\documentclass[11pt]{amsart} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{amssymb,amsmath,mathtools} \usepackage{enumitem} \usepackage[margin=2.7cm]{geometry} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{maintheorem}{Theorem} \renewcommand{\themaintheorem}{\Alph{maintheorem}} \newtheorem*{theorem*}{Theorem} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \newcommand{\cO}{\mathcal{O}} \newcommand{\GL}{\mathrm{GL}} \newcommand{\Spec}{\mathrm{Spec}} \newcommand{\Hom}{\mathrm{Hom}} \newcommand{\ad}{\mathrm{ad}} \newcommand{\Frob}{\mathrm{Frob}} \newcommand{\Ord}{\mathrm{Ord}} \newcommand{\Art}{\mathrm{Art}} \newcommand{\Ind}{\mathrm{Ind}} \newcommand{\rhobar}{\bar\rho} \newcommand{\So}{\Sigma_{\mathrm o}} \newcommand{\Sn}{\Sigma_{\mathrm n}} \newcommand{\cl}{\mathrm{cl}} \newcommand{\ps}{\mathrm{ps}} \newcommand{\ord}{\mathrm{ord}} \newcommand{\HT}{\mathrm{HT}} \newcommand{\Nm}{\mathrm N} \newcommand{\Sel}{\mathrm{Sel}} \newcommand{\St}{\mathrm{St}} \title{Fully ordinary subfamilies and the Fontaine--Mazur conjecture in the residually reducible mixed case} \author{Luis V. Dieulefait} \address{Departament de Matem\`atiques i Inform\`atica, Universitat de \break Barcelona, Gran Via de les Corts Catalanes, 585. 08007 Barcelona} \email{ldieulefait@ub.edu} \keywords{Fontaine-Mazur conjecture} \subjclass[2010]{11F80, 11F33} \begin{document} \begin{abstract} Let $F$ be a totally real field, abelian over $\Q$, in which an odd prime $p$ splits completely, and let $\rho:G_F\to\GL_2(\overline\Q_p)$ be a continuous, absolutely irreducible, totally odd representation, unramified outside finitely many places and de Rham with distinct Hodge--Tate weights at the places above $p$, with reducible residual representation. We prove that $\rho$ is modular in the mixed case, in which $\rho$ is reducible at some places above $p$ and irreducible at the others, provided that the ratio of the two residual characters extends to $G_\Q$ and its restriction to $G_{\Q_p}$ is neither $1$ nor $\omega^{\pm1}$. No assumption is made on the splitness of $\rho$ at the reducible places, on potential modularity, or on the local behaviour of $\rho$ away from $p$. A suitable irreducible component through the point of $\rho$ in a partially ordinary deformation space contains a fully ordinary subfamily of maximal dimension. A classical point of this subfamily is a regular point of the deformation space, by the vanishing of adjoint Selmer groups, and a dimension estimate for a partially ordinary Hecke algebra then identifies the component of $\rho$ with a component of the Hecke algebra. \end{abstract} \maketitle \section{Introduction} \subsection{Setting} Let $p$ be an odd prime and $F$ a totally real field, abelian over $\Q$, in which $p$ splits completely, so that $F_v=\Q_p$ for every place $v\mid p$. Let $E/\Q_p$ be a finite extension with ring of integers $\cO$, uniformizer $\varpi$ and residue field $\F$, which we enlarge when necessary. Let $\rho:G_F\to\GL_2(E)$ be a continuous, absolutely irreducible, totally odd representation, unramified outside finitely many places. Let $\Sigma_p$ be the set of places above $p$, fix a partition $\Sigma_p=\Sn\sqcup\So$, and consider, as in \cite[§1.2]{Die26}, the conditions: \begin{itemize} \item[(N)] for $v\in\Sn$, $\rho_v=\rho|_{G_{F_v}}$ is irreducible and de Rham with distinct Hodge--Tate weights; \item[(O)] for $v\in\So$, $\rho_v$ is reducible and de Rham with distinct Hodge--Tate weights; \item[(R)] $\rhobar^{ss}=\bar\chi_1\oplus\bar\chi_2$, the character $\bar\chi=\bar\chi_1\bar\chi_2^{-1}$ extends to a character of $G_\Q$, and $\bar\chi|_{G_{F_v}}\neq1$ for every $v\mid p$. \end{itemize} After enlarging $E$, irreducibility of a local representation always means absolute irreducibility. Under (O), the Jordan--Hölder character of $\rho_v$ with the smaller Hodge--Tate number is a subcharacter of $\rho_v$ \cite[Lemma~2.1]{Die26}. It is the \emph{classical character} $\psi^{\cl}_v$, the one which occurs for ordinary modular forms. Since $\bar\chi$ comes from $G_\Q$ and $p$ splits completely, $\bar\chi|_{G_{F_v}}$ is the same character of $G_{\Q_p}$ for all $v\mid p$. The case $\So=\emptyset$ is due to Pan \cite{Pan22} and Zhang \cite{Zha25}, and the case $\Sn=\emptyset$ to Pan \cite[Theorem~5.1.2]{Pan22}. In the mixed case, in which both are non-empty, the following result is proved in \cite{Die26}. \begin{theorem*}[{\cite[Theorem~A]{Die26}}] Assume \textup{(N)}, \textup{(O)} and \textup{(R)}, and that $\rho_v$ is non-split for every $v\in\So$. Then $\rho$ is modular. \end{theorem*} \subsection{Main result} \begin{maintheorem}\label{thm:main} Assume \textup{(N)}, \textup{(O)}, \textup{(R)}, and \begin{itemize} \item[(a)] $\bar\chi|_{G_{F_v}}\neq\omega^{\pm1}$ for one, equivalently every, place $v\mid p$. \end{itemize} Then $\rho$ is modular. \end{maintheorem} No assumption is made on the splitness of $\rho_v$ for $v\in\So$, on the Hodge--Tate weights beyond (N) and (O), on the degree of $F$, or on the restrictions of $\rho$ to the decomposition groups away from $p$. Condition (a) is a single condition on the restriction to $G_{\Q_p}$ of the extension of $\bar\chi$ to $G_\Q$. It says that $\rhobar^{ss}|_{G_{\Q_p}}$ is not a twist of $1\oplus\omega$, the case excluded by Kisin \cite{Kis09a} and later treated by Hu--Tan \cite{HT15} and Tung \cite{Tun21}. Its essential use is to make the reducibility ideal at the places of $\Sn$ principal (Lemma~\ref{lem:red}); it also simplifies the local smoothness and global dimension estimates of Proposition~\ref{prop:GW}. See Remark~\ref{rem:b}. \subsection{The method} Let $R^{\mathrm o}$ be the pseudo-deformation ring of $\rhobar^{ss}$ which is ordinary at the places of $\So$, with the reduction of the classical character prescribed as subcharacter, and unrestricted at the places of $\Sn$. Let $x$ be the point defined by $\rho$. Let $\T^{\mathrm o}$ be the Hecke algebra of the part of the completed cohomology of a definite quaternion algebra which is ordinary at the places of $\So$ \cite[§3.3]{Die26}. It is a quotient of $R^{\mathrm o}$, and it carries the ordinary characters at $\So$ through its diamond operators. If $x$ lies in $\Spec\T^{\mathrm o}$, then the corresponding ordinary eigenvectors exist (Proposition~\ref{prop:dimT}), and the classicality criterion \cite[Theorem~4.2]{Die26} shows that $\rho$ is modular. The proof that it does has four steps. Write $d=[F:\Q]$. \begin{enumerate} \item A Greenberg--Wiles estimate gives