Geometry of the eigencurve at CM points and trivial zeros of Katz $p$-adic $L$-functions

TL;DR

This paper explores the local geometry of the $p$-adic eigencurve at CM points, linking intersection multiplicities to $ ext{L}$-invariants and trivial zeros of Katz $p$-adic $L$-functions, extending conjectures of Gross.

Contribution

It provides a detailed analysis of the eigencurve's structure at CM points and establishes new connections between $ ext{L}$-invariants, trivial zeros, and the geometry of $p$-adic $L$-functions.

Findings

The eigencurve at CM points has 3 or 4 irreducible components.

A simple zero of the congruence ideal corresponds to a non-vanishing $ ext{L}$-invariant.

At least one of the $ ext{L}$-invariants $ ext{L}_-( ext{varphi})$ or $ ext{L}_-( ext{varphi}^{-1})$ is non-zero.

Abstract

The primary goal of this paper is to investigate the geometry of the $p$-adic eigencurve at a point $f$ corresponding to a weight one cuspidal theta series irregular at the prime number $p$. We show that $f$ belongs to exactly three or four irreducible components and study their intersection multiplicities. In particular, we show that the congruence ideal of a CM component has a simple zero at $f$ if and only if a certain anti-cyclotomic $\mathscr{L}$-invariant $\mathscr{L}_-(\varphi)$ does not vanish. Further, using Roy's Strong Six Exponential Theorem we show that at least one amongst $\mathscr{L}_-(\varphi)$ and $\mathscr{L}_-(\varphi^{-1})$ is non-zero. Combined with a divisibility proved by Hida and Tilouine, we deduce that the anti-cyclotomic Katz $p$-adic $L$-function of $\varphi$ has a simple (trivial) zero at $s=0$ if $\mathscr{L}_-(\varphi)$ is non-zero, which can be seen as an anti-cyclotomic analogue of a result of Ferrero and Greenberg. Finally, we propose a formula for the linear term of the two-variable Katz $p$-adic $L$-function of $\varphi$ at $s=0$ extending a conjecture of Gross.

Equations

$$\displaystyle L_{p}(\psi_{\mbox{-}},s_{\mathfrak{p}},s_{\bar{\mathfrak{p}}})\overset{?}{=}\big{(}\log_{p}(\bar{\mathfrak{p}})\cdot s_{\mathfrak{p}}+(\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})+\log_{p}(\bar{\mathfrak{p}}))\cdot s_{\bar{\mathfrak{p}}}\big{)}\cdot L_{p}^{*}(\psi_{\mbox{-}},0)+\text{higher order terms},$$

$$C_{\psi }^0\subset ({\zeta }_{{\psi }_{\text{mbox}-}}^{-})\subset (X),$$

$$u_{{\psi }_{\text{mbox}-}}=\sum _{g\in C}{g}^{-1}(u)\otimes {\psi }_{\text{mbox}-}(g)\in ({\mathcal{O}}_H^{\times }\otimes \overline{\mathbb{Q}})[{\psi }_{\text{mbox}-}],$$

$$({\mathcal{O}}_H^{\times }\otimes \overline{\mathbb{Q}})[{\psi }_{\text{mbox}-}]=\overline{\mathbb{Q}}\cdot u_{{\psi }_{\text{mbox}-}},\text{ and }({\mathcal{O}}_H^{\times }\otimes \overline{\mathbb{Q}})[{\psi }_{\text{mbox}-}^{\tau }]=\overline{\mathbb{Q}}\cdot \tau (u_{{\psi }_{\text{mbox}-}}).$$

$${\log }_p:{\mathcal{O}}_H^{\times }\otimes \overline{\mathbb{Q}}⟶{\overline{\mathbb{Q}}}_p\text{ , }u\otimes x\mapsto {\log }_p(u)\cdot {\iota }_p(x),$$

$$\mathcal{S}({\psi }_{\text{mbox}-})=-\frac{{\log }_p(u_{{\psi }_{\text{mbox}-}})}{{\log }_p(\tau (u_{{\psi }_{\text{mbox}-}}))}.$$

$$H^1(K,{\psi }_{\text{mbox}-})\hookrightarrow \mathrm{im}\left(H^1(K_{\mathfrak{p}},{\overline{\mathbb{Q}}}_p)\times H^1(K_{\bar{\mathfrak{p}}},{\overline{\mathbb{Q}}}_p)\to H^1({\mathrm{I}}_{\mathfrak{p}},{\overline{\mathbb{Q}}}_p)\times H^1({\mathrm{I}}_{\bar{\mathfrak{p}}},{\overline{\mathbb{Q}}}_p)\right)\simeq \mathrm{Hom}({\mathcal{O}}_{K_{\mathfrak{p}}}^{\times }\times {\mathcal{O}}_{K_{\bar{\mathfrak{p}}}}^{\times },{\overline{\mathbb{Q}}}_p)$$

$$H^1(K,{\psi }_{\text{mbox}-})\stackrel{\sim }{\to }{H}^1(H,{\overline{\mathbb{Q}}}_p)[{\psi }_{\text{mbox}-}^{\tau }]=(\mathrm{Hom}({\Gamma }_H,{\overline{\mathbb{Q}}}_p)\otimes {\psi }_{\text{mbox}-}{)}^C.$$

