On adjoint Bloch–Kato Selmer groups for $\textrm{GSp}_{2g}$

Abstract

We study the adjoint Bloch–Kato Selmer groups attached to a classical point in the cuspidal eigenvariety associated with $\textrm{GSp}_{2g}$. Our strategy is based on the study of families of Galois representations on the eigenvariety, which is inspired by the book of J. Bellaiche and G. Chenevier.

Résumé

On étudie de group de Selmer adjoint défini par Bloch et Kato attachés à un point classique dans la variété de Hecke cuspidale pour $\textrm{GSp}_{2g}$. Notre stratégie passe par l’étude de familles de représentations de Galois sur la variété de Hecke cuspidale, qui est inspirée par le livre de J. Bellaïche et G. Chenevier.

1 Introduction

In this paper, we study the adjoint Bloch–Kato Selmer groups attached to a classical point in the cuspidal eigenvariety ${{\,\mathrm{\mathcal {E}}\,}}_0$ associated with ${{\,\textrm{GSp}\,}}_{2g}$. Our strategy is based on the study of families of Galois representations on ${{\,\mathrm{\mathcal {E}}\,}}_0$, which is inspired by the book of Bellaiche and Chenevier [2].

1.1 An overview

Fix a prime number p and a positive integer $g\in {{\,\mathrm{{\textbf {Z}}}\,}}_{> 0}$. Let $X({{\,\mathrm{{\textbf {C}}}\,}})$ be the Siegel modular variety of a fixed tame level structure away from p; and let $X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}})$ be the Siegel modular variety over $X({{\,\mathrm{{\textbf {C}}}\,}})$ with an extra Iwahori level at p. We let N be the product of primes that divide the level of $X({{\,\mathrm{{\textbf {C}}}\,}})$.

On $X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}})$, one can consider the overconvergent parabolic cohomology groups $H_{{{\,\textrm{par}\,}}, \kappa }^{{{\,\textrm{tol}\,}}}$. Following the formalism in [16], one can use $H_{{{\,\textrm{par}\,}}, \kappa }^{{{\,\textrm{tol}\,}}}$ to construct the (reduced equidimensional) cuspidal eigenvariety ${{\,\mathrm{\mathcal {E}}\,}}_0$, parametrising finite-slope families of eigenclasses in the overconvergent parabolic cohomology groups. See Sects. 2.3 and 2.4 for a review.

Given a point ${{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {E}}\,}}_0$ whose weight is a dominant algebraic weight and whose slope is small enough, it is predicted by R. Langlands that there is a continuous Galois representation¹

$$\begin{aligned} \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}} \xrightarrow {\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}} {{\,\textrm{GSpin}\,}}_{2g+1}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p) \xrightarrow { {{\,\textrm{spin}\,}}} {{\,\textrm{GL}\,}}_{2^g}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p^1) \end{aligned}$$

whose characteristic polynomials of the Frobenii away from Np are equal to the Hecke polynomials away from Np. Here, ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}}$ denotes the absolute Galois group of ${{\,\mathrm{{\textbf {Q}}}\,}}$.

Let ${{\,\textrm{ad}\,}}^0 \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}$ be the (trace-0) adjoint representation of $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}$ and consider the adjoint Bloch–Kato Selmer group $H_{f}^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0 \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})$. We have the following conjecture of S. Bloch and K. Kato:

Conjecture 1

(Bloch–Kato conjecture)

• (i) The order of vanishing of the adjoint L-function $L({{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}, s)$ at $s =1$ is equal to the dimension of $H_{f}^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0 \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})$.

• (ii) The adjoint Bloch–Kato Selmer group $H_{f}^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0 \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})$ vanishes.

The aim of this paper is to show that, under certain assumptions, $H_{f}^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0 \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})$ does vanish. In particular, the following natural hypotheses are assumed to achieve our goal:

• Hypothesis 1: Roughly speaking, this hypothesis states that the aforementioned philosophy of Langlands holds true.

• Hypothesis 2: Roughly speaking, this hypothesis ensures that there exists a real finite extension L of ${{\,\mathrm{{\textbf {Q}}}\,}}$ and a generic cuspidal automorphic representation ${{\,\textrm{GL}\,}}_{2^g}({{\,\mathrm{{\textbf {A}}}\,}}_L)$ whose associated Galois representation coincide with $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{L}}$, where ${{\,\mathrm{{\textbf {A}}}\,}}_L$ is the ring of adeles of L and ${{\,\textrm{Gal}\,}}_L$ is the absolute Galois group of L.

• Hypothesis 3: This is a technical hypothesis, which ensures us to obtain a ${{\,\textrm{GSpin}\,}}_{2g+1}$-valued Galois representation with coefficients in the local eigenalgebra of ${{\,\mathrm{{\varvec{{x}}}}\,}}$ and that the chosen tame $\Gamma ^{(p)}$ implies a particular ramification type of this Galois representation at bad primes.

Theorem

(Corollary 3.26) Let ${{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {E}}\,}}_0$ whose weight is a dominant algebraic weight and whose slope is small enough. Suppose the following assumptions hold:²

• (I) Standard assumptions:

  • The point ${{\,\mathrm{{\varvec{{x}}}}\,}}$ corresponds to a p-stabilisation of an eigenclass of tame level (see Sects. 3.2 and 3.3 for more discussion).

  • Hypothesis 1 holds so that we get a ${{\,\textrm{GSpin}\,}}_{2g+1}$-valued Galois representation $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}$ attached to ${{\,\mathrm{{\varvec{{x}}}}\,}}$. We write $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}} := {{\,\textrm{spin}\,}}\circ \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}$ be the associated ${{\,\textrm{GL}\,}}_{2^g}$-valued Galois representation.

• (II) Technical assumption: Hypothesis 3 hold.

• (III) Assumptions used in the strategy of [2]:

  • The restriction $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ admits a refinement ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ that satisfies (REG) and (NCR) (see Sect. 3.1 for definitions of ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}$, (REG) and (NCR)).

  • The restriction $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ is not isomorphic to its twist by the p-adic cyclotomic character.

• (IV) Assumptions to apply [25]:

  • Hypothesis 2 holds.

  • The cuspidal automorphic representation $\pi _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ of ${{\,\textrm{GL}\,}}_{2^g}({{\,\mathrm{{\textbf {A}}}\,}}_L)$ ensured by Hypothesis 2 is regular algebraic and polarised (see, for example, [4, §2.1]).

  • The image $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}({{\,\textrm{Gal}\,}}_{L(\zeta _{p^{\infty }})})$ is enormous (see [25, Definition 2.27]).

Then

• (i) The adjoint Bloch–Kato Selmer group $H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})$ associated with $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}$ vanishes.

• (ii) There is an ‘infinitesimal $R=T$ theorem’ locally at ${{\,\mathrm{{\varvec{{x}}}}\,}}$.

Strategy of the proof We now summarise the strategy to achieve the statement:

Step 1. Using the standard assumptions in the theorem and Proposition 3.13, we construct a refined family of Galois representations $({{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}, {{\,\textrm{Det}\,}}^{{{\,\textrm{univ}\,}}}, {{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit }^{{{\,\textrm{cl}\,}}}, \{\alpha _i: i=1, \ldots , 2^g\}, \{F_i: i=1, \ldots , 2^g\})$ in Theorem 3.16.

Step 2. Following the strategy in [2] and using the assumption that $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ admits a refinement ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}$, we define global deformation problems ${{\,\mathrm{\mathscr {D}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{spin}\,}}}$ and ${{\,\mathrm{\mathscr {D}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{spin}\,}}}$. It is standard in Galois deformation theory that these two functors are pro-representable by complete noetherian local rings by $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{univ}\,}}}$ and $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{spin}\,}}}$ respectively.

Step 3. By applying a theorem of Bellaïche–Chenevier to the refined family of Galois representations in Step 1 and combining Hypothesis 3 and (III) in the theorem, we deduce in Proposition 3.25 the following statements:

• (i) There exists a canonical ring homomorphism $R^{{{\,\textrm{univ}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}\rightarrow {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$, where ${{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ is the local eigenalgebra at ${{\,\mathrm{{\varvec{{x}}}}\,}}$.

• (ii) If the adjoint Bloch–Kato Selmer group $H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})$ vanishes, then the canonical map in $R^{{{\,\textrm{univ}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}\rightarrow {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ is an isomorphism (an ‘infinitesimal $R=T$ theorem’).

Step 4. To conclude the result, the assumption (IV) and [25, Theorem 5.3] imply that $H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}) = 0$. The desired assertions then follow. $\square$

We close this introduction with the remark that one can also deduce the vanishing of the adjoint Bloch–Kato Selmer group without assuming (IV) in the theorem above but with two other (probably strong) assumptions (see Corollary 3.27). Such a statement shall allow one to obtain a (conjectural) link between the p-adic adjoint L-function $L^{{{\,\textrm{adj}\,}}}$ defined in [35] and the adjoint Bloch–Kato Selmer group (see Remark 3.28). This is, in fact, the original motivation of our study in this paper.

1.2 Conventions

Throughout this paper, we fix the following:

• $g\in {{\,\mathrm{{\textbf {Z}}}\,}}_{\ge 1}$.

• For any prime number $\ell$, we fix once and forever an algebraic closure $\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_{\ell }$ of ${{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }$ and an algebraic isomorphism ${{\,\mathrm{{\textbf {C}}}\,}}_{\ell }\simeq {{\,\mathrm{{\textbf {C}}}\,}}$, where ${{\,\mathrm{{\textbf {C}}}\,}}_{\ell }$ is the $\ell$-adic completion of $\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_{\ell }$. We write ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}$ for the absolute Galois group ${{\,\textrm{Gal}\,}}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_{\ell }/{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell })$. We also fix the $\ell$-adic absolute value on ${{\,\mathrm{{\textbf {C}}}\,}}_{\ell }$ so that $|\ell |=\ell ^{-1}$.

• We also fix an algebraic closure $\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}$ of ${{\,\mathrm{{\textbf {Q}}}\,}}$ and embeddings $\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_{\ell } \hookleftarrow \overline{{{\,\mathrm{{\textbf {Q}}}\,}}} \hookrightarrow {{\,\mathrm{{\textbf {C}}}\,}}$, which is compatible with the chosen isomorphisms ${{\,\mathrm{{\textbf {C}}}\,}}_{\ell } \simeq {{\,\mathrm{{\textbf {C}}}\,}}$. We analogously write ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}}$ for the absolute group ${{\,\textrm{Gal}\,}}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}/{{\,\mathrm{{\textbf {Q}}}\,}})$ and identify ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}$ as a (decomposition) subgroup of ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}}$.

• We fix an odd prime number $p\in {{\,\mathrm{{\textbf {Z}}}\,}}_{> 0}$.

• For $n\in {{\,\mathrm{{\textbf {Z}}}\,}}_{\ge 1}$ and any set R, we denote by $M_n(R)$ the set of n by n matrices with coefficients in R.

• The transpose of a matrix ${{\,\mathrm{\varvec{\alpha }}\,}}$ is denoted by ${{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\alpha }}\,}}$.

• For any $n\in {{\,\mathrm{{\textbf {Z}}}\,}}_{\ge 1}$, we denote by $\mathbb {1}_n$ the $n\times n$ identity matrix and denote by $\breve{\mathbb {1}}_n$ the $n\times n$ anti-diagonal matrix whose non-zero entries are 1; i.e.,

  

2 Preliminaries

In this section, we recall some preliminaries. In particular, after setting up the notations in Sect. 2.1 and recalling the Siegel modular varieties in Sect. 2.2, we briefly review the construction of the overconvergent parabolic cohomology groups and the construction of the cuspidal eigenvariety in Sects. 2.3 and 2.4.

2.1 Algebraic and p-adic groups

Let ${{\,\mathrm{{\textbf {V}}}\,}}={{\,\mathrm{{\textbf {V}}}\,}}_{{{\,\mathrm{{\textbf {Z}}}\,}}}$ be the free ${{\,\mathrm{{\textbf {Z}}}\,}}$-module ${{\,\mathrm{{\textbf {Z}}}\,}}^{2g}$ of rank 2g. By viewing elements in ${{\,\mathrm{{\textbf {V}}}\,}}$ as column vectors, we equip ${{\,\mathrm{{\textbf {V}}}\,}}$ with the symplectic pairing

$$\begin{aligned} {{\,\mathrm{{\textbf {V}}}\,}}\times {{\,\mathrm{{\textbf {V}}}\,}}\rightarrow {{\,\mathrm{{\textbf {Z}}}\,}}, \quad (v, v')\mapsto {{\,\mathrm{^{\texttt {t}}\!}\,}}v\begin{pmatrix} &{} \quad -\breve{\mathbb {1}}_g\\ \breve{\mathbb {1}}_g\end{pmatrix} v'. \end{aligned}$$

(1)

The algebraic group ${{\,\textrm{GSp}\,}}_{2g}$ (over ${{\,\mathrm{{\textbf {Z}}}\,}}$) is then defined to be the group that preserves this symplectic pairing up to units. More precisely, for any ring R,

$$\begin{aligned} {{\,\textrm{GSp}\,}}_{2g}(R)= & {} \left\{ {{\,\mathrm{\varvec{\gamma }}\,}}\in {{\,\textrm{GL}\,}}_{2g}(R): {{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}\begin{pmatrix} &{}\quad -\breve{\mathbb {1}}_g\\ \breve{\mathbb {1}}_g\end{pmatrix} {{\,\mathrm{\varvec{\gamma }}\,}}\right. \\= & {} \left. \varsigma ({{\,\mathrm{\varvec{\gamma }}\,}}) \begin{pmatrix} &{} \quad -\breve{\mathbb {1}}_g\\ \breve{\mathbb {1}}_g\end{pmatrix} \text { for some } \varsigma ({{\,\mathrm{\varvec{\gamma }}\,}})\in R^\times \right\} . \end{aligned}$$

Equivalently, for any ${{\,\mathrm{\varvec{\gamma }}\,}}=\begin{pmatrix}{{\,\mathrm{\varvec{\gamma }}\,}}_a &{}\quad {{\,\mathrm{\varvec{\gamma }}\,}}_b\\ {{\,\mathrm{\varvec{\gamma }}\,}}_c &{}\quad {{\,\mathrm{\varvec{\gamma }}\,}}_d\end{pmatrix}\in {{\,\textrm{GL}\,}}_{2g}$, ${{\,\mathrm{\varvec{\gamma }}\,}}\in {{\,\textrm{GSp}\,}}_{2g}$ if and only if

$$\begin{aligned} {{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}_a\breve{\mathbb {1}}_g{{\,\mathrm{\varvec{\gamma }}\,}}_c={{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}_c\breve{\mathbb {1}}_g{{\,\mathrm{\varvec{\gamma }}\,}}_a, \quad {{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}_b\breve{\mathbb {1}}_g{{\,\mathrm{\varvec{\gamma }}\,}}_d={{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}_d\breve{\mathbb {1}}_g{{\,\mathrm{\varvec{\gamma }}\,}}_b, \text { and }{{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}_a\breve{\mathbb {1}}_g{{\,\mathrm{\varvec{\gamma }}\,}}_d-{{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}_c\breve{\mathbb {1}}_g{{\,\mathrm{\varvec{\gamma }}\,}}_b=\varsigma ({{\,\mathrm{\varvec{\gamma }}\,}})\breve{\mathbb {1}}_g \end{aligned}$$

for some $\varsigma ({{\,\mathrm{\varvec{\gamma }}\,}})\in {{\,\mathrm{\mathbb {G}}\,}}_m$. One can easily check that ${{\,\textrm{GSp}\,}}_{2g}$ is stable under transpose. Thus, the above conditions are also equivalent to

$$\begin{aligned} {{\,\mathrm{\varvec{\gamma }}\,}}_a\breve{\mathbb {1}}_g{{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}_b={{\,\mathrm{\varvec{\gamma }}\,}}_b\breve{\mathbb {1}}_g{{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}_a, \quad {{\,\mathrm{\varvec{\gamma }}\,}}_c\breve{\mathbb {1}}_g{{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}_d={{\,\mathrm{\varvec{\gamma }}\,}}_d\breve{\mathbb {1}}_g{{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}_c,\quad \text {and }{{\,\mathrm{\varvec{\gamma }}\,}}_a\breve{\mathbb {1}}_g{{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}_d-{{\,\mathrm{\varvec{\gamma }}\,}}_b\breve{\mathbb {1}}_g{{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}_c=\varsigma ({{\,\mathrm{\varvec{\gamma }}\,}})\breve{\mathbb {1}}_g \end{aligned}$$

for some $\varsigma ({{\,\mathrm{\varvec{\gamma }}\,}})\in {{\,\mathrm{\mathbb {G}}\,}}_m$.

We shall be also considering the following algebraic and p-adic subgroups of ${{\,\textrm{GL}\,}}_g$ and ${{\,\textrm{GSp}\,}}_{2g}$:

• We consider the upper triangular Borel subgroups

  $$\begin{aligned} B_{{{\,\textrm{GL}\,}}_g}&:= \hbox { the Borel subgroup of upper triangular matrices in}\ {{\,\textrm{GL}\,}}_g\\ B_{{{\,\textrm{GSp}\,}}_{2g}}&:= \text {the Borel subgroup of upper triangular matrices in} {{\,\textrm{GSp}\,}}_{2g}. \end{aligned}$$

  The reason why we are able to consider the upper triangular Borel subgroup for ${{\,\textrm{GSp}\,}}_{2g}$ is because of the choice of the pairing in (1).

• The corresponding unipotent radicals are

  $$\begin{aligned} U_{{{\,\textrm{GL}\,}}_g}&:= \hbox { the upper triangular } g\times g \hbox {matrices whose diagonal entries are all } 1\\ U_{{{\,\textrm{GSp}\,}}_{2g}}&:= \hbox { the upper triangular } 2g\times 2g \hbox {matrices in } {{\,\textrm{GSp}\,}}_{2g} \hbox { whose diagonal}\\&\quad \hbox {entries are all }1. \end{aligned}$$

  Consequently, the maximal tori for both algebraic groups are the tori of diagonal matrices. The Levi decomposition then yields

  $$\begin{aligned} B_{{{\,\textrm{GL}\,}}_g}=U_{{{\,\textrm{GL}\,}}_g}T_{{{\,\textrm{GL}\,}}_g}\quad \text {and}\quad B_{{{\,\textrm{GSp}\,}}_{2g}}=U_{{{\,\textrm{GSp}\,}}_{2g}}T_{{{\,\textrm{GSp}\,}}_{2g}}. \end{aligned}$$

  Moreover, we denote by $U_{{{\,\textrm{GL}\,}}_g}^{{{\,\textrm{opp}\,}}}$ and $U_{{{\,\textrm{GSp}\,}}_{2g}}^{{{\,\textrm{opp}\,}}}$ the opposite unipotent radical of $U_{{{\,\textrm{GL}\,}}_g}$ and $U_{{{\,\textrm{GSp}\,}}_{2g}}$ respectively.

• To simplify the notation, for any $s\in {{\,\mathrm{{\textbf {Z}}}\,}}_{\ge 0}$, we write

  $$\begin{aligned} T_{{{\,\textrm{GL}\,}}_g, s}&:= \left\{ \begin{array}{ll} T_{{{\,\textrm{GL}\,}}_g}({{\,\mathrm{{\textbf {Z}}}\,}}_p), &{}\quad s=0 \\ \ker (T_{{{\,\textrm{GL}\,}}_g}({{\,\mathrm{{\textbf {Z}}}\,}}_p)\rightarrow T_{{{\,\textrm{GL}\,}}_g}({{\,\mathrm{{\textbf {Z}}}\,}}/p^s{{\,\mathrm{{\textbf {Z}}}\,}})), &{}\quad s>0 \end{array}\right. \\ T_{{{\,\textrm{GSp}\,}}_{2g}, s}&:= \left\{ \begin{array}{ll} T_{{{\,\textrm{GSp}\,}}_{2g}}({{\,\mathrm{{\textbf {Z}}}\,}}_p), &{}\quad s=0 \\ \ker (T_{{{\,\textrm{GSp}\,}}_{2g}}({{\,\mathrm{{\textbf {Z}}}\,}}_p)\rightarrow T_{{{\,\textrm{GSp}\,}}_{2g}}({{\,\mathrm{{\textbf {Z}}}\,}}/p^s{{\,\mathrm{{\textbf {Z}}}\,}})), &{}\quad s>0 \end{array}\right. \\ U_{{{\,\textrm{GL}\,}}_g, s}&:= \left\{ \begin{array}{ll} U_{{{\,\textrm{GL}\,}}_g}({{\,\mathrm{{\textbf {Z}}}\,}}_p), &{}\quad s=0 \\ \ker (U_{{{\,\textrm{GL}\,}}_g}({{\,\mathrm{{\textbf {Z}}}\,}}_p)\rightarrow U_{{{\,\textrm{GL}\,}}_g}({{\,\mathrm{{\textbf {Z}}}\,}}/p^s{{\,\mathrm{{\textbf {Z}}}\,}})), &{}\quad s>0 \end{array}\right. \\ U_{{{\,\textrm{GSp}\,}}_{2g}, s}&:= \left\{ \begin{array}{ll} U_{{{\,\textrm{GSp}\,}}_{2g}}({{\,\mathrm{{\textbf {Z}}}\,}}_p), &{}\quad s=0 \\ \ker (U_{{{\,\textrm{GSp}\,}}_{2g}}({{\,\mathrm{{\textbf {Z}}}\,}}_p)\rightarrow U_{{{\,\textrm{GSp}\,}}_{2g}}({{\,\mathrm{{\textbf {Z}}}\,}}/p^s{{\,\mathrm{{\textbf {Z}}}\,}})), &{}\quad s>0 \end{array}\right. . \end{aligned}$$

  The above maps are all reduction maps.