an irreducible component $C$ of $\Spec R^{\mathrm o}$ through $x$ of dimension at least $1+2d-|\So|$ (Proposition~\ref{prop:GW}). \item Under (a), the reducible locus at each place of $\Sn$ is cut out by one equation. Cutting $C$ by these equations gives an irreducible closed subset $D\subset C$ which is reducible at every place above $p$, with ordinary subcharacters chosen uniformly on $D$, of dimension $1+d$. By Pan's finiteness theorem $D$ is finite and surjective over the Iwasawa algebra, so its classical points are dense. By Pan's theorem in the fully ordinary case and Hida theory, they lie in $\Spec\T^{\mathrm o}$ (Proposition~\ref{prop:hida}). \item Let $y$ be a classical point of $D$ of sufficiently regular weight which is not of CM type. By the Greenberg--Wiles formula, the tangent space of $R^{\mathrm o}$ at $y$ has the expected dimension $2d-|\So|$ as soon as the Bloch--Kato Selmer group $H^1_f(F,\ad^0\rho_y(1))$ vanishes, and this follows from the theorem of Newton and Thorne \cite[Theorem~B]{NT23} together with global duality. Hence $R^{\mathrm o}$ is regular at $y$, $C$ is the only irreducible component of $\Spec R^{\mathrm o}$ through $y$, and $\dim C=1+2d-|\So|$ (Proposition~\ref{prop:smooth}). At the places away from $p$ where $\rho$ has Steinberg type, we use local--global compatibility at $y$ (Lemma~\ref{lem:St}). \item Every irreducible component of $\Spec\T^{\mathrm o}$ has dimension at least $1+2d-|\So|$, by an adaptation of Pan's Gelfand--Kirillov dimension argument (Proposition~\ref{prop:dimT}). Hence an irreducible component of $\Spec\T^{\mathrm o}$ through $y$ is contained in $C$, and equal to it. Thus $x\in\Spec\T^{\mathrm o}$. \end{enumerate} The deep inputs are Pan's theorem in the fully ordinary case, the theorem of Newton--Thorne, local--global compatibility for Hilbert modular forms, and the theory of ordinary parts at the places of $\So$ together with the classicality criterion of \cite[§§3--4]{Die26}. Theorem~A of \cite{Die26} is not used. There are no nice primes, no connectedness arguments, no auxiliary primes, no local indecomposability theorems, and no Taylor--Wiles patching beyond the one contained in these results. The pattern of steps (2)--(4) is that of Böckle \cite{Boc01}: one produces a modular point of $C$ at which the deformation ring is regular, and combines it with a lower bound for the dimension of the Hecke algebra. In the residually reducible case this was carried out by Deo \cite{Deo23} over $\Q$ and by Zhang \cite{Zha26}, over $\Q$ and, when $\So=\emptyset$, over abelian totally real fields in which $p$ splits completely. In particular, the cut of the unrestricted locus by the principal reducibility ideal, the use of Pan's ordinary finiteness theorem and the Newton--Thorne vanishing theorem already appear there. What is new here is the partially ordinary setting: the deformation ring $R^{\mathrm o}$ and the Hecke algebra $\T^{\mathrm o}$ with ordinary conditions at $\So$, the dimension estimate of Proposition~\ref{prop:dimT} for $\T^{\mathrm o}$, and the local analysis at the places of $\So$ and at the places of Steinberg type (Lemma~\ref{lem:St} and Proposition~\ref{prop:smooth}). \subsection*{Acknowledgments} The author acknowledges the use of Claude (Anthropic), an AI assistant, during the preparation of this work: it was used to check the relevant parts of the literature, to work out and verify technical details of the arguments and to help draft the manuscript. \subsection*{Notation} We use the conventions of \cite{Die26}: $\HT(\varepsilon)=-1$, characters of $G_{\Q_p}$ are identified with characters of $\Q_p^\times$ via the local Artin map normalized as in \cite{Pan22,Zha25}, and $\omega$ is the mod $p$ cyclotomic character. We write $G_v=G_{F_v}$, $I_v$ for its inertia subgroup, $\phi_v$ for a lift of the arithmetic Frobenius and $q_v=\Nm(v)$. For a pseudo-representation $T$ with coefficients in a ring $R$ and a prime $\mathfrak r$ of $R$, $\rho(\mathfrak r)$ is the semisimple representation over an algebraic closure of $k(\mathfrak r)$ with trace $T\bmod\mathfrak r$; for a point $z$ we also write $\rho_z$. A prime $\mathfrak r$ is \emph{irreducible} if $\rho(\mathfrak r)$ is absolutely irreducible. We write $h^i(V)=\dim H^i(V)$. \section{Setup}\label{sec:setup} \subsection{The base field}\label{subsec:field} Twisting $\rho$ by the Teichmüller lift of $\bar\chi_2^{-1}$, we may assume $\rhobar^{ss}=1\oplus\bar\chi$. Put $\chi=\det\rho$, a totally odd de Rham character lifting $\bar\chi$. We first normalize the determinant as in \cite[Remark~5.5.2]{Zha25}. Write $\chi=\alpha\varepsilon^k$ with $\alpha$ of finite order, and put $\chi_0=\widetilde{\bar\alpha}\varepsilon^k$, where $\bar\alpha=\bar\chi\omega^{-k}$ extends to $G_\Q$ by (R) and $\widetilde{\bar\alpha}$ is the Teichmüller lift. The character $\chi_0\chi^{-1}$ has finite $p$-power order, so it has a unique square root $\varphi$ of $p$-power order. Replacing $\rho$ by $\rho\otimes\varphi$, which has trivial reduction and Hodge--Tate weight $0$ and preserves (N), (O), (R), (a), local splitness and modularity, we may assume $\chi=\chi_0$. By \cite[Theorem~5.5.1 and Remark~5.5.2]{Zha25}, there is a totally real extension $F_1/F$, abelian over $\Q$, of even degree, in which $p$ splits completely, such that $\rho|_{G_{F_1}}$ satisfies the conditions of \cite[§4.1]{Zha25} recalled in \ref{F2} below. We replace $F$ by $F_1$ and $\rho$ by $\rho|_{G_{F_1}}$, which is absolutely irreducible by \cite[Lemma~4.4]{Die26}. The hypotheses of Theorem~\ref{thm:main} are preserved, since $F_{1,w}=F_v=\Q_p$ for $w\mid v\mid p$, and $\rho$ is modular if $\rho|_{G_{F_1}}$ is, by solvable descent \cite[Lemma~2.2.2]{BLGGT14}, \cite{AC89}. We keep the notation $\So$, $\Sn$ for the sets of places of $F_1$ above the original ones, write $d=[F:\Q]$, and let $\Sigma$ be the set of places above $p$ and those at which $\rho$ is ramified. Thus: \begin{enumerate}[label=(F\arabic*)] \item\label{F1} $F$ is totally real, abelian over $\Q$, of even degree $d\geq2$, and $p$ splits completely in $F$; $\rhobar^{ss}=1\oplus\bar\chi$; \item\label{F2} for $v\in\Sigma\setminus\Sigma_p$: $p\mid q_v-1$, $\chi$ is unramified at $v$, $\chi(\phi_v)\equiv1\pmod\varpi$, and there is a character $\xi_v$ of $p$-power order of $k(v)^\times$ with $\mathrm{tr}\,\rho|_{I_v}=\xi_v+\xi_v^{-1}$ \cite[Remark~4.1.2]{Zha25}. \end{enumerate} Here $\Sigma$ is chosen after the twists and the base change. For the inertial assertion