$$H^1(\mathbb{Q},{\mathrm{Ind}}_K^{\mathbb{Q}}{\psi }_{\text{mbox}-})\stackrel{\sim }{\to }{H}^1(K,{\psi }_{\text{mbox}-}\oplus {\psi }_{\text{mbox}-}^{\tau }{)}^{\tau =1}\stackrel{\sim }{\to }(\mathrm{Hom}({\Gamma }_H,{\overline{\mathbb{Q}}}_p)\otimes {\mathrm{Ind}}_K^{\mathbb{Q}}{\psi }_{\text{mbox}-}{)}^G.$$

$$0\to H^1(H,{\overline{\mathbb{Q}}}_p)[{\psi }_{\text{mbox}-}^{\tau }]\stackrel{{\mathrm{res}}_p}{\to }\mathrm{Hom}({\mathcal{O}}_{H_{\mathfrak{p}}}^{\times },{\overline{\mathbb{Q}}}_p)[{\psi }_{\text{mbox}-}^{\tau }]\times \mathrm{Hom}({\mathcal{O}}_{H_{\bar{\mathfrak{p}}}}^{\times },{\overline{\mathbb{Q}}}_p)[{\psi }_{\text{mbox}-}^{\tau }]\to \mathrm{Hom}({\mathcal{O}}_H^{\times },{\overline{\mathbb{Q}}}_p).$$

$$\sum _{g\in C}{\psi }_{\text{mbox}-}(g){\log }_p(g^{-1}\cdot )+s\cdot \sum _{g\in C}{\psi }_{\text{mbox}-}(g){\log }_p(\tau g^{-1}\cdot ),$$

$$\mathcal{L}({\psi }_{\text{mbox}-}^{\tau })=-\frac{{\sum }_{g\in C}{\psi }_{\text{mbox}-}(g){\log }_p(g^{-1}(y_0))+\mathcal{S}({\psi }_{\text{mbox}-}){\sum }_{g\in C}{\psi }_{\text{mbox}-}(g){\log }_p(\tau g^{-1}(y_0))}{{\mathrm{ord}}_{v_0}(y_0)}$$

$$1\to {\mathcal{O}}_H^{\times }\to {\mathcal{O}}_H[\frac{1}{\mathfrak{p}}{]}^{\times }\stackrel{\mathrm{ord}}{\to }\mathbb{Z}[C]$$

$$0\to \overline{\mathbb{Q}}\cdot u_{{\psi }_{\text{mbox}-}}\to ({\mathcal{O}}_H[\frac{1}{\mathfrak{p}}{]}^{\times }\otimes \overline{\mathbb{Q}})[{\psi }_{\text{mbox}-}]\stackrel{{\mathrm{ord}}_{v_0}}{\to }\overline{\mathbb{Q}}\to 0.$$

$$\mathcal{L}({\psi }_{\text{mbox}-}^{\tau })=-\frac{{\log }_p(u_{\mathfrak{p},{\psi }_{\text{mbox}-}})+\mathcal{S}({\psi }_{\text{mbox}-}){\log }_p(\tau (u_{\mathfrak{p},{\psi }_{\text{mbox}-}}))}{{\mathrm{ord}}_{v_0}(u_{\mathfrak{p},{\psi }_{\text{mbox}-}})}.$$

$${\mathcal{L}}_{\mathfrak{p}}=-\frac{{\log }_p(u_{\mathfrak{p}})}{{\mathrm{ord}}_{\mathfrak{p}}(u_{\mathfrak{p}})}.$$

$$0\to H^1(K,{\overline{\mathbb{Q}}}_p)\to \mathrm{Hom}(K_{\mathfrak{p}}^{\times }\times {\mathcal{O}}_{K_{\bar{\mathfrak{p}}}}^{\times },{\overline{\mathbb{Q}}}_p)\to \mathrm{Hom}({\mathcal{O}}_K[\frac{1}{\mathfrak{p}}{]}^{\times },{\overline{\mathbb{Q}}}_p)$$

$$0\to H^1(H,{\overline{\mathbb{Q}}}_p)\to \mathrm{Hom}(H_{v_0}^{\times }\times \prod _{v_0\ne v\mid p}{\mathcal{O}}_{H_v}^{\times },{\overline{\mathbb{Q}}}_p)\to \mathrm{Hom}({\mathcal{O}}_H[\frac{1}{v_0}{]}^{\times },{\overline{\mathbb{Q}}}_p)$$

$$\sum _{g\in C}{\psi }_{\text{mbox}-}(g){\log }_p(g^{-1}\cdot )+\mathcal{S}({\psi }_{\text{mbox}-})\sum _{g\in C}{\psi }_{\text{mbox}-}(g){\log }_p(\tau g^{-1}\cdot ).$$

$${\mathrm{ord}}_{v_0}(y_0){\mathrm{res}}_{v_0}(\eta -{\eta }_{\mathfrak{p}})({\mathrm{Frob}}_{\mathfrak{p}})=\sum _{1\ne g\in C}(1-{\psi }_{\text{mbox}-}(g)){\log }_p(g^{-1}(y_0))-\mathcal{S}({\psi }_{\text{mbox}-})\sum _{g\in C}{\psi }_{\text{mbox}-}(g){\log }_p(\tau g^{-1}(y_0)).$$