• The Iwahori subgroups are

  $$\begin{aligned} {{\,\textrm{Iw}\,}}_{{{\,\textrm{GL}\,}}_g}&: = \hbox { the preimage of} B_{{{\,\textrm{GL}\,}}_g}({{\,\mathrm{{\textbf {F}}\!}\,}}_p) \hbox {under the reduction map} {{\,\textrm{GL}\,}}_g({{\,\mathrm{{\textbf {Z}}}\,}}_p)\rightarrow {{\,\textrm{GL}\,}}_g({{\,\mathrm{{\textbf {F}}\!}\,}}_p)\\ {{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}&:= \hbox { the preimage of} B_{{{\,\textrm{GSp}\,}}_{2g}}({{\,\mathrm{{\textbf {F}}\!}\,}}_p) \hbox {under the reduction map} {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_p)\rightarrow {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {F}}\!}\,}}_p). \end{aligned}$$

  We have Iwahori decompositions for ${{\,\textrm{Iw}\,}}_{{{\,\textrm{GL}\,}}_g}$ and ${{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$

  $$\begin{aligned} {{\,\textrm{Iw}\,}}_{{{\,\textrm{GL}\,}}_g} = U_{{{\,\textrm{GL}\,}}_g, 1}^{{{\,\textrm{opp}\,}}} T_{{{\,\textrm{GL}\,}}_g, 0}U_{{{\,\textrm{GL}\,}}_g, 0} \quad \text { and }\quad {{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}} = U_{{{\,\textrm{GSp}\,}}_{2g}, 1}^{{{\,\textrm{opp}\,}}} T_{{{\,\textrm{GSp}\,}}_{2g}, 0} U_{{{\,\textrm{GSp}\,}}_{2g}, 0}. \end{aligned}$$

For later purposes, we also recall the Weyl groups of ${{\,\textrm{GSp}\,}}_{2g}$ and $H:={{\,\textrm{GL}\,}}_g\times {{\,\mathrm{\mathbb {G}}\,}}_m$ from [13, Chapter VI, §5]. Here, we view H as an algebraic subgroup of ${{\,\textrm{GSp}\,}}_{2g}$ via the embedding

$$\begin{aligned} H = {{\,\textrm{GL}\,}}_g \times {{\,\mathrm{\mathbb {G}}\,}}_m \hookrightarrow {{\,\textrm{GSp}\,}}_{2g}, \quad ({{\,\mathrm{\varvec{\gamma }}\,}}, {{\,\mathrm{\varvec{\upsilon }}\,}})\mapsto \begin{pmatrix} {{\,\mathrm{\varvec{\gamma }}\,}}&{} \\ {} &{}\quad {{\,\mathrm{\varvec{\upsilon }}\,}}\breve{\mathbb {1}}_g {{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}^{-1}\breve{\mathbb {1}}_g\end{pmatrix}. \end{aligned}$$

Consider the character group ${{\,\mathrm{\mathbb {X}}\,}}= {{\,\textrm{Hom}\,}}(T_{{{\,\textrm{GSp}\,}}_{2g}}, {{\,\mathrm{\mathbb {G}}\,}}_m)$. We have the following isomorphism

$$\begin{aligned} {{\,\mathrm{{\textbf {Z}}}\,}}^{g+1} \xrightarrow {\sim } {{\,\mathrm{\mathbb {X}}\,}}, \quad (k_1, \ldots , k_g; k_0)\mapsto \left( {{\,\textrm{diag}\,}}({{\,\mathrm{\varvec{\tau }}\,}}_1, \ldots , {{\,\mathrm{\varvec{\tau }}\,}}_g, {{\,\mathrm{\varvec{\tau }}\,}}_0{{\,\mathrm{\varvec{\tau }}\,}}_g^{-1}, \ldots , {{\,\mathrm{\varvec{\tau }}\,}}_0{{\,\mathrm{\varvec{\tau }}\,}}_1^{-1})\mapsto \prod _{i=0}^{g} {{\,\mathrm{\varvec{\tau }}\,}}_i^{k_i}\right) . \end{aligned}$$

Let $x_1, \ldots , x_g, x_0$ be the basis of ${{\,\mathrm{\mathbb {X}}\,}}$ that corresponds to the standard basis on ${{\,\mathrm{{\textbf {Z}}}\,}}^{g+1}$. Note that ${{\,\mathrm{\mathbb {X}}\,}}$ can also be viewed as the character group of the maximal torus $T_H=T_{{{\,\textrm{GL}\,}}_g}\times {{\,\mathrm{\mathbb {G}}\,}}_m$ of H via the isomorphisms $T_{{{\,\textrm{GSp}\,}}_{2g}}\simeq {{\,\mathrm{\mathbb {G}}\,}}_m^{g+1}\simeq T_{{{\,\textrm{GL}\,}}_g}\times {{\,\mathrm{\mathbb {G}}\,}}_m = T_H$.

Under the above choices of the maximal tori, we can describe the root systems of ${{\,\textrm{GSp}\,}}_{2g}$ and H explicitly

$$\begin{aligned} \Phi _{{{\,\textrm{GSp}\,}}_{2g}}&= \{\pm (x_i-x_j),\,\, \pm (x_i+x_j-x_0),\,\, \pm (2x_t-x_0): 1\le i<j\le g, 1\le t\le g\}\\ \Phi _H&= \{\pm (x_i-x_j),\,\, \pm x_g,\,\, \pm x_0: 1\le i<j\le g\}. \end{aligned}$$

Moreover, the choices of the Borel subgroups yield the description of the positive roots

$$\begin{aligned} \Phi _{{{\,\textrm{GSp}\,}}_{2g}}^+&= \{x_i-x_j,\,\, x_i+x_j-x_0,\,\, 2x_t-x_0: 1\le i<j\le g, 1\le t\le g\}\\ \Phi _H^+&= \{x_i-x_j: 1\le i<j\le g\}(=\Phi _H\cap \Phi _{{{\,\textrm{GSp}\,}}_{2g}}^+). \end{aligned}$$

The Weyl groups of ${{\,\textrm{GSp}\,}}_{2g}$ and H are defined as

$$\begin{aligned} {{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}} := N_{{{\,\textrm{GSp}\,}}_{2g}}(T_{{{\,\textrm{GSp}\,}}_{2g}})/T_{{{\,\textrm{GSp}\,}}_{2g}} \quad \text { and }\quad {{\,\textrm{Weyl}\,}}_H := N_H(T_H)/T_H, \end{aligned}$$

where $N_{{{\,\textrm{GSp}\,}}_{2g}}(T_{{{\,\textrm{GSp}\,}}_{2g}})$ (resp. $N_{H}(T_H)$) is the group of normalisers of $T_{{{\,\textrm{GSp}\,}}_{2g}}$ (resp. $T_H$) in ${{\,\textrm{GSp}\,}}_{2g}$ (resp. H). They can also be described explicitly as follows.

• We can identify ${{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$ with ${{\,\mathrm{\varvec{\Sigma }}\,}}_g < imes ({{\,\mathrm{{\textbf {Z}}}\,}}/2{{\,\mathrm{{\textbf {Z}}}\,}})^g$, where ${{\,\mathrm{\varvec{\Sigma }}\,}}_g$ denotes the permutation group on g letters. For any ${{\,\mathrm{\varvec{\tau }}\,}}= {{\,\textrm{diag}\,}}({{\,\mathrm{\varvec{\tau }}\,}}_1, \ldots , {{\,\mathrm{\varvec{\tau }}\,}}_g, {{\,\mathrm{\varvec{\tau }}\,}}_0{{\,\mathrm{\varvec{\tau }}\,}}_g^{-1}, \ldots , {{\,\mathrm{\varvec{\tau }}\,}}_0{{\,\mathrm{\varvec{\tau }}\,}}_1^{-1})\in T_{{{\,\textrm{GSp}\,}}_{2g}}$, the actions of ${{\,\mathrm{\varvec{\Sigma }}\,}}_g$ and $({{\,\mathrm{{\textbf {Z}}}\,}}/2{{\,\mathrm{{\textbf {Z}}}\,}})^{g}$ are given as

  • (i) ${{\,\mathrm{\varvec{\Sigma }}\,}}_g$ permutes ${{\,\mathrm{\varvec{\tau }}\,}}_1, \ldots , {{\,\mathrm{\varvec{\tau }}\,}}_g$,

  • (ii) the element $(\underbrace{0, \ldots , 0}_{i-1}, 1, 0, \ldots , 0)\in ({{\,\mathrm{{\textbf {Z}}}\,}}/2{{\,\mathrm{{\textbf {Z}}}\,}})^{g}$ maps ${{\,\mathrm{\varvec{\tau }}\,}}$ to

    $$\begin{aligned} {{\,\textrm{diag}\,}}({{\,\mathrm{\varvec{\tau }}\,}}_1, \ldots , {{\,\mathrm{\varvec{\tau }}\,}}_{i-1}, {{\,\mathrm{\varvec{\tau }}\,}}_0{{\,\mathrm{\varvec{\tau }}\,}}_i^{-1}, {{\,\mathrm{\varvec{\tau }}\,}}_{i+1}, \ldots ,{{\,\mathrm{\varvec{\tau }}\,}}_g, {{\,\mathrm{\varvec{\tau }}\,}}_0{{\,\mathrm{\varvec{\tau }}\,}}_g^{-1}, \ldots , {{\,\mathrm{\varvec{\tau }}\,}}_0{{\,\mathrm{\varvec{\tau }}\,}}_{i+1}^{-1}, {{\,\mathrm{\varvec{\tau }}\,}}_i, {{\,\mathrm{\varvec{\tau }}\,}}_0{{\,\mathrm{\varvec{\tau }}\,}}_{i-1}^{-1}, \ldots , {{\,\mathrm{\varvec{\tau }}\,}}_0{{\,\mathrm{\varvec{\tau }}\,}}_{1}^{-1}). \end{aligned}$$

• We can identify ${{\,\textrm{Weyl}\,}}_H$ with ${{\,\mathrm{\varvec{\Sigma }}\,}}_g$, whose action on $T_H$ is defined as the action of ${{\,\mathrm{\varvec{\Sigma }}\,}}_g$ on $T_{{{\,\textrm{GSp}\,}}_{2g}}$.

The actions of the Weyl groups on the maximal tori then induce actions on the root systems $\Phi _{{{\,\textrm{GSp}\,}}_{2g}}$ and $\Phi _H$. Following [13, Chapter VI, §5], let

$$\begin{aligned} {{\,\textrm{Weyl}\,}}^H : = \{ x\in {{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}: x(\Phi _{{{\,\textrm{GSp}\,}}_{2g}}^+)\supset \Phi _H^+\} \subset {{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}. \end{aligned}$$

The subset ${{\,\textrm{Weyl}\,}}^H$ consequently gives a system of representatives of the quotient ${{\,\textrm{Weyl}\,}}_H\backslash {{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$.

2.2 The Siegel modular varieties

Let $\Gamma ^{(p)} \subset {{\,\textrm{GSp}\,}}_{2g}(\widehat{{{\,\mathrm{{\textbf {Z}}}\,}}})$ be a neat open compact subgroup such that $\Gamma ^{(p)} = \prod _{\ell : \text { prime}} \Gamma ^{(p)}_{\ell }$, where each $\Gamma ^{(p)}_{\ell }$ is an open compact subgroup of ${{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell })$. Let ${{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}:= \{\ell \text { prime number}: \Gamma _{\ell }^{(p)} \subsetneq {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell })\} \cup \{p\}$. We shall assume this union is a disjoint union and write $N:= \prod _{\ell \in {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}\smallsetminus \{p\}}\ell$ ( and so $p\not \mid N$).

Fix a primitive N-th roots of unity $\zeta _N\in \overline{{{\,\mathrm{{\textbf {Q}}}\,}}}\subset \overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p$. Let be the category of locally noetherian schemes over ${{\,\mathrm{{\textbf {Z}}}\,}}_p[\zeta _N]$. Consider the functor

Assume that $\Gamma ^{(p)}$ is chosen so that the above functor is representable by a scheme $X_{{{\,\mathrm{{\textbf {Z}}}\,}}_p[\zeta _N]}$. Denote by $X={X_{{{\,\mathrm{{\textbf {C}}}\,}}_p}}$ the base change of $X_{{{\,\mathrm{{\textbf {Z}}}\,}}_p[\zeta _N]}$ to ${{\,\mathrm{{\textbf {C}}}\,}}_p$.

Example 2.1

Suppose $\Gamma ^{(p)} = \Gamma (N) := \ker ({{\,\textrm{GSp}\,}}_{2g}(\widehat{{{\,\mathrm{{\textbf {Z}}}\,}}})\rightarrow {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}/N{{\,\mathrm{{\textbf {Z}}}\,}}))$ for N large enough, then $\Gamma (N)$ defines the level structure asking for symplectic isomorphisms,

$$\begin{aligned} \psi _N: A[N] \xrightarrow {\sim } ({{\,\mathrm{{\textbf {Z}}}\,}}/N{{\,\mathrm{{\textbf {Z}}}\,}})^{2g}, \end{aligned}$$

i.e., isomorphisms that preserve symplectic pairings on both sides up to units, where we consider the Weil pairing on the left-hand side and the symplectic pairing induced by (1) on the right-hand side. $\square$

Fix a primitive p-th root of unity $\zeta _p\in \overline{{{\,\mathrm{{\textbf {Q}}}\,}}}\subset \overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p$, we also consider the scheme $X_{{{\,\textrm{Iw}\,}}, {{\,\mathrm{{\textbf {Q}}}\,}}_p[\zeta _N, \zeta _p]}$, parametrising tuples

$$\begin{aligned} (A, \lambda , \psi _N, {{\,\textrm{Fil}\,}}_{\bullet }), \end{aligned}$$

where $(A, \lambda , \psi _N)\in X_{{{\,\mathrm{{\textbf {Q}}}\,}}_p[\zeta _N, \zeta _p]}:=X_{{{\,\mathrm{{\textbf {Z}}}\,}}_p[\zeta _N]}\times _{{{\,\mathrm{{\textbf {Z}}}\,}}_p[\zeta _N]}{{\,\textrm{Spec}\,}}{{\,\mathrm{{\textbf {Q}}}\,}}_p[\zeta _p, \zeta _N]$ and ${{\,\textrm{Fil}\,}}_{\bullet }$ is a full filtration of A[p] such that ${{\,\textrm{Fil}\,}}_{\bullet }^{\perp } = {{\,\textrm{Fil}\,}}_{2g-\bullet }$ (with respect to the Weil pairing). Similarly, we write $X_{{{\,\textrm{Iw}\,}}}=X_{{{\,\textrm{Iw}\,}}, {{\,\mathrm{{\textbf {C}}}\,}}_p}$ the base change of $X_{{{\,\textrm{Iw}\,}}, {{\,\mathrm{{\textbf {Q}}}\,}}_p[\zeta _N, \zeta _p]}$ to ${{\,\mathrm{{\textbf {C}}}\,}}_p$. Obviously, we have the natural forgetful morphism

$$\begin{aligned} X_{{{\,\textrm{Iw}\,}}} \rightarrow X, \quad (A, \lambda , \psi _N, {{\,\textrm{Fil}\,}}_{\bullet }) \mapsto (A, \lambda , \psi _N). \end{aligned}$$

This morphism is obviously an étale morphism.

With respect to the fixed isomorphism ${{\,\mathrm{{\textbf {C}}}\,}}\simeq {{\,\mathrm{{\textbf {C}}}\,}}_p$ in the beginning, we can consider the ${{\,\mathrm{{\textbf {C}}}\,}}$-points $X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}})$. The space $X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}})$ can then be identified with the locally symmetric space

$$\begin{aligned} X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}) = {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Q}}}\,}})\backslash {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {A}}}\,}}_f)\times {{\,\mathrm{\mathbb {H}}\,}}_g/{{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}\Gamma ^{(p)}, \end{aligned}$$

where ${{\,\mathrm{\mathbb {H}}\,}}_g$ is the (disjoint) union of the Siegel upper- and lower-half plane and ${{\,\mathrm{{\textbf {A}}}\,}}_f$ is the ring of finite adèles of ${{\,\mathrm{{\textbf {Q}}}\,}}$. It is well-known that the dimension of the Siegel modular variety $X_{{{\,\textrm{Iw}\,}}}$ (as well as X) is $n_0= g(g+1)/2$.

Let M be a left ${{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_p)$-module (over some commutative ring). Thus, M also admits a left action of ${{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$ via restriction. By equipping M with a trivial action by $\Gamma ^{(p)}$, the module M naturally defines a local system on $X({{\,\mathrm{{\textbf {C}}}\,}})$ and $X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}})$ as explained in [1, §2.2] (see also [16, §2.1]). One can then consider the (Betti) cohomology groups $H^t(X({{\,\mathrm{{\textbf {C}}}\,}}), M)$ and $H^t(X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}), M)$ (resp. compactly supported cohomology groups $H^t_c(X({{\,\mathrm{{\textbf {C}}}\,}}), M)$ and $H^t_c(X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}), M)$) with coefficients in M. Then, there are natural morphisms

$$\begin{aligned} \Lambda _p: H^t(X({{\,\mathrm{{\textbf {C}}}\,}}), M)&\rightarrow H^t(X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}), M) \end{aligned}$$

(2)

$$\begin{aligned} \Lambda _p: H_c^t(X({{\,\mathrm{{\textbf {C}}}\,}}), M)&\rightarrow H_c^t(X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}), M) \end{aligned}$$

(3)

induced by the forgetful morphism.

