that every family $I$ contains at least a classical point. By the accumulation property of classical point and the irreducibility of $I$ , the classical points are a Zariski-dense subset of $I$ . Hence, the set of classical points of $\operatorname{Spec}\mathbb{I}^{\circ }$ is Zariski-dense in $\operatorname{Spec}\mathbb{I}^{\circ }$ . Every classical point of $\operatorname{Spec}\mathbb{I}^{\circ }$ lies over an arithmetic prime of $\operatorname{Spec}\unicode[STIX]{x1D6EC}_{g,h}$ . For a family $\unicode[STIX]{x1D703}:\mathbb{T}_{g,h}\rightarrow \mathbb{I}^{\circ }$ , we give the following.

4.2 The Galois representation associated with a finite slope family

Let $\unicode[STIX]{x1D705}$ be a point of ${\mathcal{W}}_{g}^{\circ }$ with $\mathbb{Z}_{p}$ -coordinates. Let $\mathbb{T}_{2,h}$ be the genus $2$ , $h$ -adapted Hecke algebra of centre $\unicode[STIX]{x1D705}$ . As before, we will leave the dependence on $\unicode[STIX]{x1D705}$ implicit. For simplicity, let $\unicode[STIX]{x1D6EC}_{h}=\unicode[STIX]{x1D6EC}_{2,h}$ , $\mathbb{T}_{h}=\mathbb{T}_{2,h}$ . We implicitly replace $\mathbb{T}_{h}$ by one of its local components. Let $\unicode[STIX]{x1D703}:\mathbb{T}_{h}\rightarrow \mathbb{I}^{\circ }$ be a finite slope family of $\operatorname{GSp}_{4}$ -eigenforms. Let $\mathbb{F}_{\mathbb{T}_{h}}$ be the residue field of $\mathbb{T}_{h}$ . The pseudocharacter $T_{2}:G_{\mathbb{Q}}\rightarrow {\mathcal{O}}({\mathcal{D}}_{2}^{N})^{\circ }$ induces pseudocharacters $T_{\mathbb{T}_{h}}:G_{\mathbb{Q}}\rightarrow \mathbb{T}_{h}$ and $\overline{T}_{\mathbb{T}_{h}}:G_{\mathbb{Q}}\rightarrow \mathbb{F}_{\mathbb{T}_{h}}$ . By [Reference RouquierRou96, Corollary 5.2] the pseudocharacter $\overline{T}_{\mathbb{T}_{h}}$ is associated with a representation $\overline{\unicode[STIX]{x1D70C}}_{\mathbb{T}_{h}}:G_{\mathbb{Q}}\rightarrow \operatorname{GL}_{4}(\overline{\mathbb{F}}_{p})$ , unique up to isomorphism. We call $\overline{\unicode[STIX]{x1D70C}}_{\mathbb{T}_{h}}$ the residual Galois representation associated with $\mathbb{T}_{h}$ . We assume from now on that

By the compactness of $G_{\mathbb{Q}}$ there exists a finite extension $\mathbb{F}^{\prime }$ of $\mathbb{F}_{\mathbb{T}_{h}}$ such that $\overline{\unicode[STIX]{x1D70C}}_{\mathbb{T}_{h}}$ is defined on $\mathbb{F}^{\prime }$ . Let $W(\mathbb{F}_{\mathbb{T}_{h}})$ and $W(\mathbb{F}^{\prime })$ be the rings of Witt vectors of $\mathbb{F}_{\mathbb{T}_{h}}$ and $\mathbb{F}^{\prime }$ , respectively. Let $\mathbb{T}_{h}^{\prime }=\mathbb{T}_{h}\otimes _{W(\mathbb{F}_{\mathbb{T}_{h}})}W(\mathbb{F}^{\prime })$ . We consider $T_{\mathbb{T}_{h}}$ as a pseudocharacter $G_{\mathbb{Q}}\rightarrow \mathbb{T}_{h}^{\prime }$ via the natural inclusion $\mathbb{T}_{h}{\hookrightarrow}\mathbb{T}_{h}^{\prime }$ . Then $T_{\mathbb{T}_{h}}$ satisfies the hypotheses of [Reference RouquierRou96, Corollary 5.2], so there exists a representation $\unicode[STIX]{x1D70C}_{\mathbb{T}_{h}^{\prime }}:G_{\mathbb{Q}}\rightarrow \operatorname{GL}_{4}(\mathbb{T}_{h}^{\prime })$ such that $\operatorname{Tr}\unicode[STIX]{x1D70C}_{T_{h}^{\prime }}=\mathbb{T}_{\mathbb{T}_{h}}$ . By Proposition 2.5, for every prime $\ell$ not dividing $Np$ we have