$$\sum _{g\in C}{\log }_p(g^{-1}(y_0))={\log }_p({\mathrm{N}}_{H/K}(y_0))=-{\mathcal{L}}_{\mathfrak{p}}{\mathrm{ord}}_{v_0}(y_0),$$

$${\mathrm{res}}_{\mathfrak{p}}(\eta )=(\mathcal{L}({\psi }_{\text{mbox}-}^{\tau })-{\mathcal{L}}_{\mathfrak{p}})\cdot {\mathrm{ord}}_{\mathfrak{p}}+{\eta }_{\mathfrak{p}}={\mathcal{L}}_{\text{mbox}-}({\psi }_{\text{mbox}-}^{\tau })\cdot {\mathrm{ord}}_{\mathfrak{p}}+({\eta }_{\mathfrak{p}}-{\eta }_{\bar{\mathfrak{p}}}),$$

$${\mathrm{ord}}_{v_0}(y_0){\mathcal{L}}_{\text{mbox}-}({\psi }_{\text{mbox}-}^{\tau })={\mathrm{res}}_{v_0}(\eta -{\eta }_{\mathfrak{p}}+{\eta }_{\bar{\mathfrak{p}}})(y_0)=$$

$$=\sum _{g\in C-\{1\}}(1-{\psi }_{\text{mbox}-}(g)){\log }_p(g^{-1}(y_0))-\mathcal{S}({\psi }_{\text{mbox}-})\sum _{g\in C}{\psi }_{\text{mbox}-}(g){\log }_p(\tau g^{-1}(y_0))-\sum _{g\in C}{\log }_p(\tau g^{-1}(y_0))=$$

$$=-{\log }_p(u_{\mathfrak{p},{\psi }_{\text{mbox}-}})-\mathcal{S}({\psi }_{\text{mbox}-}){\log }_p(\tau (u_{\mathfrak{p},{\psi }_{\text{mbox}-}}))+2{\log }_p({\mathrm{N}}_{H/K}(y_0)).$$

$${\mathrm{ord}}_{v_0}(y_0){\mathcal{L}}_{\text{mbox}-}({\psi }_{\text{mbox}-})=-{\log }_p(u_{\mathfrak{p},{\psi }_{\text{mbox}-}^{\tau }})-\mathcal{S}({\psi }_{\text{mbox}-}^{\tau }){\log }_p(\tau (u_{\mathfrak{p},{\psi }_{\text{mbox}-}^{\tau }}))+2{\log }_p({\mathrm{N}}_{H/K}(y_0)).$$

$$\left(\begin{array}{ccc}{\log }_p(u_{{\psi }_{\text{mbox}-}})& {\log }_p(u_{\mathfrak{p},{\psi }_{\text{mbox}-}})-2{\log }_p({\mathrm{N}}_{H/K}(y_0))& {\log }_p(u_{\bar{\mathfrak{p}},{\psi }_{\text{mbox}-}})\\ {\log }_p(u_{{\psi }_{\text{mbox}-}^{\tau }})& {\log }_p(u_{\bar{\mathfrak{p}},{\psi }_{\text{mbox}-}^{\tau }})& {\log }_p(u_{\mathfrak{p},{\psi }_{\text{mbox}-}^{\tau }})-2{\log }_p({\mathrm{N}}_{H/K}(y_0))\end{array}\right)$$

$$\sum _{1\ne g\in C}({\psi }_{\text{mbox}-}(g)-1)\left(a\cdot {\log }_p(g^{-1}(u))+b\cdot {\log }_p(g^{-1}(y_0)\right)+\sum _{g\in C}(c\cdot {\psi }_{\text{mbox}-}(g)+b){\log }_p(g^{-1}\tau (y_0))=0.$$

$$0\to F_{\rho }=\overline{\mathbb{Q}}{}_p{e}_1\to V_{\rho }\to V_{\rho }/F_{\rho }\to 0.$$

$$\mathrm{tr}{\rho }_A(\tau g\tau )=\mathrm{tr}{\rho }_A(g)\text{ for any }g\in {\Gamma }_K.$$

$$\dim {\mathcal{D}}_{\rho }^K({\overline{\mathbb{Q}}}_p[ϵ])=1<\dim {\mathcal{D}}_{\rho }^{\mathrm{red}}({\overline{\mathbb{Q}}}_p[ϵ])=\dim {\mathcal{D}}_{\rho }^{\mathrm{ord}}({\overline{\mathbb{Q}}}_p[ϵ])=2.$$

Full text

Introduction

Our investigations lie at the crossroad between Iwasawa theory and Hida theory. Iwasawa theory and its subsequent generalizations study $p$-adic $L$-functions of automorphic forms or motives. The Iwasawa Main Conjecture postulates that those generate the characteristic ideals of appropriately defined Greenberg–Selmer groups. In the few cases where a proof is available, comparing the orders of the trivial (or extra) zeros is quite involved and requires additional arguments. On the other hand, Hida has extended the construction from his seminal trilogy [22, 23, 24] to congruence modules for $p$-adic families of automorphic forms, and has conjectured that the corresponding characteristic power series is generated by the adjoint $p$-adic $L$-function of the family, and that it does not vanish at classical points of weight at least two. Hida theory has allowed to seek understanding of the Iwasawa Main Conjecture through the study of the geometry of $p$-adic eigenvarieties, and to detect even finer local and global geometric phenomena such as intersection numbers and ramification indices over the weight space.

