2.1 Galois deformation rings

Let ${\mathbf {E}}/{\mathbf {Q}}_p$ be a finite extension, with ring of integers ${\mathscr {O}}_E$ , uniformizer $\varpi _E$ , and residue field ${\mathbf {F}}$ , and let G be a profinite group satisfying Mazur’s condition $\Phi _p$ . The two cases we will be most interested in are $G=\operatorname {\mathrm {Gal}}_K$ and $G=\operatorname {\mathrm {Gal}}_{F,S}$ , where K is a finite extension of ${\mathbf {Q}}_p$ , and F is a number field, and S is a set of places of F.

Suppose we have a continuous homomorphism $\overline {\rho }:G\rightarrow \operatorname {\mathrm {GL}}_d({\mathbf {F}})$ . Then we may construct the universal framed deformation ring $R_{\overline {\rho }}^\square $ , which prorepresents the functor

$$\begin{align*}A\rightsquigarrow \{\rho:G\rightarrow \operatorname{\mathrm{GL}}_d(A)\mid \rho\equiv \overline{\rho}\quad\pmod{\mathfrak{m}_A}\} \end{align*}$$

on the category of complete local Noetherian ${\mathscr {O}}_E$ -algebras with residue field ${\mathbf {F}}$ , of lifts of $\overline {\rho }$ , that is, deformations of $\overline {\rho }$ together with a basis. If $\operatorname {\mathrm {End}}_G(\overline {\rho })={\mathbf {F}}$ (for example, if $\overline {\rho }$ is absolutely irreducible), we additionally have the universal (unframed) deformation ring $R_{\overline {\rho }}$ parametrizing deformations of $\rho $ .

If R is a complete local Noetherian ${\mathscr {O}}_E$ -algebra with maximal ideal $\mathfrak {m}_R$ and finite residue field, and $\psi :\operatorname {\mathrm {Gal}}_K\rightarrow R^{\times }$ is a continuous character such that $\det \overline {\rho }=\psi \mod \mathfrak {m}_R$ , there is a quotient $R\mathop {\widehat \otimes } R_{\overline {\rho }}^\square \twoheadrightarrow R_{\overline {\rho }}^{\square ,\psi }$ parametrizing lifts of $\overline {\rho }$ with determinant $\psi $ . Indeed, there is a homomorphism $R_{\det \overline {\rho }}\rightarrow R_{\overline {\rho }}^\square $ given by the determinant map, and the choice of $\psi $ defines a homomorphism $R_{\det \overline {\rho }}\rightarrow R$ ; then $R_{\overline {\rho }}^{\square ,\psi }=R\mathop {\widehat \otimes }_{R_{\det \overline {\rho }}}R_{\overline {\rho }}^\square $ . If $\operatorname {\mathrm {End}}_G(\overline {\rho })={\mathbf {F}}$ , there is similarly a quotient $R\mathop {\widehat \otimes } R_{\overline {\rho }}\twoheadrightarrow R_{\overline {\rho }}^\psi $ parametrizing deformations of $\overline {\rho }$ with determinant $\psi $ .

Now, we specialize to the arithmetic situations of interest. Let $K/{\mathbf {Q}}_p$ be a finite extension, and assume that $\operatorname {\mathrm {Hom}}(K,E)$ has cardinality $[K:{\mathbf {Q}}_p]$ . Then by [Reference Böckle, Iyengar and PaškūnasBIP23a, Corollary 3.37], $R_{\overline {\rho }}^\square $ is a complete intersection, and by [Reference Böckle, Iyengar and PaškūnasBIP23a, Corollary 4.21] the irreducible components of $\operatorname {\mathrm {Spec}} R_{\overline {\rho }}^\square $ are in bijection with the irreducible components of $\operatorname {\mathrm {Spec}} R_{\det \overline {\rho }}$ . More precisely, if $\mu :=\mu _{p^\infty }(K)$ denotes the p-power roots of unity in $K^{\times }$ , local class field theory identifies it with a subgroup of $\operatorname {\mathrm {Gal}}_K^{\mathrm {ab}}$ ; by [Reference Böckle, Iyengar and PaškūnasBIP23a, Lemma 4.1] $R_{\det \overline {\rho }}$ is a power series ring over ${\mathscr {O}}_E[\mu ]$ , so its irreducible components are in bijection with characters $\chi :\mu \rightarrow {\mathscr {O}}_E^{\times }$ . There are quotients $R_{\overline {\rho }}^\square \twoheadrightarrow R_{\overline {\rho }}^{\square ,\chi }$ parametrizing lifts of $\overline {\rho }$ whose determinant restricted to $\mu $ (via the Artin map) agrees with $\chi $ , and by [Reference Böckle, Iyengar and PaškūnasBIP23a, Corollary 4.5, Corollary 4.19] the rings $R_{\overline {\rho }}^{\square ,\chi }$ are normal domains and complete intersections. In particular, $R_{\overline {\rho }}^{\square }$ is reduced.

Let F be a number field, and let $\Sigma _p:=\{v\mid p\}$ . If $\rho :\operatorname {\mathrm {Gal}}_F\rightarrow \operatorname {\mathrm {GL}}_d({\mathbf {F}})$ is a continuous representation and v is a place of F, we let $\rho _v$ denote $\rho |_{\operatorname {\mathrm {Gal}}_{F_v}}$ . Suppose that $\overline {\rho }$ is absolutely irreducible, and let S be a finite set of places of F containing $\Sigma _p$ and the infinite places such that $\overline {\rho }$ is unramified outside S. Then we let $R_{\overline {\rho },S}$ denote the universal deformation ring parametrizing deformations unramified outside of S, and we let $R_{\overline {\rho },S}^{\square }$ denote the universal deformation ring whose A-points are deformations $\rho _A$ of $\overline {\rho }$ unramified outside of S, together with bases for $\rho _A|_{\operatorname {\mathrm {Gal}}_{F_v}}$ for each $v\in \Sigma _p$ . We also let $R_{\overline {\rho },\mathrm {loc}}^{\square }:=\otimes _{v\in \Sigma _p}R_{\overline {\rho }_v}^{\square }$ .

If $\psi :\operatorname {\mathrm {Gal}}_F\rightarrow R^{\times }$ is a continuous character as above, we let

$$ \begin{align*} R_{\overline{\rho},S}^\psi&:=R\mathop{\widehat\otimes}_{R_{\det\overline{\rho},S}}R_{\overline{\rho},S} \\ R_{\overline{\rho},S}^{\square,\psi}&:=R\mathop{\widehat\otimes}_{R_{\det\overline{\rho},S}}R_{\overline{\rho},S}^{\square} \\ R_{\overline{\rho},\mathrm{loc}}^{\square,\psi}&:=R\mathop{\widehat\otimes}_{R_{\det\overline{\rho},\mathrm{loc}}}R_{\overline{\rho},\mathrm{loc}}^{\square}. \end{align*} $$

For any place $v\in \Sigma _p$ , restriction from $\operatorname {\mathrm {Gal}}_{F,S}$ to $\operatorname {\mathrm {Gal}}_{F_v}$ defines a homomorphism $R_{\overline {\rho }_v}^\square \rightarrow R_{\overline {\rho },S}^\square $ , and so we obtain homomorphisms

$$\begin{align*}R_{\overline{\rho},\mathrm{loc}}^{\square}\rightarrow R_{\overline{\rho},S}^\square \end{align*}$$

and

$$\begin{align*}R_{\overline{\rho},\mathrm{loc}}^{\square,\psi}\rightarrow R_{\overline{\rho},S}^{\square,\psi}. \end{align*}$$

We can relate our local and global deformation rings more precisely:

Lemma 2.1.1. Suppose that $p\nmid d$ . Let $h^1$ denote the dimension (as an ${\mathbf {F}}$ -vector space) of

$$\begin{align*}\ker\,\left(H^1(\operatorname{\mathrm{Gal}}_{F,S},\operatorname{\mathrm{ad}}^0(\overline{\rho}))\rightarrow \prod_{v\in \Sigma_p}H^1(\operatorname{\mathrm{Gal}}_{F_v},\operatorname{\mathrm{ad}}^0(\overline{\rho}_v))\right), \end{align*}$$

let $\delta _F:=\dim _{{\mathbf {F}}}H^0(\operatorname {\mathrm {Gal}}_{F,S},\operatorname {\mathrm {ad}}\overline {\rho })$ , and for $v\in \Sigma _p$ let $\delta _v:=\dim _{{\mathbf {F}}}H^0(\operatorname {\mathrm {Gal}}_{F_v},\operatorname {\mathrm {ad}}\overline {\rho }_v)$ . Then $R_{\overline {\rho },S}^{\square ,\psi }$ can be topologically generated over $R_{\overline {\rho },\mathrm {loc}}^{\square ,\psi }$ by $g:=h^1+\sum _{v\in \Sigma _p}\delta _v-\delta _F$ elements.

2.2 Deformations of trianguline $(\varphi ,\Gamma )$ -modules

Trianguline $(\varphi ,\Gamma )$ -modules are those which are extensions of $(\varphi ,\Gamma )$ -modules of character type. More precisely,

As in the characteristic $0$ situation treated in [Reference Bellache and ChenevierBC09, §2.3], we may define and study deformations of trianguline $(\varphi ,\Gamma )$ -modules:

Definition 2.2.2. Let R be a finite extension of ${\mathbf {F}}_p(\!(u)\!)$ , and let D be a fixed $(\varphi ,\Gamma _{K})$ -module of rank d over $\Lambda _{R,\mathrm {rig},K}$ equipped with a triangulation $\operatorname {\mathrm {Fil}}^{\bullet } D$ with parameter $\underline \delta $ . Let $\mathcal {C}_R$ denote the category of Artin local ${\mathbf {Z}}_p$ -algebras $R'$ equipped with an isomorphism $R'/\mathfrak {m}_{R^{\prime }}\xrightarrow {\sim }R$ . The trianguline deformation functor $\mathrm {Def}_{D,\operatorname {\mathrm {Fil}}^{\bullet }}:\mathcal {C}_R\rightarrow \underline {\mathrm {Set}}$ is defined to be the set of isomorphism classes

$$\begin{align*}\mathrm{Def}_{D,\operatorname{\mathrm{Fil}}^{\bullet}}(R'):=\{(D_{R^{\prime}},\operatorname{\mathrm{Fil}}^{\bullet} D_{R^{\prime}}, \iota)\}/\sim, \end{align*}$$

where $D_{R^{\prime }}$ is a $(\varphi ,\Gamma _{K})$ -module over $\Lambda _{R^{\prime },\mathrm {rig},K}$ , $\operatorname {\mathrm {Fil}}^{\bullet } D_{R^{\prime }}$ is a triangulation, and $\iota :R\otimes _{R^{\prime }}D_{R^{\prime }}\xrightarrow {\sim }D$ is an isomorphism which also defines isomorphisms $R\otimes _{R^{\prime }}\operatorname {\mathrm {Fil}}^iD_{R^{\prime }}\xrightarrow {\sim }\operatorname {\mathrm {Fil}}^iD$ .

One of the consequences of the proof of [Reference BellovinBel23b, Proposition 5.1] is that when $d=1$ , $\mathrm {Def}_{D,\operatorname {\mathrm {Fil}}^{\bullet }}$ is formally smooth. As in the characteristic $0$ situation, the same is true for general d, so long as the parameter satisfies a certain regularity condition. Note that the regularity condition in here is slightly different than in characteristic $0$ ; the additional characters avoided in the statement of [Reference Bellache and ChenevierBC09, Proposition 2.3.10] do not make sense in characteristic p.

In order to build moduli spaces of trianguline $(\varphi ,\Gamma )$ -modules, we will use moduli spaces of characters, as in [Reference BellovinBel23a, §2.3]. If G is a commutative p-adic Lie group and $G'\subset G$ is a compact subgroup such that $G/G'$ is free and finitely generated, then we have $\widehat {G'}:=\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![G']\!]$ and the pseudorigid spaces $\widehat {G'}^{\mathrm {{an}}}$ and $\widehat {G}^{\mathrm {{an}}}:=\operatorname {\mathrm {Spa}}({\mathbf {Z}}[G/G'],{\mathbf {Z}}){\times }_{{\mathbf {Z}}}\widehat {G'}^{\mathrm {{an}}}$ . If X is a pseudorigid space, we also have the pseudorigid space $\widehat G_X$ , which represents the functor

$$\begin{align*}Y\rightsquigarrow \operatorname{\mathrm{Hom}}_{\mathrm{{cts}}}(G,{\mathscr{O}}(Y)) \end{align*}$$

on the category of adic spaces over X.

In particular, if K is a finite extension of ${\mathbf {Q}}_p$ , we will be interested in $\widehat {K^{\times }}^{\mathrm {{an}}}$ and $\widehat {(K^{\times })^d}^{\mathrm {{an}}}$ for $d\geq 1$ :

We see that $\widehat {K^{\times }}^{\mathrm {{an}}}\cong {\mathbf {G}}_m^{\operatorname {\mathrm {ad}}}{\times }_{{\mathbf {Z}}}\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![{\mathscr {O}}_K^{\times }]\!]^{\mathrm {{an}}}$ , and $\mathcal {T}^d\cong {\mathbf {G}}_m^{\operatorname {\mathrm {ad}},d}{\times }_{{\mathbf {Z}}}\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![({\mathscr {O}}_K^{\times })^d]\!]^{\mathrm {{an}}}$ . Since ${\mathscr {O}}_K^{\times }$ is compact, $\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![{\mathscr {O}}_K^{\times }]\!]^{\mathrm {{an}}}$ is a quasicompact pseudorigid space; it has a finite cover $\{U_i:=\operatorname {\mathrm {Spa}} R_i\}$ by affinoid subspaces, and ${\mathbf {G}}_{m,U_i}$ is a rising union of relative annuli $C_{U_i,h}:=\operatorname {\mathrm {Spa}} R_i\left \langle u^hT, u^hT^{-1}\right \rangle $ .

If $K={\mathbf {Q}}_p$ , then $\widehat {{\mathbf {Q}}_p^{\times }}^{\mathrm {{an}}}$ has connected components indexed by the elements of $\mu _{p-1}$ , each of which is isomorphic to $\left (\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![{\mathbf {Z}}_p]\!]\right )^{\mathrm {{an}}}\times {\mathbf {G}}_m^{\operatorname {\mathrm {ad}}}$ .

Remark 2.2.5. In the pseudorigid setting (unlike the classical rigid analytic setting), it is not true that $\widehat {G_1\times G_2}^{\mathrm {{an}}}\cong \widehat {G_1}^{\mathrm {{an}}}\times \widehat {G_2}^{\mathrm {{an}}}$ . Indeed, $\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![T_1,T_2]\!]^{\mathrm {{an}}}$ consists of all valuations which do not vanish on all three of $p, T_1, T_2$ . But

$$\begin{align*}\operatorname{\mathrm{Spa}} {\mathbf{Z}}_p[\![T_1]\!]^{\mathrm{{an}}}{\times}_{{\mathbf{Z}}_p}\operatorname{\mathrm{Spa}} {\mathbf{Z}}_p[\![T_2]\!]^{\mathrm{{an}}} \end{align*}$$

also excludes valuations vanishing at both p and $T_1$ (or both p and $T_2$ ). In particular, $\mathcal {T}^d$ is not a product of copies of $\mathcal {T}$ .