2.3 The overconvergent parabolic cohomology groups

Define

$$\begin{aligned} {{\,\mathrm{{\textbf {T}}}\,}}_0 = \left\{ ({{\,\mathrm{\varvec{\gamma }}\,}}, {{\,\mathrm{\varvec{\upsilon }}\,}})\in {{\,\textrm{Iw}\,}}_{{{\,\textrm{GL}\,}}_g} \times M_g(p{{\,\mathrm{{\textbf {Z}}}\,}}_p) : {{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}\breve{\mathbb {1}}_g{{\,\mathrm{\varvec{\upsilon }}\,}}= {{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\upsilon }}\,}}\breve{\mathbb {1}}_g{{\,\mathrm{\varvec{\gamma }}\,}}\right\} . \end{aligned}$$

Elements in ${{\,\mathrm{{\textbf {T}}}\,}}_0$ can be viewed as the left $(2g\times g)$-columns of matrices in ${{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$ as explained in [35, §2.2]. Then ${{\,\mathrm{{\textbf {T}}}\,}}_0$ admits a right action of $B_{{{\,\textrm{GL}\,}}_g, 0}$ given by the right multiplication and a left action of $\Xi : = \begin{pmatrix} {{\,\textrm{Iw}\,}}_{{{\,\textrm{GL}\,}}_g} &{}\quad M_g({{\,\mathrm{{\textbf {Z}}}\,}}_p)\\ M_g(p{{\,\mathrm{{\textbf {Z}}}\,}}_p) &{}\quad M_g({{\,\mathrm{{\textbf {Z}}}\,}}_p)\end{pmatrix}\cap {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Q}}}\,}}_p)$ by the left multiplication. Moreover, ${{\,\mathrm{{\textbf {T}}}\,}}_0$ admits a special subset

$$\begin{aligned} {{\,\mathrm{{\textbf {T}}}\,}}_{00} : = \left\{ ({{\,\mathrm{\varvec{\gamma }}\,}}, {{\,\mathrm{\varvec{\upsilon }}\,}})\in {{\,\mathrm{{\textbf {T}}}\,}}_0: {{\,\mathrm{\varvec{\gamma }}\,}}\in U_{{{\,\textrm{GL}\,}}_g, 1}^{{{\,\textrm{opp}\,}}} \right\} , \end{aligned}$$

which can be identified with $U_{{{\,\textrm{GSp}\,}}_{2g}, 1}^{{{\,\textrm{opp}\,}}}$ via

$$\begin{aligned} {{\,\mathrm{{\textbf {T}}}\,}}_{00} \xrightarrow {\sim } U_{{{\,\textrm{GSp}\,}}_{2g}, 1}^{{{\,\textrm{opp}\,}}}, \quad ({{\,\mathrm{\varvec{\gamma }}\,}}, {{\,\mathrm{\varvec{\upsilon }}\,}})\mapsto \begin{pmatrix} {{\,\mathrm{\varvec{\gamma }}\,}}&{} \\ {{\,\mathrm{\varvec{\upsilon }}\,}}&{}\quad \breve{\mathbb {1}}_g {{\,\mathrm{^{\texttt {t}}\!}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}^{-1}\breve{\mathbb {1}}_g\end{pmatrix}. \end{aligned}$$

For any affinoid ${{\,\mathrm{{\textbf {Q}}}\,}}_p$-algebra R and any p-adic weight (i.e., continuous character) $\kappa : T_{{{\,\textrm{GL}\,}}_g, 0} \rightarrow R^\times$ and any $s\in {{\,\mathrm{{\textbf {Z}}}\,}}_{>0}$, we consider the s-locally analytic functions

$$\begin{aligned} A_{\kappa }^s({{\,\mathrm{{\textbf {T}}}\,}}_0, R) := \left\{ \phi : {{\,\mathrm{{\textbf {T}}}\,}}_0\rightarrow R : \begin{array}{l} \phi ({{\,\mathrm{\varvec{\gamma }}\,}}{{\,\mathrm{\varvec{\beta }}\,}}, {{\,\mathrm{\varvec{\upsilon }}\,}}{{\,\mathrm{\varvec{\beta }}\,}}) = \kappa ({{\,\mathrm{\varvec{\beta }}\,}})\phi ({{\,\mathrm{\varvec{\gamma }}\,}}, {{\,\mathrm{\varvec{\upsilon }}\,}})\,\,\forall (({{\,\mathrm{\varvec{\gamma }}\,}}, {{\,\mathrm{\varvec{\upsilon }}\,}}), {{\,\mathrm{\varvec{\beta }}\,}})\in {{\,\mathrm{{\textbf {T}}}\,}}_0\times B_{{{\,\textrm{GL}\,}}_g, 0} \\ \phi |_{{{\,\mathrm{{\textbf {T}}}\,}}_{00}} \text { is }s\text {-locally analytic} \end{array}\right\} . \end{aligned}$$

Here, we extend $\kappa$ to a function on $B_{{{\,\textrm{GL}\,}}_g, 0}$ by setting $\kappa |_{U_{{{\,\textrm{GL}\,}}_g, 0}} = 1$ and the ‘s-locally analytic’ condition is in the sense of [16, §2 Definition] (after identifying ${{\,\mathrm{{\textbf {T}}}\,}}_{00}$ with $U_{{{\,\textrm{GSp}\,}}_{2g}, 1}^{{{\,\textrm{opp}\,}}}$). One sees immediately that we have a natural inclusion $A_{\kappa }^s({{\,\mathrm{{\textbf {T}}}\,}}_0, R)\subset A_{\kappa }^{s+1}({{\,\mathrm{{\textbf {T}}}\,}}_0, R)$.

The s-locally analytic distributions are then defined to be

$$\begin{aligned} D_{\kappa }^s({{\,\mathrm{{\textbf {T}}}\,}}_0, R) : = {{\,\textrm{Hom}\,}}_{R}^{{{\,\textrm{cts}\,}}}(A_{\kappa }^{s}({{\,\mathrm{{\textbf {T}}}\,}}_0, R), R). \end{aligned}$$

The natural inclusion $A_{\kappa }^s({{\,\mathrm{{\textbf {T}}}\,}}_0, R)\subset A_{\kappa }^{s+1}({{\,\mathrm{{\textbf {T}}}\,}}_0, R)$ then yields a natural projection $D_{\kappa }^{s+1}({{\,\mathrm{{\textbf {T}}}\,}}_0, R)\rightarrow D_{\kappa }^s({{\,\mathrm{{\textbf {T}}}\,}}_0, R)$. Consequently, we define

$$\begin{aligned}&A_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R) := \varinjlim _{s} A_{\kappa }^s({{\,\mathrm{{\textbf {T}}}\,}}_0, R)\\&D_{\kappa }^{\dagger } ({{\,\mathrm{{\textbf {T}}}\,}}_0, R) := \varprojlim _{s} D_{\kappa }^s({{\,\mathrm{{\textbf {T}}}\,}}_0, R). \end{aligned}$$

We call elements of these two modules overconvergent functions and overconvergent distributions respectively. It is also obvious that $D_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R)$ is the continuous dual of $A_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R)$.

Observe that ${{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}\subset \Xi$, thus $D_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R)$ is naturally a left ${{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$-module. Equip it with a trivial action by $\Gamma ^{(p)}$, we can consequently consider the cohomology groups (resp. compactly supported cohomology groups) $H^t(X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}), D_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R))$ (resp. $H_c^t(X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}), D_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R))$) for any $0\le t\le 2n_0$. The overconvergent parabolic cohomology group is then defined to be

$$\begin{aligned} H_{{{\,\textrm{par}\,}}}^t(X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}), D_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R)):= {{\,\textrm{image}\,}}\left( H_c^t(X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}), D_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R)) \rightarrow H^t(X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}), D_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R))\right) , \end{aligned}$$

where the map is the natural map from the compactly supported cohomology group to the cohomology group. In what follows, we will be considering the total overconvergent parabolic cohomology group

$$\begin{aligned} H_{{{\,\textrm{par}\,}}, \kappa }^{{{\,\textrm{tol}\,}}} := \oplus _{t=0}^{2n_0} H_{{{\,\textrm{par}\,}}}^t(X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}), D_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R)). \end{aligned}$$

2.4 Hecke operators and the (reduced equidimensional) cuspidal eigenvariety

Let $\ell$ be a prime number that does not divide pN. We consider the set of double cosets

$$\begin{aligned}\Upsilon _{\ell }:=\{[{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell }){{\,\mathrm{\varvec{\delta }}\,}}{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell })]: {{\,\mathrm{\varvec{\delta }}\,}}\in {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Q}}}\,}}_q)\cap M_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_q)\}. \end{aligned}$$

For any fixed ${{\,\mathrm{\varvec{\delta }}\,}}$, we have the coset decomposition

$$\begin{aligned}{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell }){{\,\mathrm{\varvec{\delta }}\,}}{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell })=\sqcup _{j} {{\,\mathrm{\varvec{\delta }}\,}}_j{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell }) \end{aligned}$$

for finitely many ${{\,\mathrm{\varvec{\delta }}\,}}_j\in {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Q}}}\,}}_{\ell })\cap M_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell })$. By letting ${{\,\mathrm{\varvec{\delta }}\,}}_j$’s act trivially on $D_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R)$, we have a left action of the double coset $[{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell }){{\,\mathrm{\varvec{\delta }}\,}}{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell })]$ on the cochain complex $C^{\bullet }_{\kappa }$ (resp. $C_{c, \kappa }^{\bullet }$) that computes the cohomology groups $H^{t}(X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}), D_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R))$ (resp. the compactly supported cohomology groups $H_c^t(X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}), D_{\kappa }^{\dagger }({{\,\mathrm{{\textbf {T}}}\,}}_0, R))$) by

$$\begin{aligned}{}[{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_q){{\,\mathrm{\varvec{\delta }}\,}}{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_q)]\cdot \sigma = \sum _{j}{{\,\mathrm{\varvec{\delta }}\,}}_j\cdot \sigma \end{aligned}$$

for any $\sigma \in C^{\bullet }_{\kappa }$ (resp. $C_{c, \kappa }^{\bullet }$). Then the Hecke algebra at $\ell$ (over ${{\,\mathrm{{\textbf {Z}}}\,}}_{p}$) is defined to be ${{\,\mathrm{\mathbb {T}}\,}}_{\ell }={{\,\mathrm{\mathbb {T}}\,}}_{{\ell }, {{\,\mathrm{{\textbf {Z}}}\,}}_p}={{\,\mathrm{{\textbf {Z}}}\,}}_p[\Upsilon _{\ell }]$. Consequently, the unramified Hecke algebra is

$$\begin{aligned} {{\,\mathrm{\mathbb {T}}\,}}^p:= \otimes _{\ell \not \mid pN}{{\,\mathrm{\mathbb {T}}\,}}_{\ell }. \end{aligned}$$

We specify out a special element ${{\,\mathrm{{\textbf {t}}}\,}}_{\ell , 0} = {{\,\textrm{diag}\,}}(\mathbb {1}_g, \ell \mathbb {1}_g)\in {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Q}}}\,}}_{\ell })\cap M_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell })$. For any $x\in {{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$, denote by $T_{\ell , 0}^x$ the Hecke operator defined by the double coset $[{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell }) (x\cdot {{\,\mathrm{{\textbf {t}}}\,}}_{\ell , 0}) {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_{\ell })]$. Following [15, §3], we define the Hecke polynomial at $\ell$ to be

$$\begin{aligned} P_{{{\,\textrm{Hecke}\,}}, \ell }(Y) : = \prod _{x\in {{\,\textrm{Weyl}\,}}^H}(Y-T_{\ell , 0}^x)\in {{\,\mathrm{\mathbb {T}}\,}}_{\ell }[Y]. \end{aligned}$$

(4)

One sees immediately that this is a polynomial of degree $2^g$.

For Hecke operators at p, consider matrices

$$\begin{aligned} {{\,\mathrm{{\textbf {u}}}\,}}_{p, i} : = \left\{ \begin{array}{ll} \begin{pmatrix}\mathbb {1}_g\\ {} &{}\quad p\mathbb {1}_g\end{pmatrix}, &{}\quad i=0 \\ \\ \begin{pmatrix} \mathbb {1}_{g-i} \\ {} &{}\quad p\mathbb {1}_{i}\\ {} &{} &{}\quad p\mathbb {1}_{i}\\ {} &{} &{} &{}\quad p^2\mathbb {1}_{g-i}\end{pmatrix},&\quad 1\le i\le g-1 \end{array}\right. . \end{aligned}$$

For any $({{\,\mathrm{\varvec{\gamma }}\,}}, {{\,\mathrm{\varvec{\upsilon }}\,}})\in {{\,\mathrm{{\textbf {T}}}\,}}_0$, write $({{\,\mathrm{\varvec{\gamma }}\,}}, {{\,\mathrm{\varvec{\upsilon }}\,}}) = ({{\,\mathrm{\varvec{\gamma }}\,}}_0, {{\,\mathrm{\varvec{\upsilon }}\,}}_0){{\,\mathrm{\varvec{\beta }}\,}}$ for some ${{\,\mathrm{\varvec{\beta }}\,}}\in B_{{{\,\textrm{GL}\,}}_g, 0}^+$ such that ${{\,\mathrm{\varvec{\gamma }}\,}}_0\in U_{{{\,\textrm{GL}\,}}_g, 1}^{{{\,\textrm{opp}\,}}}$. Then, the left action of ${{\,\mathrm{{\textbf {u}}}\,}}_{p,i}$ on ${{\,\mathrm{{\textbf {T}}}\,}}_0$ is defined by the formula

$$\begin{aligned} {{\,\mathrm{{\textbf {u}}}\,}}_{p,i}\cdot ({{\,\mathrm{\varvec{\gamma }}\,}}, {{\,\mathrm{\varvec{\upsilon }}\,}}) = ({{\,\mathrm{{\textbf {u}}}\,}}_{p, i}^{\square }{{\,\mathrm{\varvec{\gamma }}\,}}_0{{\,\mathrm{{\textbf {u}}}\,}}_{p, i}^{\square , -1}, {{\,\mathrm{{\textbf {u}}}\,}}_{p,i}^{\blacksquare }{{\,\mathrm{\varvec{\upsilon }}\,}}_{0}{{\,\mathrm{{\textbf {u}}}\,}}_{p, i}^{\square , -1}){{\,\mathrm{\varvec{\beta }}\,}}, \end{aligned}$$

where we write

$$\begin{aligned} {{\,\mathrm{{\textbf {u}}}\,}}_{p, i} = \begin{pmatrix}{{\,\mathrm{{\textbf {u}}}\,}}_{p, i}^{\square } &{} \\ {} &{}\quad {{\,\mathrm{{\textbf {u}}}\,}}_{p,i}^{\blacksquare }\end{pmatrix}. \end{aligned}$$

On the other hand, we also have a coset decomposition of ${{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}{{\,\mathrm{{\textbf {u}}}\,}}_{p,i}{{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$, given by

$$\begin{aligned} {{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}{{\,\mathrm{{\textbf {u}}}\,}}_{p, i}{{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}=\sqcup _{j}{{\,\mathrm{\varvec{\delta }}\,}}_{i, j}{{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}} \end{aligned}$$

for some ${{\,\mathrm{\varvec{\delta }}\,}}_{i, j}\in {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Q}}}\,}}_p)\cap M_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_p)$; in particular, ${{\,\mathrm{\varvec{\delta }}\,}}_{i,j} = {{\,\mathrm{\varvec{\lambda }}\,}}_{i,j}{{\,\mathrm{{\textbf {u}}}\,}}_{p, i}$ for some ${{\,\mathrm{\varvec{\lambda }}\,}}_{i,j}\in {{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$. Hence, we have the action

$$\begin{aligned}{}[{{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}} {{\,\mathrm{{\textbf {u}}}\,}}_{p,i} {{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}] \cdot \sigma : = \sum _{j}{{\,\mathrm{\varvec{\delta }}\,}}_{i,j}\cdot \sigma = \sum _{j} {{\,\mathrm{\varvec{\lambda }}\,}}_{i,j}\cdot \left( {{\,\mathrm{{\textbf {u}}}\,}}_{p, i}\cdot \sigma \right) \end{aligned}$$

for any $\sigma \in C^{\bullet }_{\kappa }$ (resp. $C_{c, \kappa }^{\bullet }$). We denote by $U_{p,i}$ the Hecke operator defined by the double coset $[{{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}} {{\,\mathrm{{\textbf {u}}}\,}}_{p,i} {{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}]$. Similarly, for any $x\in {{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$, we denote by $U_{p,i}^{x}$ the Hecke operator defined by the double coset $[{{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}} (x\cdot {{\,\mathrm{{\textbf {u}}}\,}}_{p, i}) {{\,\textrm{Iw}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}]$, whose action is similarly defined as above. Then, the Hecke algebra at p is defined to be ${{\,\mathrm{\mathbb {T}}\,}}_{p} = {{\,\mathrm{\mathbb {T}}\,}}_{p, {{\,\mathrm{{\textbf {Z}}}\,}}_p} = {{\,\mathrm{{\textbf {Z}}}\,}}_p[U_{p,i}^x: i=0, 1, \ldots , g-1, w\in {{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}]$. Consequently, the (universal) Hecke algebra is defined to be

$$\begin{aligned} {{\,\mathrm{\mathbb {T}}\,}}:= {{\,\mathrm{\mathbb {T}}\,}}^p\otimes _{{{\,\mathrm{{\textbf {Z}}}\,}}_p} {{\,\mathrm{\mathbb {T}}\,}}_p. \end{aligned}$$

There is a special Hecke operator $U_p\in {{\,\mathrm{\mathbb {T}}\,}}_p$ defined to be

$$\begin{aligned} U_p : = \prod _{i=0}^{g-1}U_{p, i}. \end{aligned}$$

Combining the discussions in [16, §2.2] and [20, §3.2],³ the operator $U_p$ defines a compact operator on $C_{\kappa }^{\bullet }$ (resp., $C_{c, \kappa }^{\bullet }$). Consequently, we consider the slope decomposition on $C_{\kappa }^{\bullet }$ (resp., $C_{c, \kappa }^{\bullet }$) with respect to the action of $U_p$, which allows us to consider the finite slope cohomology groups (resp., compactly supported cohomology groups).

Let ${{\,\mathrm{\mathcal {W}}\,}}= {{\,\textrm{Spa}\,}}({{\,\mathrm{{\textbf {Z}}}\,}}_p{{\,\mathrm{\![\![\!}\,}}T_{{{\,\textrm{GL}\,}}_g,0}{{\,\mathrm{\!]\!]}\,}}, {{\,\mathrm{{\textbf {Z}}}\,}}_p{{\,\mathrm{\![\![\!}\,}}T_{{{\,\textrm{GL}\,}}_g, 0}{{\,\mathrm{\!]\!]}\,}})_{\eta }^{{{\,\textrm{an}\,}}}$ be our weight space, where the superscript ‘$\bullet ^{{{\,\textrm{an}\,}}}$’ means that we are taking the analytic points of the adic space and the subscript ‘$\bullet _\eta$’ means that we are considering the generic fibre of the adic space. The slope decomposition on the cochain complexes $C_{\kappa }^{\bullet }$ then defines a Fredholm surface ${{\,\mathrm{\mathcal {Z}}\,}}$ over ${{\,\mathrm{\mathcal {W}}\,}}$. As the natural map $C_{c, \kappa }^{\bullet } \rightarrow C_{\kappa }^{\bullet }$ is Hecke-equivariant, the finite-slope cohomology groups and finite-slope compactly supported cohomology groups define finite-slope parabolic cohomology groups $H_{{{\,\textrm{par}\,}}, \kappa }^{{{\,\textrm{tol}\,}}, <h}$ (see [35, §3.3]).

For any slope datum $({{\,\mathrm{\mathcal {U}}\,}}, h)$ (see [16, §3.1]; in particular, ${{\,\mathrm{\mathcal {U}}\,}}\subset {{\,\mathrm{\mathcal {W}}\,}}$), denote by $\kappa _{{{\,\mathrm{\mathcal {U}}\,}}}$ the universal weight on ${{\,\mathrm{\mathcal {U}}\,}}$ and define

$$\begin{aligned} {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\textrm{par}\,}}, {{\,\mathrm{\mathcal {U}}\,}}}^{{{\,\textrm{red}\,}}, h}:= {{\,\textrm{image}\,}}\left( {{\,\mathrm{\mathbb {T}}\,}}\rightarrow {{\,\textrm{End}\,}}_{{{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {W}}\,}}}({{\,\mathrm{\mathcal {U}}\,}})}\left( H_{{{\,\textrm{par}\,}}, \kappa _{{{\,\mathrm{\mathcal {U}}\,}}}}^{{{\,\textrm{tol}\,}}, \le h}\right) \right) ^{{{\,\textrm{red}\,}}}, \end{aligned}$$

where the superscript ‘$\bullet ^{{{\,\textrm{red}\,}}}$’ stands for the maximal reduced quotient of the corresponding ring. The algebras ${{\,\mathrm{\mathbb {T}}\,}}_{{{\,\textrm{par}\,}}, {{\,\mathrm{\mathcal {U}}\,}}}^{{{\,\textrm{red}\,}}, h}$ then glue together to a coherent sheaf of ${{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {Z}}\,}}}$-algebras, denoted ${{\,\mathrm{\mathscr {T}}\,}}_{{{\,\textrm{par}\,}}}^{{{\,\textrm{red}\,}}}$. The reduced cuspidal eigenvariety is then defined to be

$$\begin{aligned} {{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{red}\,}}} := {{\,\textrm{Spa}\,}}_{{{\,\mathrm{\mathcal {Z}}\,}}}({{\,\mathrm{\mathscr {T}}\,}}_{{{\,\textrm{par}\,}}}^{{{\,\textrm{red}\,}}}, {{\,\mathrm{\mathscr {T}}\,}}_{{{\,\textrm{par}\,}}}^{{{\,\textrm{red}\,}}, \circ }), \end{aligned}$$

where the sheaf of integral elements ${{\,\mathrm{\mathscr {T}}\,}}_{{{\,\textrm{par}\,}}}^{{{\,\textrm{red}\,}}, \circ }$ is guaranteed by [20, Lemma A.3]. We finally define the (reduced equidimensional) cuspidal eigenvariety

$$\begin{aligned} {{\,\mathrm{\mathcal {E}}\,}}_0 := \text {the equidimensional locus of }{{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{red}\,}}} \end{aligned}$$

The natural map

$$\begin{aligned} {{\,\textrm{wt}\,}}: {{\,\mathrm{\mathcal {E}}\,}}_0 \rightarrow {{\,\mathrm{\mathcal {W}}\,}}\end{aligned}$$

is called the weight map.

Remark 2.2

If we work with the strict Iwahori level as in [35], then ${{\,\mathrm{\mathcal {E}}\,}}_0$ is the reduced and equidimensional part of the $p\ne 0$ locus of the cuspidal eigenvariety considered in loc. cit.. We focus on the reduced cuspidal eigenvariety due to later purposes on families of Galois representations.