Let $K$ be an imaginary quadratic field in which $p$ splits as $(p)=\mathfrak{p}\bar{\mathfrak{p}}$. Let $\psi$ be a finite order Hecke character of $K$ which does not descend to $\mathbb{Q}$ and has conductor relatively prime to $p$. We assume that $\psi(\mathfrak{p})=\psi(\bar{\mathfrak{p}})$, so that the weight $1$ theta series $\theta_{\psi}$ attached to $\psi$ has a unique $p$-stabilization, denoted by $f$, and we study the conjectural relation between the congruence power series of a Hida family containing $f$ and its adjoint $p$-adic $L$-function. In the case of the CM family $\Theta_{\psi}$, the latter is essentially given by the Katz anti-cyclotomic $p$-adic $L$-function $L_{p}^{-}(\psi_{\mbox{-}},s)$ of $\psi_{\mbox{-}}=\psi/\psi^{\tau}$, where $\tau$ denotes a complex conjugation, and Katz’s $p$-adic Kronecker Second Limit Formula implies that $L_{p}^{-}(\psi_{\mbox{-}},s)$ has a trivial zero at $s=0$. While the Iwasawa Main Conjecture relates $L_{p}^{-}(\psi_{\mbox{-}},s)$ to the characteristic series of a certain Iwasawa module over the anti-cyclotomic tower of $K$, neither the original formulation, nor its proofs by Rubin [37] and by Tilouine [38], provide a formula for the orders of their common trivial zeros.

A main theme of this paper is to describe as precisely as possible the local geometry of the eigencurve $\mathscr{E}$ at $f$ and draw some arithmetic consequences, such as a criterion for $L_{p}^{-}(\psi_{\mbox{-}},s)$ to have a simple trivial zero. Our investigations pushed us to go further and propose the following expression for the leading term of the Katz $2$-variable $p$-adic $L$-function

[TABLE]

where $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})$ is the anti-cyclotomic $\mathscr{L}$-invariant introduced in Definition 1.9, while $L_{p}^{*}(\psi_{\mbox{-}},0)$ is related via the $p$-adic Kronecker Second Limit Formula (44) to a $p$-adic logarithm of an elliptic unit, which is non-zero by the Baker–Brumer Theorem (see §4.3 for more details).

The above formula lies outside the scope of Gross’ famous conjecture [21] on the leading term at $s=0$ of $p$-adic $L$-functions of Artin Hecke characters. Indeed, the fixed field of $\psi_{\mbox{-}}$ is not a CM field, unless $\psi_{\mbox{-}}$ is quadratic, in which case we prove (1) up to a scalar in $\mathbb{Q}^{\times}$ using Gross’ factorization formula. In the general case, we provide the following evidence.

Theorem A**.**

If $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\neq 0$, then $L_{p}^{-}(\psi_{\mbox{-}},s)=L_{p}(\psi_{\mbox{-}},s,-s)$ has a simple (trivial) zero at $s=0$. Moreover, at least one amongst $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})$ and $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})$ is non-zero.

Theorem A gives an affirmative answer, for more than half of the anti-cyclotomic characters, to a long standing question raised by Hida–Tilouine in [28, p.106], as $\operatorname{ord}_{s=0}L_{p}^{-}(\varphi,s)=1$ for at least one of the characters $\varphi$ occurring in each set $\{\psi_{\mbox{-}},\psi_{\mbox{-}}^{\tau}\}$.

Let us now briefly describe our approach. The second part is proved in Proposition 1.11 using the Strong Six Exponentials Theorem, while the Strong Four Exponentials Conjecture would imply that $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\cdot\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})\neq 0$. An important observation made in Theorem 3.3 is that in addition to the two Hida families $\Theta_{\psi}$ and $\Theta_{\psi^{\tau}}$ having CM by $K$, $f$ also belongs to at least one Hida family without CM by $K$, hence lies in the closed analytic subspace $\mathscr{E}^{\perp}$ of $\mathscr{E}$, union of irreducible components having no CM by $K$. Denote by $\mathcal{T}$, resp. by $\mathcal{T}^{\perp}$, the completed local ring at $f$ of $\mathscr{E}$, resp. of $\mathscr{E}^{\perp}$. Both algebras $\mathcal{T}$ and $\mathcal{T}^{\perp}$ are finite and flat over the completed local ring $\varLambda=\overline{\mathbb{Q}}_{p}\lsem X\rsem$ of the weight space at the point corresponding to the weight of $f$. Consider the congruence ideal $C^{0}_{\psi}=\pi_{\psi}(\mathrm{Ann}_{\mathcal{T}}(\ker(\pi_{\psi})))$ attached to the $\Theta_{\psi}$-projection $\pi_{\psi}:\mathcal{T}\twoheadrightarrow\varLambda$. Letting $\zeta^{-}_{\psi_{\mbox{-}}}$ denote the $p$-adic inverse Mellin transform of $L_{p}^{-}(\psi_{\mbox{-}},s)$, seen as an element of $\varLambda$, Hida and Tilouine showed in [27, Thm. I] that

[TABLE]

thus providing an upper bound for $\operatorname{ord}_{s=0}L_{p}^{-}(\psi_{\mbox{-}},s)$ in terms of the local geometry of $\mathscr{E}$ at $f$.