in \ref{F2}: the residual representation at $v$ is trivial, so $\rho(I_v)$ is pro-$p$, hence factors through tame inertia; by quasi-unipotence and since $\chi$ is unramified at $v$, its eigencharacters are $\xi_v,\xi_v^{-1}$ of $p$-power order, and Frobenius does not interchange them when $\xi_v\neq1$, since $q_v\equiv1\pmod p$ and $p>2$. Since $p$ is odd, $\xi_v^2\neq1$ unless $\xi_v=1$. If $\xi_v\neq1$, then $\rho|_{G_v}\simeq\mu_1\oplus\mu_2$ with $\mu_1|_{I_v}=\xi_v$ and $\mu_2|_{I_v}=\xi_v^{-1}$, since these restrictions are distinct. If $\xi_v=1$, then $\rho|_{I_v}$ is unipotent and non-trivial, as $\rho$ is ramified at $v$. Hence $\rho|_{G_v}$ is of Steinberg type: its Weil--Deligne representation is $(\nu\oplus\nu|\cdot|,N)$ with $N\neq0$. Let $\Sigma_{\St}$ be the set of these places. For $v\in\Sigma_{\St}$ we fix $\sigma_v\in I_v$ mapping to a topological generator of the $p$-part of the tame inertia group. \subsection{Deformation rings}\label{subsec:def} Let $R^{\ps}$ be the universal deformation ring of the pseudo-representation $1+\bar\chi$ of $G_{F,\Sigma}$ with determinant $\chi$. For $v\mid p$, on the locus where $T|_{G_v}$ is reducible we have $T|_{G_v}=\psi+\chi\psi^{-1}$ with $\psi$ lifting $1$ or $\bar\chi|_{G_v}$, and the decomposition is unique since $\bar\chi|_{G_v}\neq1$. For $v\in\So$ let $\bar\psi_v\in\{1,\bar\chi|_{G_v}\}$ be the reduction of $\psi^{\cl}_v$. Let $\Xi$ be the finite set of tuples $\xi'=(\xi'_v)_{v\in\Sigma\setminus\Sigma_p}$ with $\xi'_v\in\{\xi_v,\xi_v^{-1}\}$. \begin{definition}\label{def:rings} \begin{enumerate} \item Let $R^{\mathrm o}$ be the quotient of $R^{\ps}$ by the conditions: \begin{itemize} \item $T|_{I_v}=\xi_v+\xi_v^{-1}$ for $v\in\Sigma\setminus\Sigma_p$; \item for $v\in\So$, $T|_{G_v}=\psi_v+\chi\psi_v^{-1}$ is reducible with $\psi_v$ lifting $\bar\psi_v$, and $T$ is $\psi_v$-ordinary in the sense of \cite[Definition~5.3.2]{Pan22} (recalled in \cite[§2.2]{Die26}). \end{itemize} \item Let $R^{\ps,\ord}$ be the quotient of $R^{\mathrm o}$ by the condition that $T|_{G_v}$ is reducible for every $v\in\Sn$. \end{enumerate} \end{definition} The point $x\in\Spec R^{\mathrm o}[1/p]$ defined by $\rho$ lies in $\Spec R^{\mathrm o}$ by \cite[Lemma~2.3(1)]{Die26}, and $\psi_v(x)=\psi^{\cl}_v$ for $v\in\So$: the two Jordan--Hölder characters of $\rho_v$ have distinct reductions, and $\psi^{\cl}_v$ is the one lifting $\bar\psi_v$. On $\Spec R^{\ps,\ord}$, for every $v\mid p$ write $T|_{G_v}=\psi_v+\chi\psi_v^{-1}$, with $\psi_v$ lifting $\bar\psi_v$ if $v\in\So$ and lifting $1$ if $v\in\Sn$. Let $\Lambda=\cO[[\prod_{v\mid p}(1+p\Z_p)]]$, a power series ring over $\cO$ in $d$ variables. Through the restrictions of the characters $\psi_v$ to $\Art(1+p\Z_p)\subset I_v$, $R^{\ps,\ord}$ is a $\Lambda$-algebra. \begin{lemma}\label{lem:finite} $R^{\ps,\ord}$ is finite over $\Lambda$. In particular, every irreducible closed subset of $\Spec R^{\ps,\ord}$ has dimension at most $1+d$, and one of dimension $1+d$ is finite over, and surjective onto, $\Spec\Lambda$. \end{lemma} \begin{proof} $R^{\ps,\ord}$ is a quotient of the ring $R^{\ps,\ord}$ of \cite[5.1.1]{Pan22}, which is finite over $\Lambda_F=\cO[[\prod_{v\mid p}\Z_p^\times]]$ by \cite[Theorem~5.1.2(1)]{Pan22} (see \cite[Theorem~4.2.3]{Zha25} for $p=3$). The hypotheses of loc.\ cit.\ hold by \ref{F1}, (R) and the oddness of $\chi$. Our $\Lambda$-structure differs from the restriction of that of loc.\ cit.\ by the automorphism induced by $\psi\mapsto\chi\psi^{-1}$ at the places of $\So$ where $\bar\psi_v=\bar\chi|_{G_v}$, and $\Lambda_F$ is finite over $\Lambda$. As $\Lambda$ is a domain of dimension $1+d$, the last assertions follow. \end{proof} The pseudo-representations which are reducible as pseudo-representations of $G_F$ form a closed subset of $\Spec R^{\ps}$ of dimension at most $2$. Indeed, they are of the form $\psi+\chi\psi^{-1}$ with $\psi$ a character of $G_{F,\Sigma}$ lifting $1$, and the pro-$p$ completion of $G^{\mathrm{ab}}_{F,\Sigma}$ has $\Z_p$-rank $1$, by Leopoldt's conjecture for the abelian field $F$. \subsection{Partially ordinary Hecke algebras}\label{subsec:hecke} We use the setting of \cite[§3]{Die26}. Let $B$ be the quaternion algebra over $F$ ramified exactly at the infinite places, $\psi=\chi\varepsilon$, and $U^p=\prod_{v\nmid p}U_v$, with $U_v=\GL_2(\cO_{F_v})$ for $v\notin\Sigma$ and $U_v$ the Iwahori subgroup for $v\in\Sigma\setminus\Sigma_p$. For $\xi'\in\Xi$ and $v\in\Sigma\setminus\Sigma_p$ we use the character $\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\mapsto\xi'_v(a/d\bmod v)$ of $U_v$, which is trivial on the centre and corresponds to the inertial pair $\xi'_v,\xi_v^{\prime-1}$ \cite[4.1.2]{Pan22}. Let $S_{\xi'}=S_{\psi,\xi'}(U^p)_{\mathfrak m}$ be the completed cohomology with these characters, localized at the Eisenstein maximal ideal of $1+\bar\chi$, and let $S^{\mathrm o}_{\xi'}=\Ord_P(S_{\xi'})$ be its ordinary part at the places of $\So$, where $P$ is the upper Borel subgroup at the places of $\So$ and the whole group at those of $\Sn$ \cite[§3.3]{Die26}. Let $\T^{\mathrm o}_{\xi'}$ be the closed image of the algebra generated by the operators $T_w$, $w\notin\Sigma$, and the full torus operators at the places of $\So$ (including the uniformizer operators $U_v$), acting on $S^{\mathrm o}_{\xi'}$, and let $\mathfrak m'$ be the maximal ideal at which the first torus character $\psi_{v,1}$ of \cite[§3.3]{Die26} reduces to $\bar\psi_v$ for every $v\in\So$. The comparison of $S^{\mathrm o}_{\xi'}$ with torsion ordinary parts and the finiteness properties of $\T^{\mathrm o}_{\xi',\mathfrak m'}$ are proved in Proposition~\ref{prop:dimT}. The ordinary Galois identities are \cite[Lemma~3.4]{Die26}: there is a pseudo-representation over $\T^{\mathrm o}_{\xi'}$ with determinant $\chi$ which is $\psi_{v,1}$-ordinary at every $v\in\So$. Since $\psi$ need not be locally constant at $p$, its proof first passes to the finite order central character $\psi_0$, the Teichmüller lift of $\bar\psi$, by the twist by $\theta\circ\mathrm{Nrd}$ with $\theta^2=\psi_0\psi^{-1}$, and then transports the identities back; $\theta$ is not assumed to be de Rham. By \ref{F2}, $\psi$, hence $\theta$, is unramified at the places of $\Sigma\setminus\Sigma_p$, so the same $U^p$ and $\xi'$ can be used. Together with local--global compatibility at the places of $\Sigma\setminus\Sigma_p$ \cite[4.1.3]{Pan22}, this gives a surjection $R^{\mathrm o}\twoheadrightarrow\T^{\mathrm