3 Families of Galois representations

In this section, we study families of Galois representations on the reduced equidimensional cuspidal eigenvariety ${{\,\mathrm{\mathcal {E}}\,}}_0$. We shall first recall several formalisms about families of Galois representations from [2]. Our main results concerning the Bloch–Kato conjecture are then proven in Sect. 3.5.

3.1 Determinants and families of representations

In this subsection, we recall several terminologies for studying families of Galois representations. Most of the materials presented in this subsection are taken from [2].

Determinants. We briefly recall the notion of ‘determinants’ from [6] and refer the readers to loc. cit. for more detailed discussions. We remark in the beginning that the notion of determinants is used to strengthen the notion of ‘pseudocharacters’ first introduced by R. Taylor in [31] and studied by other mathematicians. We also remark that determinants are equivalent to pseudocharacters in characteristic 0.

Definition 3.1

Let A be a commutative ring and R be an A-algebra (not necessarily commutative).

• (i) For any A-module M, one can view M as a functor from the category of commutative A-algebras to the category of sets, sending B to $M\otimes _A B$. Let M, N be two A-modules. Then an A-polynomial law between M and N is a natural transformation

  $$\begin{aligned} M\otimes _A B \rightarrow N\otimes _A B \end{aligned}$$

  on the category of commutative A-algebras.

• (ii) Let $P: M \rightarrow N$ be an A-polynomial law and $d\in {{\,\mathrm{{\textbf {Z}}}\,}}_{>0}$. We say P is homogeneous of dimension d if for any commutative A-algebra B, any $b\in B$ and any $x\in M\otimes _A B$, we have $P(bx) = b^d P(x)$.

• (iii) Let $P: R\rightarrow A$ be an A-polynomial law. We say P is multiplicative if, for any commutative A-algebra B, $P(1) = 1$ and $P(xy) = P(x)P(y)$ for any $x, y\in R\otimes _A B$.

• (iv) For $d\in {{\,\mathrm{{\textbf {Z}}}\,}}_{>0}$, a d-dimensional A-valued determinant on R is a multiplicative A-polynomial law $D: R\rightarrow A$ which is homogeneous of dimension d.

Example 3.2

Let G be a group and A be any ring. Let $\rho : G \rightarrow {{\,\textrm{GL}\,}}_d(A)$ be a representation of dimension d. Then

$$\begin{aligned} D: A[G] \rightarrow A, \quad G\ni \sigma \mapsto \det \rho (\sigma ) \end{aligned}$$

is an A-valued determinant of dimension d on A[G]. We also say that D is an A-valued determinant of dimension d on G.

Theorem 3.3

([6, Theorem A and Theorem B])

Let G be a group.

• (i) Let k be an algebraically closed field and let $D: k[G] \rightarrow k$ be a determinant of dimension d. Then, there exists a unique (up to isomorphism) semisimple representation $\rho : G \rightarrow {{\,\textrm{GL}\,}}_d(k)$ such that for any $\sigma \in G$, we have

  $$\begin{aligned} \det (1+X\rho (\sigma )) = D(1+X\sigma ) \in k[X]. \end{aligned}$$

  In particular, $\det \rho = D$.

• (ii) Let A be an henselian local ring with algebraically closed residue field k, $D: A[G]\rightarrow A$ be a determinant of dimension d and let $\rho$ be the semisimple representation attached to $D\otimes _A k$ in (i). Suppose $\rho$ is irreducible, then there exists a unique (up to isomorphism) representation $\widetilde{\rho }: G \rightarrow {{\,\textrm{GL}\,}}_d(A)$ such that

  $$\begin{aligned} \det (1+X\widetilde{\rho }(\sigma )) = D(1+X\sigma ) \in A[X] \end{aligned}$$

  for any $\sigma \in G$.

Refinements of crystalline representations. We recall the notion of ‘refinements’ of crystalline representations from [2, §2.4]. Let L be a finite extension of ${{\,\mathrm{{\textbf {Q}}}\,}}_p$ and let V be an n-dimensional L-representation of ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}$. Assume that V is crystalline. Also assume that the crystalline Frobenius $\varphi = \varphi _{{{\,\textrm{cris}\,}}}$ acting on ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(V)$ has all eigenvalues living in $L^\times$.

Definition 3.4

([2, §2.4.1]) A refinement of V is the data of a full $\varphi$-stable L-filtration

$$\begin{aligned} {{\,\mathrm{\mathbb {F}}\,}}_{\bullet } : 0 = {{\,\mathrm{\mathbb {F}}\,}}_{0} \subsetneq {{\,\mathrm{\mathbb {F}}\,}}_1 \subsetneq \cdots \subsetneq {{\,\mathrm{\mathbb {F}}\,}}_{n-1}\subsetneq {{\,\mathrm{\mathbb {F}}\,}}_{n} = {{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(V). \end{aligned}$$

Suppose ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }$ is a refinement of V, one sees immediately that it determines two orderings:

• (Ref 1) An ordering $(\varphi _1, \ldots , \varphi _n)$ of the eigenvalues of $\varphi$ by the formula

  $$\begin{aligned} \det (X - \varphi |_{{{\,\mathrm{\mathbb {F}}\,}}_i}) = \prod _{j=1}^i (X - \varphi _j). \end{aligned}$$

  Notice that if the $\varphi _j$’s are all distinct, then such an ordering of eigenvalues of $\varphi$ conversely determines the refinement.

• (Ref 2) An ordering $(a_1, \ldots , a_n)$ of Hodge–Tate weights of V. More precisely, the jumps of the Hodge filtration of ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(V)$ induced on ${{\,\mathrm{\mathbb {F}}\,}}_i$ are $(a_1, \ldots , a_i)$.

Definition 3.5

([2, Definition 2.4.5]) Suppose the Hodge–Tate weights $a_1< \cdots < a_n$ of V are all distinct. Let ${{\,\mathrm{\mathbb {F}}\,}}$ be a refinement of V and let ${{\,\textrm{Fil}\,}}^{\bullet } {{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(V)$ be the Hodge filtration of ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(V)$. We say ${{\,\mathrm{\mathbb {F}}\,}}$ is non-critical if, for all $1\le i \le n$, we have

$$\begin{aligned} {{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(V) = {{\,\mathrm{\mathbb {F}}\,}}_i \oplus {{\,\textrm{Fil}\,}}^{a_i+1} {{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(V). \end{aligned}$$

Recall the Robba ring

$$\begin{aligned} {{\,\mathrm{\mathcal {R}}\,}}_{L} := \left\{ f(X) = \sum _{i\in {{\,\mathrm{{\textbf {Z}}}\,}}} t_n(X-1)^n\in L{{\,\mathrm{\![\![\!}\,}}X {{\,\mathrm{\!]\!]}\,}}: \begin{array}{c} f(X)\hbox { converges on some annulus of}\ {{\,\mathrm{{\textbf {C}}}\,}}_p \\ \text {of the form }r(f) \le |X-1| \le 1 \end{array}\right\} . \end{aligned}$$

Here the norm $|\cdot |$ is the p-adic norm on ${{\,\mathrm{{\textbf {C}}}\,}}_p$ with the normalisation $|p| = 1/p$. Let $\Gamma = {{\,\mathrm{{\textbf {Z}}}\,}}_p^\times$. The theory of $(\varphi , \Gamma )$-modules yields an equivalence of categories between the category finite-dimensional L-representations of ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}$ and the category of étale $(\varphi , \Gamma )$-modules over ${{\,\mathrm{\mathcal {R}}\,}}_{L}$ (see, for example, [2, §2.2]). In particular, we have a $(\varphi , \Gamma )$-module ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{rig}\,}}}(V)$ over ${{\,\mathrm{\mathcal {R}}\,}}_{L}$ associated with V.

Proposition 3.6

([2, Proposition 2.4.1 and Proposition 2.4.7]) Let ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }$ be a refinement of V.

• (i) Then ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }$ determines a unique filtration ${{\,\textrm{Fil}\,}}_{\bullet } {{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{rig}\,}}}(V)$ of length n, i.e., a triangulation of ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{rig}\,}}}(V)$. Consequently, ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }$ determines a unique collection of continuous characters $\delta _i: {{\,\mathrm{{\textbf {Q}}}\,}}_p^\times \rightarrow L^\times$ via the isomorphism

  $$\begin{aligned} {{\,\textrm{Fil}\,}}_{i}{{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{rig}\,}}}(V)/{{\,\textrm{Fil}\,}}_{i-1}{{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{rig}\,}}}(V) \simeq {{\,\mathrm{\mathcal {R}}\,}}_{L}(\delta _i) \end{aligned}$$

  given by [2, Proposition 2.3.1]. Here, the tuple $\delta = (\delta _1, \ldots , \delta _n)$ is called the parameter of V.

• (ii) Moreover, suppose the Hodge–Tate weight of V are all distinct $h_1< \cdots < h_n$. Then, ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }$ is non-critical if and only if the sequence Hodge–Tate weights $(a_1, \ldots , a_n)$ associated with ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }$ in (Ref 2) is increasing, i.e., $a_i = h_i$ for all $i=1, \ldots , n$.

Remark 3.7

The theory of $(\varphi , \Gamma )$-modules can be worked out for local artinian ${{\,\mathrm{{\textbf {Q}}}\,}}_p$-algebras (see, for example, [2, §2]). Thus, it makes sense to consider the following deformation functors. Let be the category of local artinian ${{\,\mathrm{{\textbf {Q}}}\,}}_p$-algebras whose residue field is isomorphic to L. Then, we define the (local) trianguline deformation functor

We will also denote the above deformation functor by ${{\,\mathrm{\mathscr {D}}\,}}_{V, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }}$ as the triangulation ${{\,\textrm{Fil}\,}}_{\bullet } D_{{{\,\textrm{rig}\,}}}(V)$ is uniquely determined by ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }$. In fact, we will confuse the refinement ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }$ with the triangulation ${{\,\textrm{Fil}\,}}_{\bullet }D_{{{\,\textrm{rig}\,}}}(V)$ in what follows.

Families of representations. Here, we collect some terminologies introduced in [2, §5] that will be needed in the later subsections. Note that the terminology of ‘pseudocharacters’ is used in op. cit. since the notion of ‘determinants’ was not yet discovered. In what follows, we shall adapt everything with the notion of determinants.

Let G be a topological group with a continuous group homomorphism ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}\rightarrow G$, e.g., $G= {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}}$ with the natural inclusion ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}\hookrightarrow {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}}$. Therefore, any (continuous) representation $\rho$ of G induces a (continuous) representation of ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}$, denoted by $\rho |_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$.

By a family of representations, we mean a datum $({{\,\mathrm{\mathcal {X}}\,}}, D)$, where ${{\,\mathrm{\mathcal {X}}\,}}$ is a reduced separated rigid analytic variety (viewed as an adic space) over ${{\,\textrm{Spa}\,}}({{\,\mathrm{{\textbf {Q}}}\,}}_p, {{\,\mathrm{{\textbf {Z}}}\,}}_p)$ and a continuous determinant $D: {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}}({{\,\mathrm{\mathcal {X}}\,}})[G] \rightarrow {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}}({{\,\mathrm{\mathcal {X}}\,}})$. The dimension of this family is understood to be the dimension of the determinant D, denoted by n. For any ${{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}$, let $k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ be the residue field of ${{\,\mathrm{{\varvec{{x}}}}\,}}$, then we have the specialisation

$$\begin{aligned} D|_{{{\,\mathrm{{\varvec{{x}}}}\,}}} : G \xrightarrow {D} {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}}({{\,\mathrm{\mathcal {X}}\,}}) \rightarrow k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}. \end{aligned}$$

(5)

Applying Theorem 3.3 (i), we see that $D|_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ is nothing but the determinant of a (unique up to isomorphism) continuous semisimple representation $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}: G \rightarrow {{\,\textrm{GL}\,}}_n(\overline{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}})$.

Definition 3.8

([2, Definition 4.2.3]) A refined family of representations of dimension n is a datum $({{\,\mathrm{\mathcal {X}}\,}}, D, {{\,\mathrm{\mathcal {Q}}\,}}, \{\alpha _i: i=1, \ldots , n\}, \{F_i: i=1, \ldots , n\}),$ where

• (a) $({{\,\mathrm{\mathcal {X}}\,}}, D)$ is a family of representations of dimension n,

• (b) ${{\,\mathrm{\mathcal {Q}}\,}}\subset {{\,\mathrm{\mathcal {X}}\,}}$ is a Zariski dense subset,

• (c) $\alpha _i\in {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}}({{\,\mathrm{\mathcal {X}}\,}})$ is an analytic function for $i=1, \ldots , n$,

• (d) $F_i\in {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}}({{\,\mathrm{\mathcal {X}}\,}})$ is an analytic function for $i=1, \ldots , n$,

such that

• (i) For every ${{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}$, the Hodge–Tate–Sen weights⁴ for $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ are $\alpha _1({{\,\mathrm{{\varvec{{x}}}}\,}})$, ..., $\alpha _n({{\,\mathrm{{\varvec{{x}}}}\,}})$.

• (ii) For each $y\in {{\,\mathrm{\mathcal {Q}}\,}}$, the representation $\rho _{{{\,\mathrm{{\varvec{{y}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ is crystalline (so that $\alpha _i({{\,\mathrm{{\varvec{{y}}}}\,}})$’s are integers) and we have $\alpha _1({{\,\mathrm{{\varvec{{y}}}}\,}})< \cdots < \alpha _n({{\,\mathrm{{\varvec{{y}}}}\,}})$.

• (iii) For each ${{\,\mathrm{{\varvec{{y}}}}\,}}\in {{\,\mathrm{\mathcal {Q}}\,}}$, the eigenvalues of the crystalline Frobenius $\varphi$ on ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(\rho _{{{\,\mathrm{{\varvec{{y}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}})$ are distinct and are $(p^{\alpha _1({{\,\mathrm{{\varvec{{y}}}}\,}})} F_1({{\,\mathrm{{\varvec{{y}}}}\,}}), \ldots , p^{\alpha _n({{\,\mathrm{{\varvec{{y}}}}\,}})}F_n({{\,\mathrm{{\varvec{{y}}}}\,}}))$.

• (iv) For any $C\in {{\,\mathrm{{\textbf {Z}}}\,}}_{> 0}$, define

  $$\begin{aligned} {{\,\mathrm{\mathcal {Q}}\,}}_{C} := \left\{ {{\,\mathrm{{\varvec{{y}}}}\,}}\in {{\,\mathrm{\mathcal {Q}}\,}}: \begin{array}{l} \alpha _{i+1}({{\,\mathrm{{\varvec{{y}}}}\,}}) - \alpha _{i}({{\,\mathrm{{\varvec{{y}}}}\,}})> C(\alpha _{i}({{\,\mathrm{{\varvec{{y}}}}\,}}) - \alpha _{i-1}({{\,\mathrm{{\varvec{{y}}}}\,}})) \text { for }i=2, \ldots , n-1 \\ \alpha _2({{\,\mathrm{{\varvec{{y}}}}\,}}) - \alpha _1({{\,\mathrm{{\varvec{{y}}}}\,}}) > C \end{array} \right\} . \end{aligned}$$

  We request that ${{\,\mathrm{\mathcal {Q}}\,}}_C$ accumulates at any point of ${{\,\mathrm{\mathcal {Q}}\,}}$ for any C. In other words, for any ${{\,\mathrm{{\varvec{{y}}}}\,}}\in {{\,\mathrm{\mathcal {Q}}\,}}$ and any $C\in {{\,\mathrm{{\textbf {Z}}}\,}}_{>0}$, there is a basis of affinoid neighbourhoods ${{\,\mathrm{\mathcal {U}}\,}}$ of ${{\,\mathrm{{\varvec{{x}}}}\,}}$ such that ${{\,\mathrm{\mathcal {U}}\,}}\cap {{\,\mathrm{\mathcal {Q}}\,}}_C$ is Zariski dense in ${{\,\mathrm{\mathcal {U}}\,}}$.

• (*) For each $i=1, \ldots , n$, there is a continuous character ${{\,\mathrm{{\textbf {Z}}}\,}}_p^\times \rightarrow {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}}({{\,\mathrm{\mathcal {X}}\,}})^\times$ whose derivative at 1 is the map $\alpha _i$ and whose evaluation at any point ${{\,\mathrm{{\varvec{{y}}}}\,}}\in {{\,\mathrm{\mathcal {Q}}\,}}$ is the elevation to the $\alpha _i({{\,\mathrm{{\varvec{{y}}}}\,}})$-th power.

Let $({{\,\mathrm{\mathcal {X}}\,}}, D, {{\,\mathrm{\mathcal {Q}}\,}}, \{\alpha _i: i=1, \ldots , n\}, \{F_i: i=1, \ldots , n\})$ be a refined family of dimension n. We fix a point ${{\,\mathrm{{\varvec{{y}}}}\,}}\in {{\,\mathrm{\mathcal {Q}}\,}}$. Then $\rho _{{{\,\mathrm{{\varvec{{y}}}}\,}}}$ admits a natural refinement ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{y}}}}\,}}}$ given by the ordering of distinct eigenvalues

$$\begin{aligned} (p^{\alpha _1({{\,\mathrm{{\varvec{{y}}}}\,}})} F_1({{\,\mathrm{{\varvec{{y}}}}\,}}), \ldots , p^{\alpha _n({{\,\mathrm{{\varvec{{y}}}}\,}})} F_n({{\,\mathrm{{\varvec{{y}}}}\,}})) \end{aligned}$$

of the crystalline Frobenius acting on ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(\rho _y|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}})$ ( [2, Definition 4.2.4]). We assume that $\rho _{{{\,\mathrm{{\varvec{{y}}}}\,}}}$ is irreducible and it satisfies the following two conditions:

• (REG) The refinement ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{y}}}}\,}}}$ is regular, i.e., for any $i=1, \ldots , n$, $p^{\alpha _1({{\,\mathrm{{\varvec{{y}}}}\,}}) + \cdots + \alpha _i({{\,\mathrm{{\varvec{{y}}}}\,}})}F_1({{\,\mathrm{{\varvec{{y}}}}\,}}) \cdots F_i({{\,\mathrm{{\varvec{{y}}}}\,}})$ is an eigenvalue of the crystalline Frobenius $\varphi$ acting on ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(\wedge ^i {\rho }_{{{\,\mathrm{{\varvec{{y}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}})$ of multiplicity one.

• (NCR) The refinement ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{y}}}}\,}}}$ is non-critical.

Since $\rho _{{{\,\mathrm{{\varvec{{y}}}}\,}}}$ is assumed to be irreducible, Theorem 3.3 (ii) implies that there is a unique continuous representation

$$\begin{aligned} \rho _{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}}: G \rightarrow {{\,\textrm{GL}\,}}_n({{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}}) \end{aligned}$$

such that $\rho _{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}} \otimes _{{{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}}} k_{{{\,\mathrm{{\varvec{{y}}}}\,}}} = \rho _{{{\,\mathrm{{\varvec{{y}}}}\,}}}$ and so $\det \rho _{{{\,\mathrm{{\varvec{{y}}}}\,}}}$ coincides with the composition $G\xrightarrow {D} {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}}({{\,\mathrm{\mathcal {X}}\,}}) \rightarrow {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}}$. Following [2, §4.4], we define a continuous character $\delta _{{{\,\mathrm{{\varvec{{y}}}}\,}}}: {{\,\mathrm{{\textbf {Q}}}\,}}_p^\times \rightarrow ({{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}}^\times )^n$ by setting

$$\begin{aligned} \delta _{{{\,\mathrm{{\varvec{{y}}}}\,}}}(p) = (F_{1, {{\,\mathrm{{\varvec{{y}}}}\,}}}, \ldots , F_{n, {{\,\mathrm{{\varvec{{y}}}}\,}}})\quad \text { and }\quad \delta _{{{\,\mathrm{{\varvec{{y}}}}\,}}}|_{{{\,\mathrm{{\textbf {Z}}}\,}}_p^\times } = (\alpha _{1, {{\,\mathrm{{\varvec{{y}}}}\,}}}^{-1}, \ldots , \alpha _{n, {{\,\mathrm{{\varvec{{y}}}}\,}}}^{-1}), \end{aligned}$$

(6)

where $F_{i, {{\,\mathrm{{\varvec{{y}}}}\,}}}$ and $\alpha _{i, {{\,\mathrm{{\varvec{{y}}}}\,}}}$ are the images of $F_i$ and $\alpha _i$ in ${{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}}$ respectively.