To prove the first part of Theorem A it then suffices to show that $C^{0}_{\psi}=(X)$ if and only if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\neq 0$, which is a consequence of the main result of this paper.

Theorem B** (Thm. 4.8).**

The $\varLambda$-algebra $\mathcal{T}$ is isomorphic to $\varLambda\times_{\overline{\mathbb{Q}}_{p}}({\mathcal{T}}^{\perp}\times_{\overline{\mathbb{Q}}_{p}[X]/(X^{r-1})}\varLambda)$, where

• •

if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})\cdot\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})=0$, then $r\geq 3$ and $\mathcal{T}^{\perp}=\varLambda[Z]/(Z^{2}-X^{r})$,

• •

if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})+\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})=0$, then $r=2$ and $\mathcal{T}^{\perp}=\overline{\mathbb{Q}}_{p}\lsem X^{1/e}\rsem\times_{\overline{\mathbb{Q}}_{p}}\overline{\mathbb{Q}}_{p}\lsem X^{1/e}\rsem$ for some $e\geq 2$,

• •

in all other cases we have $r=2$ and $\mathcal{T}^{\perp}=\varLambda\times_{\overline{\mathbb{Q}}_{p}}\varLambda$.

If $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\neq 0$, then the first projection in the fiber product description of $\mathcal{T}$ above is given by $\pi_{\psi}$, hence both $\zeta_{\psi_{\mbox{-}}}^{-}$ and the characteristic power series of the congruence module attached to $\Theta_{\psi}$ have a simple zero at $X=0$, as conjectured by Hida–Tilouine [27, p.192]. The conjecture was proven by Tilouine [38] and Mazur–Tilouine [34], under some technical hypotheses, when $\psi_{\mbox{-}}(\mathfrak{p})\neq 1$. We would like to emphasize that cases with a trivial zero like ours lie beyond the reach of the methods used in loc. cit. as the ordinary deformation functor is not representable and, more importantly, $C^{0}_{\psi}$ and $C^{1}_{\psi}=\ker(\pi_{\psi})/\ker(\pi_{\psi})^{2}$ are not equal, as the Hecke algebra $\mathcal{T}$ is not a local complete intersection. Theorem B also allows us to establish a conjecture of Darmon, Lauder and Rotger [12] on the dimension of the generalized eigenspace of $p$-adic modular forms corresponding to a weight one cusform with CM, which was previously only known for $\psi_{\mbox{-}}$ quadratic by the work [31] of Lee (see Corollary 5.5).

Let us now explain some of the ideas and techniques that go in the proof of Theorem B. As the $2$-dimensional $\Gamma_{\mathbb{Q}}$-representation $\rho_{f}=\operatorname{Ind}^{\mathbb{Q}}_{K}\psi$ is locally scalar at $p$, neither its Mazur’s $p$-ordinary deformation functor is representable, nor its deformation to $\mathcal{T}$ has a $p$-ordinary filtration. In contrast with the $p$-regular setting considered in [3], where $\mathcal{T}$ is isomorphic to the $p$-ordinary deformation ring of $\rho_{f}$, we need to resort to new methods. First, we divide the difficulties and study separately the components having no CM by $K$. As the restriction to $\Gamma_{K}$ of the $\mathcal{T}^{\perp}$-valued pseudo-character is generically irreducible and lifts $\psi+\psi^{\tau}$, and as $\operatorname{Ext}_{\Gamma_{K}}^{1}(\psi^{\tau},\psi)$ and $\operatorname{Ext}_{\Gamma_{K}}^{1}(\psi,\psi^{\tau})$ are both $1$-dimensional generated by $\rho$ and $\rho^{\prime}$, respectively, a Ribet style argument shows the existence of pair $(\rho_{\mathcal{T}^{\perp}},\rho_{\mathcal{T}^{\perp}}^{\prime})$ of $\mathfrak{p}$-ordinary $\Gamma_{K}$-deformations lifting $(\rho,\rho^{\prime})$. This leads naturally to a universal deformation $\varLambda$-algebra $\mathcal{R}^{\perp}$ classifying pairs of $\mathfrak{p}$-ordinary deformations of $(\rho,\rho^{\prime})$ sharing the same $\tau$-invariant traces and rank one $\mathfrak{p}$-unramified quotients. We show that $\mathcal{R}^{\perp}\simeq\overline{\mathbb{Q}}_{p}\lsem X^{1/e}\rsem$ is a discrete valuation ring endowed with flat morphism $\mathcal{R}^{\perp}\hookrightarrow\mathcal{T}^{\perp}$ of $\varLambda$-algebras (see Thm. 3.7). Furthermore, the reducibility ideal of the $2$-dimensional pseudo-character carried by $\mathcal{R}^{\perp}$ is generated by an element of valuation $r\geqslant 2$, with $r=2$ if and only if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})\cdot\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\neq 0$. In §3.4 we determine all possible extensions to $\Gamma_{\mathbb{Q}}$ of the pseudo-character carried by $\mathcal{R}^{\perp}$ and deduce that $\mathcal{R}^{\perp}[Z]/(Z^{2}-X^{r/e})\simeq\mathcal{T}^{\perp}$ as $\varLambda$-algebras. With all the components of $\mathscr{E}$ containing $f$ determined in §3, §4.1 is devoted to the study of the CM ideal, measuring the congruences between a component having CM by $K$ and a component without CM by $K$. Finally in §4.2 we determine the ideal $C^{0}_{\psi}$ governing the congruences between $\Theta_{\psi}$ and all other families.