(6) $$\begin{eqnarray}\operatorname{Tr}(T_{\mathbb{T}_{h}})(\operatorname{Frob}_{\ell })=\unicode[STIX]{x1D713}_{2}(T_{\ell ,2}^{(2)})|_{{\mathcal{D}}_{2,B_{h}}^{M,h}}.\end{eqnarray}$$

In particular, $\operatorname{Tr}(T_{\mathbb{T}_{h}})(\operatorname{Frob}_{\ell })$ is an element of $\mathbb{T}_{h}$ . As $\mathbb{T}_{h}$ is complete, Chebotarev’s theorem implies that $T_{\mathbb{T}_{h}}(g)$ is an element of $\mathbb{T}_{h}$ for every $g\in G_{\mathbb{Q}}$ . By a theorem of Carayol [Reference CarayolCar94, Théorème 1] there exists a representation $\unicode[STIX]{x1D70C}_{\mathbb{T}_{h}}:G_{\mathbb{Q}}\rightarrow \operatorname{GL}_{4}(\mathbb{T}_{h})$ that is isomorphic to $\unicode[STIX]{x1D70C}_{\mathbb{T}_{h}}$ over  $\mathbb{T}_{h}^{\prime }$ .

The morphism $\unicode[STIX]{x1D703}:\mathbb{T}_{h}\rightarrow \mathbb{I}^{\circ }$ induces a morphism $\operatorname{GL}_{4}(\mathbb{T}_{h})\rightarrow \operatorname{GL}_{4}(\mathbb{I}^{\circ })$ that we still denote by  $\unicode[STIX]{x1D703}$ . Let $\unicode[STIX]{x1D70C}_{\mathbb{I}^{\circ }}:G_{\mathbb{Q}}\rightarrow \operatorname{GL}_{4}(\mathbb{I}^{\circ })$ be the representation defined by $\unicode[STIX]{x1D70C}_{\mathbb{I}^{\circ }}=\unicode[STIX]{x1D703}\circ \,\unicode[STIX]{x1D70C}_{\mathbb{T}_{h}}$ . Recall that we set $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D703}}=\unicode[STIX]{x1D703}(\unicode[STIX]{x1D713}_{2}|_{{\mathcal{D}}_{2,B_{h}}^{M,h}}):{\mathcal{H}}_{2}^{M}\rightarrow \mathbb{I}^{\circ }$ . Let

$$\begin{eqnarray}\mathbb{I}_{\operatorname{Tr}}^{\circ }=\unicode[STIX]{x1D6EC}_{h}[\{\operatorname{Tr}(\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D703}}(g))\}\text{}_{g\in G_{\mathbb{Q}}}].\end{eqnarray}$$

As $\unicode[STIX]{x1D6EC}_{h}\subset \mathbb{I}_{\operatorname{Tr}}^{\circ }\subset \mathbb{I}^{\circ }$ , the ring $\mathbb{I}_{\operatorname{Tr}}^{\circ }$ is a finite $\unicode[STIX]{x1D6EC}_{h}$ -algebra. In particular, $\mathbb{I}_{\operatorname{Tr}}^{\circ }$ is complete. We keep our usual notation for the reduction modulo an ideal $\mathfrak{P}$ of $\mathbb{I}_{\operatorname{Tr}}^{\circ }$ . We say that a point $\mathfrak{P}$ of $\operatorname{Spec}\mathbb{I}_{\operatorname{Tr}}^{\circ }$ is classical if it lies under a classical point of $\operatorname{Spec}\mathbb{I}^{\circ }$ .