Theorem 3.9

([2, Theorem 4.4.1]) For any ideal ${{\,\mathrm{\mathfrak {I}}\,}}\subsetneq {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}}$ of cofinite length, $\rho _{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}} \otimes _{{{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}}} {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}}/{{\,\mathrm{\mathfrak {I}}\,}}$ is a trianguline deformation of $(\rho _{{{\,\mathrm{{\varvec{{y}}}}\,}}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{y}}}}\,}}})$, i.e., it belongs to ${{\,\mathrm{\mathscr {D}}\,}}_{\rho _{{{\,\mathrm{{\varvec{{y}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{y}}}}\,}}}}({{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}}/{{\,\mathrm{\mathfrak {I}}\,}})$ (defined in Remark 3.7), whose parameter is $\delta _{{{\,\mathrm{{\varvec{{y}}}}\,}}} \otimes {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {X}}\,}}, {{\,\mathrm{{\varvec{{y}}}}\,}}}/{{\,\mathrm{\mathfrak {I}}\,}}$.

3.2 Galois representations for ${{\,\textrm{GSp}\,}}_{2g}$

Given a dominant weight $k = (k_1, \ldots , k_g)\in {{\,\mathrm{{\textbf {Z}}}\,}}_{\ge 0}^g$, recall the ${{\,\textrm{GSp}\,}}_{2g}$-representations

$$\begin{aligned} {{\,\mathrm{{\textbf {V}}}\,}}_{{{\,\textrm{GSp}\,}}_{2g}, k}^{{{\,\textrm{alg}\,}}}&= \left\{ \phi : {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Q}}}\,}}_p)\rightarrow {{\,\mathrm{{\textbf {Q}}}\,}}_p: \begin{array}{l} \phi \text { is a polynomial function } \\ \phi ({{\,\mathrm{\varvec{\gamma }}\,}}{{\,\mathrm{\varvec{\beta }}\,}}) = k({{\,\mathrm{\varvec{\beta }}\,}})\phi ({{\,\mathrm{\varvec{\gamma }}\,}}) \,\, \forall ({{\,\mathrm{\varvec{\gamma }}\,}}, {{\,\mathrm{\varvec{\beta }}\,}})\in {{\,\textrm{GSp}\,}}_{2g} \\ ({{\,\mathrm{{\textbf {Q}}}\,}}_p)\times B_{{{\,\textrm{GSp}\,}}_{2g}}({{\,\mathrm{{\textbf {Q}}}\,}}_p) \end{array}\right\} \\ {{\,\mathrm{{\textbf {V}}}\,}}_{{{\,\textrm{GSp}\,}}_{2g}, k}^{{{\,\textrm{alg}\,}}, \vee }&= {{\,\textrm{Hom}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}({{\,\mathrm{{\textbf {V}}}\,}}_{{{\,\textrm{GSp}\,}}_{2g}, k}^{{{\,\textrm{alg}\,}}}, {{\,\mathrm{{\textbf {Q}}}\,}}_p). \end{aligned}$$

The representation ${{\,\mathrm{{\textbf {V}}}\,}}_{{{\,\textrm{GSp}\,}}_{2g}, k}^{{{\,\textrm{alg}\,}}}$ is equipped with a right ${{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Q}}}\,}}_p)$-action by the formula

$$\begin{aligned} {{\,\mathrm{\varvec{\gamma }}\,}}\cdot \phi ({{\,\mathrm{\varvec{\gamma }}\,}}') = \phi ({{\,\mathrm{\varvec{\gamma }}\,}}{{\,\mathrm{\varvec{\gamma }}\,}}') \end{aligned}$$

for any $\phi \in {{\,\mathrm{{\textbf {V}}}\,}}_{{{\,\textrm{GSp}\,}}_{2g}, k}^{{{\,\textrm{alg}\,}}}$, ${{\,\mathrm{\varvec{\gamma }}\,}}, {{\,\mathrm{\varvec{\gamma }}\,}}'\in {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Q}}}\,}}_p)$. Hence, ${{\,\mathrm{{\textbf {V}}}\,}}_{{{\,\textrm{GSp}\,}}_{2g}, k}^{{{\,\textrm{alg}\,}}, \vee }$ is equipped with a left ${{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Q}}}\,}}_p)$-action and consequently induces a local system on both $X_{{{\,\textrm{Iw}\,}}^+}({{\,\mathrm{{\textbf {C}}}\,}})$ and $X({{\,\mathrm{{\textbf {C}}}\,}})$. We abuse the notation and use the same symbol to denote such local system. In particular, we can consider the parabolic cohomology group

$$\begin{aligned} H_{{{\,\textrm{tame}\,}},{{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}} := \oplus _{t=0}^{2n_0} H_{{{\,\textrm{par}\,}}}^t(X({{\,\mathrm{{\textbf {C}}}\,}}), {{\,\mathrm{{\textbf {V}}}\,}}_{{{\,\textrm{GSp}\,}}_{2g}, k}^{{{\,\textrm{alg}\,}}, \vee }). \end{aligned}$$

Note that the double cosets $[{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_p) (x\cdot {{\,\mathrm{{\textbf {u}}}\,}}_{p,i}){{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_p)]$ acts on $H_{{{\,\textrm{tame}\,}}, {{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}}$ for any $x\in {{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$. We denote by

$$\begin{aligned} {{\,\mathrm{\mathbb {T}}\,}}^{{{\,\textrm{tame}\,}}}&:= {{\,\mathrm{\mathbb {T}}\,}}^p \otimes _{{{\,\mathrm{{\textbf {Z}}}\,}}_p} {{\,\mathrm{{\textbf {Z}}}\,}}_p\big [[{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_p) (x\cdot {{\,\mathrm{{\textbf {u}}}\,}}_{p,i}){{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_p)]:\\&\qquad i=0, 1, \ldots , g-1, x\in {{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}\big ]. \end{aligned}$$

In particular, it makes sense to consider the Hecke polynomial $P_{{{\,\textrm{Hecke}\,}}, p}(Y)$ at p in this case and is defined as in (4).⁵

Hypothesis 1

For any ${{\,\mathrm{\mathbb {T}}\,}}^{{{\,\textrm{tame}\,}}}$-eigenclass $[\mu ]\in H_{{{\,\textrm{tame}\,}}, {{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}}$ with eigensystem $\lambda _{[\mu ]}: {{\,\mathrm{\mathbb {T}}\,}}^{{{\,\textrm{tame}\,}}} \rightarrow \overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p$, there exists a (continuous) Galois representation

$$\begin{aligned} \rho _{[\mu ]}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}} \xrightarrow {\rho _{[\mu ]}^{{{\,\textrm{spin}\,}}}} {{\,\textrm{GSpin}\,}}_{2g+1}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p)\xrightarrow {{{\,\textrm{spin}\,}}} {{\,\textrm{GL}\,}}_{2^g}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p^5) \end{aligned}$$

such that

• (i) The representation $\rho _{[\mu ]}$ is unramified outside pN and

  $$\begin{aligned} {{\,\mathrm{char.poly}\,}}({{\,\textrm{Frob}\,}}_{\ell })(Y) = \lambda _{[\mu ]}(P_{{{\,\textrm{Hecke}\,}}, \ell }(Y)) := \prod _{x\in {{\,\textrm{Weyl}\,}}^H} (Y - \lambda _{[\mu ]}(T_{\ell , 0}^x)) \end{aligned}$$

  for any $\ell \not \mid pN$, where ${{\,\mathrm{char.poly}\,}}({{\,\textrm{Frob}\,}}_{\ell })(Y)$ stands for the characteristic polynomial of the Frobenius at $\ell$ and $P_{{{\,\textrm{Hecke}\,}}, \ell }(Y)$ is the Hecke polynomial defined in (4). Moreover, the coefficients of these two polynomials are algebraic integers over ${{\,\mathrm{{\textbf {Q}}}\,}}$.

• (ii) The representation $\rho _{[\mu ]}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ is crystalline with Hodge–Tate weights⁶

  $$\begin{aligned} (a_1, \ldots , a_{2^g}) = (0, a_g', \ldots , a_1', a_g'+a_{g-1}', \ldots , a_2'+a_1', \ldots , a_g'+\cdots +a_1'^6), \end{aligned}$$

  where $a_i' = (g+1-i)+k_i$. Let $\varphi = \varphi _{{{\,\textrm{cris}\,}}}$ be the crystalline Frobenius acting on ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(\rho _{[\mu ]}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}})$, we moreover have

  $$\begin{aligned} {{\,\mathrm{char.poly}\,}}(\varphi )(Y) = \lambda _{[\mu ]} (P_{{{\,\textrm{Hecke}\,}}, p}(Y)), \end{aligned}$$

  where ${{\,\mathrm{char.poly}\,}}(\varphi )(X)$ is the characteristic polynomial of $\varphi$ acting on ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(\rho _{[\mu ]}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}})$, and the coefficients of these two polynomials are algebraic integers over ${{\,\mathrm{{\textbf {Q}}}\,}}$. We order the eigenvalues of $\varphi$ so they satisfy

  $$\begin{aligned} (\varphi _1, \ldots , \varphi _{2^g}) = \varphi _1(1, \varphi _2', \ldots , \varphi _{g+1}', \varphi _2'\varphi _3', \ldots , \varphi _{g}'\varphi _{g+1}', \ldots , \varphi _{2}'\cdots \varphi _{g+1}') \end{aligned}$$

  for some $(\varphi _2', \ldots , \varphi _{g+1}')$. The order of the later tuple is chosen similarly as the Hodge–Tate weights. In particular, $\varphi _2, \ldots , \varphi _{g+1}$ are divisible by $\varphi _1$ and the $2^g$ eigenvalues of $\varphi$ depend only on $\varphi _1, \ldots , \varphi _{g+1}$.

Remark 3.10

Recall that ${{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}$ is the finite set of prime numbers which divides pN. Let ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}}$ be the Galois group of the maximal extension of ${{\,\mathrm{{\textbf {Q}}}\,}}$ which is unramified outside ${{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}$. Therefore, the representation $\rho _{[\mu ]}$ in Hypothesis 1 can be regarded as a Galois representation of ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}}$.

Remark 3.11

Evidently, Hypothesis 1 comes from Global Langlands Correspondence. We comment briefly on this hypothesis.

• (i) When $g\le 2$, Hypothesis 1 (i) is well-known (see, for example, [34]). The work of Kret and Shin [22] gave a positive answer to Hypothesis 1 (i) under some conditions on the automorphic representations for general g. Although their result is not completely unconditional, it suggests that Hypothesis 1 is reasonable to assume (but could be difficult to prove in general).

• (ii) Hypothesis 1 (ii) is also well-studied when $g\le 2$. In particular, Urban proved the case for $g=2$ in [33], result deduced from Scholl’s motive for modular forms [28]. For general g, the property is expected if Hypothesis 2 below holds (see, for example, [27, Theorem 2.1 and Corollary 2.2]).

By [22, Lemma 0.1] and under the assumption of Hypothesis 1, we know that given a ${{\,\mathrm{\mathbb {T}}\,}}^{{{\,\textrm{tame}\,}}}$-eigenclass $[\mu ]$ as above, $\rho _{[\mu ]}$ factors as

$$\begin{aligned} \rho _{[\mu ]}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}} \xrightarrow {\rho _{[\mu ]}^{{{\,\textrm{spin}\,}}}} {{\,\textrm{GSpin}\,}}_{2g+1}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p) \xrightarrow {{{\,\textrm{spin}\,}}}\textrm{GS}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p) \rightarrow {{\,\textrm{GL}\,}}_{2^g}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p), \end{aligned}$$

where

$$\begin{aligned} \textrm{GS} = \left\{ \begin{array}{ll} {{\,\textrm{GO}\,}}_{2^g}, &{}\quad \text { if }g(g+1)/2 \text { is even} \\ {{\,\textrm{GSp}\,}}_{2^g}, &{}\quad \text { if }g(g+1)/2 \text { is odd} \end{array}\right. \end{aligned}$$

and the last arrow is nothing but the natural inclusion. Define

Then, the inclusions

$$\begin{aligned} {{\,\textrm{GSpin}\,}}_{2g+1}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p) \hookrightarrow \textrm{GS}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p) \hookrightarrow {{\,\textrm{GL}\,}}_{2^g}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p) \end{aligned}$$

induces ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}}$-equivariant inclusions

$$\begin{aligned} {{\,\textrm{ad}\,}}\rho _{[\mu ]}^{{{\,\textrm{spin}\,}}} \hookrightarrow {{\,\textrm{ad}\,}}\rho _{[\mu ]} \hookrightarrow \mathfrak {gl}_{2^g}, \end{aligned}$$

which then further induces inclusions of the Galois cohomology groups

$$\begin{aligned} H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}}, {{\,\textrm{ad}\,}}\rho _{[\mu ]}^{{{\,\textrm{spin}\,}}}) \hookrightarrow H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}}, {{\,\textrm{ad}\,}}\rho _{[\mu ]}) \hookrightarrow H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}}, \mathfrak {gl}_{2^g}). \end{aligned}$$

On the other hand, let $\mathfrak {sl}_{2^g}$ be the trace-zero part of $\mathfrak {gl}_{2^g}$ and let

$$\begin{aligned} {{\,\textrm{ad}\,}}^{0}\rho _{[\mu ]} := {{\,\textrm{ad}\,}}\rho _{[\mu ]} \cap \mathfrak {sl}_{2^g}\quad \text { and }\quad {{\,\textrm{ad}\,}}^0\rho _{[\mu ]}^{{{\,\textrm{spin}\,}}} := {{\,\textrm{ad}\,}}\rho _{[\mu ]}^{{{\,\textrm{spin}\,}}} \cap \mathfrak {sl}_{2^g}. \end{aligned}$$

Note that the decomposition $\mathfrak {gl}_{2^g} = \mathfrak {sl}_{2^g} \oplus \mathfrak {gl}_1$ is ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}}$-equivariant, we thus have a commutative diagram

(7)

where the arrows are all inclusions.

Under the assumption of Hypothesis 1, one obtains a $2^g$-dimensional Galois representation for each eigenclass $[\mu ]$. It is then a natural question to ask whether the attached Galois representation admits an associated cuspidal automorphic representation of ${{\,\textrm{GL}\,}}_{2^g}$. The answer to this question is expected to be affirmative, which we state as the next hypothesis.

Hypothesis 2

(The potential spin functoriality) Given a ${{\,\mathrm{\mathbb {T}}\,}}^{{{\,\textrm{tame}\,}}}$-eigenclass $[\mu ]\in H_{{{\,\textrm{tame}\,}}, {{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}}$, there exists a finite real extension $L\subset \overline{{{\,\mathrm{{\textbf {Q}}}\,}}}$ of ${{\,\mathrm{{\textbf {Q}}}\,}}$ with $\rho _{[\mu ]}|_{{{\,\textrm{Gal}\,}}_L}$ being irreducible and a generic cuspidal automorphic representation $\pi _{[\mu ]}$ of ${{\,\textrm{GL}\,}}_{2^g}({{\,\mathrm{{\textbf {A}}}\,}}_{L})$, where ${{\,\mathrm{{\textbf {A}}}\,}}_{L}$ is the ring of adles of L, such that

• $\pi _{[\mu ]}$ is unramified outside the places above ${{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}$ and

• the Galois representation associated with $\pi _{[\mu ]}$ is isomorphic to $\rho _{[\mu ]}|_{{{\,\textrm{Gal}\,}}_{L}}$.

Remark 3.12

We should remark that Kret and Shin verify the above hypothesis in [22, Theorem C] under some stronger conditions than the ones they verify Hypothesis 1.

On the other hand, we also write

$$\begin{aligned} H_{{{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}} := \oplus _{t=0}^{2n_0} H_{{{\,\textrm{par}\,}}}^t(X_{{{\,\textrm{Iw}\,}}^+}({{\,\mathrm{{\textbf {C}}}\,}}), {{\,\mathrm{{\textbf {V}}}\,}}_{{{\,\textrm{GSp}\,}}_{2g}, k}^{{{\,\textrm{alg}\,}}, \vee }). \end{aligned}$$

The forgetful map $X_{{{\,\textrm{Iw}\,}}}({{\,\mathrm{{\textbf {C}}}\,}}) \rightarrow X({{\,\mathrm{{\textbf {C}}}\,}})$ then induces a morphism (see also (2) and (3))

$$\begin{aligned} \Lambda _p : H_{{{\,\textrm{tame}\,}}, {{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}} \rightarrow H_{{{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}}. \end{aligned}$$

(8)

Observe that this morphism is ${{\,\mathrm{\mathbb {T}}\,}}^p$-equivariant. Moreover, we have slope decomposition on the latter space with respect to the action of $U_p$ since it is a finite-dimensional ${{\,\mathrm{{\textbf {Q}}}\,}}_p$-vector space. Thus, for each $h\in {{\,\mathrm{{\textbf {Q}}}\,}}_{>0}$, we write

$$\begin{aligned} H_{{{\,\textrm{tame}\,}}, {{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}, \le h} := {{\,\textrm{image}\,}}\left( H_{{{\,\textrm{tame}\,}}, {{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}} \xrightarrow {\Lambda _p} H_{{{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}} \twoheadrightarrow H_{{{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}, \le h}\right) , \end{aligned}$$

where $H_{{{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}, \le h}$ is the ‘$\le h$’ part of $H_{{{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}}$ under the action of $U_p$. Thus, for any ${{\,\mathrm{\mathbb {T}}\,}}^{{{\,\textrm{tame}\,}}}$-eigenclass $[\mu ]$ in $H_{{{\,\textrm{tame}\,}}, {{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}}$, its image in $H_{{{\,\textrm{tame}\,}}, {{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}, \le h}$ can be decomposed as a sum of ${{\,\mathrm{\mathbb {T}}\,}}$-eigenclasses. We call any of these factors a p-stabilisation of $[\mu ]$.

It is a natural question asking how the eigenvalues of a ${{\,\mathrm{\mathbb {T}}\,}}^{{{\,\textrm{tame}\,}}}$-eigenclass interact with the eigenvalues of its p-stabilisations. The following statement is due to Harron–Jorza.

Proposition 3.13

([17, Lemma 17])

• (i) Let $[\mu ]$ be a ${{\,\mathrm{\mathbb {T}}\,}}^{{{\,\textrm{tame}\,}}}$-eigenclass with eigensystem $\lambda _{[\mu ]}$ in $H_{{{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}, \le h}$. Then, there exist $2^gg!$ p-stabilisations $[\mu ]^{(p)}$, indexed by ${{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$.

• (ii) Chose a bijection of sets $\iota : \{1, 2, \ldots , 2^g\} \xrightarrow {\sim } {{\,\textrm{Weyl}\,}}^H$ so that $\lambda _{[\mu ]}\left( T_{p, 0}^{\iota (i)}\right) = \varphi _i$, where $T_{p, 0}^{\iota (i)}$ is the Hecke operator defined by $[{{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_p) (\iota (i)\cdot {{\,\mathrm{{\textbf {u}}}\,}}_{p,0}) {{\,\textrm{GSp}\,}}_{2g}({{\,\mathrm{{\textbf {Z}}}\,}}_p)]$ acting on $H_{{{\,\textrm{tame}\,}}, {{\,\textrm{par}\,}}, {{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}}$ and $\varphi _i$ is the i-th eigenvalue of the crystalline Frobenius associated with $\rho _{[\mu ]}$.⁷ Denote by $\lambda _{i} = \lambda _{[\mu ]}(T_{p, 0}^{\iota (i)})$ and let $[\mu ]^{(p)}$ be any of the p-stablisation of $[\mu ]$ with Hecke eigensystem $\lambda _{[\mu ]}^{(p)}$. Then, there exists a constant $\vartheta \in {{\,\mathrm{{\textbf {Q}}}\,}}$ (depending only on g) such that, for $i=1, \ldots , g+1$,

  $$\begin{aligned} \lambda _{[\mu ]}^{(p)} (U_{p,0}^{\iota (i)})= p^{\vartheta }\cdot p^{-(g+1-i)} \lambda _1 \prod _{j=1}^{g}(\lambda _{j+1}/\lambda _1)^{a_{\nu (j)}\text { or }1-a_{\nu (j)}}, \end{aligned}$$

  where

  • the index of $[\mu ]^{(p)}$ is $(\epsilon , \nu )\in {{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}} = {{\,\mathrm{\varvec{\Sigma }}\,}}_g < imes ({{\,\mathrm{{\textbf {Z}}}\,}}/2{{\,\mathrm{{\textbf {Z}}}\,}})^{g}$ and

    $$\begin{aligned} a_{\nu (j)} = \left\{ \begin{array}{ll} 1, &{}\quad \nu (j) = i\\ 0, &{}\quad \text {otherwise} \end{array}\right. ; \end{aligned}$$

  • the exponent depends on whether $\epsilon (\nu (j)) = 0 \text { or }1\in {{\,\mathrm{{\textbf {Z}}}\,}}/2{{\,\mathrm{{\textbf {Z}}}\,}}$.