Consider the functor $\mathcal {S}_d^\square $ on pseudorigid spaces defined via

$$\begin{align*}X\rightsquigarrow \{(D,\operatorname{\mathrm{Fil}}^{\bullet} D,\underline\delta,\underline\nu)\}/\sim, \end{align*}$$

where D is a trianguline $(\varphi ,\Gamma _K)$ -module with filtration $\operatorname {\mathrm {Fil}}^{\bullet } D$ and regular parameter $\underline \delta \in \mathcal {T}_{\mathrm {reg}}^d$ , and $\underline \nu $ is a sequence of trivializations $\nu _i:\operatorname {\mathrm {gr}}^iD\xrightarrow {\sim }\Lambda _{X,\mathrm {rig},K}$ . There is a natural transformation $\mathcal {S}_d^\square \rightarrow \mathcal {T}_{\mathrm {reg}}^d$ given on X-points by

$$\begin{align*}(D,\operatorname{\mathrm{Fil}}^{\bullet} D,\underline\delta,\underline\nu)\rightsquigarrow \underline\delta. \end{align*}$$

Exactly as in [Reference ChenevierChe13, Théorème 3.3] and [Reference Hellmann and SchraenHS16, Theorem 2.4], we have the following:

One proves by induction on d that if D is a trianguline $(\varphi ,\Gamma _K)$ -module over X with parameter $\underline \delta \in (\mathcal {T}_{\mathrm {reg}})^d$ , then $H_{\varphi ,\Gamma _K}^1(D)$ is a vector bundle over X of rank $d[K:{\mathbf {Q}}_p]$ (the regularity assumption ensures that $H_{\varphi ,\Gamma _K}^0(D)=H_{\varphi ,\Gamma _K}^2(D)=0$ ). Now, $\mathcal {S}_1^\square =\mathcal {T}=\mathcal {T}_{\mathrm {reg}}^1$ , so $\mathcal {S}_1^\square $ is representable and is smooth of the correct dimension over $\mathcal {T}_{\mathrm {reg}}^1$ . Then one may proceed by induction on d again and construct $\mathcal {S}_d^\square $ as the moduli space of extensions of the universal $(\varphi ,\Gamma _K)$ -module of character type $\Lambda _{\mathcal {T},\mathrm {rig},K}(\delta _{\mathrm {univ}})$ by the universal object $D_{d-1,\mathrm {univ}}$ over $\mathcal {S}_{d-1}^\square $ . For a specified regular parameter $\underline \delta =(\delta _1,\ldots ,\delta _d)\in \mathcal {T}_{\mathrm {reg}}^d(X)$ , the fiber $\mathcal {S}_d^\square |_{\underline \delta }$ is equal to $\operatorname {\mathrm {Ext}}^1(\Lambda _{X,\mathrm {rig},K}(\delta _d),D_{d-1,\mathrm {univ}}|_{(\delta _1,\ldots ,\delta _{d-1})})=H_{\varphi ,\Gamma _K}^1(D_{d-1,\mathrm {univ}}|_{(\delta _1,\ldots ,\delta _{d-1})}(\delta _d^{-1}))$ . This is a rank- $(d-1)$ vector bundle over X, and the claim follows.

We also introduce variants of $\mathcal {S}_d^\square $ with families of fixed determinant and weights. More precisely, suppose X is a pseudorigid space and we have a continuous character $\delta _{\det }:K^{\times }\rightarrow {\mathscr {O}}(X)^{\times }$ and a d-tuple of continuous characters $\underline \kappa :=(\kappa _1,\ldots ,\kappa _d):{\mathscr {O}}_K^{\times }\rightarrow {\mathscr {O}}(X)^{\times }$ . We say that $\delta _{\det }$ and $\underline \kappa $ are compatible if $\delta _{\det }|_{{\mathscr {O}}_K^{\times }}=\kappa _1\cdots \kappa _d$ . If $\delta _{\det }$ and $\underline \kappa $ are compatible, we consider the functors $\mathcal {S}_d^{\square ,\delta _{\det }}$ and $\mathcal {S}_d^{\square ,\delta _{\det },\underline \kappa }$ on pseudorigid spaces over X defined via

$$\begin{align*}Y\rightsquigarrow \{(D,\operatorname{\mathrm{Fil}}^{\bullet} D,\underline\delta)\in \mathcal{S}_d^\square(Y) \mid \delta_1\cdots\delta_d=\delta_{\det}\}/\sim \end{align*}$$

and

$$\begin{align*}Y\rightsquigarrow \{(D,\operatorname{\mathrm{Fil}}^{\bullet} D,\underline\delta,\underline\nu)\in \mathcal{S}_d^\square(Y) \mid \delta_i|_{{\mathscr{O}}_K^{\times}}=\kappa_i\text{ for all }i, \delta_1\cdots\delta_d=\delta_{\det}\}/\sim. \end{align*}$$

Proof. Set $Y:=\widehat {({\mathscr {O}}_K^{\times })^d}^{\mathrm {{an}}}$ . Then there is a morphism $\mathcal {T}^d\rightarrow {\mathbf {G}}_{m,Y}$ , given by $\underline \delta \mapsto \left (\delta _1|_{{\mathscr {O}}_K^{\times }},\ldots ,\delta _d|_{{\mathscr {O}}_K^{\times }},\delta _1(\varpi _K)\cdots \delta _d(\varpi _K)\right )$ , and it is smooth of relative dimension $d-1$ . The choice of $\delta _{\det }$ and $\underline \kappa $ define a morphism $X\rightarrow {\mathbf {G}}_{m,Y}$ , and we have a pullback square

Then the result follows from Proposition 2.2.7.

2.3 Structure of trianguline varieties

Let $K/{\mathbf {Q}}_p$ be a finite extension, and let $\overline {\rho }:\operatorname {\mathrm {Gal}}_K\rightarrow \operatorname {\mathrm {GL}}_d(k)$ be a continuous representation, where k is a finite field containing the residue field of K. The fiber product $(\operatorname {\mathrm {Spa}} R_{\overline {\rho }}^{\square })^{\mathrm {{an}}}{\times }_{\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p}\mathcal {T}^d$ exists as a pseudorigid space, and it is contained in the fiber product

$$\begin{align*}{\mathbf{G}}_m^{\operatorname{\mathrm{ad}},d}{\times}_{\mathbf{Z}} \widehat{({\mathscr{O}}_K^{\times})^d} \times \operatorname{\mathrm{Spa}}(R_{\overline{\rho}}^{\square})^{\mathrm{{an}}} \end{align*}$$

(with complement of codimension $\geq 2$ if $d\geq 2$ ). Let $X_{\mathrm {{tri}},\overline {\rho }}^{\square }$ be the Zariski closure in the latter of the set of maximal points $x=\{(\rho _x,\underline \delta _x)\}$ , where $\rho _x$ is a (framed) lift of $\overline {\rho }$ and $\underline \delta _x\in \mathcal {T}_{\mathrm {reg}}^d(L)$ is a regular parameter of $D_{\mathrm {rig}}(\rho _x)$ .

Let R be a complete local Noetherian ${\mathbf {Z}}_p$ -algebra with finite residue field. Fix an d-tuple of characters $\underline \kappa :=(\kappa _1,\ldots ,\kappa _d)$ , where $\kappa _i:{\mathscr {O}}_K^{\times }\rightarrow {\mathscr {O}}(X)^{\times }$ and $X:=\left (\operatorname {\mathrm {Spa}} R\right )^{\mathrm {{an}}}$ , and fix a character $\psi :\operatorname {\mathrm {Gal}}_K\rightarrow R^{\times }$ . Over the pseudorigid space X, a character $\psi :\operatorname {\mathrm {Gal}}_K\rightarrow {\mathscr {O}}(X)^{\times }$ corresponds to a rank- $1\ (\varphi ,\Gamma )$ -module of the form $D_{\mathrm {rig}}(\delta _{\psi })$ , for some character $\delta _{\psi }:K^{\times }\rightarrow {\mathscr {O}}(X)^{\times }$ . If $\delta _{\psi }$ and $\underline \kappa $ are compatible, we may define

$$\begin{align*}X_{\mathrm{{tri}},\overline{\rho}}^{\square,\psi,\underline\kappa}\subset {\mathbf{G}}_m^{\operatorname{\mathrm{ad}},d}{\times}_{{\mathbf{Z}}}\widehat{({\mathscr{O}}_K^{\times})^d}\times (\operatorname{\mathrm{Spa}} R_{\overline{\rho}}^{\square,\psi})^{\mathrm{{an}}} \end{align*}$$

to be the Zariski closure of the set of maximal points $x=\{(\rho _x,\underline \delta _x)\}$ , where $\rho _x$ is a framed lift of $\overline {\rho }$ with determinant $\psi $ and $\underline \delta _x\in \mathcal {T}_{\mathrm {reg}}^d(L)$ is a regular parameter of $D_{\mathrm {rig}}(\rho _x)$ such that $\delta _i|_{{\mathscr {O}}_K^{\times }}=\kappa _i$ .

In order to study the structure of $X_{\mathrm {{tri}},\overline {\rho }}^{\square }$ and $X_{\mathrm {{tri}},\overline {\rho }}^{\square ,\psi ,\underline \kappa }$ , we will need to know something about the essential image of the functor from Galois representations to $(\varphi ,\Gamma )$ -modules. We refer the reader to [Reference BellovinBel23b] for details on definitions of pseudorigid overconvergent period rings and the construction of $(\varphi ,\Gamma )$ -modules in the pseudorigid setting. However, we note here that $\Lambda _{R,[0,b],K}$ is the coordinate ring of a closed annulus over $\operatorname {\mathrm {Spa}} R$ , $\Lambda _{R,(0,b],K}$ is the ring of global functions on a half-open annulus over $\operatorname {\mathrm {Spa}} R$ , and $\Lambda _{R,{\mathrm {rig}},K}:=\varprojlim _{b\rightarrow 0}\Lambda _{R,(0,b],K}$ . As in the work of [Reference Cherbonnier and ColmezCC98] and [Reference Berger and ColmezBC08], $(\varphi ,\Gamma )$ -modules attached to Galois representations are constructed over $\Lambda _{R,[0,b],K}$ for some $b>0$ (which depends in subtle ways on the representation).

Proof. We need to show that if D is a projective $(\varphi ,\Gamma _K)$ -module over a pseudoaffinoid algebra $R'$ , and $I\subset R'$ is a square-zero ideal such that $(R'/I)\otimes _{R^{\prime }}D$ arises from a family of Galois representations, then D also arises from a family of Galois representations. Indeed, we have a short exact sequence

$$\begin{align*}0\rightarrow ID\rightarrow D\rightarrow (R'/I)\otimes_{R^{\prime}}D\rightarrow 0. \end{align*}$$

By assumption, $D':=(R'/I)\otimes _{R^{\prime }}D$ arises from a family of $\operatorname {\mathrm {Gal}}_K$ representations $M'$ over $R'/I$ , and since

$$\begin{align*}D":=ID \cong I\otimes_{R^{\prime}} D \cong (R'/\operatorname{\mathrm{ann}}_{R^{\prime}}I)\otimes_{R^{\prime}/I}D \end{align*}$$

it arises from a family of $\operatorname {\mathrm {Gal}}_K$ representations $M"$ over $R'/\operatorname {\mathrm {ann}}_{R^{\prime }}I$ . Since D has a model $D_b$ over $\Lambda _{R^{\prime },(0,b],K}$ , we have a commutative diagram

By construction, $\widetilde \Lambda _{R^{\prime },(0,b]}\otimes _{R^{\prime }}D"\cong \widetilde \Lambda _{R^{\prime },(0,b]}\otimes \left (\widetilde \Lambda _{R_0',[0,b]}\otimes _{R_0'}M_0"\right )$ and $\widetilde \Lambda _{R^{\prime },(0,b]}\otimes _{R^{\prime }}D'\cong \widetilde \Lambda _{R^{\prime },(0,b]}\otimes \left (\widetilde \Lambda _{R_0',[0,b]}\otimes _{R_0'}M_0'\right )$ , for some integral models $M_0"$ and $M_0'$ (perhaps after localizing on $\operatorname {\mathrm {Spa}} R'$ and shrinking b). Therefore, we have quasi-isomorphisms

$$ \begin{align*} [M"]&\xrightarrow\sim [\widetilde\Lambda_{R^{\prime},[0,b]}\otimes_{R_0'}M_0"\xrightarrow{\varphi-1}\widetilde\Lambda_{R^{\prime},[0,b/p]}\otimes_{R_0'}M_0"] \\ &\xrightarrow\sim [\widetilde\Lambda_{R^{\prime},(0,b]}\otimes_{R^{\prime}}D" \xrightarrow{\varphi-1}\widetilde\Lambda_{R^{\prime},(0,b/p]}\otimes_{R^{\prime}}D"] \end{align*} $$

and

$$ \begin{align*} [M']&\xrightarrow\sim [\widetilde\Lambda_{R^{\prime},[0,b]}\otimes_{R_0'}M_0'\xrightarrow{\varphi-1}\widetilde\Lambda_{R^{\prime},[0,b/p]}\otimes_{R_0'}M_0'] \\ &\xrightarrow\sim [\widetilde\Lambda_{R^{\prime},(0,b]}\otimes_{R^{\prime}}D'\xrightarrow{\varphi-1}\widetilde\Lambda_{R^{\prime},(0,b/p]}\otimes_{R^{\prime}}D']. \end{align*} $$

Then the snake lemma implies that we have an exact sequence

$$\begin{align*}0\rightarrow M"\rightarrow \left(\widetilde\Lambda_{R^{\prime},\mathrm{rig},K}\otimes D\right)^{\varphi=1}\rightarrow M'\rightarrow 0 \end{align*}$$

of $R'$ -modules equipped with continuous $R'$ -linear actions of $\operatorname {\mathrm {Gal}}_K$ , with $M'$ finite projective over $R'/I$ and $M"\cong (R'/\operatorname {\mathrm {ann}}_{R^{\prime }}I)\otimes _{R^{\prime }/I}M'$ . It follows that $M:=\left (\widetilde \Lambda _{R^{\prime },\mathrm {rig},K}\otimes D\right )^{\varphi =1}$ is a projective $R'$ -module of the same rank and $D_{\mathrm {rig},K}(M)=D$ .

In [Reference Breuil, Hellmann and SchraenBHS17, §2.2], the authors show that the characteristic $0$ locus $X_{\mathrm {{tri}},\overline {\rho }}^{\square ,\mathrm {rig}}$ of the trianguline variety is equidimensional of dimension $d^2+[K:{\mathbf {Q}}_p]\frac {d(d+1)}{2}$ , and generically smooth. We note that if $\psi :\operatorname {\mathrm {Gal}}_K\rightarrow {\mathscr {O}}_E^{\times }$ is a crystalline character, where $E/{\mathbf {Q}}_p$ is a finite extension and ${\mathscr {O}}_E$ is its ring of integers, then an identical argument shows that the rigid analytic locus $X_{\mathrm {{tri}},\overline {\rho }}^{\square ,\psi ,\mathrm {rig}}\subset X_{\mathrm {{tri}},\overline {\rho }}^{\square ,\psi }$ is equidimensional of dimension $d^2-1+[K:{\mathbf {Q}}_p]\frac {(d+2)(d-1)}{2}$ (indeed, [Reference Böckle, Iyengar and PaškūnasBIP23b, Theorem 1.2] provides the necessary crystalline lifts with fixed determinant).