By Proposition 2.5 we have $P_{\operatorname{char}}(\operatorname{Tr}(\unicode[STIX]{x1D70C}_{\mathbb{I}^{\circ }}))(\operatorname{Frob}_{\ell })=\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D703}}(P_{\operatorname{min}}(t_{\ell ,2}^{(2)};X))$ , so we deduce that $\mathbb{I}_{\operatorname{Tr}}^{\circ }=\unicode[STIX]{x1D6EC}_{h}[\{\operatorname{Tr}(\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D703}}(g))\}\text{}_{g\in G_{\mathbb{Q}}}]$ . As the traces of $\unicode[STIX]{x1D70C}_{\mathbb{I}^{\circ }}$ belong to $\mathbb{I}_{\operatorname{Tr}}^{\circ }$ , another application of Carayol’s theorem [Reference CarayolCar94, Théorème 1] provides us with a representation

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D703}}:G_{\mathbb{Q}}\rightarrow \operatorname{GL}_{4}(\mathbb{I}_{\operatorname{Tr}}^{\circ })\end{eqnarray}$$

that is isomorphic to $\unicode[STIX]{x1D70C}_{\mathbb{I}^{\circ }}$ over $\mathbb{I}^{\circ }$ . Thanks to the following lemma we can attach to $\unicode[STIX]{x1D703}$ a symplectic representation.

Thanks to the lemma, up to replacing $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D703}}$ by a conjugate representation, we can suppose that it takes values in $\operatorname{GSp}_{4}(\mathbb{I}_{\operatorname{Tr}}^{\circ })$ . We call $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D703}}:G_{\mathbb{Q}}\rightarrow \operatorname{GSp}_{4}(\mathbb{I}_{\operatorname{Tr}}^{\circ })$ the Galois representation associated with the family $\unicode[STIX]{x1D703}:\mathbb{T}_{h}\rightarrow \mathbb{I}_{\operatorname{Tr}}^{\circ }$ . In the following, we work mainly with this representation, so we denote it simply by $\unicode[STIX]{x1D70C}$ . We write $\mathbb{F}$ for the residue field of $\mathbb{I}_{\operatorname{Tr}}^{\circ }$ and $\overline{\unicode[STIX]{x1D70C}}:G_{\mathbb{Q}}\rightarrow \operatorname{GSp}_{4}(\mathbb{F})$ for the residual representation associated with $\unicode[STIX]{x1D70C}$ .

Remark 4.8. Let $\unicode[STIX]{x1D700}$ be the Nebentypus of the family $\unicode[STIX]{x1D703}$ . By interpolating the determinants of the classical specializations of $\unicode[STIX]{x1D70C}$ , we obtain

$$\begin{eqnarray}\det \unicode[STIX]{x1D70C}(g)=\unicode[STIX]{x1D700}(g)(u^{-3}(1+T_{1})(1+T_{2}))^{2\log (\unicode[STIX]{x1D712}(g))/\!\log (u)}\in \unicode[STIX]{x1D6EC}_{2,h}\end{eqnarray}$$

for every $g\in G_{\mathbb{Q}}$ .

5.1 Lifting conjugate self-twists from classical points to families

Keep the notation as above. Let $P_{\text{}\underline{k}}\subset \unicode[STIX]{x1D6EC}_{h}$ be any non-critical arithmetic prime, as in Definition 4.5. The representation $\unicode[STIX]{x1D70C}$ can be reduced modulo $P_{\text{}\underline{k}}\mathbb{I}_{\operatorname{Tr}}^{\circ }$ to a representation $\unicode[STIX]{x1D70C}_{P_{\text{}\underline{k}}}:G_{\mathbb{Q}}\rightarrow \operatorname{GSp}_{4}(\mathbb{I}_{\operatorname{Tr}}^{\circ }/P_{\text{}\underline{k}}\mathbb{I}_{\operatorname{Tr}}^{\circ })$ . Let $\widetilde{\unicode[STIX]{x1D70E}}\in \unicode[STIX]{x1D6E4}$ and let $\widetilde{\unicode[STIX]{x1D702}}:G_{\mathbb{Q}}\rightarrow (\mathbb{I}_{\operatorname{Tr}}^{\circ })^{\times }$ be the character associated with $\widetilde{\unicode[STIX]{x1D70E}}$ . The automorphism $\widetilde{\unicode[STIX]{x1D70E}}$ induces a ring automorphism $\widetilde{\unicode[STIX]{x1D70E}}_{P_{\text{}\underline{k}}}$ of $\mathbb{I}_{\operatorname{Tr}}^{\circ }/P_{\text{}\underline{k}}\mathbb{I}_{\operatorname{Tr}}^{\circ }$ . The character $\widetilde{\unicode[STIX]{x1D702}}:G_{\mathbb{Q}}\rightarrow \mathbb{I}_{\operatorname{Tr}}^{\circ }$ induces a character $\widetilde{\unicode[STIX]{x1D702}}_{P_{\text{}\underline{k}}}:G_{\mathbb{Q}}\rightarrow (\mathbb{I}_{\operatorname{Tr}}^{\circ }/P_{\text{}\underline{k}}\mathbb{I}_{\operatorname{Tr}}^{\circ })^{\times }$ , satisfying