3.3 Families of Galois representations on the cuspidal eigenvariety

The goal of this section is to construct families of Galois representations on a sublocus of the cuspidal eigenvariety ${{\,\mathrm{\mathcal {E}}\,}}_0$ under the assumption of Hypothesis 1.

For any dominant algebraic weight $k\in {{\,\mathrm{{\textbf {Z}}}\,}}_{>0}^g$, recall from [1, Theorem 6.4.1] that there is $h_k\in {{\,\mathrm{{\textbf {R}}}\,}}_{>0}$ such that for any $h\in {{\,\mathrm{{\textbf {Q}}}\,}}_{>0}$ with $h<h_k$, we have a canonical isomorphism

$$\begin{aligned} H_{{{\,\textrm{par}\,}}, k}^{{{\,\textrm{tol}\,}}, \le h} \xrightarrow {\sim } H_{{{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}, \le h}. \end{aligned}$$

We then define the p-stabilised classical locus of ${{\,\mathrm{\mathcal {E}}\,}}_0$ to be the locus ${{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}\subset {{\,\mathrm{\mathcal {E}}\,}}_0$, containing those ${{\,\mathrm{{\varvec{{x}}}}\,}}$ with the following conditions:

• ${{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}}) = k\in {{\,\mathrm{{\textbf {Z}}}\,}}_{>0}^g$ is a dominant algebraic weight;

• there exists $h< h_k$ such that ${{\,\mathrm{{\varvec{{x}}}}\,}}$ corresponds to a p-stabilisation of slope $\le h$ of a ${{\,\mathrm{\mathbb {T}}\,}}^{{{\,\textrm{tame}\,}}}$-eigenclass $[\mu ]$ in $H_{{{\,\textrm{tame}\,}}, {{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}}$;

• the Galois representation $\rho _{[\mu ]}^{{{\,\textrm{spin}\,}}}$ attached to $[\mu ]$ (by Hypothesis 1) is irreducible.

Consequently, we define

$$\begin{aligned} {{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}} := \text { the Zariski closure of} {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}} in {{\,\mathrm{\mathcal {E}}\,}}_0. \end{aligned}$$

Remark 3.14

We do not expect every classical point in ${{\,\mathrm{\mathcal {E}}\,}}_0$ corresponds to an irreducible Galois representation due to the endoscopy theory of automorphic forms. As we will be only interested in classical points that correspond to irreducible Galois representations, we do not lose information if we only consider ${{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}$.

Proposition 3.15

Assume the truthfulness of Hypothesis 1.

• (i) For any ${{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}$, there is an associated Galois representation

  $$\begin{aligned} \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}}\xrightarrow {\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}} {{\,\textrm{GSpin}\,}}_{2g+1}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p) \xrightarrow {{{\,\textrm{spin}\,}}} {{\,\textrm{GL}\,}}_{2^g}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p) \end{aligned}$$

  that satisfies the properties in Hypothesis 1.

• (ii) There is a universal determinant

  $$\begin{aligned} {{\,\textrm{Det}\,}}^{{{\,\textrm{univ}\,}}}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}} \rightarrow {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}}^+({{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}) \end{aligned}$$

  of dimension $2^g$ such that, for any ${{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}$, the specialisation ${{\,\textrm{Det}\,}}^{{{\,\textrm{univ}\,}}}|_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ (notation as in (5)) coincides with $\det \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$.

Proof

The first assertion is easy. Let ${{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}$. It corresponds to a p-stabilisation class $[\mu ]^{(p)}\in H_{{{\,\textrm{tame}\,}}, {{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}, \le h}$. That is, there is a ${{\,\mathrm{\mathbb {T}}\,}}^{{{\,\textrm{tame}\,}}}$-eigenclass $[\mu ]\in H_{{{\,\textrm{tame}\,}}, {{\,\textrm{par}\,}}, k}^{{{\,\textrm{alg}\,}}, {{\,\textrm{tol}\,}}}$ such that $[\mu ]^{(p)}$ is a p-stabilisation of $[\mu ]$. By Hypothesis 1, the class $[\mu ]$ is associated with a Galois representation with desired properties. Then, we define $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}} := \rho _{[\mu ]}^{{{\,\textrm{spin}\,}}}$ and $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}} := \rho _{[\mu ]}$.

For the second assertion, we follow the proof of [5, Proposition 7.1.1] (see also [6, Example 2.32]). Consider the morphism

$$\begin{aligned} \Phi : {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}}^+({{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}) \rightarrow \prod _{{{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}}{{\,\mathrm{{\textbf {C}}}\,}}_p, \quad f\mapsto (f({{\,\mathrm{{\varvec{{x}}}}\,}}))_{{{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}}. \end{aligned}$$

Equipped $\prod _{{{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}}{{\,\mathrm{{\textbf {C}}}\,}}_p$ with the product topology, one sees that $\Phi$ is continuous. We claim that $\Phi ({{\,\mathrm{\mathscr {O}}\,}}^+_{{{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}}({{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}))$ is homeomorphic to ${{\,\mathrm{\mathscr {O}}\,}}^+_{{{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}}({{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}})$ and is closed in $\prod _{{{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}}{{\,\mathrm{{\textbf {C}}}\,}}_p$. Indeed, since ${{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}$ is Zariski dense in the reduced space ${{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}$, the map $\Phi$ is injective. Apply [20, Corollary 5.4.4], we know that ${{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}}^+({{\,\mathrm{\mathcal {E}}\,}}_{0}^{{{\,\textrm{irr}\,}}})$ is compact and so $\Phi ({{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}}^+({{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}))$ is closed in $\prod _{{{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}}{{\,\mathrm{{\textbf {C}}}\,}}_p$.

On the other hand, we have a continuous map

$$\begin{aligned} {{\,\textrm{Det}\,}}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}} \rightarrow \prod _{{{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}} {{\,\mathrm{{\textbf {C}}}\,}}_p, \quad \sigma \mapsto (\det \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}(\sigma ))_{{{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}}. \end{aligned}$$

One checks easily that ${{\,\textrm{Det}\,}}$ is a determinant of dimension $2^g$, in fact, the determinant of a representation ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}}\rightarrow {{\,\textrm{GL}\,}}_{2^g}(\prod _{{{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}}{{\,\mathrm{{\textbf {C}}}\,}}_p)$. Hypothesis 1 and ${{\,\textrm{image}\,}}\Phi$ being closed in $\prod _{{{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}}{{\,\mathrm{{\textbf {C}}}\,}}_p$ imply that ${{\,\textrm{image}\,}}{{\,\textrm{Det}\,}}\subset {{\,\textrm{image}\,}}\Phi$. Hence, we define

$$\begin{aligned} {{\,\textrm{Det}\,}}^{{{\,\textrm{univ}\,}}}:= \Phi ^{-1}\circ {{\,\textrm{Det}\,}}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}} \rightarrow {{\,\mathrm{\mathscr {O}}\,}}^+_{{{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}}({{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}). \end{aligned}$$

Since $\Phi$ is injective and ${{\,\textrm{Det}\,}}$ is a determinant of dimension $2^g$, ${{\,\textrm{Det}\,}}^{{{\,\textrm{univ}\,}}}$ is as desired. $\square$

Theorem 3.16

There exists a subset ${{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit }^{{{\,\textrm{cl}\,}}}\subset {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}$ which is Zariski dense in ${{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}$, $2^g$ analytic functions $\alpha _1, \ldots , \alpha _{2^g}\in {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}}({{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}})$ and $2^g$ analytic functions $F_1, \ldots , F_{2^g}\in {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}}({{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}})$ such that

$$\begin{aligned} ({{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}, {{\,\textrm{Det}\,}}^{{{\,\textrm{univ}\,}}}, {{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit }^{{{\,\textrm{cl}\,}}}, \{\alpha _i: i=1, \ldots , 2^g\}, \{F_i: i=1, \ldots , 2^g\}) \end{aligned}$$

is a refined family of Galois representations.

Proof

For any p-adic weight $\kappa = (\kappa _1, \ldots , \kappa _g)$, define an ordering of functions on ${{\,\mathrm{{\textbf {Z}}}\,}}_p^\times$ via

$$\begin{aligned} (\alpha _1, \ldots , \alpha _{2^g}):= & {} (0 , \alpha _g', \ldots , \alpha _1', \alpha _g' + \alpha _{g-1}', \ldots , \alpha _g'+\alpha _1', \alpha _{g-1}'+\alpha _{g-2}', \ldots , \alpha _2'\\ {}{} & {} +\alpha _1', \ldots , \alpha _g'+\cdots + \alpha _1'), \end{aligned}$$

where $\alpha _i' = (g+1-i)+\kappa _i$ is the character $a\mapsto \kappa _i(a)a^{g+1-i}$ for every $a\in {{\,\mathrm{{\textbf {Z}}}\,}}_p^\times$. We can view $\alpha _j$’s as functions on ${{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}$ by composing with the weight map ${{\,\textrm{wt}\,}}$. Obviously from this definition, for any ${{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}$, the functions $\alpha _j$’s provide an ordering of the Hodge–Tate weight of $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ in Hypothesis 1 (iii).

Define

$$\begin{aligned} {{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit }^{{{\,\textrm{cl}\,}}} := \left\{ {{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}: \begin{array}{l} 0 = \alpha _1({{\,\mathrm{{\varvec{{x}}}}\,}})< \alpha _2({{\,\mathrm{{\varvec{{x}}}}\,}})< \cdots < \alpha _{2^g}({{\,\mathrm{{\varvec{{x}}}}\,}}) \\ \text {eigenvalues of the crystalline Frobenius acting on}\\ {{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}({\rho }_{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}) \text { are distinct} \end{array}\right\} . \end{aligned}$$

Observe that ${{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit }^{{{\,\textrm{cl}\,}}}$ is Zariski dense in ${{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}$ since ${{\,\mathrm{\mathcal {X}}\,}}^{{{\,\textrm{cl}\,}}}$ is Zariski dense in ${{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}$ and the first condition defining ${{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit }^{{{\,\textrm{cl}\,}}}$ is an open condition on weights while the second condition is an open condition on ${{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}$. We claim that ${{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit }^{{{\,\textrm{cl}\,}}}$ satisfies condition (iv) in Definition 3.8. That is, for any $C\in {{\,\mathrm{{\textbf {Z}}}\,}}_{>0}$, we have to show that the set

$$\begin{aligned} {{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit , C}^{{{\,\textrm{cl}\,}}} : = \left\{ {{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit }^{{{\,\textrm{cl}\,}}}: \begin{array}{l} \alpha _{i+1}({{\,\mathrm{{\varvec{{x}}}}\,}}) - \alpha _{i}({{\,\mathrm{{\varvec{{x}}}}\,}})> C (\alpha _{i}({{\,\mathrm{{\varvec{{x}}}}\,}}) - \alpha _{i-1}({{\,\mathrm{{\varvec{{x}}}}\,}})) \text { for }i=2, \ldots , 2^g-1 \\ \alpha _2({{\,\mathrm{{\varvec{{x}}}}\,}}) - \alpha _1({{\,\mathrm{{\varvec{{x}}}}\,}}) >C \end{array}\right\} \end{aligned}$$

satisfies that, for any basis of affinoid neighbourhoods ${{\,\mathrm{\mathcal {V}}\,}}$ of ${{\,\mathrm{{\varvec{{x}}}}\,}}$, ${{\,\mathrm{\mathcal {V}}\,}}\cap {{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit , C}^{{{\,\textrm{cl}\,}}}$ is Zariski dense in ${{\,\mathrm{\mathcal {V}}\,}}$. However, this follows from that the condition defining ${{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit , C}^{{{\,\textrm{cl}\,}}}$ is an open condition on the weights.

Now, for any ${{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit }^{{{\,\textrm{cl}\,}}}$, the associated representation $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ is crystalline at p. Let $\varphi _1({{\,\mathrm{{\varvec{{x}}}}\,}}), \ldots , \varphi _{2^g}({{\,\mathrm{{\varvec{{x}}}}\,}})$ be eigenvalues of the crystalline Frobenius $\varphi = \varphi _{{{\,\textrm{cris}\,}}}$ acting on ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}})$. The order of the eigenvalues $\varphi _i$’s is defined so that it defines a non-critical refinement on $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$. This is achievable by applying Proposition 3.6 (ii). Define

$$\begin{aligned} F_i({{\,\mathrm{{\varvec{{x}}}}\,}}) := \varphi _i({{\,\mathrm{{\varvec{{x}}}}\,}})/p^{\alpha _i({{\,\mathrm{{\varvec{{x}}}}\,}})} \in {{\,\mathrm{{\textbf {C}}}\,}}_p. \end{aligned}$$

We claim that the collection $\{(F_i({{\,\mathrm{{\varvec{{x}}}}\,}}))_{i=1, \ldots , 2^g}\}_{{{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {G}}\,}}_{\heartsuit }^{{{\,\textrm{cl}\,}}}}$ glue to $2^g$ analytic functions $(F_1, \ldots , F_{2^g})$ in ${{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}}({{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}})$. Let $\lambda _{{{\,\mathrm{{\varvec{{x}}}}\,}}}: {{\,\mathrm{\mathbb {T}}\,}}^{{{\,\textrm{tame}\,}}} \rightarrow \overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p$ be the eigensystem corresponds to ${{\,\mathrm{{\varvec{{x}}}}\,}}$. Consider

$$\begin{aligned} p^{\vartheta }p^{\kappa _{i}'}F_i := \hbox { image of the operator } U_{p,0}^{\iota (i)} \hbox { in } {{\,\mathrm{\mathscr {O}}\,}}_{{{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}}({{\,\mathrm{\mathcal {E}}\,}}_0^{{{\,\textrm{irr}\,}}}), \end{aligned}$$

where

$$\begin{aligned} (\kappa _1', \ldots , \kappa _{2^g}')= & {} (0, \kappa _g, \ldots , \kappa _1, \kappa _g+\kappa _{g-1}, \ldots , \kappa _{g}+\kappa _{1}, \kappa _{g-1}+\kappa _{g-2}, \ldots , \kappa _{2}\\{} & {} +\kappa _1, \ldots , \kappa _g+\cdots +\kappa _1) \end{aligned}$$

and $(\kappa _1, \ldots , \kappa _g) = {{\,\textrm{wt}\,}}$ is the weight map. Then, Hypothesis 1 (ii) and Proposition 3.13 imply the desired result (see also [2, Proposition 7.5.13]). $\square$

Remark 3.17

Recall that we have ordered the eigenvalues of the crystalline Frobenius $\varphi$ so that they satisfy

$$\begin{aligned} (\varphi _1, \ldots , \varphi _{2^g}) = \varphi _1(1, \varphi _2', \ldots , \varphi _{g+1}', \varphi _2'\varphi _3', \ldots , \varphi _{g}'\varphi _{g+1}', \ldots , \varphi _{2}'\cdots \varphi _{g+1}'). \end{aligned}$$

On the other hand, recall that ${{\,\textrm{Weyl}\,}}^H$ is a set of representatives of ${{\,\textrm{Weyl}\,}}_H\backslash {{\,\textrm{Weyl}\,}}_{{{\,\textrm{GSp}\,}}_{2g}}$, where ${{\,\textrm{Weyl}\,}}_{H} \simeq {{\,\mathrm{\varvec{\Sigma }}\,}}_g$. Observe that ${{\,\textrm{diag}\,}}(\mathbb {1}_g, p\mathbb {1}_g)$ is stable under the action of ${{\,\mathrm{\varvec{\Sigma }}\,}}_g$, thus the action of ${{\,\textrm{Weyl}\,}}^H$ on $T_{p, 0}$ only depends on the action of $({{\,\mathrm{{\textbf {Z}}}\,}}/2{{\,\mathrm{{\textbf {Z}}}\,}})^g$. Combining everything together, we have the relation

$$\begin{aligned} (F_1, \ldots , F_{2^g}) = F_1(1, F_2', \ldots , F_{g+1}', F_{2}'F_{3}', \ldots , F_{g}'F_{g+1}', \ldots , F_2'\cdots F_{g+1}'). \end{aligned}$$

In particular, $F_2, \ldots , F_{g+1}$ are divisible by $F_1$.

3.4 Local and global Galois deformations

We keep the notations in the previous subsection. Fix ${{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {X}}\,}}_{\heartsuit }^{{{\,\textrm{cl}\,}}}$ with ${{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}}) = k = (k_1, \ldots , k_g)$ and we write

$$\begin{aligned} \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}} \xrightarrow {\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}} {{\,\textrm{GSpin}\,}}_{2g+1}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p) \xrightarrow {{{\,\textrm{spin}\,}}} {{\,\textrm{GL}\,}}_{2^g}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p) \end{aligned}$$

for the Galois representation attached to ${{\,\mathrm{{\varvec{{x}}}}\,}}$, given by Proposition 3.15. We fix a large enough finite field extension $k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ of ${{\,\mathrm{{\textbf {Q}}}\,}}_p$ such that $k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ contains the residue field at ${{\,\mathrm{{\varvec{{x}}}}\,}}$ and $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}$ takes values in ${{\,\textrm{GSpin}\,}}_{2g+1}(k_{{{\,\mathrm{{\varvec{{x}}}}\,}}})$. We also assume that $k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ contains all eigenvalues of the Frobenii.

Let now be the category of local artinian $k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$-algebras whose residue field is $k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$. We denote by ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ the refinement of $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ induced by the refined family defined in Theorem 3.16. We also denote by $\delta = (\delta _1, \ldots , \delta _{2^g})$ the parameter attached to the triangulation associated with ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}$. Notice that the relation of the eigenvalues of crystalline Frobenius and the Hodge–Tate weight implies that the parameter $\delta$ satisfies

$$\begin{aligned} (\delta _1, \ldots , \delta _{2^g})= & {} \delta _1(1, \delta _2', \ldots , \delta _{g+1}', \delta _{2}'\delta _{3}',\\ {}{} & {} \ldots , \delta _{g}'\delta _{g+1}', \ldots , \delta _{2}'\cdots \delta _{g+1}') \end{aligned}$$

for some continuous characters $\delta _2', \ldots , \delta _{g+1}'$ such that $\delta _i = \delta _1\delta _i'$ for all $i=2, \ldots , g+1$.