Unfortunately, we cannot rule out components of $X_{\mathrm {{tri}},\overline {\rho }}^{\square }$ or $X_{\mathrm {{tri}},\overline {\rho }}^{\square ,\psi }$ supported entirely in characteristic p, and so to deduce the same result in the pseudorigid setting, we need to repeat a large part of the argument in a neighborhood of the characteristic p fiber.

Proof. The proofs of the first two parts are very similar to that of [Reference Breuil, Hellmann and SchraenBHS17, Théorème 2.6], and we will prove them simultaneously. By construction, there is a universal framed deformation $\rho _{\mathrm {univ}}:\operatorname {\mathrm {Gal}}_K\rightarrow \operatorname {\mathrm {GL}}_d(R_{\overline {\rho }}^\square )$ of $\overline {\rho }$ , and we may pull it back to $X_{\mathrm {{tri}},\overline {\rho }}^\square $ (resp. $X_{\mathrm {{tri}},\overline {\rho }}^{\square ,\psi ,\underline \kappa }$ ). Then for any irreducible open affinoid $X\subset X\rightarrow X_{\mathrm {{tri}},\overline {\rho }}^{\square }$ (resp. $X_{\mathrm {{tri}},\overline {\rho }}^{\square ,\psi ,\underline \kappa }$ ), by [Reference BellovinBel23a, Corollary 5.10] there is a sequence of blow-ups and normalizations $f:\widetilde X\rightarrow X$ and an open subspace $U\subset \widetilde X$ containing the characteristic p locus such that $f^\ast \rho _{\mathrm {univ}}|_U$ is trianguline with parameters $f^\ast \underline \delta $ . Shrinking U if necessary, we may assume that $f^\ast \underline \delta $ is regular (indeed, the preimage of $\mathcal {T}_{\mathrm {reg}}^d$ in U is open, and by construction U contains a Zariski dense set of points corresponding to trianguline representations with regular parameters). Furthermore, there is a Zariski-dense and open subspace $V\subset X_{\mathrm {{tri}},\overline {\rho }}^{\square }$ (resp. $X_{\mathrm {{tri}},\overline {\rho }}^{\square ,\psi ,\underline \kappa }$ ) such that $f^{-1}(V)\subset U$ and f defines an isomorphism $f^{-1}(V)\xrightarrow \sim V$ .

Over U, the $(\varphi ,\Gamma _K)$ -module $D:=D_{\mathrm {rig},K}(f^\ast \rho _{\mathrm {univ}})$ is equipped with an increasing filtration $\operatorname {\mathrm {Fil}}^{\bullet } D$ such that $\operatorname {\mathrm {gr}}^iD\cong \Lambda _{U,\mathrm {rig},K}(f^\ast \delta _i)\otimes \mathscr {L}_i$ for some line bundle $\mathscr {L}_i$ on U. We may therefore construct a ${\mathbf {G}}_{m,U}^d$ -torsor $U^\square \rightarrow U$ trivializing each of the $\mathscr {L}_i$ ; since $U^\square $ carries the data $(D,\operatorname {\mathrm {Fil}}^{\bullet } D, f^\ast \underline \delta , \underline \nu )$ , where $\underline \nu $ is the set of trivializations $\nu _i:\operatorname {\mathrm {gr}}^iD\xrightarrow \sim \Lambda _{U,\mathrm {rig},K}(f^\ast \delta _i)$ , there is a morphism $U^\square \rightarrow \mathcal {S}_d^\square $ .

Let $V^\square \subset U^\square $ denote the pullback of $U^\square \rightarrow U$ to V. We claim that $V^\square \rightarrow \mathcal {S}_d^\square $ is smooth of relative dimension $d^2$ . To see this, suppose we have a pseudoaffinoid algebra $R'$ , a morphism $\operatorname {\mathrm {Spa}} R'\rightarrow \mathcal {S}_d^\square $ and a square-zero ideal $I\subset R'$ such that the composition $\operatorname {\mathrm {Spa}} R'/I\hookrightarrow \operatorname {\mathrm {Spa}} R'\rightarrow \mathcal {S}_d^\square $ is in the image of $V^\square $ . Then there is a ring of definition $R_0'\subset R'/I$ such that the homomorphism $R_{\overline {\rho }}^\square \rightarrow R'/I$ factors through $R_0'$ ; we let $M_0'\cong R_0^{\prime \oplus d}$ be the pullback of the universal framed deformation to $R_0'$ and we let $M':=R'/I\otimes _{R_0'}M_0'$ .

By Lemma 2.3.1, there is a $\operatorname {\mathrm {Gal}}_K$ -representation M over $R'$ such that $(R'/I)\otimes _{R^{\prime }}M\xrightarrow {\sim }M'$ . It follows that $M_0'$ and its basis lift to a free module $M_0$ over some ring of definition $R_0'\subset R'$ such that $R'\otimes _{R_0'}M_0=M$ . Moreover, $M'$ is residually a lift of $\overline {\rho }$ at every maximal point of $\operatorname {\mathrm {Spa}} R'$ , so M is as well. By [Reference Wang-EricksonWE18, Theorem 3.8], $M_0$ corresponds to a $\operatorname {\mathrm {Spa}} R_0'$ -point of $\operatorname {\mathrm {Spf}} R_{\overline {\rho }}^\square $ , and by construction M corresponds to a $\operatorname {\mathrm {Spa}} R'$ -point of $X_{\mathrm {{tri}},\overline {\rho }}^{\square }$ deforming $M'$ . Since $M'$ corresponds to a $\operatorname {\mathrm {Spa}} (R'/I)$ -point of the Zariski-open subspace $V\subset X_{\mathrm {{tri}},\overline {\rho }}^{\square }$ , the image of the morphism $\operatorname {\mathrm {Spa}} R'\rightarrow X_{\mathrm {{tri}},\overline {\rho }}^{\square }$ also lands in V. Since D is trianguline with regular parameters and trivialized quotients, the morphism $\operatorname {\mathrm {Spa}} R'\rightarrow V$ lifts to a morphism $\operatorname {\mathrm {Spa}} R'\rightarrow V^\square $ .

The claim that $V^\square \rightarrow \mathcal {S}_d^\square $ has relative dimension $d^2$ follows because ‘changing the framing’ makes $V^\square $ a $\left (\operatorname {\mathrm {GL}}_d\right )^{\mathrm {{an}}}$ -torsor over its image in $\mathcal {S}_d^\square $ .

Now, we can compute the dimension. By Proposition 2.2.7, we see that $V^\square $ is equidimensional of dimension $d^2 + \frac {d(d-1)}{2}[K:{\mathbf {Q}}_p] + d[K:{\mathbf {Q}}_p] + d$ (resp. $d^2 + \frac {d(d-1)}{2}[K:{\mathbf {Q}}_p] + d[K:{\mathbf {Q}}_p] + d + \dim \operatorname {\mathrm {Spa}} R^{\mathrm {{an}}}$ ). Since $V^\square \rightarrow V$ is a ${\mathbf {G}}_{m,V}^d$ -torsor, it follows that V is equidimensional of dimension $d^2+\frac {d(d+1)}{2}[K:{\mathbf {Q}}_p]$ (resp. $d^2 + \frac {d(d-1)}{2}[K:{\mathbf {Q}}_p] + d[K:{\mathbf {Q}}_p] +\dim \operatorname {\mathrm {Spa}} R^{\mathrm {{an}}}$ . Finally, $V\subset X$ is Zariski-dense, so we are done.

For the last part, we define $V^{\square ,\psi ,\underline \kappa }$ via the pullback

where $Y:=\widehat {({\mathscr {O}}_K^{\times })^d}$ and the morphism $\operatorname {\mathrm {Spa}} R^{\mathrm {{an}}}\rightarrow {\mathbf {G}}_{m,Y}$ is given by $\underline \kappa $ and $\delta _\psi $ . Since $V^\square \rightarrow {\mathbf {G}}_{m,Y}$ is smooth, its image is open, and the preimage in $\operatorname {\mathrm {Spa}} R^{\mathrm {{an}}}$ is open, as well.

Now, we consider a global setup. Let F be a number field, and suppose that $\overline {\rho }:\operatorname {\mathrm {Gal}}_{F}\rightarrow \operatorname {\mathrm {GL}}_d({\mathbf {F}})$ is an absolutely irreducible representation, unramified outside a finite set of primes S.

Then the homomorphisms

$$\begin{align*}R_{\overline{\rho}_v}^\square\rightarrow R_{\overline{\rho},S}^\square \end{align*}$$

for each $v\mid p$ induce a morphism

$$\begin{align*}\left(\operatorname{\mathrm{Spa}} R_{\overline{\rho},S}^\square\right)^{\mathrm{{an}}}\times \prod_{v\mid p}\mathcal{T}^d\rightarrow \prod_{v\mid p}\left(\left(\operatorname{\mathrm{Spa}} R_{\overline{\rho}_v}^\square\right)^{\mathrm{{an}}}\times\mathcal{T}^d\right) \end{align*}$$

and we define $X_{\mathrm {{tri}},\overline {\rho },S}^\square $ to be the preimage of $\prod _{v\mid p}X_{\mathrm {{tri}},\overline {\rho }_v}^\square $ .

If R is a complete local Noetherian ${\mathbf {Z}}_p$ -algebra with maximal ideal $\mathfrak {m}_R$ and finite residue field, and $\psi :\operatorname {\mathrm {Gal}}_F\rightarrow R^{\times }$ is a continuous character such that $\det \overline {\rho } = \psi \mod \mathfrak {m}_R$ , the homomorphisms

$$\begin{align*}R_{\overline{\rho}_v}^{\square,\psi_v}\rightarrow R_{\overline{\rho},S}^{\square,\psi} \end{align*}$$

and

$$\begin{align*}R_{\overline{\rho},\mathrm{loc}}^{\square,\psi}\rightarrow R_{\overline{\rho},S}^{\square,\psi} \end{align*}$$

induce a sequence of morphisms

where $\psi _v:=\psi |_{\operatorname {\mathrm {Gal}}_{F_v}}$ . We define $X_{\mathrm {{tri}},\overline {\rho },S}^{\square ,\psi }$ and $X_{\mathrm {{tri}},\overline {\rho },\mathrm {loc}}^{\square ,\psi }$ to be the preimages of $\prod _{v\mid p}X_{\mathrm {{tri}},\overline {\rho }_v}^{\square ,\psi _v}$ in $\left (\operatorname {\mathrm {Spa}} R_{\overline {\rho },S}^{\square ,\psi }\right )^{\mathrm {{an}}}\times \prod _{v\mid p}\mathcal {T}^d$ and $\left (\operatorname {\mathrm {Spa}} R_{\overline {\rho },\mathrm {loc}}^{\square ,\psi }\right )^{\mathrm {{an}}}\times \prod _{v\mid p}\mathcal {T}^d$ , respectively.

If we additionally have d-tuples of characters $\underline \kappa _v:=(\kappa _{v,1},\ldots ,\kappa _{v,d})$ , where $\kappa _{v,i}:{\mathscr {O}}_{F_v}^{\times }\rightarrow {\mathscr {O}}(X)^{\times }$ is a continuous character, and we set $X:=\left (\operatorname {\mathrm {Spa}} R\right )^{\mathrm {{an}}}$ , we may form the spaces

In particular, suppose we have fixed a neat level $K=K^pI$ , as in Sections 3 and 4, and consider the ring $R={\mathbf {Z}}_p[\![T_0/\overline {Z(K)}]\!]$ corresponding to integral weight space. Since $T_0=\prod _{v\mid p}(\operatorname {\mathrm {Res}}_{{\mathscr {O}}_{F_v}/{\mathbf {Z}}_p}T_v)({\mathbf {Z}}_p)$ , we have homomorphisms ${\mathbf {Z}}_p[\![T_v({\mathscr {O}}_{F_v})]\!]\rightarrow R$ , and hence morphisms $\operatorname {\mathrm {Spa}} R\rightarrow \operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![T_v({\mathscr {O}}_{F_v})]\!]$ . Suppose we have a determinant character $\psi :\operatorname {\mathrm {Gal}}_F\rightarrow R^{\times }$ , and a set of weights $\underline \kappa _v:=(\kappa _{v,1},\ldots ,\kappa _{v,d}):{\mathscr {O}}_{F_v}^{\times }\rightarrow {\mathscr {O}}(\mathscr {W}_F)^{\times }$ for each $v\mid p$ such that $\psi |_{\operatorname {\mathrm {Gal}}_{F_v}}$ and $\underline \kappa _v$ are compatible for all v and such that $\psi _v$ and $\underline \kappa _v$ factor through ${\mathbf {Z}}_p[\![T_v({\mathscr {O}}_{F_v})]\!]\rightarrow R$ for all v, that is, they depend only on the projection to $\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![T_v({\mathscr {O}}_{F_v})]\!]$ .

Proof. Viewing $\psi _v$ as a character $\operatorname {\mathrm {Gal}}_{F_v}\rightarrow {\mathbf {Z}}_p[\![T_v({\mathscr {O}}_{F_v})]\!]^{\times }$ and viewing $\underline \kappa _v=(\kappa _{v,1},\ldots ,\kappa _{v,d})$ as a d-tuple of characters ${\mathscr {O}}_{F_v}^{\times }\rightarrow {\mathbf {Z}}_p[\![T_v({\mathscr {O}}_{F_v})]\!]^{\times }$ , we have a pullback diagram

The right vertical morphism has relative dimension

$$\begin{align*}\sum_{v\mid p}\left(d^2-1 + [F_v:{\mathbf{Q}}_p]\frac{d(d-1)}{2}\right) = \lvert\Sigma_p\rvert(d^2-1)+[F:{\mathbf{Q}}]\frac{d(d-1)}{2} \end{align*}$$

over an open subspace of $\prod _{v\mid p}\left (\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![T_v({\mathscr {O}}_{F_v})]\!]\right )^{\mathrm {{an}}}$ , so the morphism $X_{\mathrm {{tri}},\overline {\rho },\mathrm {loc}}^{\square ,\psi ,\underline \kappa }\rightarrow \mathscr {W}_F$ does, as well.

The case we will be most interested in is the case where $F/{\mathbf {Q}}$ is cyclic and totally split at p, and $d=2$ . In that case, $X_{\mathrm {{tri}},\overline {\rho }_v}^{\square ,\psi _v,\underline \kappa _v}\rightarrow \operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![T_v({\mathscr {O}}_{F_v})]\!]^{\mathrm {{an}}}$ has relative dimension $4$ over an open subspace of $\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![T_v({\mathscr {O}}_{F_v})]\!]^{\mathrm {{an}}}$ for each $v\mid p$ , and hence $X_{\mathrm {{tri}},\overline {\rho },\mathrm {loc}}^{\square ,\psi ,\underline \kappa }\rightarrow \mathscr {W}_F$ has relative dimension $4[F:{\mathbf {Q}}]$ over an open subspace of $\mathscr {W}_F$ .

2.4 Trianguline deformation rings

We have constructed the trianguline varieties $X_{\mathrm {{tri}},\overline {\rho }}^\square $ and $X_{\mathrm {{tri}},\overline {\rho }}^{\square ,\psi ,\underline \kappa }$ as subspaces of the (nonquasicompact) pseudorigid space ${\mathbf {G}}_m^{\operatorname {\mathrm {ad}},d}{\times }_{{\mathbf {Z}}}\widehat {({\mathscr {O}}_K^{\times })^d}\times \left (\operatorname {\mathrm {Spa}} R_{\overline {\rho }}^\square \right )^{\mathrm {{an}}}$ . However, the advantage of working with general pseudorigid spaces is that we can construct integral models, so long as we bound the slope.