While the deformation to $\mathcal{T}$ is not $p$-ordinary, its localization at any generic point is, motivating the use in §5 of a universal “generically” $p$-ordinary deformation ring $\mathcal{R}^{\triangle}$ of $\rho_{f}$. As $\mathcal{T}$ is not Gorenstein (see Prop. 5.4) the following result lies beyond the scope of the Taylor–Wiles method and its proof resorts to a novel approach to modularity.

Theorem C**.**

There exists an isomorphism of non-Gorenstein local $\varLambda$-algebras $\mathcal{R}^{\triangle}_{\operatorname{red}}\simeq\mathcal{T}$, where $\mathcal{R}^{\triangle}_{\operatorname{red}}$ is the nilreduction of $\mathcal{R}^{\triangle}$.

In closing, let us specify the above results in the simplest case of a quadratic character $\psi_{\mbox{-}}$ defining a biquadratic extension $H$ of $\mathbb{Q}$ in which $p$ splits completely. Denote by $K^{\prime}$ (resp. $F$) the other imaginary quadratic field (resp. the unique real quadratic field) contained in $H$. In that case we have $\mathcal{T}\simeq\varLambda\times_{\overline{\mathbb{Q}}_{p}}\varLambda\times_{\overline{\mathbb{Q}}_{p}}\varLambda\times_{\overline{\mathbb{Q}}_{p}}\varLambda$ and, in addition to the families $\Theta_{\psi}$ and $\Theta_{\psi^{\tau}}=\Theta_{\psi}\otimes\epsilon_{F}$ having CM by $K$, $f$ is contained in families $\Theta_{\psi^{\prime}}$ and $\Theta_{\psi^{\prime}{}^{\tau}}=\Theta_{\psi^{\prime}}\otimes\epsilon_{F}$ having CM by $K^{\prime}$. Both $p$-adic $L$-functions $L_{p}^{-}(\psi_{\mbox{-}},s)$ and $L_{p}^{-}(\psi_{\mbox{-}}^{\prime},s)$ then have simple zeros at $s=0$.

Contents

1. 1 Slopes and $\mathscr{L}$-invariants

2. 1.1 The slope of $\psi_{\mbox{-}}$

3. 1.2 A Galois theoretic interpretation of the slope

4. 1.3 $\mathscr{L}$-invariants

5. 1.4 A Galois theoretic interpretation of the $\mathscr{L}$-invariants

6. 1.5 An interlude on the Six Exponentials Theorem

7. 1.6 Non-vanishing of $\mathscr{L}$-invariants

8. 2 Galois deformations

9. 2.1 Ordinary deformations of $\rho$

10. 2.2 Reducible and CM deformations of $\rho$

11. 2.3 Non-CM deformations of $\rho$

12. 2.4 Tangent spaces

13. 3 Components of the eigencurve containing $f$

14. 3.1 Components with CM by K containing $f$

15. 3.2 Existence of a component without CM by K containing $f$

16. 3.3 The reducibility ideal of $\mathcal{R}^{\perp}$

17. 3.4 Components without CM by K containing $f$

18. 4 Geometry of the eigencurve and congruence ideals

19. 4.1 Iwasawa cohomology and the CM ideal

20. 4.2 The congruence ideal

21. 4.3 The Katz $p$-adic $L$-function at $s=0$

22. 5 An $R=T$ modularity lifting theorem for non-Gorenstein local rings

23. 5.1 Ordinary framed deformations of a $2$-dimensional trivial representation

24. 5.2 Generically ordinary deformations of $\rho_{f}$

25. 5.3 Modularity in the non-Gorenstein case

1. Slopes and $\mathscr{L}$-invariants

We denote by $\Gamma_{L}=\operatorname{Gal}(\bar{L}/L)$ the absolute Galois group of a perfect field $L$. We consider the field of algebraic numbers $\overline{\mathbb{Q}}$ as a subfield of $\mathbb{C}$ which determines a complex conjugation $\tau\in\Gamma_{\mathbb{Q}}$. We let $\epsilon_{L}$ denote the Dirichlet character of a quadratic extension $L/\mathbb{Q}$.

Denote by $H$ the splitting field of the anti-cyclotomic character $\psi_{\mbox{-}}=\psi/\psi^{\tau}\neq\mathbf{1}$, where for any function $h$ on $\Gamma_{K}$ we let $h^{\tau}(\cdot)=h(\tau\cdot\tau)$. Recall that $\psi$ has conductor relatively prime to $p$ and $\psi_{\mbox{-}}(\mathfrak{p})=1$, i.e., $p$ splits completely in $H$. We fix an embedding $\iota_{p}:\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{p}$ which determines an embedding $\Gamma_{\mathbb{Q}_{p}}\hookrightarrow\Gamma_{K}$ and a place $v_{0}$ of $H$ above the place $\mathfrak{p}$ of $K$. We let $\mathrm{I}_{\mathfrak{p}}$ denote the inertia subgroup of $\Gamma_{K_{\mathfrak{p}}}\simeq\Gamma_{\mathbb{Q}_{p}}$.