o}_{\xi',\mathfrak m'}$ sending $\psi_v$ to $\psi_{v,1}$ for $v\in\So$, surjective because $\T^{\mathrm o}_{\xi'}$ is topologically generated by the $T_w$ and the torus operators. A prime of $R^{\mathrm o}$ is \emph{o-pro-modular} if it lies in the image of $\Spec\T^{\mathrm o}_{\xi',\mathfrak m'}$ for some $\xi'\in\Xi$. An o-pro-modular prime thus records both the trace and the ordinary characters at the places of $\So$. \section{Local results and a lower bound for $C$}\label{sec:GW} \begin{lemma}\label{lem:red} Let $v\in\Sn$, and let $R^{\ps}_v$ be the deformation ring of the pseudo-representation $(1+\bar\chi)|_{G_v}$ with determinant $\chi|_{G_v}$. There is $f_v\in R^{\ps}_v$ such that a deformation of $(1+\bar\chi)|_{G_v}$ with values in a field is reducible if and only if it kills $f_v$. \end{lemma} \begin{proof} Since $\bar\chi|_{G_v}\neq1$, the universal Cayley--Hamilton algebra over $R^{\ps}_v$ is a generalized matrix algebra $\left(\begin{smallmatrix}R^{\ps}_v&B\\C&R^{\ps}_v\end{smallmatrix}\right)$. A pseudo-representation with values in a field is reducible if and only if it kills the image $BC$ of $B\otimes C$ \cite[§1.3--1.5]{BC09}. The minimal numbers of generators of $B$ and $C$ are at most $h^1(G_{\Q_p},\bar\chi)$ and $h^1(G_{\Q_p},\bar\chi^{-1})$ \cite[Theorem~1.5.5]{BC09}, \cite[Lemma~2.5]{Deo}. By the local Euler characteristic formula, (R) and (a), $h^1(G_{\Q_p},\bar\chi^{\pm1})=1+h^0(\bar\chi^{\pm1})+h^0(\bar\chi^{\mp1}\omega)=1$. Hence $B=R^{\ps}_vb$ and $C=R^{\ps}_vc$, and $f_v=bc$ works. Compare \cite[Lemma~5.3.3]{Zha26}, where the same principal ideal is used to cut out the ordinary locus. \end{proof} \begin{lemma}\label{lem:h0} Let $G$ be a group, $k$ a field, $\alpha\neq\beta$ and $\eta\neq1$ characters $G\to k^\times$, and $r=\left(\begin{smallmatrix}\alpha&*\\0&\beta\end{smallmatrix}\right)$ a non-split extension. If $H^0(G,\ad^0r\otimes\eta)\neq0$, then $\alpha\beta^{-1}=\eta^{-1}$. \end{lemma} \begin{proof} A non-zero invariant is a $G$-equivariant map $X:V\to V\otimes\eta$, where $V=k^2$ with the action $r$. The representations $V$ and $V\otimes\eta$ have unique stable lines, both equal to the first basis line $L$, with characters $\alpha$ and $\alpha\eta$. If $X$ were invertible, it would map $L$ to $L$, so $\alpha=\alpha\eta$, which is impossible. Hence $X$ has rank $1$, its kernel and image are stable lines, so both are $L$. Then $X$ is a multiple of $\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)$, on which $G$ acts through $r$ by $\alpha\beta^{-1}$, so $\alpha\beta^{-1}=\eta^{-1}$. \end{proof} \begin{proposition}\label{prop:GW} There is an irreducible component $C$ of $\Spec R^{\mathrm o}$ containing $x$ with $\dim C\geq1+2d-|\So|$. \end{proposition} \begin{proof} We follow the proof of \cite[Lemma~5.4.1]{Zha25}. Let $\mathfrak r$ be the prime of $R^{\mathrm o}$ defined by $x$. It is one-dimensional and irreducible. After enlarging $\cO$, choose a $G_{F,\Sigma}$-stable lattice $\rho^\circ$ in $\rho$ whose reduction $\rhobar_b$ is a non-split extension of $\bar\chi$ by $1$ \cite[Lemma~2.13]{SW99}. By Lemma~\ref{lem:h0} with $\eta=\omega$, applied to $\alpha\beta^{-1}=\bar\chi^{-1}$, and by (a), we have $H^0(G_{F,\Sigma},\ad^0\rhobar_b(1))=0$; here $\omega|_{G_F}\neq1$, since $F\cap\Q(\zeta_p)$ is unramified at $p$ by \ref{F1}, hence equal to $\Q$. For $v\in\So$, saturate in $\rho^\circ$ the stable line of $\rho_v$ with character $\psi^{\cl}_v$, and let $\bar L_v$ be its reduction. Let $R_b$ be the deformation ring of $\rhobar_b$ with determinant $\chi$ which is ordinary along $\bar L_v$ for $v\in\So$ (that is, admits a stable direct summand of rank one lifting $\bar L_v$), unrestricted at $\Sn$, and with $\mathrm{tr}|_{I_v}=\xi_v+\xi_v^{-1}$ for $v\in\Sigma\setminus\Sigma_p$. The local framed rings are as follows. \begin{itemize} \item For $v\in\So$: the ordinary ring $R^\Delta_v\simeq R^B_v[[n]]$ of \cite[5.3.8(1)]{Pan22}. Here $R^B_v$ parametrizes lifts to the upper Borel subgroup. Its obstruction group $H^2(G_{\Q_p},\mathfrak b^0)$ is dual to $H^0(G_{\Q_p},(\mathfrak b^0)^\vee(1))$, whose Jordan--Hölder factors are $\omega$ and $\omega\bar\chi^{\pm1}$, both non-trivial by (a). Hence $R^\Delta_v$ is formally smooth over $\cO$ of relative dimension $5$ (compare \cite[Lemma~5.3.9]{Pan22}). \item For $v\in\Sn$: the unrestricted framed ring, formally smooth of relative dimension $6$, since the Jordan--Hölder factors of $\ad^0\rhobar_b(1)$ are $\omega$, $\omega\bar\chi$ and $\omega\bar\chi^{-1}$, all non-trivial by (a). \item For $v\in\Sigma\setminus\Sigma_p$: the ring $R_{v,\xi_v}$ of \cite[4.2.1]{Pan22}, equidimensional of dimension $4$ and $\cO$-flat \cite[Lemma~4.2.10]{Pan22}. \end{itemize} By \cite[Corollary~2.3.5]{CHT08}, as in \cite[5.4.2]{Pan22}, the framed version of $R_b$ is a quotient of a power series ring in $g$ variables over the completed tensor product $R_{\mathrm{loc}}$ of these rings by $r$ relations, with $g-r=|\Sigma|-1-d-h^0(G_{F,\Sigma},\ad^0\rhobar_b(1))=|\Sigma|-1-d$. Removing the $4|\Sigma|-1$ framing variables, which is possible since $\rhobar_b$ is a non-split extension of distinct characters, so that its endomorphisms are scalars, every minimal prime $Q$ of $R_b$ satisfies \[ \dim R_b/Q\geq1+5|\So|+6|\Sn|+3(|\Sigma|-d)+(|\Sigma|-1-d)-(4|\Sigma|-1)=1+2d-|\So|. \] Let $\mathfrak r_b$ be the prime of $R_b$ defined by $\rho^\circ$, and choose a minimal prime $Q\subset\mathfrak r_b$. Since the reducible locus is closed and does not contain $\mathfrak r_b$, the representation at $Q$ is irreducible. By \cite[Proposition~2.2.5(2)]{Zha25} (that is, \cite[Corollary~2.3.6]{Pan22}), applied to the inverse image of $Q$ in the deformation ring of $\rhobar_b$ without local conditions, the pull-back $P$ of $Q$ to $R^{\mathrm o}$ satisfies $\dim R^{\mathrm o}/P\geq\dim R_b/Q\geq1+2d-|\So|$. As $P\subset\mathfrak r$, any irreducible component of $\Spec R^{\mathrm o}$ containing $V(P)$ has the required properties. \end{proof} We fix such a component $C$, with its reduced structure. Since $x\in C$ has characteristic $0$, $C$ is flat over $\cO$, and $C[1/p]$ is irreducible. \section{Places of Steinberg type}\label{sec:St} \begin{lemma}\label{lem:St} Let $v\in\Sigma_{\St}$ and put $P_v=(1+q_v)^2\chi(\phi_v)-q_v\,T(\phi_v)^2$, a function on $\Spec R^{\mathrm o}$. \begin{enumerate} \item $P_v$ vanishes on $C$. \item Let $y\in C[1/p]$ be a closed point such that $\rho_y$ is irreducible and arises from a cuspidal automorphic representation $\pi$ of $\GL_2(\mathbb A_F)$. Then $\pi_v$ is a twist of the Steinberg representation, $\rho_y|_{G_v}$ is of Steinberg type, and $H^0(G_v,\ad^0\rho_y(1))=0$. \end{enumerate} \end{lemma} \begin{proof} For a closed point $z$ of $\Spec R^{\mathrm o}[1/p]$, $\rho_z|_{I_v}$ has trace $2$, hence is unipotent, and it factors through the tame inertia group since the wild inertia group is pro-$\ell$. If $a,b$ are the eigenvalues of $\rho_z(\phi_v)$, then $ab=\chi(\phi_v)$ and \[ P_v(z)=(1+q_v)^2ab-q_v(a+b)^2=(a-q_vb)(b-q_va). \] (1) Let $Z_N\subset C[1/p]$ be the closed subset of points $z$ with $T_z(\sigma_vg)=T_z(g)$ for all $g\in G_{F,\Sigma}$. If $\rho_z$ is irreducible, then $\rho_z(G_{F,\Sigma})$ spans $M_2$, and the trace form is non-degenerate, so $z\in Z_N$ if and only if $\rho_z(\sigma_v)=1$, that is, if and only if the monodromy of $\rho_z|_{G_v}$ vanishes. Hence $x\notin Z_N$. Let $U\subset C[1/p]$ be the open subset of irreducible points not in $Z_N$. It contains $x$, so it is dense in $C[1/p]$. For $z\in U$, the Weil--Deligne representation of $\rho_z|_{G_v}$ has non-zero monodromy, so the ratio of the eigenvalues of $\rho_z(\phi_v)$ is $q_v^{\pm1}$, and $P_v(z)=0$. As $C[1/p]$ is reduced and Jacobson, $P_v$ vanishes on $C[1/p]$, hence on $C$, which is $\cO$-flat. (2) By (1), the semisimplification of $\rho_y|_{G_v}$ is $\nu_1\oplus\nu_2$ with $\nu_1,\nu_2$ unramified and $\nu_1\nu_2^{-1}=\varepsilon^{\pm1}$. By local--global compatibility up to semisimplification \cite{Car86,Tay89,Jar97}, the Langlands parameter of $\pi_v$ has semisimplification a sum of two characters with ratio $|\cdot|^{\pm1}$. As $\pi_v$ is irreducible and generic, it is a twist of the Steinberg representation, which is square integrable. Since $\pi$ is then square integrable at a finite place, full local--global compatibility at $v$, including the monodromy operator, holds by Carayol's theorem \cite[Théorème~(A)]{Car86}. Hence $\rho_y|_{G_v}$ is a non-split extension $\left(\begin{smallmatrix}\nu\varepsilon&*\\0&\nu\end{smallmatrix}\right)$. By Lemma~\ref{lem:h0} with $\eta=\varepsilon|_{G_v}$, a non-zero element of $H^0(G_v,\ad^0\rho_y(1))$ would force $\varepsilon=\varepsilon^{-1}$ on $G_v$, which is false since $\varepsilon(\phi_v)=q_v$ and $q_v^2\neq1$. \end{proof} \section{Classical points of the fully ordinary subfamily}\label{sec:classical} \subsection{The subfamily} Let $A_C=\cO(C)=R^{\mathrm o}/\mathfrak q$, where $\mathfrak q$ is the minimal prime of $R^{\mathrm o}$ defining $C$; it is a complete local domain of dimension at least $1+2d-|\So|$ by Proposition~\ref{prop:GW}. For $v\in\Sn$, the restriction $T|_{G_v}$ of the universal pseudo-representation is a deformation of $(1+\bar\chi)|_{G_v}$ with determinant $\chi|_{G_v}$. By the universal property of $R^{\ps}_v$, it is induced by a unique local homomorphism $r_v:R^{\ps}_v\to R^{\ps}$. Denote also by $f_v\in A_C$ the image of the element $f_v$ of Lemma~\ref{lem:red} under the composite \[ R^{\ps}_v\xrightarrow{\ r_v\ }R^{\ps}\twoheadrightarrow R^{\mathrm o}\twoheadrightarrow R^{\mathrm o}/\mathfrak q=A_C . \] By Lemma~\ref{lem:red}, a point $z$ of $C$ lies in $V(f_v)$ if and only if $T_z|_{G_v}$ is reducible. The element $f_v$ lies in the maximal ideal of $A_C$, since the residual pseudo-representation is reducible. By Krull's height theorem, a minimal prime $\mathfrak D$ of $(f_v)_{v\in\Sn}$ satisfies \[ \dim A_C/\mathfrak D\geq\dim A_C-|\Sn|\geq1+2d-|\So|-|\Sn|=1+d. \] Let $D=\Spec A_C/\mathfrak D$, with its reduced structure. Its points are reducible at every place above $p$: at the places of $\So$ because $C\subset\Spec R^{\mathrm o}$, and at those of $\Sn$ by Lemma~\ref{lem:red}. Hence $D\subset\Spec R^{\ps,\ord}$, and by Lemma~\ref{lem:finite}, $\dim D=1+d$ and $D$ is finite over, and surjective onto, $\Spec\Lambda$. Since $1+d>2$, the irreducible points form a dense open subset $D^{\mathrm{irr}}$ of $D$. \begin{lemma}\label{lem:sign} For every $v\in\Sn$ there is $\psi_{v,\mathrm s}\in\{\psi_v,\chi\psi_v^{-1}\}$ such that $T$ is $\psi_{v,\mathrm s}$-ordinary on $D$. For $v\in\So$ put $\psi_{v,\mathrm s}=\psi_v$. \end{lemma} \begin{proof} For $z\in D^{\mathrm{irr}}$ and $v\in\Sn$, $\rho_z|_{G_v}$ is reducible, so one of $\psi_v(z)$, $\chi\psi_v(z)^{-1}$ is a subcharacter. By \cite[Lemma~2.3]{Die26}, $T_z$ is $\psi_v$-ordinary or $\chi\psi_v^{-1}$-ordinary. Both conditions are closed, and $D^{\mathrm{irr}}$ is irreducible. \end{proof} \begin{definition}\label{def:good} A closed point $y$ of $D[1/p]$ is \emph{good} if: \begin{enumerate} \item for every $v\mid p$, $\psi_{v,\mathrm s}(y)$ is de Rham, of Hodge--Tate number $h_v$, and $\HT(\chi\psi_{v,\mathrm s}(y)^{-1})\geq h_v+2$; \item $\rho_y$ is absolutely irreducible and not induced from a character of an index two subgroup of $G_F$. \end{enumerate} \end{definition} \begin{lemma}\label{lem:dense} Good points are Zariski dense in $D$. \end{lemma} \begin{proof} The characters $\psi_{v,\mathrm s}|_{\Art(1+p\Z_p)}$ are, up to the automorphism of Lemma~\ref{lem:finite}, the coordinates of $D\to\Spec\Lambda$. Their restrictions to $\Art(\Z_p^\times)$ differ from them by a fixed character of finite order. Hence condition (1) holds at every point above the set of points of $\Spec\Lambda[1/p]$ at which $\psi_{v,\mathrm s}|_{1+p\Z_p}$ is $u\mapsto u^{-h_v}$ with $h_v\in\Z$ and $2h_v\leq\HT(\chi)-2$. This set is Zariski dense in $\Spec\Lambda$. Since $D\to\Spec\Lambda$ is finite and surjective and $D$ is irreducible of the same dimension, its inverse image is dense in $D$: if it were contained in a proper closed subset, the image of that subset would be closed of dimension $<1+d$ and would contain a dense set. The points with $\rho_z$ reducible form a proper closed subset. So do the induced points, since only finitely many quadratic extensions $K/F$ unramified outside $\Sigma$ occur, and $\rho_z\simeq\rho_z\otimes\eta_{K/F}$ is a closed condition on traces. It is proper because a point $z$ satisfying (1) is not induced. Indeed, if $\rho_z\simeq\Ind_{G_K}^{G_F}\theta$, then $\rhobar^{ss}\simeq\rhobar^{ss}\otimes\eta_{K/F}$, so $\eta_{K/F}\equiv\bar\chi$ and, by (R), no place above $p$ splits in $K$. For $v\mid p$, $\rho_z|_{G_v}$ is then induced from a quadratic extension of $\Q_p$ and is reducible, so it is $\mu\oplus\mu\eta$ with $\eta$ quadratic. This contradicts the distinctness of the Hodge--Tate numbers in (1). \end{proof} \subsection{Good points are o-pro-modular} \begin{proposition}\label{prop:hida} Let $y$ be a good point. Then: \begin{enumerate} \item $\rho_y$ is the Galois representation of a regular algebraic cuspidal automorphic representation $\pi$ of $\GL_2(\mathbb A_F)$, not of CM type, ordinary at every $v\mid p$; \item for every $v\in\So$, $\psi_v(y)$ is the classical character of $\rho_y|_{G_v}$, that is, the Jordan--Hölder character with the smaller Hodge--Tate number; \item $y$ is o-pro-modular. \end{enumerate} \end{proposition} \begin{proof} For $v\mid p$, $\rho_y|_{G_v}$ is an extension of $\chi\psi_{v,\mathrm s}(y)^{-1}$ by $\psi_{v,\mathrm s}(y)$, by Lemma~\ref{lem:sign} and \cite[Lemma~2.3]{Die26}. It is de Rham by \cite[Lemma~2.1]{Die26}, since both characters are de Rham and the subcharacter has the smaller Hodge--Tate number. This proves (2), as $\psi_{v,\mathrm s}=\psi_v$ for $v\in\So$. By \cite[Theorem~5.1.2(2)]{Pan22} (see \cite[Theorem~4.2.3]{Zha25} for $p=3$), which allows $\rho_y|_{G_v}$ to be split, $\rho_y$ comes from a twist of a Hilbert modular form. This gives $\pi$ as in (1), and $\pi$ is not of CM type since $\rho_y$ is not induced. For (3), choose at each $v\in\So$ the nearly ordinary refinement of $\pi_v$ corresponding to the Jordan--Hölder character of smaller Hodge--Tate number, using the weight-normalized torus operators at sufficiently deep $p$-power level \cite[§5.2.1]{Pan22}. By ordinary local--global compatibility \cite[§5.2.2]{Pan22}, \cite[Theorem~2]{Wil88}, the full torus eigencharacter, including its value on a uniformizer, is $\psi_v(y)$; the distinct Hodge--Tate numbers select this character even when $\rho_y|_{G_v}$ is split, and the argument applies to ramified as well as unramified local characters. By local--global compatibility at the places of $\Sigma\setminus\Sigma_p$, $\pi_v$ has nonzero vectors under the Iwahori subgroup with character $\xi'_v$, for some $\xi'\in\Xi$. As $d$ is even, the Jacquet--Langlands correspondence gives automorphic forms on $B^\times$. Their ordinary stabilization at the places of $\So$ gives, by Hida theory \cite{Hid89}, \cite[§5.2]{Pan22} and the comparison with torsion ordinary parts of Proposition~\ref{prop:dimT} (see also \cite[Lemma~3.3]{Die26}), an eigenvector in $S^{\mathrm o}_{\xi'}\otimes E$ with Hecke eigenvalues $\mathrm{tr}\,\rho_y(\Frob_w)$ and diamond characters $\psi_v(y)$, $v\in\So$. At the places of $\Sn$ it contributes locally algebraic vectors \cite[3.2.10]{Pan22}. Hence $y$ is o-pro-modular. \end{proof} \subsection{Good points are regular} \begin{proposition}\label{prop:smooth} Let $y$ be a good point and $\mathfrak p_y$ the corresponding prime of $R^{\mathrm o}$. Then the local ring $R^{\mathrm o}_{\mathfrak p_y}$ is regular of dimension $2d-|\So|$. Consequently $C$ is the unique irreducible component of $\Spec R^{\mathrm o}$ containing $y$, and $\dim C=1+2d-|\So|$. \end{proposition} \begin{proof} Let $k=k(y)$ and $V=\ad^0\rho_y$. Since $y\in C$, $\dim R^{\mathrm o}_{\mathfrak p_y}\geq\dim C-1\geq2d-|\So|$, as $C$ is the spectrum of an $\cO$-flat complete local domain and $\mathfrak p_y$ has coheight $1$. It therefore suffices to show that the tangent space $t_y$ of $R^{\mathrm o}_{\mathfrak p_y}$, a $k$-vector space, has dimension at most $2d-|\So|$. A regularity statement of the same kind at ordinary regular automorphic points, also based on \cite{NT23}, is \cite[Corollary~5.3.9 and Lemma~5.3.11]{Zha26}. \emph{Step 1: $t_y$ as a Selmer group.} Elements of $t_y$ are continuous homomorphisms $R^{\mathrm o}\to k[\epsilon]$ lifting $y$ (cf.\ \cite[§2.3]{Kis09b}), that is, pseudo-representations $G_{F,\Sigma}\to k[\epsilon]$ with determinant $\chi$ lifting $\mathrm{tr}\,\rho_y$ and satisfying the conditions of Definition~\ref{def:rings}. As $\rho_y$ is absolutely irreducible, these are traces of deformations of $\rho_y$ to $k[\epsilon]$, unique up to conjugation \cite[Theorem~1.4.4]{BC09}. For $v\in\So$, the $\psi_v$-ordinarity of the trace is equivalent, at the irreducible prime $\mathfrak p_y$, to the existence of a $G_v$-stable direct summand of rank one with character lifting $\psi_v(y)$ \cite[Lemma~5.3.3 and 5.3.8(2)]{Pan22}. Hence $t_y$ is contained in the Selmer group $H^1_{\mathcal L}(F,V)$ of classes unramified outside $\Sigma$ with local conditions: \begin{itemize} \item $\mathcal L_v=\mathrm{im}\big(H^1(G_v,\mathfrak b_v^0)\to H^1(G_v,V)\big)$ for $v\in\So$, where $\mathfrak b_v^0\subset V$ is the stabilizer of the $\psi_v(y)$-line; \item $\mathcal L_v=H^1(G_v,V)$ for $v\in\Sigma\setminus\So$. \end{itemize} \emph{Step 2: local terms.} Let $v\mid p$ and write $\rho_y|_{G_v}$ as an extension of $\psi'=\chi\psi_{v,\mathrm s}(y)^{-1}$ by $\psi=\psi_{v,\mathrm s}(y)$, and put $m=\HT(\psi')-\HT(\psi)\geq2$ (Definition~\ref{def:good}(1)). The Jordan--Hölder factors of $V(1)$ have Hodge--Tate numbers $-1$, $-m-1$ and $m-1$, all non-zero, so $H^0(G_v,V(1))=0$. For $v\in\Sn$ this gives $h^1(G_v,V)-h^0(G_v,V)=3$. For $v\in\So$, $Q_v=V/\mathfrak b^0_v\simeq\psi'\psi^{-1}$ has Hodge--Tate number $m>0$, so $H^0(G_v,Q_v)=0$, $\mathcal L_v\simeq H^1(G_v,\mathfrak b^0_v)$ and $H^0(G_v,\mathfrak b^0_v)=H^0(G_v,V)$. The factors of $(\mathfrak b_v^0)^\vee(1)$ have Hodge--Tate numbers $-1$ and $m-1$, so $H^2(G_v,\mathfrak b^0_v)=0$, and $\dim\mathcal L_v-h^0(G_v,V)=2$ by the local Euler characteristic formula. Moreover $D_{\mathrm{dR}}(Q_v)/\mathrm{Fil}^0=0$, so $H^1_f(G_v,Q_v)=0$ \cite[Corollary~3.8.4]{BK90}, and by functoriality $H^1_f(G_v,V)\subset\ker\big(H^1(G_v,V)\to H^1(G_v,Q_v)\big)=\mathcal L_v$. Let $v\in\Sigma\setminus\Sigma_p$. If $\xi_v\neq1$, then $\rho_y|_{G_v}\simeq\mu_1\oplus\mu_2$ with $\mu_1\mu_2^{-1}$ ramified, so the factors $\varepsilon$, $\varepsilon\mu_1\mu_2^{-1}$, $\varepsilon\mu_2\mu_1^{-1}$ of $V(1)$ are non-trivial and $H^0(G_v,V(1))=0$. If $v\in\Sigma_{\St}$, then $H^0(G_v,V(1))=0$ by Lemma~\ref{lem:St}(2). In both cases $h^1(G_v,V)-h^0(G_v,V)=h^2(G_v,V)=0$. \emph{Step 3: the global estimate.