We will apply this to find formal models for pieces of our trianguline varieties. Recall that, when K is a finite extension of ${\mathbf {Q}}_p$ and $\overline {\rho }$ is a representation of $\operatorname {\mathrm {Gal}}_K$ , we defined $X_{\mathrm {{tri}},\overline {\rho }}^\square $ and $X_{\mathrm {{tri}},\overline {\rho }}^{\square ,\psi ,\underline \kappa }$ as analytic subspaces of ${\mathbf {G}}_m^{\operatorname {\mathrm {ad}}}\times \operatorname {\mathrm {Spa}}{\mathbf {Z}}_p[\![({\mathscr {O}}_K^{\times })^d]\!]\times \left (\operatorname {\mathrm {Spa}} R_{\overline {\rho }}^{\square }\right )^{\mathrm {{an}}}$ and ${\mathbf {G}}_m^{\operatorname {\mathrm {ad}}}\times \operatorname {\mathrm {Spa}}{\mathbf {Z}}_p[\![({\mathscr {O}}_K^{\times })^d]\!]\times \left (\operatorname {\mathrm {Spa}} R\mathop {\widehat \otimes } R_{\overline {\rho }}^{\square }\right )^{\mathrm {{an}}}$ , respectively. By construction, $\operatorname {\mathrm {Spa}}{\mathbf {Z}}_p[\![({\mathscr {O}}_K^{\times })^d]\!]\times \left (\operatorname {\mathrm {Spa}} R_{\overline {\rho }}^{\square }\right )^{\mathrm {{an}}}$ has a quasicompact integral model, but ${\mathbf {G}}_m^{\operatorname {\mathrm {ad}}}\times \operatorname {\mathrm {Spa}}{\mathbf {Z}}_p[\![({\mathscr {O}}_K^{\times })^d]\!]\times \left (\operatorname {\mathrm {Spa}} R_{\overline {\rho }}^{\square }\right )^{\mathrm {{an}}}\times \left (\operatorname {\mathrm {Spa}} R_{\overline {\rho }}^{\square }\right )^{\mathrm {{an}}}$ does not; in particular, it is not equal to the analytic locus of $\operatorname {\mathrm {Spa}} R_{\overline {\rho }}^{\square }\mathop {\widehat \otimes }{\mathbf {Z}}_p[\![({\mathscr {O}}_K^{\times })^d]\!]\left \langle T,T^{-1}\right \rangle $ (and similarly for ${\mathbf {G}}_m^{\operatorname {\mathrm {ad}}}\times \operatorname {\mathrm {Spa}}{\mathbf {Z}}_p[\![({\mathscr {O}}_K^{\times })^d]\!]\times \left (\operatorname {\mathrm {Spa}} R\mathop {\widehat \otimes } R_{\overline {\rho }}^{\square }\right )^{\mathrm {{an}}}$ ).

In order to construct integral models of annuli, we begin with an illustrative example.

Example 2.4.1. Suppose $R={\mathbf {Z}}_p[\![u]\!]$ and h is an integer. We may cover $Y:=\operatorname {\mathrm {Spa}}\left ({\mathbf {Z}}_p[\![u]\!]\right )^{\mathrm {{an}}}$ with the open affinoid subspaces $U_1:=\operatorname {\mathrm {Spa}}\left ({\mathbf {Q}}_p\left \langle \frac u p\right \rangle \right )$ and $U_2:=\operatorname {\mathrm {Spa}}\left ({\mathbf {Z}}_p[\![u]\!]\left \langle \frac p u\right \rangle \left [\frac 1 u\right ]\right )$ ; their intersection is the circle $U_1\cap U_2=\operatorname {\mathrm {Spa}}\left ({\mathbf {Q}}_p\left \langle \frac u p, \frac p u\right \rangle \right )$ .

The annulus $C_{U_1,h}$ is affinoid, with coordinate ring

$$\begin{align*}{\mathbf{Q}}_p\left\langle \frac u p,p^hT,T_2\right\rangle/(TT_2-p^h) = {\mathbf{Q}}_p\left\langle \frac u p,T,T_1,T_2\right\rangle/(T_1-p^hT,TT_2-p^h). \end{align*}$$

Restricting to $U_1\cap U_2$ , we obtain an affinoid with coordinate ring

$$\begin{align*}{\mathbf{Z}}_p[\![u]\!]\left\langle \frac u p,\frac p u,T,T_1,T_2\right\rangle\left[\frac 1 u\right]/(T_1-u^h\left(\frac p u\right)^hT,TT_2-u^h\left(\frac p u\right)^h). \end{align*}$$

Writing $T_1':=\left (\frac u p\right )^hT_1$ and $T_2':=\left (\frac u p\right )^hT_2$ , we get

$$\begin{align*}{\mathbf{Z}}_p[\![u]\!]\left\langle \frac u p,\frac p u,T,T_1',T_2'\right\rangle\left[\frac 1 u\right]/(T_1'-u^hT,TT_2'-u^h) \end{align*}$$

which is also the restriction of $C_{U_2,h}$ to $U_1\cap U_2$ .

The affinoid spaces $C_{U_i,h}$ and $C_{U_1\cap U_2,h}$ have integral models, compatible with gluing. Thus, we see that $C_{Y,h}$ in this case admits a formal model which lives over the blow-up $\mathrm {Bl}_{(p,u)}R$ .

Returning to the general case, we may choose a ${\mathbf {Z}}_p$ -basis for the torsion-free part of ${\mathscr {O}}_K^{\times }$ and corresponding coordinates on $\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![{\mathscr {O}}_K^{\times }]\!]^{\mathrm {{an}}}$ . Then we may consider relative annuli over $\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![\widehat {{\mathscr {O}}_K^{\times }}]\!]$ ; as above, these annuli glue to a space $\mathcal {T}_h$ which has an integral model $\mathfrak {T}_h$ over the blow-up $\mathrm {Bl}_I{\mathbf {Z}}_p[\![{\mathscr {O}}_K^{\times }]\!]$ (where $I=\cap \mathfrak {m}$ is the intersection of the maximal ideals of ${\mathbf {Z}}_p[\![{\mathscr {O}}_K^{\times }]\!]$ ). Similarly, given some integer $d\geq 1$ , we may define relative annuli $\mathcal {T}_h^d\subset \mathcal {T}^d$ over $\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![({\mathscr {O}}_K^{\times })^d]\!]^{\mathrm {{an}}}$ , which have integral models $\mathfrak {T}_h^d$ .

Now, we may set

$$\begin{align*}X_{\mathrm{{tri}},\overline{\rho},\leq h}^\square:= X_{\mathrm{{tri}},\overline{\rho}}^\square\cap \left(\operatorname{\mathrm{Spa}} R_{\overline{\rho}}^\square\times\mathcal{T}_h^d\right)^{\mathrm{{an}}} \end{align*}$$

and

$$\begin{align*}X_{\mathrm{{tri}},\overline{\rho},\leq h}^{\square,\psi,\underline\kappa}:= X_{\mathrm{{tri}},\overline{\rho}}^{\square,\psi,\underline\kappa}\cap \left(\operatorname{\mathrm{Spa}} R\mathop{\widehat\otimes} R_{\overline{\rho}}^\square\times \mathcal{T}_h^d\right)^{\mathrm{{an}}}. \end{align*}$$

When F is a totally real field and $\overline {\rho }$ is a representation of $\operatorname {\mathrm {Gal}}_F$ unramified outside a finite set of places S, we may similarly define bounded global trianguline varieties $X_{\mathrm {{tri}},\overline {\rho },S,\leq h}^\square $ and $X_{\mathrm {{tri}},\overline {\rho },S,\leq h}^{\square ,\psi ,\underline \kappa }$ as subspaces of $\left (\operatorname {\mathrm {Spa}} R_{\overline {\rho }}^\square \times \prod _{v\mid p}\mathcal {T}_h^d\right )^{\mathrm {{an}}}$ and $\left (\operatorname {\mathrm {Spa}} R\mathop {\widehat \otimes } R_{\overline {\rho }}^\square \times \prod _{v\mid p}\mathcal {T}_h^d\right )^{\mathrm {{an}}}$ , respectively.

Now, we restrict to the case $K={\mathbf {Q}}_p$ , and we choose coordinates $z_1,\ldots ,z_d$ on each component of $\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![({\mathbf {Z}}_p^{\times })^{d}]\!]^{\mathrm {{an}}}$ . Write $h=a/b$ , where $a,b$ are nonnegative relatively prime integers. We will construct an integral model of $X_{\mathrm {{tri}},\overline {\rho },\leq h}^{\square }$ using Corollary A.0.2.

For $z\in \{p,z_1,\ldots ,z_d\}$ , we get an affinoid $U_z:=\operatorname {\mathrm {Spa}} R_z$ , where $R_z:={\mathbf {Z}}_p[\![({\mathbf {Z}}_p^{\times })^{d}]\!]\left \langle \frac {p}{z},\frac {z_1}{z},\ldots ,\frac {z_d}{z}\right \rangle \left [\frac 1 z\right ]$ with ring of integers $R_{z,0}$ , inside $\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![({\mathbf {Z}}_p^{\times })^{d}]\!]^{\mathrm {{an}}}$ . Then the restriction of $\mathcal {T}_h^d$ to $U_z$ has the presentation

$$\begin{align*}\operatorname{\mathrm{Spa}} {\mathbf{Z}}_p[\![({\mathbf{Z}}_p^{\times})^{d}]\!]\left\langle \frac{p}{z},\frac{z_1}{z},\ldots,\frac{z_d}{z}, z^aT_1^{\pm b},\ldots, z^aT_d^{\pm b}\right\rangle\left[\frac 1 z\right]. \end{align*}$$

Over this space, there is a d-tuple $\delta _1,\ldots ,\delta _d:{\mathbf {Q}}_p^{\times }\rightrightarrows R_z^{\times }$ , where $\delta _i(p)=T_i$ and $(\delta _i|_{{\mathbf {Z}}_p^{\times }})$ is the restriction of the universal character on $({\mathbf {Z}}_p^{\times })^d$ .

Given an affinoid $\operatorname {\mathrm {Spa}} R\subset \left (\operatorname {\mathrm {Spa}} R_{\overline {\rho }}^\square \right )^{\mathrm {{an}}}$ with pseudouniformizer $u\in R$ , there is a $(\varphi ,\Gamma )$ -module $D_R$ of rank d over $\operatorname {\mathrm {Spa}} R$ . To study the bounded trianguline variety, we first study morphisms $D_R\rightarrow \Lambda _{R,\mathrm {rig},{\mathbf {Q}}_p}(\delta _d)$ . Equivalently, we consider the twist $D_R(\delta _d^{-1})$ over the (nonquasicompact) space $\operatorname {\mathrm {Spa}} R\times \operatorname {\mathrm {Spa}} R_z\left \langle z^hT_d^{\pm 1}\right \rangle $ and consider morphisms $D_R(\delta _d^{-1})\rightarrow \Lambda _{R,\mathrm {rig},{\mathbf {Q}}_p}$ to the trivial rank- $1\ (\varphi ,\Gamma )$ -module.

We wish to first consider the closure $\overline Z_R$ of

$$\begin{align*}Z_R:=\{(\rho_x,\delta_{d,x})\mid \text{there is a surjective map }D_R\rightarrow \Lambda_{\kappa(x),\mathrm{rig},{\mathbf{Q}}_p}(\delta_{d,x})\} \end{align*}$$

in $\operatorname {\mathrm {Spa}} R_0\mathop {\widehat \otimes } R_{z,0}\left \langle z^aT_i^b,T_i'\right \rangle /(T_i^bT_i'-z^a)$ .

There is a nonzero morphism at precisely the points $x\in \operatorname {\mathrm {Spa}} R\times \operatorname {\mathrm {Spa}} R_z$ , where $H^0(D_R^\vee (\delta _d)_x)$ is nonvanishing; equivalently (by Tate duality), at precisely the points where $H^2(D_R(\delta _d^{-1}\chi _{\mathrm {{cyc}}})_x)$ is nonvanishing. We write $Z_R'$ for the support of $H^2(D_R(\delta _d^{-1}\chi _{\mathrm {{cyc}}})_x)$ .

The closure of $Z_R$ is a priori defined as a subspace of the locus

$$\begin{align*}\{ \lvert z\rvert^h \leq \lvert T_i\rvert \leq \lvert z\rvert^{-h} \} \subset \operatorname{\mathrm{Spa}} R\times\operatorname{\mathrm{Spa}} R_{z,0}\times{\mathbf{G}}_m \end{align*}$$

We first claim that there is a closed subspace of

$$\begin{align*}\operatorname{\mathrm{Spa}} R\times \operatorname{\mathrm{Spa}} R_{z,0}\left\langle z^aT_i^b,T_i'\right\rangle/(T_i^bT_i'-z^a) \end{align*}$$

whose restriction to this subspace is our original closure.

By [Reference BellovinBel23a, Proposition 5.2] (more precisely, by the proof of the corresponding result [Reference Kedlaya, Pottharst and XiaoKPX14, Proposition 3.3] in characteristic $0$ ), $H^2(D_R(\delta _d^{-1}\chi _{\mathrm {{cyc}}}))$ vanishes if $T_d$ is sufficiently u-adically small. Here, ‘sufficiently small’ depends only on $D_R$ , not on the twist by $\delta _d|_{{\mathbf {Z}}_p^{\times }}$ . Thus, $Z_R'$ is contained in the locus $\{T_d\geq u^N\}$ for some $N\gg 0$ . Since $\operatorname {\mathrm {Spa}} R\mathop {\widehat \otimes } R_{z,0}\left \langle z^aT_i^b,T_i'\right \rangle /(T_i^bT_i'-z^a)$ is covered by the loci $\{\lvert T_d\rvert \geq \lvert u^{2N}\rvert \}$ and $\{\lvert T_d\rvert \leq \lvert u^{2N}\rvert \}$ , the closure of $Z_R$ is well-behaved in $\operatorname {\mathrm {Spa}} R\mathop {\widehat \otimes } R_{z,0}\left \langle z^aT_i^b,T_i'\right \rangle /(T_i^bT_i'-z^a)$ .

We next consider the intersection of $Z_R$ with

$$\begin{align*}\{T=T_d\}\subset\operatorname{\mathrm{Spa}} R\left\langle u^NT^{\pm 1}\right\rangle\times \operatorname{\mathrm{Spa}} R_{z,0}\left\langle z^aT_d^{\pm b}\right\rangle, \end{align*}$$

that is, to the locus where $T_d$ is not very u-adically large. We may apply [Reference BellovinBel23a, Corollary 5.3] to the universal twist of $D_R$ over this space, and we conclude that $Z_R'$ is contained in the subspace $\{\lvert z^{N'}\rvert \leq \lvert u\rvert \}$ for some $N'\gg 0$ . Since $Z_R\subset Z_R'$ , the same is true of $Z_R$ and its Zariski closure $\overline Z_R$ .