Local Galois deformations at p. We shall consider two deformation problems at p:

• (i) The deformation problem

  

  sending each to the isomorphism classes of representations $\rho _A^{{{\,\textrm{spin}\,}}} : {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p} \rightarrow {{\,\textrm{GSpin}\,}}_{2g+1}(A)$ with a triangulation ${{\,\textrm{Fil}\,}}_{\bullet } {{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{rig}\,}}}({{\,\textrm{spin}\,}}\circ \rho _A^{{{\,\textrm{spin}\,}}})$ such that

  • $\rho _{A}^{{{\,\textrm{spin}\,}}} \otimes _A k_{{{\,\mathrm{{\varvec{{x}}}}\,}}} \simeq \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$;

  • $({{\,\textrm{spin}\,}}\circ \rho _A^{{{\,\textrm{spin}\,}}}, {{\,\textrm{Fil}\,}}_{\bullet }{{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{rig}\,}}}({{\,\textrm{spin}\,}}\circ \rho ^{{{\,\textrm{spin}\,}}}_A))\in {{\,\mathrm{\mathscr {D}}\,}}_{\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}(A)$ and write $\delta _A = (\delta _{A, 1}, \ldots , \delta _{A, 2^g})$ for the associated parameter;

  • the parameter $\delta _A$ satisfies

    $$\begin{aligned} (\delta _{A, 1}, \ldots , \delta _{A, 2^g})= & {} \delta _{A, 1}( 1, \delta _{A, 2}', \ldots , \delta _{A, g+1}', \delta _{A, 2}'\delta _{A, 3}', \ldots , \delta _{A, 2}'\delta _{A, g+1}', \delta _{A, 3}'\delta _{A, 4}', \ldots ,\\{} & {} \delta _{A, g}'\delta _{A, g+1}', \ldots , \delta _{A,2}'\cdots \delta _{A, g+1}') \end{aligned}$$

    for some continuous characters $\delta _{A, 2}', \ldots , \delta _{A, g+1}'$;

  • $\det {{\,\textrm{spin}\,}}\circ \rho _A^{{{\,\textrm{spin}\,}}} = \det \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$

• (ii) The deformation problem

  

  sending each to the isomorphism classes of representations $\rho _A^{{{\,\textrm{spin}\,}}}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}\rightarrow {{\,\textrm{GSpin}\,}}_{2g+1}(A)$ such that

  • $\rho _A^{{{\,\textrm{spin}\,}}} \otimes _A k_{{{\,\mathrm{{\varvec{{x}}}}\,}}} \simeq \rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$;

  • the $(\varphi , \Gamma )$-module ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{rig}\,}}}({{\,\textrm{spin}\,}}\circ \rho ^{{{\,\textrm{spin}\,}}}_A)$ is crystalline in the sense of [2, Definition 2.2.10] whose eigenvalues $(\varphi _{A, 1}, \ldots , \varphi _{A, 2^g})$ of the crystalline Frobenius satisfy

    $$\begin{aligned} (\varphi _{A, 1}, \ldots , \varphi _{A, 2^g})&= \varphi _{A, 1}(1, \varphi _{A, 2}', \ldots , \varphi _{A, g+1}', \varphi _{A, 2}'\varphi _{A, 3}', \ldots , \varphi _{A, g}'\varphi _{A, g+1}',\\&\quad \ldots , \varphi _{A, 2}'\cdots \varphi _{A, g+1}'), \end{aligned}$$

    order chosen the same as for $\varphi _i$’s;

  • $\det {{\,\textrm{spin}\,}}\circ \rho _A^{{{\,\textrm{spin}\,}}} = \det \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$

Consider

$$\begin{aligned} L_p' := \ker \left( H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}) \rightarrow H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}} \otimes _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} {{\,\mathrm{{\textbf {B}}}\,}}_{{{\,\textrm{cris}\,}}})\right) , \end{aligned}$$

where ${{\,\mathrm{{\textbf {B}}}\,}}_{{{\,\textrm{cris}\,}}}$ is Fontaine’s ring of crystalline periods. It is well-known that $L_p'$ defines the tangent space of the crystalline deformation problem for $\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ with fixed determinant. Consequently, the tangent space ${{\,\mathrm{\mathscr {D}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f, p}^{{{\,\textrm{spin}\,}}}(k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}[\varepsilon ])$, where $\varepsilon$ is a variable such that $\varepsilon ^2 = 0$, of ${{\,\mathrm{\mathscr {D}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f, p}^{{{\,\textrm{spin}\,}}}$ defines a subspace of $L_p'$. Thus, we define

$$\begin{aligned} L_p := {{\,\mathrm{\mathscr {D}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f, p}^{{{\,\textrm{spin}\,}}}(k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}[\varepsilon ]) \subset L_p'. \end{aligned}$$

(9)

Local Galois deformations at N. For any $\ell | N$, we consider the following deformation problem

sending each to the isomorphism classes of representations $\rho _A^{{{\,\textrm{spin}\,}}}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }} \rightarrow {{\,\textrm{GSpin}\,}}_{2g+1}(A)$ such that

• $\rho _A^{{{\,\textrm{spin}\,}}} \otimes _{A}k_{{{\,\mathrm{{\varvec{{x}}}}\,}}} \simeq \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}}$;

• $\rho _{A}^{{{\,\textrm{spin}\,}}}|_{I_{\ell }} \simeq \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}|_{I_{\ell }} \otimes _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}}A$

• $\det {{\,\textrm{spin}\,}}\circ \rho _A^{{{\,\textrm{spin}\,}}} = \det \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}}$

Here, $I_{\ell }\subset {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}$ denotes the inertia subgroup. Then, one sees that the tangent space ${{\,\mathrm{\mathscr {D}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, \ell }(k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}[\varepsilon ])$ of ${{\,\mathrm{\mathscr {D}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, \ell }$ is a $k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$-subspace of $H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}})$. We consequently define

$$\begin{aligned} L_{\ell } := {{\,\mathrm{\mathscr {D}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, \ell }(k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}[\varepsilon ]) \subset H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}). \end{aligned}$$

(10)

We learnt the following lemma from P. Allen.

Lemma 3.18

Under the assumption of Hypothesis 2, we have

$$\begin{aligned} L_{\ell } = H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}). \end{aligned}$$

Proof

Let

$$\begin{aligned} H_{\textrm{unr}}^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}) := \ker \left( H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}) \rightarrow H^1(I_{\ell }, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})\right) \end{aligned}$$

By definition, we see that $H_{\textrm{unr}}^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}) \subset L_{\ell }$. Thus, it is enough to show that

$$\begin{aligned} H_{\textrm{unr}}^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}) = H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}). \end{aligned}$$

First of all, observe that

$$\begin{aligned} H_{\textrm{unr}}^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}) = H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}/I_{\ell }, ({{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})^{I_{\ell }}) \end{aligned}$$

by definition. Note that ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}/I_{\ell } \simeq \widehat{{{\,\mathrm{{\textbf {Z}}}\,}}}$. Hence, one deduces from the discussion in [29, Chapter XIII, §1] that

$$\begin{aligned} \dim _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} H_{\textrm{unr}}^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})&= \dim _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}/I_{\ell }, ({{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})^{I_{\ell }})\\&= \dim _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} H^0({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}/I_{\ell }, ({{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})^{I_{\ell }})\\&= \dim _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} H^0({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}). \end{aligned}$$

By applying the local Euler characteristic, the desired equation will follow once we show

$$\begin{aligned} H^2({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}) = 0. \end{aligned}$$

By Tate duality, it is equivalent to show

$$\begin{aligned} H^0({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}(1)) = 0. \end{aligned}$$

Let L be the real extension of ${{\,\mathrm{{\textbf {Q}}}\,}}$ as in Hypothesis 2, we claim that for any place v in L sitting above $\ell$, we have

$$\begin{aligned} H^0({{\,\textrm{Gal}\,}}_{L_{v}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}(1)) = 0, \end{aligned}$$

where ${{\,\textrm{Gal}\,}}_{L_v} = {{\,\textrm{Gal}\,}}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_{\ell }/L_v)$ is the absolute Galois group of $L_v$. However, under the assumption of Hypothesis 2, the desired vanishing follows from [4, Lemma 1.3.2] and the discussion around (7).

Finally, observe that the restriction map

$$\begin{aligned} {{\,\textrm{Res}\,}}: H^0({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}(1)) \rightarrow H^0({{\,\textrm{Gal}\,}}_{L_{v}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}(1)) \end{aligned}$$

is an injection since $k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ is of characteristic zero so that

$$\begin{aligned} \textrm{Corres}\circ {{\,\textrm{Res}\,}}= \text { multiplication by }[L_v : {{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }] \end{aligned}$$

is an injection. The assertion then follows. $\square$

Global Galois deformations. Consider the following two global deformation functors:

• (i) The deformation problem

  

  sending each to isomorphism classes of representations $\rho ^{{{\,\textrm{spin}\,}}}_A: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}} \rightarrow {{\,\textrm{GSpin}\,}}_{2g+1}(A)$ and triangulation ${{\,\textrm{Fil}\,}}_{\bullet }{{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{rig}\,}}}({{\,\textrm{spin}\,}}\circ \rho ^{{{\,\textrm{spin}\,}}}_A|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}})$ such that

  • $\rho ^{{{\,\textrm{spin}\,}}}_A\otimes _A k_{{{\,\mathrm{{\varvec{{x}}}}\,}}} \simeq \rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$

  • $\det {{\,\textrm{spin}\,}}\circ \rho _A^{{{\,\textrm{spin}\,}}} = \det \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$

  • $({{\,\textrm{spin}\,}}\circ \rho ^{{{\,\textrm{spin}\,}}}_{A}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}, {{\,\textrm{Fil}\,}}_{\bullet }{{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{rig}\,}}}({{\,\textrm{spin}\,}}\circ \rho _A|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}))\in {{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}, p}(A)$

  • $\rho ^{{{\,\textrm{spin}\,}}}_A|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{{\ell }}}}\in {{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, \ell }(A)$ for $\ell \in {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}$

• (ii) The deformation problem

  

  sending each to isomorphism classes of representations $\rho ^{{{\,\textrm{spin}\,}}}_A: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}} \rightarrow {{\,\textrm{GSpin}\,}}_{2g+1}(A)$ such that

  • $\rho ^{{{\,\textrm{spin}\,}}}_A \otimes _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}} \simeq \rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$

  • $\det {{\,\textrm{spin}\,}}\circ \rho _A = \det \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$

  • $\rho ^{{{\,\textrm{spin}\,}}}_A|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}\in {{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f, p}(A)$

  • $\rho ^{{{\,\textrm{spin}\,}}}_A|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}}\in {{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, \ell }(A)$ for $\ell \in {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}$.

Lemma 3.19

Keep the above notations.

• (i) The deformation problems ${{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}$ and ${{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}$ are pro-representable. Denote by $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}$ and $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{univ}\,}}}$ the complete noetherian local rings that represent these two deformation functors respectively.

• (ii) Suppose ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ is non-critical, then ${{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}$ is a subfunctor of ${{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}$.

Proof

Since $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ is absolutely irreducible, the first assertion follows from standard Galois deformation theory (see, for example, [23, §4] and [19, Proposition 3.7 & Proposition 3.8]). The second assertion is an immediate consequence of [2, Proposition 2.5.8]. Notice that our deformation problems are slightly different from the ones considered in op. cit. and [19]. In fact, one sees easily that our deformation problems are subfunctors of the deformation problems considered therein. Their results imply ours since ${{\,\textrm{spin}\,}}: {{\,\textrm{GSpin}\,}}_{2g+1} \rightarrow {{\,\textrm{GL}\,}}_{2^g}$ is a closed immersion, the conditions we required on the relations of the parameters and the fixed determinant of the deformations are closed conditions and they are stable under isomorphisms, i.e., they satisfy the definition of ‘deformation problems’ (see, for example, [23, Definition 4.1]). $\square$

The Bloch–Kato Selmer group associated with ${{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ is defined to be

$$\begin{aligned} H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}) := \ker \left( H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}) \xrightarrow {{{\,\textrm{Res}\,}}} \prod _{\ell \in {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}\cup \{p\}} \frac{H^1({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}})}{L_{\ell }}\right) , \end{aligned}$$

(11)

where $L_{\ell }$ are as defined in (9) and (10).

Proposition 3.20

The tangent space ${{\,\mathrm{\mathscr {D}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{spin}\,}}}(k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}[\varepsilon ])$ of ${{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}$ can naturally be identified with the Bloch–Kato Selmer group $H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}})$.

Proof

This is follows from standard Galois deformation theory (see, for example, [19, Proposition 3.7]) and the definition of $L_p$ and $L_{\ell }$ (see (9) and (10)). $\square$

3.5 The adjoint Bloch–Kato Selmer groups

We keep the notations and assumptions in the previous subsection. We further assume the following

• the refinement ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ of $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ satisfies (REG) and (NCR);⁸

• the representation $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ is not isomorphic to its twist by the p-adic cyclotomic character.

Lemma 3.21

Denote by ${{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}} := \widehat{{{\,\mathrm{\mathscr {O}}\,}}}_{{{\,\mathrm{\mathcal {E}}\,}}_0^{}, {{\,\mathrm{{\varvec{{x}}}}\,}}}$ the completed local ring at ${{\,\mathrm{{\varvec{{x}}}}\,}}$. Then, for any ideal of cofinite length ${{\,\mathrm{\mathfrak {I}}\,}}\subset {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ there exists a Galois representation

$$\begin{aligned} \rho _{{{\,\mathrm{\mathfrak {I}}\,}}}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}} \rightarrow {{\,\textrm{GL}\,}}_{2^g}({{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/{{\,\mathrm{\mathfrak {I}}\,}}) \end{aligned}$$

such that

• (i) $\rho _{{{\,\mathrm{\mathfrak {I}}\,}}} \otimes _{{{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}} \simeq \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$

• (ii) $\rho _{{{\,\mathrm{\mathfrak {I}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}} \in {{\,\mathrm{\mathscr {D}}\,}}_{\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}, p}({{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/{{\,\mathrm{\mathfrak {I}}\,}})$

Proof

The first assertion is a consequence of Theorem 3.3. The second assertion is a consequence of Theorem 3.9. $\square$

Hypothesis 3

Consider the Galois representation $\rho _{{{\,\mathrm{\mathfrak {I}}\,}}}$ in Lemma 3.21 for any ideal of cofinite length ${{\,\mathrm{\mathfrak {I}}\,}}\subset {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$. We assume

• (i) The Galois representation $\rho _{{{\,\mathrm{\mathfrak {I}}\,}}}$ factors as

  $$\begin{aligned} \rho _{{{\,\mathrm{\mathfrak {I}}\,}}}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}} \xrightarrow {\rho _{{{\,\mathrm{\mathfrak {I}}\,}}}^{{{\,\textrm{spin}\,}}}} {{\,\textrm{GSpin}\,}}_{2g+1}({{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/{{\,\mathrm{\mathfrak {I}}\,}}) \xrightarrow {{{\,\textrm{spin}\,}}} {{\,\textrm{GL}\,}}_{2^g}({{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/{{\,\mathrm{\mathfrak {I}}\,}}). \end{aligned}$$

• (ii) The Galois representation $\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{\mathfrak {I}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}\in {{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}, p}({{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/{{\,\mathrm{\mathfrak {I}}\,}})$.

• (iii) The tame level structure $\Gamma ^{(p)}$ implies that the Galois representation $\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{\mathfrak {I}}\,}}}$ satisfies

  $$\begin{aligned} \rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{\mathfrak {I}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }}} \in {{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, \ell }({{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/{{\,\mathrm{\mathfrak {I}}\,}}) \end{aligned}$$

  for any $\ell | N$.

Remark 3.22

We remark that the above hypothesis is safe to assume:

• (i) The first two conditions are natural. When $g=1$, the conditions are trivial. When $g=2$, ${{\,\textrm{GSpin}\,}}_{5}$ is isomorphic to ${{\,\textrm{GSp}\,}}_{4}$. In this case, the proof of [15, Lemma 4.3.3] implies the conditions.

• (ii) Roughly speaking, the third condition in the hypothesis means that the level structure on the automorphic side determines the ramification type on the Galois side. This condition is inspired by the Taylor–Wiles method. When $g=1$, the classical example is the work of R. Taylor and A. Wiles in [32]. In loc. cit., they showed that if one considers the Hecke algebra on the space of weight-2 modular forms of a certain level, then the Galois representation with coefficients in the local Hecke algebra satisfies certain Galois deformation problem. For higher-rank groups, one sees, for example, such a relation in [15, §4.3] for ${{\,\textrm{GSp}\,}}_4$ and [7, §3.4] for ${{\,\textrm{GL}\,}}_n$ over CM fields.

Lemma 3.23

Denote by $R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}$ the complete local ring at ${{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})$ and so we have a natural homomorphism $R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})} \rightarrow {{\,\mathrm{{\textbf {Q}}}\,}}_p\rightarrow k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$, where the first map is given by quotienting the maximal ideal and the second map is the natural inclusion. Then, $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}$ admits an action of $R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}$ and

$$\begin{aligned} R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}\otimes _{R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}} = R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{univ}\,}}}. \end{aligned}$$

Proof

Let us first explain the action of $R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}$ on $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}$. For any , observe that we have a natural morphism

$$\begin{aligned}{} & {} {{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}(A) \rightarrow {{\,\textrm{Hom}\,}}_{{{\,\textrm{cts}\,}}}(T_{{{\,\textrm{GL}\,}}_g,1}, A^\times ), \\{} & {} \rho _A^{{{\,\textrm{spin}\,}}}\mapsto ((\delta _{A,g+1}')^{-1}|_{{{\,\mathrm{{\textbf {Z}}}\,}}_p^\times }-g, (\delta _{A, g}')^{-1}|_{{{\,\mathrm{{\textbf {Z}}}\,}}_p^{\times }} - (g-1), \ldots , (\delta _{A,2}')^{-1}|_{{{\,\mathrm{{\textbf {Z}}}\,}}_p^\times }-1). \end{aligned}$$

Under this map, the image of $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}$ is exactly $k = (k_1, \ldots , k_g)$ by (6). Consequently, there is a natural morphism

$$\begin{aligned} {{\,\mathrm{{\textbf {Z}}}\,}}_p{{\,\mathrm{\![\![\!}\,}}T_{{{\,\textrm{GL}\,}}_g, 1}{{\,\mathrm{\!]\!]}\,}}\rightarrow R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}, \end{aligned}$$

which factors through $R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}$.

Since the refinement ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ satisfies (REG), together with the relation of parameters and the condition of fixed determinant, the desired isomorphism follows from the constant weight lemma ([2, Proposition 2.5.4]), i.e., the crystalline deformations of $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ are of constant Hodge–Tate weight, of which being the same as $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$. $\square$

Lemma 3.24

Denote by $H_{{{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}})$ the tangent space ${{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}(k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}[\varepsilon ])$ of ${{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}$. We have an exact sequence

$$\begin{aligned} 0 \rightarrow H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}) \rightarrow H_{{{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}) \rightarrow k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}^g. \end{aligned}$$

Proof

Following [2, Proposition 7.6.4], we expect an exact sequence

$$\begin{aligned} 0 \rightarrow H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}) \rightarrow H_{{{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}) \rightarrow k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{2^g}. \end{aligned}$$

The first map is clear while the second map is defined as follows. For any , we have

$$\begin{aligned} {{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}(A) \rightarrow {{\,\textrm{Hom}\,}}_{{{\,\textrm{cts}\,}}}({{\,\mathrm{{\textbf {Q}}}\,}}_p^\times , A^\times )^{2^g}, \quad \rho _A\mapsto (\delta _{A, 1}, \ldots , \delta _{A, 2^g}). \end{aligned}$$

Composing with the derivative at 1, we obtain a morphism

$$\begin{aligned} {{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}(A) \rightarrow A^{2^g}. \end{aligned}$$

That is, we obtain

$$\begin{aligned} \partial : {{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}} \rightarrow \widehat{{{\,\mathrm{\mathbb {G}}\,}}_m^{2^g}}. \end{aligned}$$

The second map is then defined to be $\partial (k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}[\varepsilon ])$. Lemma 3.23 shows that $H_{f}^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}) = \ker \partial (k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}[\varepsilon ])$.

Recall that the local condition of ${{\,\mathrm{\mathscr {D}}\,}}^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}$ at p requires a relation of the parameters and a fixed determinant. Thus, the image of $\partial (k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}[\varepsilon ])$ lies in a subspace of dimension g, depending only on the continuous characters $\delta _{A, 2}'$, ..., $\delta _{A, g+1}'$. $\square$

Proposition 3.25

Retain the notation in Lemma 3.21 and assume Hypothesis 3 holds.

• (i) There exists a canonical ring homomorphism $R^{{{\,\textrm{univ}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}\rightarrow {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$.

• (ii) If the adjoint Bloch–Kato Selmer group $H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})$ vanishes, then the canonical map in (i) is an isomorphism $R^{{{\,\textrm{univ}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}\simeq {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ (an ‘infinitesimal $R=T$ theorem’).

Proof

By Lemma 3.21 and Hypothesis 3, for any ideal ${{\,\mathrm{\mathfrak {I}}\,}}\subset {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ of cofinite length, there is a canonical ring homomorphism

$$\begin{aligned} R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}} \rightarrow {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/{{\,\mathrm{\mathfrak {I}}\,}}. \end{aligned}$$

This ring homomorphism is surjective due to the fact that the characteristic polynomials of the Frobenii under $\rho _{{{\,\mathrm{\mathfrak {I}}\,}}}$ are given by the Hecke polynomials. Consequently, one obtains a canonical morphism

$$\begin{aligned} R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}} \rightarrow {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}} = \varprojlim _{{{\,\mathrm{\mathfrak {I}}\,}}\text { : cofinite length}} {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/{{\,\mathrm{\mathfrak {I}}\,}}\end{aligned}$$

with dense image. Since $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}$ is complete, the canonical morphism $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}} \rightarrow {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ is surjective.

Finally, if $H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})$ vanishes, then the exact sequence in Lemma 3.24 implies that

$$\begin{aligned} \dim _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} H_{{{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho ^{{{\,\textrm{spin}\,}}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}) \le g. \end{aligned}$$

Since $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}$ is a local noetherian ring, its Krull dimension is bounded by the dimension of its tangent space ([30, Section 00KD]), i.e., $\dim R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}} \le g$. Moreover, we also know from loc. cit. that the equality holds if and only if $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}$ is regular. However, since ${{\,\mathrm{\mathcal {E}}\,}}_0$ is equidimensional and finite over ${{\,\mathrm{\mathcal {W}}\,}}$, we know that $\dim {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}} = \dim {{\,\mathrm{\mathcal {W}}\,}}= g$. Therefore,

$$\begin{aligned} g \ge \dim R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}} \ge \dim {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}} = g \end{aligned}$$

and $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}$ is regular of dimension g. To conclude the proof, suppose ${{\,\mathrm{\mathfrak {a}}\,}}= \ker (R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}} \rightarrow {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}})$ is non-zero and so we can identify ${{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ with $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}/{{\,\mathrm{\mathfrak {a}}\,}}$. Since $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}$ is a regular local ring, it is a domain ([30, Lemma 00NP]). We then obtain a contradiction

$$\begin{aligned} g = \dim R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}} > \dim R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}/{{\,\mathrm{\mathfrak {a}}\,}}= \dim {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}} = g. \end{aligned}$$

$\square$

Due to the nice property stated in Proposition 3.25, we will from now on assume the truthfulness of Hypothesis 3.