} Let $\mathcal L^\perp_v\subset H^1(G_v,V(1))$ be the annihilators under local Tate duality, and $H^1_{\mathcal L^\perp}(F,V(1))$ the dual Selmer group, with unramified conditions at the places outside $\Sigma$. By Step~2, $\mathcal L^\perp_v=0$ for $v\in\Sigma\setminus\So$, and $\mathcal L^\perp_v\subset H^1_f(G_v,V)^\perp=H^1_f(G_v,V(1))$ for $v\in\So$ \cite[Proposition~3.8]{BK90}. Hence $H^1_{\mathcal L^\perp}(F,V(1))\subset H^1_f(F,V(1))$. By the Greenberg--Wiles formula \cite[Theorem~2.19]{DDT97}, using $H^0(F,V)=H^0(F,V(1))=0$ and $h^0(G_v,V)=1$ for the $d$ real places $v$, \[ h^1_{\mathcal L}(F,V)-h^1_{\mathcal L^\perp}(F,V(1))=\sum_{v\in\Sigma\setminus\Sigma_\infty}\big(\dim\mathcal L_v-h^0(G_v,V)\big)-d=2|\So|+3|\Sn|-d=2d-|\So|. \] The same formula, applied to the Bloch--Kato conditions, which are mutually orthogonal, gives \[ h^1_f(F,V)-h^1_f(F,V(1))=\sum_{v\mid p}\dim D_{\mathrm{dR}}(V)/\mathrm{Fil}^0-d=0, \] using $\dim H^1_f(G_v,V)=h^0(G_v,V)+\dim D_{\mathrm{dR}}(V)/\mathrm{Fil}^0$ for $v\mid p$ and $\dim H^1_f(G_v,V)=h^0(G_v,V)$ for $v\nmid p$ \cite[Corollary~3.8.4]{BK90}, and $\dim D_{\mathrm{dR}}(V)/\mathrm{Fil}^0=1$ for $v\mid p$, since the Hodge--Tate weights of $\rho_y|_{G_v}$ are distinct. By \cite[Theorem~B]{NT23}, $H^1_f(F,\ad\rho_y)=0$, as $\pi$ is regular algebraic and not of CM type. Hence $H^1_f(F,V)=0$, so $H^1_f(F,V(1))=0$ and $H^1_{\mathcal L^\perp}(F,V(1))=0$. Therefore $\dim t_y\leq h^1_{\mathcal L}(F,V)=2d-|\So|$. \emph{Conclusion.} $R^{\mathrm o}_{\mathfrak p_y}$ is a regular local ring, hence a domain, so exactly one minimal prime $\mathfrak q$ of $R^{\mathrm o}$ is contained in $\mathfrak p_y$. Since $y\in C$, $C=\Spec R^{\mathrm o}/\mathfrak q$. As $R^{\mathrm o}/\mathfrak q$ is a complete local domain, it is catenary, and $\dim C=\dim R^{\mathrm o}_{\mathfrak p_y}+1=1+2d-|\So|$. \end{proof} \section{The dimension of partially ordinary Hecke algebras}\label{sec:dimT} \begin{proposition}\label{prop:dimT} Let $\xi'\in\Xi$. Every minimal prime $\mathfrak p$ of $\T^{\mathrm o}_{\xi',\mathfrak m'}$ has characteristic $0$, and $\dim\T^{\mathrm o}_{\xi',\mathfrak m'}/\mathfrak p\geq1+2d-|\So|$. Moreover $S^{\mathrm o}_{\xi'}/\varpi^a\simeq\Ord_P(S_{\xi'}/\varpi^a)$ for every $a\geq1$, and every closed point $\mathfrak P$ of $\T^{\mathrm o}_{\xi',\mathfrak m'}[1/p]$ occurs: $(S^{\mathrm o}_{\xi'}\otimes E)[\mathfrak P]\neq0$, after enlarging $E$. \end{proposition} \begin{proof} This is \cite[Theorem~3.6.1]{Pan22}, which gives the bound $1+2d$ for the full Hecke algebra, with completed cohomology replaced by its ordinary part at $\So$; we give the details of the changes. Write $n=|\Sn|$, $o=|\So|$, $\T=\T^{\mathrm o}_{\xi',\mathfrak m'}$ and $M=(S^{\mathrm o}_{\xi'}\otimes E/\cO)^\vee$, localized at $\mathfrak m'$. For $p=3$, one uses \cite[Theorem~3.3.3 and Corollary~3.3.10]{Zha25} instead of \cite[Theorems~3.5.5 and~3.6.1]{Pan22}, which applies since the blocks at the places of $\Sn$ are generic by (R) and (a). \emph{Freeness and reduction.} Put $V=S_{\psi,\xi'}(U^p,E/\cO)_{\mathfrak m}$ and $W=\Ord_PV$. Choose sufficiently small compact open subgroups at $p$, with an Iwahori decomposition at the places of $\So$; the upper unipotent subgroup $N_0$ may be taken as small as necessary without changing $\Ord_P$ \cite[Lemma~3.1.11]{Eme10a}. After a continuous determinant twist trivializing the central character on the compact centres (it exists since $p$ is odd, commutes with $\Ord_P$ and changes each ordinary operator by a unit), and identifying the compact quotients by the centres with determinant-one complements via $g\mapsto(\det g)^{-1/2}g$, the compact group has an Iwahori decomposition $I=\bar N_*HN_0$ with \[ H=(K_{\mathrm n}/Z_{\mathrm n})\times(T_{\mathrm o}/Z_{\mathrm o}),\qquad\dim H=3n+o, \] where $K_{\mathrm n}$ is compact open in $\prod_{v\in\Sn}\GL_2(\Q_p)$, $T_{\mathrm o}$ is a compact open subgroup of the torus at $\So$, $Z_{\mathrm n},Z_{\mathrm o}$ are their scalar subgroups, and $H$ is chosen uniform pro-$p$. By \cite[Proposition~3.2.4 and §3.7]{Pan22}, $V|_I$ is a direct summand of $C^\infty(I,E/\cO)^r$. As a right $HN_0$-space $I=\bar N_*\times HN_0$, so this module is smoothly coinduced from the divisible, hence injective, $\cO$-module $C^\infty(\bar N_*,E/\cO)$; hence $V|_{HN_0}$ is injective, and $V^{N_0}$ is injective over $H$, since taking $N_0$-invariants is right adjoint to the exact inflation from $H$. The finite modules $V[\varpi^a]^{\bar N_iH_jN_0}$ exhaust $V^{N_0}$ and are stable under the commuting operators $U_v$, $v\in\So$ \cite[Lemma~3.3.2]{Eme10a}, which commute with $H$; the Fitting projections $\lim_kU_v^{k!}$ are compatible, and their product is an $H$-equivariant idempotent $e$ of $V^{N_0}$ whose image is the canonical image of $W$ \cite[Lemmas~3.1.5--3.1.7 and Theorem~3.3.3]{Eme10a}. Hence $W$ is an injective direct summand of $V^{N_0}$. By admissibility $W^\vee$ is finitely generated over $\cO[[H]]$, and by injectivity it is projective, hence free, as $\cO[[H]]$ is local. In particular $W$ is divisible, so left exactness and $S^{\mathrm o}_{\xi'}=T_\varpi W$ \cite[Definition~3.4.1]{Eme10a} give \[ S^{\mathrm o}_{\xi'}/\varpi^a\simeq W[\varpi^a]=\Ord_P(S_{\xi'}/\varpi^a). \] Localization at $\mathfrak m'$ is a direct summand, so $M/\varpi\simeq\Lambda'^{\oplus r}$ with $\Lambda'=\F[[H]]$. \emph{The relative block module.} Let $\mathcal B_{\mathrm n}$ be the product of the blocks of $\prod_{v\in\Sn}\GL_2(\Q_p)$ attached to the residual pseudo-representation, with projective generator $P_{\mathcal B_{\mathrm n}}$ and endomorphism ring $E_{\mathcal B_{\mathrm n}}$, and put $\mathbf m^{\mathrm o}=\Hom(P_{\mathcal B_{\mathrm n}},M)$, so that $M\simeq\mathbf m^{\mathrm o}\widehat\otimes_{E_{\mathcal B_{\mathrm n}}}P_{\mathcal B_{\mathrm n}}$, the equivalence being exact, faithful and compatible with the torus action. By admissibility, applied after quotienting by the augmentation ideal of the torus, and topological Nakayama, $\mathbf m^{\mathrm o}$ is finite over $E_{\mathcal B_{\mathrm n}}\widehat\otimes\cO[[T_{\mathrm o}/Z_{\mathrm o}]]$ (the relative form of \cite[Proposition~A.1.4 and Corollary~3.3.7]{Zha25}), hence over \[ A_0=\Big(\widehat\bigotimes_{v\in\Sn}R^{\ps}_v\Big)\widehat\otimes_\cO\cO[[T_{\mathrm o}/Z_{\mathrm o}]], \] since $E_{\mathcal B_{\mathrm n}}$ is finite over its centre. The local centres at $\Sn$ commute with $N_0$, the $U_v$ and their projectors, so local--global compatibility passes to ordinary parts \cite[Lemma~3.5]{Die26}, and the action of $A_0$ factors through $\T$. As $\mathbf m^{\mathrm o}$ is faithful over $\T$, we get $\T\hookrightarrow\mathrm{End}_{A_0}(\mathbf m^{\mathrm o})$, so $\T$ is finite over the noetherian ring $A_0$. Since $\mathbf m^{\mathrm o}$ is $\cO$-torsion free, $\varpi$ is a non-zero-divisor on $\T$, and the minimal primes of $\T$ have characteristic $0$. \emph{The dimension estimate.