On the other hand, if $T_d$ is u-adically large, say, if $T_d\geq u^{-1}$ , then we have the pair of inequalities

$$\begin{align*}\lvert u^{-1}\rvert\leq T_d\leq \lvert z\rvert^{-h}. \end{align*}$$

This implies we are over the (affinoid) subspace

$$\begin{align*}\{\lvert z\rvert^h \leq \lvert u\rvert\}\subset \operatorname{\mathrm{Spa}} R\times \operatorname{\mathrm{Spa}} R_{z,0}\left\langle z^aT_d^{\pm b}\right\rangle. \end{align*}$$

This lets us study the points consisting of a Galois representation together with a first step in a triangulation and their Zariski-closure; in order to proceed by induction and study the points consisting of Galois representations together with a full triangulation, we will need the following lemma:

Proof. After replacing $\operatorname {\mathrm {Spa}} R$ with a connected component of its normalization, we may assume that $\operatorname {\mathrm {Spa}} R$ is normal and irreducible. Using [Reference Kedlaya, Pottharst and XiaoKPX14, Corollary 6.3.6(2)], there is a proper birational morphism $f:X_{R}\rightarrow \operatorname {\mathrm {Spa}} R$ such that $H^i(f^\ast D^\vee )$ is flat for $i=0$ and has Tor-dimension at most $1$ for $i=1,2$ . For any $x\in X$ , we have an exact sequence

$$\begin{align*}0\rightarrow k_x\otimes H^0\left(f^\ast D^\vee\right)\rightarrow H^0\left(k_x\otimes f^\ast D_{R}^\vee\right)\rightarrow \operatorname{\mathrm{Tor}}_1^{{\mathscr{O}}_X}\left(H^1(f^\ast D^\vee),k_x\right)\rightarrow 0 \end{align*}$$

(where we have used the low-degree exact sequences coming from the base-change spectral sequence cf. [Reference BellovinBel23a, Corollary 3.12] and the assumption that $H^i(f^\ast D^\vee )$ has Tor-dimension at most $1$ for $i=1,2$ ). Since we assumed that $H^0(k(x)\otimes _RD^\vee )$ is nonzero at a Zariski-dense set of maximal points $x\in \operatorname {\mathrm {Spa}} R$ , we see that $H^0(f^\ast D^\vee )$ is projective of nonzero rank.

Let $g:Y_{R}\rightarrow X_R$ be the projective space $\underline{\mathrm{Proj}}\left (\operatorname {\mathrm {Sym}} H^0(f^\ast D^\vee \right )^\vee )$ over $X_R$ . Since $g:Y_R\rightarrow X_R$ is flat, we have $g^\ast H^i(f^\ast D)\xrightarrow \sim H^i(g^\ast f^\ast D)$ for all i, and moreover, $g^\ast f^\ast D$ retains the property that $H^0(g^\ast f^\ast D)$ is flat (of nonzero rank) and $H^i(g^\ast f^\ast D)$ has Tor-dimension at most $1$ for $i=1,2$ .

Over $Y_R$ , there is a universal quotient $g^\ast H^0(f^\ast D^\vee )^\vee \twoheadrightarrow {\mathscr {O}}_{Y_R}(1)$ , which induces an injection ${\mathscr {O}}_{Y_R}(-1)\rightarrow g^\ast H^0(f^\ast D^\vee )$ with projective cokernel. If we consider the composition

$$\begin{align*}\Lambda_{Y_R,\mathrm{rig},{\mathbf{Q}}_p}\otimes{\mathscr{O}}_{Y_R}(-1)\rightarrow \Lambda_{Y_R,\mathrm{rig},{\mathbf{Q}}_p}\otimes g^\ast H^0(f^\ast D^\vee) \rightarrow g^\ast f^\ast D^\vee, \end{align*}$$

we may again dualize to obtain a morphism $\lambda :g^\ast f^\ast D\rightarrow {\mathscr {O}}_{Y_R}(1)\otimes \Lambda _{Y_R,\mathrm {rig},{\mathbf {Q}}_p}$ .

There is a finite affinoid cover $\{\operatorname {\mathrm {Spa}} R_j'\}$ of $Y_R$ trivializing ${\mathscr {O}}_{Y_R}(1)$ ; we let $\lambda _j$ denote the restriction of $\lambda $ to $\operatorname {\mathrm {Spa}} R_j'$ . For any $x\in \operatorname {\mathrm {Spa}} R_j'$ , we again have an exact sequence

$$\begin{align*}0\rightarrow k_x\otimes H^0\left(g^\ast f^\ast D_R^\vee\right)\rightarrow H^0\left(k_x\otimes g^\ast f^\ast D_{R}^\vee\right)\rightarrow \operatorname{\mathrm{Tor}}_1^{{\mathscr{O}}_X}\left(H^1(g^\ast f^\ast D_R^\vee),k_x\right)\rightarrow 0. \end{align*}$$

This implies in particular that the specialization of $\lambda _j$ is nonzero. If x has characteristic-p residue field, this implies that the specialization of $\lambda _j$ is surjective. As in the proof of [Reference BellovinBel23a, Lemma 5.7], this implies that there is an affinoid subdomain $V_j=\{\lvert p\rvert \leq \lvert u^{r_j}\rvert \}\subset \operatorname {\mathrm {Spa}} R_j'$ containing the locus $\{p=0\}$ over which $\lambda _j$ is surjective.

Let $N:=\max \{r_j\}$ and set $U_1:=\{\lvert p\rvert \leq \lvert u^N\rvert \}\subset \operatorname {\mathrm {Spa}} R$ . Then the preimage $(f\circ g)^{-1}\left (U_1\right )$ is contained in $\cup _j V_j$ . We will set $\widetilde U_1:=(f\circ g)^{-1}\left (U_1\right )$ . Then by construction, $\pi _1:\widetilde U_1\rightarrow U$ is surjective and

$$\begin{align*}\lambda|_{\widetilde U_1}:\pi_1^\ast D\rightarrow \pi_1^\ast{\mathscr{O}}_{Y_R}(1)|_{U_1} \end{align*}$$

is surjective, so its kernel is a family of $(\varphi ,\Gamma )$ -modules of rank $d-1$ .

On the other hand, set $U_2:=\{\lvert u^N\rvert \leq \lvert p\rvert \}\subset \operatorname {\mathrm {Spa}} R$ . Then the preimage $(f\circ g)^{-1}\left (U_2\right )$ is quasicompact and contained in the characteristic- $0$ locus of $Y_R$ , so we may apply the techniques of the proof of [Reference Kedlaya, Pottharst and XiaoKPX14, Theorem 6.3.9]. More precisely, we let $h:U_2'\rightarrow (f\circ g)^{-1}\left (U_2\right )$ be a proper birational morphism so that $H^i((f\circ g\circ h)^\ast D/t)$ is flat for $i=0$ and has Tor-dimension at most $1$ for $i=1, 2$ (again using [Reference Kedlaya, Pottharst and XiaoKPX14, Corollary 6.3.6(2)]). This lets us deduce that $h^\ast \lambda |_{U_2'}$ is surjective away from a proper Zariski-closed subspace, and locally on $U_2'$ , its cokernel is killed by a power of t. Then we make a further blow-up $\widetilde U_2\rightarrow U_2'$ such that over $U_2$ , the kernel of $\lambda $ is a family of $(\varphi ,\Gamma )$ -modules of rank $d-1$ , as desired.

This permits us to use induction to deduce the following:

Corollary 2.4.5. Let R be a pseudoaffinoid algebra with pseudouniformizer $u\in R$ , and let D be a family of rank- $d\ (\varphi ,\Gamma )$ -modules over R. Consider the Zariski closure Z of the locus in $\operatorname {\mathrm {Spa}} R_0\times \operatorname {\mathrm {Spa}} R_{z,0}\times {\mathbf {G}}_{m,R}^d$ corresponding to points $x=(D_x,\underline \delta _x)$ , where $\lvert \delta _{i,x}(p)^{\pm 1}\rvert \leq \lvert z\rvert ^{-h}$ for all i, and $\underline \delta _x$ is a regular parameter of $D_x$ . Then there is some $N'\gg 0$ such that

$$\begin{align*}Z\subset \{\lvert z^{N'}\rvert \leq \lvert u\rvert\text{ for all }i\}\subset \operatorname{\mathrm{Spa}} R\times \operatorname{\mathrm{Spa}} R_z\left\langle z^hT_i^{\pm 1}\right\rangle. \end{align*}$$

This is precisely the condition we need to apply Corollary A.0.2, so the closure we are interested in is well-behaved in $\operatorname {\mathrm {Spf}} R_0\mathop {\widehat \otimes } R_{z,0}\left \langle z^aT_i^b,T_i'\right \rangle /(T_i^bT_i'-z^a)$ .

Letting $\operatorname {\mathrm {Spa}} R$ range over a (finite) cover of $\left (\operatorname {\mathrm {Spa}} R_{\overline {\rho }}^\square \right )^{\mathrm {{an}}}$ , we obtain a closed subspace of $\operatorname {\mathrm {Spf}} R_{\overline \rho }^\square \times \operatorname {\mathrm {Spa}} R_{z,0}\left \langle z^aT_i^b,T_i'\right \rangle /(T_i^bT_i'-z^a)$ . Letting z range over $\{p,z_1,\ldots ,z_d\}$ , in turn, we may glue to get a closed subspace of $\operatorname {\mathrm {Spf}} R_0\times \mathfrak {T}_h^d$ , yielding the desired integral models of pieces of trianguline varieties:

3.1 Definitions

We briefly recall the construction of extended eigenvarieties in the two cases of interest to us. Fix a number field F and a reductive group ${{\mathrm{{H}}}}$ over F which is split at all places above p; then we define ${\mathbf {G}}:=\operatorname {\mathrm {Res}}_{F/{\mathbf {Q}}}{{\mathrm{{H}}}}$ . If we choose split models ${{\mathrm{{H}}}}_{{\mathscr {O}}_{F_v}}$ over ${\mathscr {O}}_{F_v}$ for each place $v\mid p$ , along with split maximal tori and Borel subgroups $\operatorname {\mathrm {T}}_v\subset {\mathbf {B}}_v\subset {{\mathrm{{H}}}}_{{\mathscr {O}}_{F_v}}$ , we obtain an integral model ${\mathbf {G}}_{{\mathbf {Z}}_p}:=\prod _{v\mid p}{{\mathrm{{H}}}}_{{\mathscr {O}}_{F_v}}$ of ${\mathbf {G}}$ , as well as closed subgroup schemes

$$\begin{align*}\operatorname{\mathrm{T}}:=\prod_{v\mid p}\operatorname{\mathrm{Res}}_{{\mathscr{O}}_{F_v}/{\mathbf{Z}}_p}T_v\subset {\mathbf{B}}:=\prod_{v\mid p}\operatorname{\mathrm{Res}}_{{\mathscr{O}}_{F_v}/{\mathbf{Z}}_p}{\mathbf{B}}_v. \end{align*}$$

Let $T_0:=\operatorname {\mathrm {T}}({\mathbf {Z}}_p)$ , and let the Iwahori subgroup $I\subset {\mathbf {G}}_{{\mathbf {Z}}_p}({\mathbf {Z}}_p)$ be the preimage of ${\mathbf {B}}({\mathbf {F}}_p)$ under the reduction map ${\mathbf {G}}_{{\mathbf {Z}}_p}({\mathbf {Z}}_p)\rightarrow {\mathbf {G}}_{{\mathbf {Z}}_p}({\mathbf {F}}_p)$ .

We choose a tame level by choosing compact open subgroups $K_\ell \subset {\mathbf {G}}({\mathbf {Q}}_\ell )$ for each prime $\ell \neq p$ , such that $K_\ell =\mathcal {G}({\mathbf {Z}}_\ell )$ for almost all primes $\ell $ (where $\mathcal {G}$ is some reductive model of ${\mathbf {G}}$ over ${\mathbf {Z}}[1/M]$ for some integer M). Then we put $K^p:=\prod _{\ell \neq p}K_\ell $ and $K:=K^pI$ ; we assume throughout that K contains an open normal subgroup $K'$ such that $[K:K']$ is prime to p and

(3.1.1) $$ \begin{align} x^{-1}D^{\times} x\cap K' \subset {\mathscr{O}}_F^{\times, +}\quad \text{ for all } x\in ({\mathbf{A}}_{F,f}\otimes_F D)^{\times} \end{align} $$

which is the neatness hypothesis of [Reference Johansson and NewtonJN19b].Footnote ¹ If ${\mathbf {Z}}$ denotes the center of ${\mathbf {G}}$ , we let $Z(K):={\mathbf {Z}}({\mathbf {Q}})\cap K$ and let $\overline {Z(K)}\subset T_0$ denote its p-adic closure. We also let $K_\infty \subset {\mathbf {G}}({\mathbf {R}})$ be a maximal compact and connected subgroup at infinity, and let $Z_\infty ^\circ \subset Z_\infty =:{\mathbf {Z}}({\mathbf {R}})$ denote the identity component.

Finally, let $\Sigma \subset T_0$ be the kernel of some splitting of the inclusion $T_0\subset \operatorname {\mathrm {T}}({\mathbf {Q}}_p)$ ; there are then certain submonoids $\Sigma ^{\mathrm {cpt}}\subset \Sigma ^+\subset \Sigma $ , and we fix some $t\in \Sigma ^{\mathrm {cpt}}$ .

In the cases of interest to us, F will be a totally real field, completely split at p, and ${{\mathrm{{H}}}}$ will be either $\operatorname {\mathrm {GL}}_2$ or the reductive group $\underline D^{\times }$ corresponding to the units of a totally definite quaternion algebra over F split at every place above p. Fixing isomorphisms $D_v\xrightarrow {\sim }\operatorname {\mathrm {Mat}}_2(F_v)$ for each place v where D is split, we may define integral models of ${{\mathrm{{H}}}}_v$ via ${{\mathrm{{H}}}}_{{\mathscr {O}}_{F_v}}(R_0):=\left (R_0\otimes \operatorname {\mathrm {Mat}}_2({\mathscr {O}}_{F_v})\right )^{\times }$ for all ${\mathscr {O}}_{F_v}$ -algebras $R_0$ (whether ${{\mathrm{{H}}}}=\operatorname {\mathrm {GL}}_2$ or $\underline D^{\times }$ ). In either case, we let ${\mathbf {B}}_v\subset {{\mathrm{{H}}}}_{{\mathscr {O}}_{F_v}}$ be the standard upper-triangular Borel and we let $\operatorname {\mathrm {T}}_v\subset {\mathbf {B}}_v$ be the standard diagonal maximal torus.

For either choice of ${{\mathrm{{H}}}}$ , the adelic subgroup $K(N)\subset \left ({\mathbf {A}}_{F,f}\otimes {{\mathrm{{H}}}}(F)\right )^{\times }$ of full level N is neat for $N\geq 3$ such that N is prime to the finite places v where ${{\mathrm{{H}}}}_v\neq \operatorname {\mathrm {GL}}_2$ . Thus, if we assume $p\geq 5$ , we may take $K^p$ arbitrary.