Corollary 3.26

Suppose Hypothesis 1, Hypothesis 2, and Hypothesis 3 hold. Assume the following also hold:

• The cuspidal automorphic representation $\pi _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ of ${{\,\textrm{GL}\,}}_{2^g}({{\,\mathrm{{\textbf {A}}}\,}}_L)$ associated with $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ as in Hypothesis 2 is regular algebraic and polarised (see, for example, [4, §2.1]).

• The image $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}({{\,\textrm{Gal}\,}}_{L(\zeta _{p^{\infty }})})$ is enormous (see [25, Definition 2.27]).

Then

• (i) $H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}) = 0$; and

• (ii) $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}} \simeq {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$.

Proof

By the discussion around (7), we have

$$\begin{aligned} H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}) \subset H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}). \end{aligned}$$

However, the latter space vanishes by [25, Theorem 5.3] and so we conclude by Proposition 3.25. $\square$

We conclude this paper with another situation when one can also deduce the vanishing of the adjoint Bloch–Kato Selmer group.

Corollary 3.27

Suppose Hypothesis 1 and Hypothesis 3 hold. Suppose the weight map is étale at ${{\,\mathrm{{\varvec{{x}}}}\,}}$ and suppose the canonical morphism $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}} \rightarrow {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ is an isomorphism. Then,

$$\begin{aligned} H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}) = 0. \end{aligned}$$

Proof

Observe the following sequence of isomorphisms

$$\begin{aligned} \Omega _{{{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}}^1\otimes _{{{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}&\simeq \Omega _{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}/R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}}^1\otimes _{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}\\&\simeq \Omega _{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}/R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}}^1 \widehat{\otimes }_{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}} R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{univ}\,}}} \otimes _{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{univ}\,}}}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}\\&\simeq \Omega _{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}/R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}}^1 \widehat{\otimes }_{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}} R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}} \otimes _{R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}} \otimes _{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{univ}\,}}}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}\\&\simeq \Omega _{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}}/R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}}^1 \otimes _{R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}} \otimes _{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{univ}\,}}}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}\\&\simeq \Omega _{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{univ}\,}}}/k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^1 \otimes _{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{univ}\,}}}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}. \end{aligned}$$

Here, the first isomorphism follows from the assumption $R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, {{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^{{{\,\textrm{univ}\,}}} \simeq {{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ and the third and the final isomorphism follows from Lemma 3.23. Therefore, we have

$$\begin{aligned} \dim _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})&= \dim _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} {{\,\textrm{Hom}\,}}_{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}}(\Omega _{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{univ}\,}}}/k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}}^1 \otimes _{R_{{{\,\mathrm{{\varvec{{x}}}}\,}}, f}^{{{\,\textrm{univ}\,}}}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}, k_{{{\,\mathrm{{\varvec{{x}}}}\,}}})\\&= \dim _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} {{\,\textrm{Hom}\,}}_{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}}(\Omega _{{{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}}^1\otimes _{{{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}, k_{{{\,\mathrm{{\varvec{{x}}}}\,}}})\\&= \dim _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} \Omega _{{{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}}^1\otimes _{{{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}\\&\le {{\,\textrm{length}\,}}_{{{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}}\Omega _{{{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}}^1. \end{aligned}$$

However, since the weight map is étale at ${{\,\mathrm{{\varvec{{x}}}}\,}}$, ${{\,\textrm{length}\,}}_{{{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}}\Omega _{{{\,\mathrm{\mathbb {T}}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}}/R_{{{\,\textrm{wt}\,}}({{\,\mathrm{{\varvec{{x}}}}\,}})}}^1 = 0$. We then conclude the result. $\square$

Remark 3.28

Suppose we are now working with the strict Iwahori level Siegel modular variety and suppose the p-adic adjoint L-function $L^{{{\,\textrm{adj}\,}}}$ in [35] is defined at ${{\,\mathrm{{\varvec{{x}}}}\,}}$. Suppose we are also in the situation of Corollary 3.27. Then, by [35, Theorem 4.3.5], we then have

$$\begin{aligned} {{\,\textrm{ord}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}} L^{{{\,\textrm{adj}\,}}} = 0 = \dim _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}). \end{aligned}$$

Such a relation then (conjecturally) justifies the name of $L^{{{\,\textrm{adj}\,}}}$. More generally, in light of the Bloch–Kato conjecture (Conjecture 1), we expect that, if ${{\,\mathrm{{\varvec{{x}}}}\,}}$ is a smooth point,

$$\begin{aligned} {{\,\textrm{ord}\,}}_{{{\,\mathrm{{\varvec{{x}}}}\,}}} L^{{{\,\textrm{adj}\,}}} = \dim _{k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}} H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0 \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}). \end{aligned}$$

In particular, since $H_f^1({{\,\mathrm{{\textbf {Q}}}\,}}, {{\,\textrm{ad}\,}}^0 \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}})$ is expected to vanish, it seems fair to expect that, if ${{\,\mathrm{{\varvec{{x}}}}\,}}$ is a smooth point with small slope and at which $L^{{{\,\textrm{adj}\,}}}$ is defined, the weight map is étale at ${{\,\mathrm{{\varvec{{x}}}}\,}}$. When $g=1$, this is [3, Theorem 2.16].

3.6 Appendix: Examples

In our main result (Corollary 3.26), many assumptions are made. The purpose of this appendix is to provide examples as evidence that we did not make vacuum assumptions. For the convenience of the readers, we recall the suppositions:

• The point ${{\,\mathrm{{\varvec{{x}}}}\,}}\in {{\,\mathrm{\mathcal {E}}\,}}_0$ corresponds to a p-stabilisation of an eigenclass of tame level.

• Hypothesis 1 holds. In particular, one can attach a ${{\,\textrm{GSpin}\,}}_{2g+1}$-valued Galois representation $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}$ to ${{\,\mathrm{{\varvec{{x}}}}\,}}$, which is crystalline when restricting to ${{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}$. And we let $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}} := {{\,\textrm{spin}\,}}\circ \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}}$.

• Hypothesis 2 holds, i.e., the potential spin functoriality holds. Moreover, the cuspidal automorphic representation $\pi _{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ of ${{\,\textrm{GL}\,}}_{2^g}({{\,\mathrm{{\textbf {A}}}\,}}_L)$ is regular algebraic and polarised.

• Hypothesis 3 holds, i.e., the Galois representation valued in the Hecke algebra satisfies the desired deformation conditions.

• The restriction $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ admits a refinement ${{\,\mathrm{\mathbb {F}}\,}}_{\bullet }^{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ that satisfies (REG) and (NCR).

• The restriction $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ is not isomorphic to its twist by the p-adic cyclotomic character.

• The image $\rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}({{\,\textrm{Gal}\,}}_{L(\zeta _{p^{\infty }})})$ is enormous.

In what follows, we discuss examples for $g=1, 2$. In these cases, Hypothesis 1 is well-understood by mathematicians (see Remark 3.11) and so we will skip the discussions. Note also that Hypothesis 2 is trivial when $g=1$.

Example 3.29

Our first example concerns $g=1$ and suppose ${{\,\mathrm{{\varvec{{x}}}}\,}}$ corresponds to a p-stabilisation of a weight-k normalised newform $f = \sum _{n>0}a_nq^n$ of level $\Gamma (N)$ with $p\not \mid N$ and $p > k \ge 2$. We assume that f is not a CM form. The Hecke polynomial of f at p is given by

$$\begin{aligned} Y^2 - a_pY + p^{k-1}. \end{aligned}$$

We assume that the two roots $\alpha$, $\beta$ are distinct.

In this case, by the result in [12], we know that the associated Galois representation

$$\begin{aligned} \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}} = \rho _f: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}} \rightarrow {{\,\textrm{GL}\,}}_2(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p) \end{aligned}$$

of f is irreducible and of Hodge–Tate weight $(0, k-1)$ at p. Moreover, $\rho _f|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ is crystalline by [28, Theorem 1.2.4].

Let’s now check the conditions imposed on $\rho _f$. First of all, it is easy to see that $\rho _f|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ is not isomorphic to its twist by the p-adic cyclotomic character. Moreover, since $\rho _f|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ is a 2-dimensional crystalline representation, it satisfies (NCR) by [2, Remark2.4.6]. To check (REG), note that the characteristic polynomial of the crystalline Frobenius $\varphi$ is equal to the Hecke polynomial at p ( [28, Theorem 1.2.4]). Since $\alpha \ne \beta$, $\alpha$ and $\alpha \beta = p^{k-1}$ are eigenvalues of the crystalline Frobenii on ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(\rho _{f}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}})$ and ${{\,\mathrm{{\textbf {D}}}\,}}_{{{\,\textrm{cris}\,}}}(\wedge ^2 \rho _{f}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}})$ respectively with multiplicity one. Finally, combining the result in [25, Example 2.3.4] and [26], we know that $\rho _f({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}(\zeta _{p^{\infty }})})$ is enormous.

It remains to check Hypothesis 3. The first point in Hypothesis 3 is trivial. Additionally, the second point follows from that the Galois representations of finite-slope overconvergent eigenforms are triangulline ([21, Theorem 6.3] and [8, Proposition 4.3]). Finally, by the discussions in [10, §3.2], we know that the deformation valued in the Hecke algebra is minimally ramified at $\ell |N$ and hence the last point in Hypothesis 3. $\square$

Example 3.30

In this example, we let $g=2$ and suppose ${{\,\mathrm{{\varvec{{x}}}}\,}}$ corresponds to a p-stabilisation of a discrete series cuspidal automorphic representation $\pi _{{{\,\textrm{GSp}\,}}_4}$ of ${{\,\textrm{GSp}\,}}_{4}({{\,\mathrm{{\textbf {A}}}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}})$ which is spherical at p and of cohomological weight $k = (k_1, k_2)$ with $k_1 \ge k_2 \ge 0$. We assume $\pi _{{{\,\textrm{GSp}\,}}_4}$ is neither CAP nor endoscopic.

In this case, the Galois representation

$$\begin{aligned} \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}}^{{{\,\textrm{spin}\,}}} = \rho _{\pi _{{{\,\textrm{GSp}\,}}_4}}: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}} \rightarrow {{\,\textrm{GSpin}\,}}_5(k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}) \simeq {{\,\textrm{GSp}\,}}_4(k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}) \end{aligned}$$

associated to $\pi _{{{\,\textrm{GSp}\,}}_4}$ is irreducible, where $k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}$ is a large enough finite extension of ${{\,\mathrm{{\textbf {Q}}}\,}}_p$. The Hodge–Tate weight of $\rho _{\pi _{{{\,\textrm{GSp}\,}}_4}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ is $(0, k_2+1, k_1+2, k_1+k_2+3)$. Moreover, $\rho _{\pi _{{{\,\textrm{GSp}\,}}_4}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ is crystalline and its characteristic polynomial of the crystalline Frobenius coincides with the Hecke polynomial at p. We impose the following assumptions on the Galois representation $\rho _{\pi _{{{\,\textrm{GSp}\,}}_4}}$:

• We assume that the Hecke polynomial at p is decomposed as

  $$\begin{aligned} P_{{{\,\textrm{Hecke}\,}}, p} = (Y-\alpha )(Y-\beta )(Y-\gamma )(Y-\delta ) \end{aligned}$$

  with distinct roots $\alpha$, $\beta$, $\gamma$, $\delta$. Note that all the roots are of Weil weight $k_1+k_2-3$ ([34, Theorem 1]).

• Let ${{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}$ be the finite set, consisting of primes at which $\pi _{{{\,\textrm{GSp}\,}}_4}$ is not spherical.⁹ We assume that if $\ell \in {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}$, then the restriction of the residual representation $\overline{\rho }_{\pi _{{{\,\textrm{GSp}\,}}_4}}|_{I_{\ell }}$ is absolutely irreducible and $p\not \mid \ell ^{12}-1$.

Note that the spin representation ${{\,\textrm{spin}\,}}: {{\,\textrm{GSpin}\,}}_5 \rightarrow {{\,\textrm{GL}\,}}_4$ is nothing but the natural embedding of ${{\,\textrm{GSp}\,}}_4 \hookrightarrow {{\,\textrm{GL}\,}}_4$. Following the discussion in [11, §2], we know that there is a cuspidal automorphic representation $\pi _{{{\,\textrm{GL}\,}}_4}$ of ${{\,\textrm{GL}\,}}_4({{\,\mathrm{{\textbf {A}}}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}})$, which is regular algebraic and polarised,¹⁰ such that the associated Galois representation $\pi _{{{\,\textrm{GL}\,}}_4}$ is

$$\begin{aligned} \rho _{{{\,\mathrm{{\varvec{{x}}}}\,}}} = \rho _{\pi _{{{\,\textrm{GL}\,}}_4}} : {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}} \xrightarrow {\rho _{\pi _{{{\,\textrm{GSp}\,}}_4}}} {{\,\textrm{GSp}\,}}_4(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p) \hookrightarrow {{\,\textrm{GL}\,}}_4(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_p). \end{aligned}$$

Let’s now check the conditions on $\rho _{\pi _{{{\,\textrm{GL}\,}}_4}}$. First of all, one sees that $\rho _{\pi _{{{\,\textrm{GL}\,}}_4}}|_{{{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}_p}}$ is not isomorphic to its twist by the p-adic cyclotomic character by comparing the Hodge–Tate weights on both sides. Next, since $0< k_2+1< k_1+2 < k_1+k_2+3$, we can apply [2, Proposition 2.4.7] and know that (NCR) is satisfied. Moreover, since $\alpha$, $\beta$, $\gamma$, $\delta$ are distinct but with the same Weil weight, we see that (REG) is also satisfied.

We show that $\rho _{\pi _{{{\,\textrm{GL}\,}}_4}}({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}(\zeta _{p^{\infty }})})$ is enormous. First, note that if the Zariski closure of $\rho _{\pi _{{{\,\textrm{GL}\,}}_4}}({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}(\zeta _p^{\infty })})$ in ${{\,\textrm{GL}\,}}_4$ contains ${{\,\textrm{Sp}\,}}_4$, then it is enormous by [25, Lemma 2.33]. Using the strategy in [op. cit, Example 2.34], it is enough to show that the Zariski closure of $\rho _{\pi _{{{\,\textrm{GL}\,}}_4}}({{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}})$ in ${{\,\textrm{GL}\,}}_4$ contains ${{\,\textrm{Sp}\,}}_4$. However, since $\pi _{{{\,\textrm{GSp}\,}}_4}$ is neither CAP nor endoscopic, the desired result follows from the discussion in [18, §9.3.4].

Finally, we check Hypothesis 3. The first point holds by the argument of [15, Lemma 4.3.3]. The second point holds due to the fact that the Galois representations of finite-slope overconvergent Siegel modular forms of genus 2 are triangulline ( [9, Theorem 13.3]). For the third point, we first remark that, by [15, Lemma 4.3.6], $\rho _{\pi _{{{\,\textrm{GSp}\,}}_4}}(I_{\ell }) = \rho _{\pi _{{{\,\textrm{GL}\,}}_4}}(I_{\ell })$ is finite of order prime to p. Hence, we can verify Hypothesis 3 (iii) by the following lemma:

Lemma 3.31

Let and let $\rho _A: {{\,\textrm{Gal}\,}}_{{{\,\mathrm{{\textbf {Q}}}\,}}} \rightarrow {{\,\textrm{GSp}\,}}_4(A)$ be a representation such that $\rho _A \otimes _{A}k_{{{\,\mathrm{{\varvec{{x}}}}\,}}} \simeq \rho _{\pi _{{{\,\textrm{GSp}\,}}_4}}$. Then, for $\ell \in {{\,\mathrm{\texttt {S}_{{{\,\textrm{bad}\,}}}}\,}}$, $\rho _A(I_{\ell })$ is finite of order prime to p. In particular, we have

$$\begin{aligned} \rho _A(I_{\ell }) \simeq \rho _{\pi _{{{\,\textrm{GSp}\,}}_4}}(I_{\ell }). \end{aligned}$$

Proof

The proof of this lemma is basically [15, Lemma 4.3.3].

Note first that $\ker ({{\,\textrm{GSp}\,}}_4(A) \rightarrow {{\,\textrm{GSp}\,}}_4(k_{{{\,\mathrm{{\varvec{{x}}}}\,}}}))$ is a locally pro-p group. Hence, it suffices to show that $\rho _A(I_{\ell })$ is finite of order prime to p.

To show this, we make a further reduction. Let $I_{\ell }^{(\ell )}$ be the pro-$\ell$ Sylow subgroup of $I_{\ell }$. That is, $I_{\ell }^{(\ell )}$ is the Galois group of the maximal tamely ramified extension ${{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }^{\textrm{tame}}$ of ${{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }$. Since $I_{\ell }^{(\ell )}$ is pro-$\ell$ and ${{\,\textrm{GSp}\,}}_4(A)$ is locally pro-p, the image $\rho _A(I_{\ell }^{(\ell )})$ is finite. Let $\widetilde{F}$ be the finite Galois extension of ${{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }^{\textrm{tame}}$ defined by $\ker \rho _A|_{I_{\ell }^{(\ell )}}$. Then, there exists a finite Galois extension F of ${{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }^{\textrm{unr}}$ such that $\widetilde{F} = F{{\,\mathrm{{\textbf {Q}}}\,}}_{\ell }^{\textrm{tame}}$. Moreover, if we let $I_{F, \ell } := {{\,\textrm{Gal}\,}}(\overline{{{\,\mathrm{{\textbf {Q}}}\,}}}_{\ell }/F)$ and $I_{F, \ell }^{(\ell )} := \ker \rho _A|_{I_{\ell }^{(\ell )}}$, then

$$\begin{aligned} I_{\ell }/I_{\ell }^{(\ell )} \simeq I_{F, \ell }/I_{F, \ell }^{(\ell )} \end{aligned}$$

and so it suffice to show know that $\rho _A(I_{F, \ell })$ is finite, providing $\rho _A|_{I_{F, \ell }^{(\ell )}}$ being trivial.

Recall that

$$\begin{aligned} I_{F, \ell }/I_{F, \ell }^{(\ell )} \simeq I_{\ell }/I_{\ell }^{(\ell )} \simeq \prod _{q\ne \ell }{{\,\mathrm{{\textbf {Z}}}\,}}_{q}(1). \end{aligned}$$

Therefore, via the isomorphisms above, we only need to show $\rho _A({{\,\mathrm{{\textbf {Z}}}\,}}_p(1))$ is trivial. We prove this in the following two steps:

Let $\xi \in {{\,\mathrm{{\textbf {Z}}}\,}}_p(1)$ be a topological generator. We first claim that $\rho _A(\xi )$ is unipotent. Suppose $\rho _A(\xi )$ is not unipotent, then it would admit an eigenvalue $\epsilon \ne 1$. By conjugating with the $\rho _A({{\,\textrm{Frob}\,}}_{\ell })$, we see that $\rho _A(\xi )$ and $\rho _A(\xi ^{\ell })$ have same eigenvalues. By iterating such a process, we learn that $\{\epsilon , \epsilon ^{\ell }, \epsilon ^{\ell ^2}, \epsilon ^{\ell ^3}\}$ is a subset of eigenvalues of $\rho _A(\xi )$ while $\{\epsilon ^{\ell }, \epsilon ^{\ell ^2}, \epsilon ^{\ell ^3}, \epsilon ^{\ell ^4}\}$ is a subset of eigenvalues of $\rho _A(\xi ^{\ell })$. Comparing these two sets, one deduces the identity

$$\begin{aligned} \epsilon = \epsilon ^{\ell ^{12}}. \end{aligned}$$

In particular, $\epsilon$ is a root of unity. On the other hand, since $\xi$ is a topological generator of the pro-p group, $\epsilon$ can only be a p-power root of unity. Thus, we have

$$\begin{aligned} p| \ell ^{12} -1, \end{aligned}$$

which contradicts to the assumption that $p\not \mid \ell ^{12} - 1$.

Finally, we claim that $\rho _A(\xi ) = 1$. If $\rho _A(\xi ) \ne 1$, then it would fix a subspace V of $A^4$, which is stable under the action of $I_{\ell }$. On the other hand, since $\overline{\rho }_{\pi _{{{\,\textrm{GSp}\,}}_4}}|_{I_{\ell }}$ is irreducible, $\rho _{\pi _{{{\,\textrm{GSp}\,}}_{4}}}|_{I_{\ell }}$ is irreducible and so is $\rho _A|_{I_{\ell }}$. The existence of V then contradicts the irreducibility. $\square$