1.1. The slope of $\psi_{\mbox{-}}$

The dihedral group $G=\Gamma_{H/\mathbb{Q}}$ is a semi-direct product of the cyclic group $C=\Gamma_{H/K}$ with $\{1,\tau\}$. The product is direct if and only if the character $\psi_{\mbox{-}}$ is quadratic, in which case $G$ is the Klein group.

Given any global unit $u\in\mathcal{O}_{H}^{\times}$ we consider the element

[TABLE]

where $[\psi_{\mbox{-}}]$ denotes the $\psi_{\mbox{-}}$-isotypic part for the natural left action of $C$.

Minkowski’s proof of Dirichlet’s Unit Theorem shows that the left $G$-module $\mathcal{O}_{H}^{\times}\otimes\mathbb{Q}$ is isomorphic to $(\mathrm{Ind}_{\{1,\tau\}}^{G}\mathbf{1})/(\mathbb{Q}\cdot\mathbf{1})$ (see [3, (7)]), hence the $2$-dimensional odd representation $\operatorname{Ind}^{\mathbb{Q}}_{K}\psi_{\mbox{-}}$ occurs with multiplicity one in it. By the Frobenius Reciprocity Theorem $(\mathcal{O}_{H}^{\times}\otimes\overline{\mathbb{Q}})[\psi_{\mbox{-}}]$ is a line, and similarly $(\mathcal{O}_{H}^{\times}\otimes\overline{\mathbb{Q}})[\psi_{\mbox{-}}^{\tau}]$ is a line containing $\tau(u_{\psi_{\mbox{-}}})=\sum\limits_{g\in C}\tau g^{-1}(u)\otimes\psi_{\mbox{-}}(g)$. Minkowski’s Theorem guarantees the existence of $u\in\mathcal{O}_{H}^{\times}$ such that $\{g(u)|g\in C\backslash\{1\}\}$ is a basis of $\mathcal{O}_{H}^{\times}\otimes\mathbb{Q}$, called a Minkowski unit, to which one can additionally impose to be fixed by $\tau$. As $\mathcal{O}_{K}^{\times}$ is finite, the only non-trivial relation between $\{g(u)|g\in C\}$ is given by the kernel of the relative norm $\mathrm{N}_{H/K}$, hence for any Minkowski unit $u$ one has $u_{\psi_{\mbox{-}}}\neq 0$. We record those useful facts as a lemma.

Lemma 1.1**.**

There exists $u\in\mathcal{O}_{H}^{\times}$, such that

[TABLE]

The Baker–Brumer Theorem [7] demonstrates the injectivity of the $\overline{\mathbb{Q}}$-linear homomorphism

[TABLE]

allowing to define, for $u$ as in Lemma 1.1, the following $p$-adic regulator, that we call the slope

[TABLE]

Lemma 1.2**.**

The slope $\mathscr{S}(\psi_{\mbox{-}})$ does not depend on the choice of $u$ as in Lemma 1.1. Moreover $\mathscr{S}(\psi_{\mbox{-}})\notin\overline{\mathbb{Q}}$, unless $\psi_{\mbox{-}}$ is quadratic, in which case $\mathscr{S}(\psi_{\mbox{-}})=-1$. Finally $\mathscr{S}(\psi_{\mbox{-}}^{\tau})=\mathscr{S}(\psi_{\mbox{-}})^{-1}$.

Proof.

The value of $\mathscr{S}(\psi_{\mbox{-}})$ remains clearly unchanged if we replace $u$ by $g(u)$ for any $g\in C$. The independence then follows from the fact that any unit lies in the $\mathbb{Q}$-linear span of these. If $\psi_{\mbox{-}}^{\tau}\neq\psi_{\mbox{-}}$, then $\mathscr{S}(\psi_{\mbox{-}})\notin\overline{\mathbb{Q}}$ by (2). Finally, taking $u$ to be a Minkowski unit fixed by $\tau$, one finds that $\tau(u_{\psi_{\mbox{-}}})=u_{\psi_{\mbox{-}}^{\tau}}$ which, in view of the definition (3), simultaneously shows that $\mathscr{S}(\psi_{\mbox{-}}^{\tau})=\mathscr{S}(\psi_{\mbox{-}})^{-1}$ and further, when $\psi_{\mbox{-}}$ is quadratic, that $\mathscr{S}(\psi_{\mbox{-}})=-1$.∎

1.2. A Galois theoretic interpretation of the slope

Let $\eta_{\mathfrak{p}}$ be the unique element of $\ker(\operatorname{H}^{1}\left(K,\overline{\mathbb{Q}}_{p})\to\operatorname{H}^{1}(\mathrm{I}_{\bar{\mathfrak{p}}},\overline{\mathbb{Q}}_{p})\right)$ whose image in $\mathrm{im}\left(\operatorname{H}^{1}(K_{\mathfrak{p}},\overline{\mathbb{Q}}_{p})\to\operatorname{H}^{1}(\mathrm{I}_{\mathfrak{p}},\overline{\mathbb{Q}}_{p})\right)\simeq\operatorname{Hom}(\mathcal{O}_{K_{\mathfrak{p}}}^{\times},\overline{\mathbb{Q}}_{p})$ is given by $\log_{p}$. Note that $\eta_{\mathfrak{p}}$ and $\eta_{\bar{\mathfrak{p}}}=\eta_{\mathfrak{p}}^{\tau}$ form a basis of $\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})$.