For either choice of ${{\mathrm{{H}}}}$ , we define $\Sigma _v^+:=\left \{\left (\begin {smallmatrix}\varpi _v^{a_1} & 0 \\ 0 & \varpi _v^{a_2}\end {smallmatrix}\right ) \mid a_2\geq a_1\right \}$ and $\Delta _v:=I_v\Sigma _v^+I_v$ . Similarly, we define $\Sigma ^+:=\prod _{v\mid p}\Sigma _v^+$ and $\Delta _p:=I\Sigma ^+I=\prod _{v\mid p}\Delta _v$ . Then we fix $U_{\varpi _v}:=\left [I_v\left (\begin {smallmatrix}1 & \\ & \varpi _v\end {smallmatrix}\right )I_v\right ]\in I_v\backslash {{\mathrm{{H}}}}(F_v)/I_v$ and $U_p:=\prod _{v\mid p}U_{\varpi _v}$ .

For each prime $\ell \neq p$ , we fix a monoid $\Delta _\ell \subset {\mathbf {G}}({\mathbf {Q}}_\ell )$ containing $K_\ell $ , which is equal to ${\mathbf {G}}({\mathbf {Q}}_\ell )$ when $K_\ell =\mathcal {G}({\mathbf {Z}}_\ell )$ such that $(\Delta _\ell , K_\ell )$ is a Hecke pair and the Hecke algebra $\mathbb {T}(\Delta _\ell , K_\ell )$ over ${\mathbf {Z}}_p$ is commutative. Then we define $\Delta ^p:=\prod _{\ell \neq p}^\prime \Delta _\ell $ and $\Delta :=\Delta ^p\Delta _p$ . We write $\mathbb {T}(\Delta ^p,K^p):=\otimes _{\ell \neq p}\mathbb {T}(\Delta _\ell , K_\ell )$ and $\mathbb {T}(\Delta ,K):=\otimes _\ell \mathbb {T}(\Delta _\ell , K_\ell )$ for the corresponding global Hecke algebras.

A weight is a continuous homomorphism $\kappa :T_0\rightarrow R^{\times }$ which is trivial on $Z(K)$ , where R is a pseudoaffinoid algebra over ${\mathbf {Z}}_p$ . We define weight space $\mathscr {W}$ via

$$\begin{align*}\mathscr{W}(R):=\{\kappa\in\operatorname{\mathrm{Hom}}_{\mathrm{cts}}(T_0, R^{\times}) \mid \kappa|_{Z(K)}=1\}. \end{align*}$$

It can be written explicitly as the analytic locus of $\operatorname {\mathrm {Spa}}\left ({\mathbf {Z}}_p[\![T_0/\overline {Z(K)}]\!],{\mathbf {Z}}_p[\![T_0/\overline {Z(K)}]\!]\right )$ . Then $\mathscr {W}$ is equidimensional of dimension $1+[F:{\mathbf {Q}}]+\mathfrak {d}$ , where $\mathfrak {d}$ is the defect in Leopoldt’s conjecture for F and p.

The next step is to construct a sheaf of Hecke modules over weight space such that $U_p$ acts compactly and admits a Fredholm determinant. We will actually use two such sheaves. If $\kappa :T_0\rightarrow R^{\times }$ is a weight, then [Reference Johansson and NewtonJN16] construct certain modules of analytic functions $\mathcal {A}_{\kappa }^r$ and distributions $\mathcal {D}_{\kappa }^r$ . Here, $r\in (r_{\kappa },1)$ , where $r_{\kappa }\in [1/p,1)$ . When $r_{\kappa }\in (1/p,1)$ , they also construct $\mathcal {A}_{\kappa }^{<r}$ and $\mathcal {D}_{\kappa }^{<r}$ so that $\mathcal {D}_{\kappa }^r$ is the dual of $\mathcal {A}_{\kappa }^{<r}$ and $\mathcal {A}_{\kappa }^r$ is the dual of $\mathcal {D}_{\kappa }^{<r}$ . As in [Reference Hansen and NewtonHN17], we fix augmented Borel–Serre complexes $C_{\bullet }^{\mathrm {BM}}(K,-)$ and $C_{c}^{\bullet }(K,-)$ for Borel–Moore homology and compactly supported cohomology, respectively, and we consider

$$\begin{align*}C_{\bullet}^{\mathrm{BM}}(K,\mathcal{A}_{\kappa}^r) \end{align*}$$

as well as

$$\begin{align*}C_c^{\bullet}(K,\mathcal{D}_{\kappa}^r)\quad\text{ and }\quad C_c^{\bullet}(K,\mathcal{D}_{\kappa}^{<r}). \end{align*}$$

Now, $\mathcal {A}_{\kappa }^r$ and $\mathcal {D}_{\kappa }^r$ are potentially orthonormalizable, so $C_\ast ^{\mathrm {BM}}(K,\mathcal {A}_{\kappa }^r):=\oplus _i C_i^{\mathrm {BM}}(K,\mathcal {A}_{\kappa }^r)$ and $C_c^\ast (K,\mathcal {D}_{\kappa }^r):=\oplus _iC_c^i(K,\mathcal {D}_{\kappa }^r)$ are, as well. Since $U_p$ acts compactly on $\mathcal {A}_{\kappa }^r$ and $\mathcal {D}_{\kappa }^r$ , this implies that there are Fredholm determinants $F_{\kappa }^{r,\prime }$ and $F_{\kappa }^r$ for its action on $C_\ast ^{\mathrm {BM}}(K,\mathcal {A}_{\kappa }^r)$ and $C_c^\ast (K,\mathcal {D}_{\kappa }^r)$ , respectively.

It turns out that $F_{\kappa }^{r,\prime }$ and $F_{\kappa }^r$ are independent of r by [Reference Johansson and NewtonJN16, Proposition 4.1.2]; we set $\mathscr {D}_{\kappa }:=\varprojlim _r\mathcal {D}_{\kappa }^r$ and $\mathscr {A}_{\kappa }:=\varinjlim _r\mathcal {A}_{\kappa }^r$ , and we write $F_{\kappa }$ and $F_{\kappa }^\prime $ for the Fredholm determinants of $U_p$ on $C_c^\ast (K,\mathscr {D}_{\kappa })$ and $C_\ast ^{\mathrm {BM}}(K,\mathscr {A}_{\kappa })$ , respectively. Then $F_{\kappa }$ and $F_{\kappa }^\prime $ define spectral varieties $\mathscr {Z}\subset {\mathbf {A}}_{\mathscr {W}_F}^1$ and $\mathscr {Z}^\prime \subset {\mathbf {A}}_{\mathscr {W}_F}^1$ . We let $\pi :\mathscr {Z}\rightarrow \mathscr {W}_F$ and $\pi ':\mathscr {Z}'\rightarrow \mathscr {W}_F$ be the projection on the first factor; they are flat morphisms of pseudorigid spaces.

By [Reference Johansson and NewtonJN16, Theorem 2.3.2], $\mathscr {Z}$ has a cover by open affinoid subspaces V such that $U:=\pi (V)$ is an open affinoid subspace of $\mathscr {W}_F$ and $\pi |_V:V\rightarrow U$ is finite of constant degree. This implies that over such a V, F factors as $F_V=Q_VS_V$ where $Q_V$ is a multiplicative polynomial of degree $\deg \pi |_V$ , $S_V$ is a Fredholm series, and $Q_V$ and $S_V$ are relatively prime.

If such a factorization exists, we may make $C_c^{\bullet }(K,\mathscr {D}_V)$ into a complex of ${\mathscr {O}}_{\mathscr {Z}}$ -modules by letting T act via $U_p^{-1}$ . Then the assignment $V\mapsto \ker Q_V^\ast (U_p)\subset C^{\bullet }(K,\mathscr {D}_{V})$ defines a bounded complex $\mathscr {K}^{\bullet }$ of coherent ${\mathscr {O}}_{\mathscr {Z}}$ -modules, where $Q_V^\ast (T):=T^{\deg Q_V}Q_V(1/T)$ . If $V=\pi ^{-1}(U)$ , where $(U,h)$ is a slope datum, then $\mathscr {K}^{\bullet }$ is the slope- $\leq h$ subcomplex of $C_c^{\bullet }(K,\mathscr {D}_V)$ . We set

$$\begin{align*}\mathscr{M}_c^\ast := \oplus_i H^i(\mathscr{K}) \end{align*}$$

which is a coherent sheaf on $\mathscr {Z}$ .

Such factorizations exist locally, by an extension of a result of [Reference Ash and StevensAS]:

Since spectral varieties are flat over weight space, we will be able to use the following result to show that slope factorizations exist:

We further observe that we have inclusions $\mathcal {D}_{\kappa }^r\subset \mathcal {D}_{\kappa }^{<r}\subset \mathcal {D}_{\kappa }^s$ for any $r_{\kappa }\leq s<r$ . Thus, the fact that $F_{\kappa }^r=F_{\kappa }^s$ implies that $\mathscr {M}_c^\ast = \oplus _i H_c^i(K,\mathscr {D}_{\kappa }^{<r})_{\leq h}$ for any $r>r_{\kappa }$ .

We may carry out the same procedure for the action of $U_p$ on $C_\ast ^{\mathrm {BM}}(K,\mathscr {A}_{\kappa })$ and obtain a coherent sheaf $\mathscr {M}_\ast ^{\mathrm {BM}}=\oplus _iH_i^{\mathrm {BM}}(K,\mathscr {A}_{\kappa })_{\leq h}$ on $\mathscr {Z}^{\prime }$ . Let $\mathbb {T}$ denote either $\mathbb {T}(\Delta ^p,K^p)$ or $\mathbb {T}(\Delta ,K)$ . Both $\mathscr {M}_c^\ast $ and $\mathscr {M}_\ast ^{\mathrm {BM}}$ are Hecke modules, so we have constructed eigenvariety data $(\mathscr {Z},\mathscr {M}_c^\ast ,\mathbb {T},\psi )$ and $(\mathscr {Z}^\prime ,\mathscr {M}_\ast ^{\mathrm {BM}},\mathbb {T},\psi ')$ (where $\psi :\mathbb {T}\rightarrow \operatorname {\mathrm {End}}_{{\mathscr {O}}_{\mathscr {Z}}}(\mathscr {M}_c^\ast )$ and $\psi ':\mathbb {T}\rightarrow \operatorname {\mathrm {End}}_{{\mathscr {O}}_{\mathscr {Z}^\prime }}(\mathscr {M}_\ast ^{\mathrm {BM}})$ give the Hecke-module structures).

Finally, we may construct eigenvarieties from the eigenvariety data. Let $\mathscr {T}$ and $\mathscr {T}^\prime $ denote the sheaves of ${\mathscr {O}}_{\mathscr {Z}}$ -algebras generated by the images of $\psi $ and $\psi '$ , respectively; in particular, if $\mathscr {Z}_{U,h}\subset \mathscr {Z}$ is an open affinoid corresponding to the slope datum $(U,h)$ , then

$$\begin{align*}\mathscr{T}(\mathscr{Z}_{U,h}) = \operatorname{\mathrm{im}}\left({\mathscr{O}}(\mathscr{Z}_{U,h})\otimes_{{\mathbf{Z}}_p}\mathbb{T}\rightarrow \operatorname{\mathrm{End}}_{{\mathscr{O}}(\mathscr{Z}_{U,h})}\left(H_c^\ast(K,\mathscr{D}_{U}\right)_{\leq h}\right)=:\mathbb{T}_{U,h} \end{align*}$$

and

$$\begin{align*}\mathscr{T}^\prime(\mathscr{Z}'_{U,h}) = \operatorname{\mathrm{im}}\left({\mathscr{O}}(\mathscr{Z}'_{U,h})\otimes_{{\mathbf{Z}}_p}\mathbb{T}\rightarrow \operatorname{\mathrm{End}}_{{\mathscr{O}}(\mathscr{Z}'_{U,h})}\left(H_\ast^{\mathrm{BM}}(K,\mathscr{A}_{U}\right)_{\leq h}\right)=:\mathbb{T}_{U,h}'. \end{align*}$$

Then we set

$$\begin{align*}\mathscr{X}_{{\mathbf{G}}}^{\mathbb{T}}:=\underline{\mathrm{Spa}} \mathscr{T} \end{align*}$$

and

$$\begin{align*}\mathscr{X}_{{\mathbf{G}}}^{\mathbb{T},\prime} :=\underline{\mathrm{Spa}} \mathscr{T}^\prime \end{align*}$$

and we have finite morphisms $q:\mathscr {X}_{{\mathbf {G}}}\rightarrow \mathscr {Z}$ and $q':\mathscr {X}_{{\mathbf {G}}}^\prime \rightarrow \mathscr {Z}^\prime $ , and ${\mathbf {Z}}_p$ -algebra homomorphisms $\phi _{\mathscr {X}}:\mathbb {T}\rightarrow {\mathscr {O}}(\mathscr {X}_{{\mathbf {G}}}^{\mathbb {T}})$ and $\phi _{\mathscr {X}'}:\mathbb {T}\rightarrow {\mathscr {O}}(\mathscr {X}_{{\mathbf {G}}}^{\mathbb {T},\prime })$ . If the choice of Hecke operators is clear from context, we will drop $\mathbb {T}$ from the notation.

If $\mathbb {T}=\mathbb {T}(\Delta ,K)$ , then unlike [Reference Johansson and NewtonJN16], we are adding the Hecke operators $U_{\varpi _v}$ at places $v\mid p$ to our Hecke algebras (and hence to the coordinate rings of our eigenvarieties), not just the controlling operator $U_p$ .

3.2 The middle-degree eigenvariety

When $F={\mathbf {Q}}$ and ${\mathbf {G}}={{\mathrm{{H}}}}=\operatorname {\mathrm {GL}}_2$ , for any fixed slope h such that $C_c^{\bullet }(K,\mathscr {D}_{\kappa })$ has a slope- $\leq h$ decomposition, the complex $C_c^{\bullet }(K,\mathscr {D}_{\kappa })_{\leq h}$ has cohomology only in degree $1$ , and $H_c^1(K,\mathscr {D}_{\kappa })_{\leq h}$ is projective. As a result, the eigencurve is reduced and equidimensional, and classical points are very Zariski-dense. For a general totally real field F, the situation is more complicated. The complex $C_c^{\bullet }(K,\mathscr {D}_{\kappa })_{\leq h}$ lives in degrees $[0,2d]$ , and we are still primarily interested in the degree-d cohomology; indeed, the discussion of [Reference HarderHar87, §3.6] shows that cuspidal cohomological automorphic forms contribute only to middle degree cohomology in the classical finite-dimensional classical analogue. However, there is no reason to expect the other cohomology groups to vanish.

Instead, following [Reference Bergdall and HansenBH17] we will sketch the construction of an open subspace $\mathscr {X}_{\operatorname {\mathrm {GL}}_2/F,\mathrm {mid}}\subset \mathscr {X}_{\operatorname {\mathrm {GL}}_2/F}$ , where $H_c^i(K,\mathscr {D}_{\kappa })$ vanishes for $i\neq d$ ; by [Reference Bergdall and HansenBH17, Theorem B.0.1], all classical points of $\mathscr {X}_{\operatorname {\mathrm {GL}}_2/F}$ whose associated Galois representation have sufficiently large residual image lie in $\mathscr {X}_{\operatorname {\mathrm {GL}}_2/F,\mathrm {mid}}$ . The cohomology and base change result [Reference Johansson and NewtonJN16, Theorem 4.2.1] shows that the locus where $H_c^i(K,\mathscr {D}_{\kappa })=0$ for $i\geq d+1$ is open, but we need to use the homology complexes $C_{\bullet }^{\mathrm {BM}}(K,\mathcal {A}_{\kappa })$ to control $H_c^i(K,\mathscr {D}_{\kappa })$ for $i\leq d-1$ .