(8) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{P_{\text{}\underline{k}}}^{\widetilde{\unicode[STIX]{x1D70E}}_{P_{\text{}\underline{k}}}}\cong \widetilde{\unicode[STIX]{x1D702}}_{P_{\text{}\underline{k}}}\otimes \unicode[STIX]{x1D70C}_{P_{\text{}\underline{k}}}.\end{eqnarray}$$

As $P_{\text{}\underline{k}}$ is non-critical, $\mathbb{I}^{\circ }$ is étale over $\unicode[STIX]{x1D6EC}_{h}$ at $P_{\text{}\underline{k}}$ , so $\mathbb{I}_{\operatorname{Tr}}^{\circ }$ is also étale over $\unicode[STIX]{x1D6EC}_{h}$ at $P_{\text{}\underline{k}}$ . In particular, $P_{\text{}\underline{k}}$ is a product of distinct primes in $\mathbb{I}_{\operatorname{Tr}}^{\circ }$ ; denote them by $\mathfrak{P}_{1},\mathfrak{P}_{2},\ldots ,\mathfrak{P}_{d}$ . As $\widetilde{\unicode[STIX]{x1D70E}}_{P_{\text{}\underline{k}}}$ is an automorphism of $\mathbb{I}_{\operatorname{Tr}}^{\circ }/P_{\text{}\underline{k}}\mathbb{I}_{\operatorname{Tr}}^{\circ }\cong \prod _{i=1}^{d}\mathbb{I}_{\operatorname{Tr}}^{\circ }/\mathfrak{P}_{i}$ , there is a permutation $s$ of the set $\{1,2,\ldots ,d\}$ and isomorphisms $\widetilde{\unicode[STIX]{x1D70E}}_{\mathfrak{P}_{i}}:\mathbb{I}_{\operatorname{Tr}}^{\circ }/\mathfrak{P}_{i}\rightarrow \mathbb{I}_{\operatorname{Tr}}^{\circ }/\mathfrak{P}_{s(i)}$ for $i=1,2,\ldots ,d$ such that $\widetilde{\unicode[STIX]{x1D70E}}|_{\mathbb{I}_{\operatorname{Tr}}^{\circ }/\mathfrak{P}_{i}}$ factors through $\widetilde{\unicode[STIX]{x1D70E}}_{\mathfrak{P}_{i}}$ . The character $\widetilde{\unicode[STIX]{x1D702}}_{\widetilde{\unicode[STIX]{x1D70E}}_{P_{\text{}\underline{k}}}}$ can be written as a product $\prod _{i=1}^{d}\widetilde{\unicode[STIX]{x1D702}}_{\mathfrak{P}_{i}}$ for some characters $\widetilde{\unicode[STIX]{x1D702}}_{\mathfrak{P}_{i}}:G_{\mathbb{Q}}\rightarrow (\mathbb{I}_{\operatorname{Tr}}^{\circ }/\mathfrak{P}_{i})^{\times }$ . From the equivalence (8), we deduce that

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{\mathfrak{P}_{i}}^{\widetilde{\unicode[STIX]{x1D70E}}_{\mathfrak{P}_{i}}}\cong \widetilde{\unicode[STIX]{x1D702}}_{\mathfrak{P}_{s(i)}}\otimes \unicode[STIX]{x1D70C}_{\mathfrak{P}_{s(i)}}.\end{eqnarray}$$

The goal of this subsection is to prove that if we are given, for a single value of $i$ , data $s(i)$ , $\widetilde{\unicode[STIX]{x1D70E}}_{\mathfrak{P}_{i}}$ and $\widetilde{\unicode[STIX]{x1D702}}_{\mathfrak{P}_{i}}$ satisfying the isomorphism above, then there exists an element of $\unicode[STIX]{x1D6E4}$ giving rise to $\widetilde{\unicode[STIX]{x1D70E}}_{\mathfrak{P}_{i}}$ and $\widetilde{\unicode[STIX]{x1D702}}_{\mathfrak{P}_{s_{i}}}$ via reduction modulo $P_{\text{}\underline{k}}$ . This result is an analogue of [Reference LangLan16, Theorem 3.1]. We state it precisely in the following proposition.