} Put $A=A_0/\varpi$. Pan's dimension formula \cite[Proposition~3.6.12]{Pan22}, with the torus variables kept in the central coefficient ring, gives \[ \dim_{\Lambda'}(N\widehat\otimes_{E_{\mathcal B_{\mathrm n}}}P_{\mathcal B_{\mathrm n}})\leq\dim_AN+n \] for finitely generated modules $N$ over $(E_{\mathcal B_{\mathrm n}}/\varpi)\widehat\otimes\F[[T_{\mathrm o}/Z_{\mathrm o}]]$, with the same proof \cite[Lemmas~3.6.8--3.6.11]{Pan22}: if $\dim_AN=0$, the torus acts through a finite quotient and $N\widehat\otimes P_{\mathcal B_{\mathrm n}}$ is dual to a smooth representation of finite length of $\prod_{v\in\Sn}\GL_2(\Q_p)$, of Gelfand--Kirillov dimension at most $n$ \cite[Lemma~3.6.14]{Pan22}, and the induction on a central parameter is unchanged. Let $\mathfrak p$ be a minimal prime of $\T$. Then $\mathbf m^{\mathrm o}[\mathfrak p]\neq0$, and it is $\varpi$-saturated by torsion-freeness, so $N=\mathbf m^{\mathrm o}[\mathfrak p]/\varpi\hookrightarrow\mathbf m^{\mathrm o}/\varpi$ is non-zero. Its image $N\widehat\otimes P_{\mathcal B_{\mathrm n}}$ is a non-zero submodule of $M/\varpi\simeq\Lambda'^{\oplus r}$; as $\Lambda'$ is a domain of dimension $3n+o$, it has dimension $3n+o$, and so $\dim_AN\geq2n+o$. Since $A$ acts on $N$ through $\T/(\mathfrak p,\varpi)$, \[ \dim\T/\mathfrak p=1+\dim\T/(\mathfrak p,\varpi)\geq1+2n+o=1+2d-o. \] \emph{Occurrence.} Let $\mathfrak P$ be a closed point of $\T[1/p]$ and $\mathfrak q$ its contraction to $\T$. As $\mathbf m^{\mathrm o}$ is finite and faithful, $\mathbf m^{\mathrm o}\otimes_\T k(\mathfrak P)\neq0$, and $\mathbf m^{\mathrm o}/\mathfrak q\mathbf m^{\mathrm o}$ is finite over $\cO$. After enlarging $\cO$, its quotient by its (bounded) $\cO$-torsion has a non-zero torsion-free quotient $Q$ on which $\T$ acts through $\mathfrak P$. By the block equivalence, $Q\widehat\otimes P_{\mathcal B_{\mathrm n}}$ is a non-zero compact torsion-free quotient of $M/\mathfrak qM$, and its continuous dual gives $(S^{\mathrm o}_{\xi'}\otimes E)[\mathfrak P]\neq0$. \end{proof} \section{Proof of Theorem~\ref{thm:main}}\label{sec:proof} We keep the setting of Section~\ref{sec:setup} and the component $C$ of Proposition~\ref{prop:GW}. By Lemma~\ref{lem:dense} there is a good point $y\in D\subset C$. By Proposition~\ref{prop:hida}(3), $y$ is the image of a prime of $\T^{\mathrm o}_{\xi',\mathfrak m'}$ for some $\xi'\in\Xi$. Let $W$ be an irreducible component of $\Spec\T^{\mathrm o}_{\xi',\mathfrak m'}$ containing it, viewed as a closed subset of $\Spec R^{\mathrm o}$ through the surjection $R^{\mathrm o}\twoheadrightarrow\T^{\mathrm o}_{\xi',\mathfrak m'}$. Then $W=V(\mathfrak p_W)$, where $\mathfrak p_W$ is the inverse image in $R^{\mathrm o}$ of a minimal prime of $\T^{\mathrm o}_{\xi',\mathfrak m'}$; it is an irreducible closed subset of $\Spec R^{\mathrm o}$ containing $y$, though a priori not an irreducible component of it. Let $\mathfrak q'\subset\mathfrak p_W$ be a minimal prime of $R^{\mathrm o}$. Since $y\in W$, we have $\mathfrak q'\subset\mathfrak p_W\subset\mathfrak p_y$. By Proposition~\ref{prop:smooth}, $\mathfrak q$ is the unique minimal prime of $R^{\mathrm o}$ contained in $\mathfrak p_y$, so $\mathfrak q'=\mathfrak q$ and $W\subset C$. By Propositions~\ref{prop:dimT} and~\ref{prop:smooth}, \[ \dim W\geq1+2d-|\So|=\dim C, \] so $W=C$. In particular $x\in W$: there is a maximal ideal $\mathfrak P$ of $\T^{\mathrm o}_{\xi',\mathfrak m'}[1/p]$ above $x$, with $\psi_{v,1}(\mathfrak P)=\psi_v(x)=\psi^{\cl}_v$ for $v\in\So$. Let $\mathfrak p$ be the corresponding maximal ideal of the big Hecke algebra and $\mu=\bigotimes_{v\in\So}\mu_v$ the character of the torus at $\So$ given by $\mathfrak P$, so that $\mu_{v,1}\circ\Art^{-1}=\psi^{\cl}_v$. By Proposition~\ref{prop:dimT}, $(S^{\mathrm o}_{\xi'}\otimes E)[\mathfrak P]\neq0$, and since $\Ord_P$ is left exact this says that $\Ord_P((S_{\xi'}\otimes E)[\mathfrak p])[T_{\mathrm o}=\mu]\neq0$. Then $\rho(\mathfrak p)\simeq\rho$ is irreducible, and the hypotheses of \cite[Theorem~4.2]{Die26}, including this occurrence of $\mu$, hold by (N) and (O). Hence $\rho$ is modular over the field of §\ref{subsec:field}, and over the original field by solvable descent. This proves Theorem~\ref{thm:main}. \qed \section{Remarks} \begin{remark}\label{rem:b} The essential use of condition (a) is Lemma~\ref{lem:red}, and through it the equality $\dim D=1+d$. In Proposition~\ref{prop:GW}, (a) also gives the vanishing of $H^0(G_{F,\Sigma},\ad^0\rhobar_b(1))$ and the smoothness of the local rings, but there one could argue as in \cite[Lemma~5.4.1]{Zha25} with weaker information. If $\bar\chi|_{G_{\Q_p}}=\omega^{\pm1}$ and $p>3$, the reducible locus at a place of $\Sn$ is cut out by two equations: $R^{\ps}_v\simeq\cO[[x_0,x_1,y_0,y_1]]/(x_0y_1-x_1y_0)$ and the reducible quotient is $R^{\ps}_v/(x_0,x_1)$ \cite[Lemma~7.4.4]{Pan22}. Then $D$ only has dimension at least $1+|\So|$. It lies in the fully ordinary locus, of dimension at most $1+d$, but its map to $\Spec\Lambda$ need not be dominant, so the density argument for de Rham points of Section~\ref{sec:classical} does not apply. \end{remark} \begin{remark}\label{rem:bcheck} Since $\rhobar^{ss}|_{G_v}\simeq\bar\chi_1|_{G_v}\oplus\bar\chi_2|_{G_v}$ for every $v\mid p$, condition (a) can be checked at a single place above $p$, of $\Sn$ or of $\So$, from the local behaviour of $\rho$ there. It suffices that the ratio of the two residual characters restricted to the inertia group $I_v$ is not $\omega^{\pm1}$. Condition (a) excludes some natural cases. If $\rho_v$ is crystalline with consecutive Hodge--Tate weights, the ratio is $\omega^{\pm1}$ on inertia, and (a) can only hold through the unramified parts of the two characters. If $\rho_v$ is of Steinberg type with consecutive Hodge--Tate weights, then $\rhobar_v^{ss}$ is a twist of $1\oplus\omega$, and (a) fails. On the other hand, there are cases in which (a) can be verified directly. \begin{itemize} \item \emph{Fontaine--Laffaille case.} Assume that $\rho_v$ is crystalline with Hodge--Tate weights $\{0,k-1\}$, twisted by a character, with $2\leq k\leq p-1$. By Fontaine--Laffaille theory \cite{FL82}, if $\rhobar_v$ is reducible, then the characters of $\rhobar^{ss}_v|_{I_v}$ are, up to a common twist, $1$ and $\omega^{k-1}$. Hence (a) holds as soon as $k-1\not\equiv\pm1\pmod{p-1}$, that is, for $3\leq k\leq p-2$. \item \emph{Principal series types.} Assume that $\rho_v$ is potentially crystalline of tame principal series type, with Hodge--Tate weights $\{0,k-1\}$ up to twist, becoming crystalline over a tamely ramified extension of ramification index $e$ with $e(k-1)