As in [Reference Bergdall and HansenBH17], the key points are a base change result for Borel–Moore homology, and a universal coefficients theorem relating it to compactly supported cohomology:

The proof uses both the fact that $\mathcal {D}_{\kappa }^{<r}$ is the continuous dual of $\mathcal {A}_{\kappa }^r$ and the fact that $H_c^i(K,\mathcal {D}_{\kappa }^{<r})_{\leq h}=H_c^i(K,\mathcal {D}_{\kappa }^r)_{\leq h}$ for all $r>r_{\kappa }$ .

Proposition 3.2.2. If $(U,h)$ is a slope datum, then we have a natural commuting diagram

Thus, we have a morphism $\tau :\mathscr {X}_{\operatorname {\mathrm {GL}}_2/F}^{\mathrm {{red}}}\rightarrow \mathscr {X}_{\operatorname {\mathrm {GL}}_2/F}^\prime $ and a closed immersion $i:\mathscr {X}_{\operatorname {\mathrm {GL}}_2/F}^{\mathrm {{red}}}\hookrightarrow \mathscr {X}_{\operatorname {\mathrm {GL}}_2/F}$ .

Definition 3.2.3.

$$\begin{align*}\mathscr{X}_{\operatorname{\mathrm{GL}}_2/F,\mathrm{mid}}:=\mathscr{X}_{\operatorname{\mathrm{GL}}_2/F}\smallsetminus\left[\left(\cup_{j=d+1}^{2d}\operatorname{\mathrm{supp}}(\mathscr{M}_c^j)\right)\cup\left(\cup_{j=0}^{d-1}\operatorname{\mathrm{supp}}(i_\ast\tau^\ast\mathscr{M}_j^{\mathrm{BM}}\right)\right]. \end{align*}$$

By construction, a point $x\in \mathscr {X}_{\operatorname {\mathrm {GL}}_2/F}$ of weight $\lambda _x$ lies in the Zariski-open subspace $\mathscr {X}_{\operatorname {\mathrm {GL}}_2/F,\mathrm {mid}}\subset \mathscr {X}_{\operatorname {\mathrm {GL}}_2/F}$ if and only if $H_c^j(K,k_x\otimes \mathscr {D}_{\lambda _x})_{\mathfrak {m}_x}=0$ for all $j\neq d$ (where $\mathfrak {m}_x$ is the maximal ideal of the Hecke algebra corresponding to x).

3.3 Jacquet–Langlands

The classical Jacquet–Langlands correspondence lets us transfer automorphic forms between $\operatorname {\mathrm {GL}}_2$ and quaternionic algebraic groups. Over ${\mathbf {Q}}$ , this correspondence was interpolated in [Reference ChenevierChe05] to give a closed immersion of eigencurves $\mathscr {X}_{D^{\times }/{\mathbf {Q}}}^{\mathrm {rig}}\hookrightarrow \mathscr {X}_{\operatorname {\mathrm {GL}}_2/{\mathbf {Q}}}^{\mathrm {rig}}$ ; this interpolation was given for general totally real fields in [Reference BirkbeckBir19]. We give the corresponding result for extended eigenvarieties. However, as we have elected to work with the eigenvariety for $\operatorname {\mathrm {GL}}_2/F$ constructed in [Reference Johansson and NewtonJN16] via overconvergent cohomology, instead of the eigenvariety constructed from Hilbert modular forms, we will never get an isomorphism of eigenvarieties, even when $[F:{\mathbf {Q}}]$ is even.

Let D be a totally definite quaternion algebra over F, split at every place above p, and let $\mathfrak {d}_D$ be its discriminant. For any ideal $\mathfrak {n}\subset {\mathscr {O}}_F$ with $(\mathfrak {d}_D,\mathfrak {n})=1$ , we define the subgroup $K_1^{\underline D^{\times }}(\mathfrak {n})\subset ({\mathscr {O}}_D\otimes \widehat {{\mathbf {Z}}})^{\times }$

$$\begin{align*}K_1^{\underline D^{\times}}(\mathfrak{n}):=\left\{g\in ({\mathscr{O}}_D\otimes \widehat{{\mathbf{Z}}})^{\times} \mid g\equiv \left(\begin{smallmatrix}\ast & \ast \\ 0 & 1\end{smallmatrix}\right)\quad\pmod{\mathfrak{n}}\right\}. \end{align*}$$

We may define a similar subgroup $K_1^{\operatorname {\mathrm {GL}}_2/F}(\mathfrak {n})\subset \operatorname {\mathrm {Res}}_{{\mathscr {O}}_F/{\mathbf {Z}}_p}\operatorname {\mathrm {GL}}_2(\widehat {{\mathbf {Z}}})$ .

A classical algebraic weight is a tuple $(k_\sigma )\in {\mathbf {Z}}_{\geq 2}^{\Sigma _\infty }$ together with a tuple $(v_\sigma )\in {\mathbf {Z}}^{\Sigma _\infty }$ such that $(k_\sigma )+(v_\sigma ) = (r,\ldots ,r)$ for some $r\in {\mathbf {Z}}$ , where $\Sigma _\infty $ is the set of embeddings $F\hookrightarrow {\mathbf {R}}$ . Set $e_1:=(\frac {r+k_\sigma }{2})$ and $e_2:=(\frac {r-k_\sigma }{2})$ , and define characters $\kappa _i:F^{\times }\rightarrow {\mathbf {R}}^{\times }$ for $i=1,2$ via

$$\begin{align*}\kappa_i(x) = \prod_{\sigma\in\Sigma_\infty}\sigma(x)^{e_{i,\sigma}} \end{align*}$$

Then $(\kappa _1,\kappa _2)$ is a character on $\operatorname {\mathrm {T}}({\mathbf {Z}})$ which is trivial on a finite-index subgroup of the center $Z_G({\mathbf {Z}})={\mathscr {O}}_F^{\times }$ .

Then we have the classical Jacquet–Langlands correspondence:

Theorem 3.3.1. Let $\kappa $ be a classical weight, and let $\mathfrak {n}\subset {\mathscr {O}}_F$ be an ideal such that $(\mathfrak {n},\mathfrak {d}_D)=1$ . There is a Hecke-equivariant isomorphism of spaces of cusp forms

$$\begin{align*}S_{\kappa}^{\underline D^{\times}}(K_1^{\underline D^{\times}}(\mathfrak{n}))\xrightarrow\sim S_{\kappa}^{\mathfrak{d}_D-\mathrm{new}}(K_1^{\operatorname{\mathrm{GL}}_2/F}(\mathfrak{n}\mathfrak{d}_D)). \end{align*}$$

We will interpolate this correspondence to a closed immersion $\mathscr {X}_{\underline D^{\times }}\hookrightarrow \mathscr {X}_{\operatorname {\mathrm {GL}}_2/F}$ , where the source has tame level $K_1^{\underline D^{\times }}(\mathfrak {n})$ and the target has tame level $K_1^{\operatorname {\mathrm {GL}}_2/F}(\mathfrak {n})$ . We use the interpolation theorem of [Reference Johansson and NewtonJN19a]:

We remark that in the presence of integral structures, we can make a sharper statement:

Proof. As in the proof of [Reference Johansson and NewtonJN19a, Theorem 3.2.1], one reduces to the case where $R_0:=R_{1,0}=R_{2,0}$ and $\mathbb {T}:=\mathbb {T}_1=\mathbb {T}_2$ , and one considers the actions of $\mathbb {T}_1\oplus \mathbb {T}_2$ on $M_{1,0}\oplus M_{2,0}$ . Then we have quotients

$$\begin{align*}R_{3,0}:=\operatorname{\mathrm{im}}\left(R_0\otimes\mathbb{T}\rightarrow \operatorname{\mathrm{End}}_{R_0}(M_{1,0}\oplus M_{2,0})\right)\twoheadrightarrow R_{i,0}'. \end{align*}$$

Since $\overline {\mathscr {X}}\subset \mathscr {X}_2$ and $\operatorname {\mathrm {Spf}} R_{3,0}$ is separated, we have $\overline {\mathscr {X}}_0\subset \operatorname {\mathrm {Spf}} R_{2,0}'$ , as desired.

We take $\mathscr {Z}_1=\mathscr {Z}_2=\mathscr {W}_F\times {\mathbf {G}}_m$ . In order to define $\mathbb {T}=\mathbb {T}_1=\mathbb {T}_2$ , we set

$$\begin{align*}\Delta_{v} = \begin{cases} \operatorname{\mathrm{GL}}_2(F_v) &\mbox{ if } v\nmid p\mathfrak{d}_D\mathfrak{n} \\ {K_1^{\underline D^{\times}}(\mathfrak{n})}_v &\mbox{ if } v\mid \mathfrak{d}_D\mathfrak{n}. \end{cases} \end{align*}$$

For $v\mid p$ , we take $\Delta _{v}$ as in §3.1. In other words, $\mathbb {T}$ is the commutative ${\mathbf {Z}}_p$ -algebra generated by $T_v:=[K_v\left ( \begin {smallmatrix}1 & \\ & \varpi _v\end {smallmatrix} \right )K_v]$ and $S_v:=[K_v\left ( \begin {smallmatrix}\varpi _v & \\ & \varpi _v\end {smallmatrix} \right )K_v]$ for $v\nmid p\mathfrak {d}_D\mathfrak {n}$ and $U_{\varpi _v}$ for $v\mid p$ .

However, we cannot immediately combine this interpolation theorem with the Jacquet–Langlands correspondence because our choice of weight space means that classical weights may not be Zariski dense unless Leopoldt’s conjecture is true. More precisely, given a classical algebraic weight, we constructed a character on $\operatorname {\mathrm {T}}({\mathbf {Z}})$ trivial on a finite-index subgroup of ${\mathscr {O}}_F^{\times }$ , and conversely, characters on $\operatorname {\mathrm {T}}({\mathbf {Z}})$ trivial on a finite-index subgroup of ${\mathscr {O}}_F^{\times }$ yield classical algebraic weights. This equivalence relies on Dirichlet’s unit theorem.

This means that there are two natural definitions of p-adic families of weights, $\mathscr {W}_F'=\operatorname {\mathrm {Spa}} {\mathbf {Z}}_p[\![(\operatorname {\mathrm {Res}}_{{\mathscr {O}}_F/{\mathbf {Z}}_p}{\mathbf {G}}_m)\times {\mathbf {Z}}_p^{\times }]\!]^{\mathrm {{an}}}$ interpolating classical algebraic weights, and $\mathscr {W}_F$ interpolating characters on $T_0$ , and the equivalence of those two definitions depends on Leopoldt’s conjecture.

Fortunately, the gap between these weight spaces can be controlled: There is a closed embedding $\mathscr {W}_F'\hookrightarrow \mathscr {W}_F$ , and the twisting action by characters on ${\mathscr {O}}_{F,p}^{\times }/\overline {{\mathscr {O}}_F^{\times ,+}}$ defines a surjective map

$$\begin{align*}\widehat{{\mathscr{O}}_{F,p}^{\times}/\overline{{\mathscr{O}}_F^{\times,+}}}\times\mathscr{W}_F'\rightarrow \mathscr{W}_F^{\mathrm{rig}}. \end{align*}$$

We say that a weight $\lambda \in \mathscr {W}_F^{\mathrm {rig}}(\overline {{\mathbf {Q}}}_p)$ is twist classical if it is in the $\widehat {{\mathscr {O}}_{F,p}^{\times }/\overline {{\mathscr {O}}_F^{\times ,+}}}(\overline {{\mathbf {Q}}}_p)$ -orbit of a classical weight. Then twist classical weights are very Zariski dense in $\mathscr {W}_F$ .

In addition, we may define a twisting action on Hecke modules, as in [Reference Bergdall and HansenBH17]. Let $\operatorname {\mathrm {Gal}}_{F,p}$ denote the Galois group of the maximal abelian extension of F unramified away from p and $\infty $ , and let $\eta :\operatorname {\mathrm {Gal}}_{F,p}\rightarrow \overline {{\mathbf {Q}}}_p^{\times }$ be a continuous character. Global class field theory implies that $\operatorname {\mathrm {Gal}}_{F,p}$ fits into an exact sequence

$$\begin{align*}1\rightarrow {\mathscr{O}}_{F,p}^{\times}/\overline{{\mathscr{O}}_F^{\times,+}}\rightarrow \operatorname{\mathrm{Gal}}_{F,p}\rightarrow \mathrm{Cl}_F^+\rightarrow 1, \end{align*}$$

where $\mathrm {Cl}_F^+$ is the narrow class group of F (and hence finite). Suppose M is an R-module equipped with an R-linear left $\Delta _p$ -action. Then we may define a new left $\Delta _p$ -module $M(\eta ):=M\otimes \eta ^{-1}|_{{\mathscr {O}}_{F,p}^{\times }}$ , where the action of $g\in \Delta _p$ is given by

$$\begin{align*}g\cdot m = \left(\eta^{-1}|_{{\mathscr{O}}_{F,p}^{\times}}(\det g \cdot p^{-\sum_{v\mid p}v(\det g)})\right)\cdot (g\cdot m). \end{align*}$$

In particular, $\mathscr {D}_{\kappa }(\eta )\cong \mathscr {D}_{\eta ^{-1}\cdot \kappa }$ by [Reference Bergdall and HansenBH17, Lemma 5.5.2], and there is an isomorphism

$$\begin{align*}\mathrm{tw}_{\eta}:H_c^\ast(K,\mathscr{D}_{\kappa})\xrightarrow\sim H_c^\ast(K,\mathscr{D}_{\eta^{-1}\cdot\kappa}. \end{align*}$$

Suppose $x\in \mathscr {X}_{\underline D^{\times }}(\overline {{\mathbf {Q}}}_p)$ is a point with $\mathrm {wt}(x)=:\lambda $ , corresponding to the system of Hecke eigenvalues $\psi _x:\mathbb {T}\rightarrow \overline {{\mathbf {Q}}}_p$ . Then we define a new system of Hecke eigenvalues via

$$\begin{align*}\mathrm{tw}_\eta(\psi_x)(T) = \begin{cases} \eta(\varpi_v)\psi_x(T) & \mbox{ if } v\nmid p\mathfrak{d}_D\mathfrak{n} \mbox{ and } T=T_v \\ \eta(\varpi_v)^2\psi_x(T) & \mbox{ if } v\nmid p\mathfrak{d}_D\mathfrak{n} \mbox{ and } T=S_v \\ \eta(\varpi_v)\psi_x(T) & \mbox{ if } v\mid p. \end{cases} \end{align*}$$

Then it follows from [Reference Bergdall and HansenBH17, Proposition 5.5.5] that $\mathrm {tw}_\eta (\psi _x)$ corresponds to a point $\mathrm {tw}_\eta (x)\in \mathscr {X}_{\underline D^{\times }}$ of weight $\eta ^{-1}|_{{\mathscr {O}}_{F,p}^{\times }}\cdot \kappa $ .

We say that a point $x\in \mathscr {X}_{\underline D^{\times }}(\overline {{\mathbf {Q}}}_p)$ is twist classical if it is in the $\widehat {\operatorname {\mathrm {Gal}}_{F,p}}(\overline {{\mathbf {Q}}}_p)$ -orbit of a point corresponding to a classical system of Hecke eigenvalues.