Proposition 5.4. Let $i,j\in \{1,2,\ldots ,d\}$ . Let $\unicode[STIX]{x1D70E}:\mathbb{I}_{\operatorname{Tr}}^{\circ }/\mathfrak{P}_{i}\rightarrow \mathbb{I}_{\operatorname{Tr}}^{\circ }/\mathfrak{P}_{j}$ be a ring isomorphism and let $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70E}}:G_{\mathbb{Q}}\rightarrow (\mathbb{I}_{\operatorname{Tr}}^{\circ }/\mathfrak{P}_{j})^{\times }$ be a character satisfying

(9) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{\mathfrak{P}_{i}}^{\unicode[STIX]{x1D70E}}\cong \unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70E}}\otimes \unicode[STIX]{x1D70C}_{\mathfrak{P}_{j}}.\end{eqnarray}$$

Then there exists $\widetilde{\unicode[STIX]{x1D70E}}\in \unicode[STIX]{x1D6E4}$ with associated character $\widetilde{\unicode[STIX]{x1D702}}:G_{\mathbb{Q}}\rightarrow (\mathbb{I}_{\operatorname{Tr}}^{\circ })^{\times }$ such that, via the construction of the previous paragraph, $s(i)=j$ , $\widetilde{\unicode[STIX]{x1D70E}}_{\mathfrak{P}_{i}}=\unicode[STIX]{x1D70E}$ and $\widetilde{\unicode[STIX]{x1D702}}_{\mathfrak{P}_{j}}=\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D70E}}$ .

To prove the proposition, we first lift $\unicode[STIX]{x1D70E}$ to an automorphism $\unicode[STIX]{x1D6F4}$ of a deformation ring for $\overline{\unicode[STIX]{x1D70C}}$ and then we show that $\unicode[STIX]{x1D6F4}$ descends to a conjugate self-twist for $\unicode[STIX]{x1D70C}$ . This strategy is the same as that of the proof of [Reference LangLan16, Theorem 3.1], but there are various complications that have to be taken care of. We refer to [Reference ContiCon16, §4.4] for the technical lemmas that generalize Lang’s result, and we report here only the core elements of the proof.

Before proving Proposition 5.4 we give a corollary. Let $\mathfrak{P}\in \{\mathfrak{P}_{1},\mathfrak{P}_{2},\ldots ,\mathfrak{P}_{d}\}$ . Let $\unicode[STIX]{x1D70C}_{\mathfrak{P}}:G_{\mathbb{Q}}\rightarrow \operatorname{GSp}_{4}(\mathbb{I}_{\operatorname{Tr}}^{\circ }/\mathfrak{P})$ be the reduction of $\unicode[STIX]{x1D70C}$ modulo $\mathfrak{P}$ and let $\unicode[STIX]{x1D6E4}_{\mathfrak{P}}$ be the group of conjugate self-twists for $\unicode[STIX]{x1D70C}_{\mathfrak{P}}$ over $\mathbb{Z}_{p}$ . Let $\unicode[STIX]{x1D6E4}(\mathfrak{P})=\{\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6E4}\,|\,\unicode[STIX]{x1D70E}(\mathfrak{P})=\mathfrak{P}\}$ . The reduction of the elements of $\unicode[STIX]{x1D6E4}(\mathfrak{P})$ modulo $\mathfrak{P}$ defines a morphism of groups $\unicode[STIX]{x1D6E4}(\mathfrak{P})\rightarrow \unicode[STIX]{x1D6E4}_{\mathfrak{P}}$ . Choosing $\mathfrak{P}_{i}=\mathfrak{P}_{j}=\mathfrak{P}$ in Proposition 5.4 gives the following result.