Proof. Recall that $\mathscr {X}_{\underline D^{\times }}$ admits a cover by affinoid pseudorigid spaces of the form $\operatorname {\mathrm {Spa}} \mathscr {T}(\mathscr {Z}_{U,h})$ , where $\pi :\mathscr {Z}_{U,h}\rightarrow U$ is finite of constant degree, and

$$\begin{align*}\mathscr{T}(\mathscr{Z}_{U,h}) = \operatorname{\mathrm{im}}\left({\mathscr{O}}(\mathscr{Z}_{U,h})\otimes_{{\mathbf{Z}}_p}\mathbb{T}^p\rightarrow \operatorname{\mathrm{End}}_{{\mathscr{O}}(\mathscr{Z}_{U,h})}(H_c^\ast(K,\mathscr{D}_U)_{\leq h}\right). \end{align*}$$

We write $U=\operatorname {\mathrm {Spa}} R$ for some pseudoaffinoid algebra R over ${\mathbf {Z}}_p$ . We will show that $\operatorname {\mathrm {Spec}}\mathscr {T}(\mathscr {Z}_{U,h})\rightarrow \operatorname {\mathrm {Spec}} R$ carries irreducible components surjectively onto irreducible components, and we will construct a Zariski dense set of maximal points $W_{U,h}^{\mathrm {tw-cl}}\subset U$ such that the points of $\mathrm {wt}^{-1}(W_{U,h}^{\mathrm {tw-cl}})$ are twist classical. By [Reference ChenevierChe04, Lemme 6.2.8], this implies the desired result.

To see that irreducible components of $\operatorname {\mathrm {Spec}}\mathscr {T}(\mathscr {Z}_{U,h})$ map surjectively onto irreducible components of $\operatorname {\mathrm {Spec}} R$ , we observe that D is totally definite, so the associated Shimura manifold is a finite set of points and $H_c^\ast (K,\mathscr {D}_U)$ vanishes outside degree $0$ . The base change spectral sequence of [Reference Johansson and NewtonJN16, Theorem 4.2.1] implies that the formation of $H^0(K,\mathscr {D}_U)_{\leq h}$ commutes with arbitrary base change on U, which implies that $H^0(K,\mathscr {D}_U)_{\leq h}$ is flat. Then [Reference ChenevierChe04, Lemme 6.2.10] implies that $\operatorname {\mathrm {Spec}}\mathscr {T}(\mathscr {Z}_{U,h})\rightarrow \operatorname {\mathrm {Spec}} R$ carries irreducible components surjectively onto irreducible components, as desired.

Thus, it remains to construct $W_{U,h}^{\mathrm {tw-cl}}$ . Birkbeck proved a ‘small slope implies classical’ result [Reference BirkbeckBir19, Theorem 4.3.7] and constructed a set $W_{U,h}^{\mathrm {cl}}$ Zariski dense in $U\cap \mathscr {W}_F'$ such that the points of $\mathrm {wt}^{-1}(W_{U,h}^{\mathrm {cl}})$ are classical (see the proof of [Reference BirkbeckBir19, Theorem 6.1.9]). Setting $W_{U,h}^{\mathrm {tw-cl}}$ to be the $\widehat {{\mathscr {O}}_{F,p}^{\times }/\overline {{\mathscr {O}}_F^{\times ,+}}}(\overline {{\mathbf {Q}}}_p)$ -orbit of $W_{U,h}^{\mathrm {cl}}$ , [Reference Bergdall and HansenBH17, Lemma 6.3.1] implies that points of $\mathrm {wt}^{-1}(W_{U,h}^{\mathrm {tw-cl}})$ are twist classical, and we are done.

As a corollary, we deduce that $\mathscr {X}_{\underline D^{\times }}$ has no components supported entirely in characteristic p:

We may use similar arguments to show that $\mathscr {X}_{\underline D^{\times }}$ is reduced:

Proof. We first show that $\mathscr {X}_{\underline D^{\times }}^{\mathrm {rig}}$ is reduced. By [Reference Johansson and NewtonJN16, Proposition 6.1.2] (which adapts [Reference ChenevierChe05, Proposition 3.9]), it is enough to find a Zariski dense set of twist classical weights $W_{U,h}^{{\mathrm{ss}}}\subset U\subset \mathscr {W}_F^{\mathrm {rig}}$ for each slope datum $(U,h)$ such that $\mathscr {M}(\mathscr {Z}_{U,h})_{\kappa }$ is a semisimple Hecke module for all $\kappa \in W_{U,h}^{{\mathrm{ss}}}$ . Birkbeck [Reference BirkbeckBir19, Lemma 6.1.12] constructed sets $W_{U,h}^{\prime ,{\mathrm{ss}}}$ Zariski dense in $U\cap \mathscr {W}_F^{\prime ,\mathrm {rig}}$ with this property, and we will again use twisting by p-adic characters to construct $W_{U,h}^{{\mathrm{ss}}}$ .

If $\eta :{\mathscr {O}}_{F,p}^{\times }/\overline {{\mathscr {O}}_F^{\times ,+}}\rightarrow \overline {{\mathbf {Q}}}_p^{\times }$ is a character, we have an isomorphism

$$\begin{align*}\mathrm{tw}_\eta:H_c^\ast(K,\mathscr{D}_{\kappa})\xrightarrow\sim H_c^\ast(K,\mathscr{D}_{\eta^{-1}\cdot\kappa}. \end{align*}$$

By [Reference Bergdall and HansenBH17, Proposition 5.5.5], $\mathrm {tw}_\eta $ is Hecke-equivariant up to scalars, so $\mathscr {M}(\mathscr {Z}_{U,h})_{\kappa }$ is a semisimple Hecke module if and only if $\mathscr {M}(\mathscr {Z}_{\eta ^{-1}\cdot U,h})_{\eta ^{-1}\cdot \kappa }$ is. Thus, we may take $W_{U,h}^{{\mathrm{ss}}}$ to be the $\widehat {{\mathscr {O}}_{F,p}^{\times }/\overline {{\mathscr {O}}_F^{\times ,+}}}(\overline {{\mathbf {Q}}}_p)$ -orbit of $\cup _{U'}W_{U',h}^{\prime ,{\mathrm{ss}}}$ , as $(U',h)$ varies through slope data, and we see that $\mathscr {X}_{\underline D^{\times }}^{\mathrm {rig}}$ is reduced.

Now, let $X\subset \mathscr {X}_{\underline D^{\times }}$ be an open affinoid subspace, and let $\{X_i\}$ be an open affinoid cover of the rigid analytic locus $X^{\mathrm {rig}}\subset X$ . Since $X\smallsetminus X^{\mathrm {rig}}$ contains no open subset of X, the natural map

$$\begin{align*}{\mathscr{O}}(X)\rightarrow \prod_i{\mathscr{O}}(X_i) \end{align*}$$

is injective. Each ${\mathscr {O}}(X_i)$ is reduced, so ${\mathscr {O}}(X)$ is, as well.

Now, the Jacquet–Langlands correspondence for eigenvarieties follows immediately:

In particular, if $[F:{\mathbf {Q}}]$ is even, we can find D split at all finite places and ramified at all infinite places. Then we may take in particular $\mathfrak {n}={\mathscr {O}}_F$ to obtain a morphism of eigenvarieties of tame level $1$ .

3.4 Cyclic base change

Fix an integer $N\in {\mathbf {N}}$ , and let S be a finite set of primes containing every prime dividing $pN$ . For any number field F, we again let $K_F^p\subset \operatorname {\mathrm {GL}}_2({\mathbf {A}}_F)$ be the compact open subgroup given by

$$\begin{align*}K_F^p:=\left\{g\in\operatorname{\mathrm{GL}}_2({\mathbf{A}}_F)\mid g\equiv\left(\begin{smallmatrix}\ast & \ast \\ 0 & 1\end{smallmatrix}\right)\quad\pmod N\right\} \end{align*}$$

and we let $K_F:=K_F^pI$ . We also define the Hecke algebra

$$\begin{align*}\mathbb{T}_F^S:=\mathbb{T}_{\operatorname{\mathrm{GL}}_2/F}^S:=\otimes_{v\notin S}\mathbb{T}(\operatorname{\mathrm{GL}}_2(F_v),\operatorname{\mathrm{GL}}_2({\mathscr{O}}_{F_v})). \end{align*}$$

There is a homomorphism $\sigma _F^S:\mathbb {T}_F^S\rightarrow \mathbb {T}_{{\mathbf {Q}}}^S$ induced by unramified local Langlands and restriction of Weil representations from $W_F$ to $W_{\mathbf {Q}}$ .

Similarly, there is a morphism of weight spaces $\mathscr {W}_{{\mathbf {Q}},0}\hookrightarrow \mathscr {W}_{\mathbf {Q}}\rightarrow \mathscr {W}_F$ induced by the norm map $T_{F,0}\rightarrow T_{{\mathbf {Q}},0}$ .

In the special case where $F/{\mathbf {Q}}$ is cyclic, the classical base change map produces cuspidal automorphic representations of $\operatorname {\mathrm {GL}}_2({\mathbf {A}}_F)$ from certain cuspidal automorphic representations of $\operatorname {\mathrm {GL}}_2({\mathbf {A}}_{{\mathbf {Q}}})$ , and [Reference Johansson and NewtonJN19a] interpolated it to a morphism of eigenvarieties:

Theorem 3.4.1 [Reference Johansson and NewtonJN19a, Theorem 4.3.1].Footnote ²

There is a finite morphism

$$\begin{align*}\mathscr{X}_{\operatorname{\mathrm{GL}}_2/{\mathbf{Q}},\mathrm{cusp},F\mathrm{ - ncm}}^{S}\rightarrow \mathscr{X}_{\operatorname{\mathrm{GL}}_2/F}^S \end{align*}$$

lying over $\mathscr {W}_{{\mathbf {Q}}}\rightarrow \mathscr {W}_F$ and compatible with the homomorphism $\sigma _F^S$ .

Here, the source includes only cuspidal components with a Zariski-dense set of forms without complex multiplication by an imaginary quadratic subfield of F.

Let $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})=\left \langle \tau \right \rangle $ . Then a cuspidal automorphic representation $\pi $ of $\operatorname {\mathrm {GL}}_2({\mathbf {A}}_F)$ is in the image of the base change map if and only if $\pi \circ \tau \cong \pi $ . In particular, the systems of Hecke eigenvalues of base-changed representations must be fixed by $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})$ .

This characterization of the image of the classical base change map permits us to prove automorphy lifting theorems by passing to a more convenient solvable extension. We therefore wish to characterize the image of the interpolated base change map when F is totally real and completely split at p (so that the ‘F - ncm’ condition is vacuous). We further assume that $[F:{\mathbf {Q}}]$ is prime to p.

We will study the ‘ $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})$ -fixed locus’ in the $\operatorname {\mathrm {GL}}_{2/F}$ -eigenvariety $\mathscr {X}_{\operatorname {\mathrm {GL}}_2/F}^{S,\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})}$ and show that it is the image of the cyclic base change map; Xiang [Reference XiangXia18] used a similar idea to construct p-adic families of essentially self-dual automorphic representations.

We first observe that $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})$ acts on $\operatorname {\mathrm {GL}}_{2/F}$ , stabilizing $\operatorname {\mathrm {T}}\subset {\mathbf {B}}$ and I, and also stabilizing the tame level $K_F^p$ . Next, observe that $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})$ acts on $\mathbb {T}_F^S$ via $(\tau \cdot T)(g)=T(\tau ^{-1}(g))$ for all $T\in \mathbb {T}_F^S$ and $g\in \operatorname {\mathrm {GL}}_2({\mathbf {A}}_{F,f})$ . Then for any $\delta \in \Delta $ , $(\tau \cdot [K_F\delta K_F])(g) = [K_F\tau ^{-1}(\delta )K_F](g)$ ; in particular, $\tau \cdot U_{\varpi _v}=U_{\tau (v)}$ , and hence $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})$ fixes $U_p$ . Similarly, we have an action of $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})$ on $\mathscr {W}_{\mathbf {Q}}$ given via $(\tau \cdot \lambda )(g) = \lambda (\tau ^{-1}(g))$ ; the image of $\mathscr {W}_{{\mathbf {Q}}}$ in $\mathscr {W}_F$ is the diagonal locus, that is, exactly the $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})$ -fixed locus.

Since $U_p$ is fixed by $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})$ , we see that if $\kappa $ is a weight fixed by $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})$ , then the Fredholm determinant $F_{\kappa }(T)$ of the action of $U_p$ on $C^{\bullet }(K_F,\mathscr {D}_{\kappa })$ is fixed by $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})$ . Thus, we have a spectral variety $\mathscr {Z}^{\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})}\subset \mathscr {W}_F^{\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})}\times {\mathbf {A}}^{1,\mathrm {an}}$ over $\mathscr {W}_F^{\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})}$ .

Proof. We may assume $T=[K_F\delta K_F]$ for some $\delta \in \Delta $ . Then $\tau \cdot [K_F\delta K_F] = [K_F\tau (\delta )K_F]$ , and the corresponding morphism

$$\begin{align*}C^{\bullet}(K_F,\mathscr{D}_{\kappa}) \rightarrow C^{\bullet}(\tau(\delta) K_F\tau(\delta)^{-1}, \mathscr{D}_{\kappa}) \end{align*}$$

is induced by the conjugation map $\tau (\delta )K_F\tau (\delta )^{-1}\rightarrow K_F$ and the map $\mathscr {D}_{\kappa }\rightarrow \mathscr {D}_{\kappa }$ given by $d\mapsto \tau (\delta )\cdot d$ . But $\tau (\delta ) K_F\tau (\delta )^{-1} = \tau \left (\delta \tau ^{-1}(K_F) \delta ^{-1}\right )$ , so we may factor the conjugation map as

$$\begin{align*}\tau(\delta) K_F\tau(\delta)^{-1}\xrightarrow{\tau^{-1}} \delta \tau^{-1}(K_F) \delta^{-1} \rightarrow \tau^{-1}(K_F) \xrightarrow{\tau} K_F. \end{align*}$$

Similarly, $d\mapsto \tau (\delta )\cdot d$ factors as $\tau \circ T\circ \tau ^{-1}$ , so our morphism of complexes also factors as desired.

We may restrict $\mathscr {M}_c^\ast $ to $\mathscr {Z}^{\operatorname {\mathrm {Gal}}(F/Q)}$ ; we denote this restriction by $\mathscr {H}^\ast $ and by abuse of notation, we again use $\mathscr {T}$ to denote the sheaf generated by the image of $\mathbb {T}_F^S$ in $\mathscr {E}nd_{\mathscr {Z}^{\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})}}\left (\mathscr {H}^\ast \right )$ . Then the slice of the eigenvariety $\mathscr {X}_{\operatorname {\mathrm {GL}}_2/F}^S$ over $\mathscr {W}_F^{\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})}$ is, by definition, $\underline{\mathrm{Spa}} \mathscr {T}$ .

Both $\mathbb {T}(\Delta ^p,K_F^p)$ and $\operatorname {\mathrm {End}}_{{\mathscr {O}}(V)}\left (\mathscr {H}^\ast \right )$ have actions of $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})$ , and Lemma 3.4.4 implies that they are compatible. Thus, $\mathscr {T}(V)$ and $\mathscr {X}_{\operatorname {\mathrm {GL}}_2/F}^S|_{\mathscr {W}_F^{\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})}}$ have actions of $\operatorname {\mathrm {Gal}}(F/{\mathbf {Q}})$ .