Proposition 1.3**.**

The image of the natural restriction homomorphism

[TABLE]

is a line generated by $(\log_{p},\mathscr{S}(\psi_{\mbox{-}})\log_{p})$. In particular both maps $\operatorname{H}^{1}(K,\psi_{\mbox{-}})\to\operatorname{H}^{1}(\mathrm{I}_{\mathfrak{p}},\overline{\mathbb{Q}}_{p})$ and $\operatorname{H}^{1}(K,\psi_{\mbox{-}})\to\operatorname{H}^{1}(\mathrm{I}_{\bar{\mathfrak{p}}},\overline{\mathbb{Q}}_{p})$ are injective.

Proof.

The inflation-restriction exact sequence (see [3, (8)]) yields an isomorphism

[TABLE]

Using Shapiro’s Lemma the vector spaces in (4) are further isomorphic to

[TABLE]

As Leopoldt’s Conjecture holds for cyclic extensions of imaginary quadratic fields, the dimension of the latter equals $\dim(\operatorname{Ind}^{\mathbb{Q}}_{K}\psi_{\mbox{-}})^{\tau=-1}=1$.

Taking the $\psi_{\mbox{-}}^{\tau}$-isotypic part of [3, (6)] for the left $C$-action yields an exact sequence

[TABLE]

As $\operatorname{Hom}(\mathcal{O}_{H_{\mathfrak{p}}}^{\times},\overline{\mathbb{Q}}_{p})$, resp. $\operatorname{Hom}(\mathcal{O}_{H_{\bar{\mathfrak{p}}}}^{\times},\overline{\mathbb{Q}}_{p})$, is the regular representation of $C$, its $(\psi_{\mbox{-}}^{\tau})$-component is $1$-dimensional, generated by $\sum_{g\in C}\psi_{\mbox{-}}(g)\log_{p}(g^{-1}\cdot)$, resp. $\sum_{g\in C}\psi_{\mbox{-}}(g)\log_{p}(\tau g^{-1}\cdot)$. It follows that $\operatorname{H}^{1}(K,\psi_{\mbox{-}})\xrightarrow{\sim}\operatorname{H}^{1}(H,\overline{\mathbb{Q}}_{p})[\psi_{\mbox{-}}^{\tau}]$ has a basis whose restriction to $(\mathcal{O}_{H}\otimes\mathbb{Z}_{p})^{\times}$ is given by

[TABLE]

for some $s\in\overline{\mathbb{Q}}_{p}$. The triviality on global units (in particular on $u_{\psi_{\mbox{-}}}$) implies, in view of Lemma 1.2, that $s=\mathscr{S}(\psi_{\mbox{-}})$.∎

1.3. $\mathscr{L}$-invariants

As $\mathfrak{p}$ splits completely in $H$, the set of places of $H$ above $\mathfrak{p}$ consists of $g(v_{0})$ for $g\in C$. Choose $y_{0}\in\mathcal{O}_{H}[\tfrac{1}{v_{0}}]^{\times}$ having a non-zero valuation $\operatorname{ord}_{v_{0}}(y_{0})$ at $v_{0}$. Then

[TABLE]

is independent of the choice of $y_{0}$, as by Lemma 1.2 multiplying $y_{0}$ by an element of $\mathcal{O}_{H}^{\times}$ does not affect its value. Another description of $\mathscr{L}(\psi_{\mbox{-}}^{\tau})$ can be given using the exact sequence

[TABLE]

of $C$-modules, where the last map is obtained by taking valuations at all places $v$ of $H$ dividing $\mathfrak{p}$. Taking the $\psi_{\mbox{-}}$-isotypic part for the left action of $C$ yields a short exact sequence

[TABLE]

Letting $u_{\mathfrak{p},\psi_{\mbox{-}}}\in(\mathcal{O}_{H}[\tfrac{1}{\mathfrak{p}}]^{\times}\otimes\overline{\mathbb{Q}})[\psi_{\mbox{-}}]$ be any element such that $\operatorname{ord}_{v_{0}}(u_{\mathfrak{p},\psi_{\mbox{-}}})\neq 0$, for example $u_{\mathfrak{p},\psi_{\mbox{-}}}=\sum_{g\in C}g^{-1}(y_{0})\otimes\psi_{\mbox{-}}(g)$, one easily checks that

[TABLE]

Finally taking $u_{\mathfrak{p}}$ to be any non-torsion element of $\mathcal{O}_{K}[\tfrac{1}{\mathfrak{p}}]^{\times}$, for example $\mathrm{N}_{H/K}(y_{0})=\sum_{g\in C}g^{-1}(y_{0})$, one defines

[TABLE]

1.4. A Galois theoretic interpretation of the $\mathscr{L}$-invariants

Denote by $\eta$ a representative of the canonical generator $[\eta]$ of $\operatorname{H}^{1}(K,\psi_{\mbox{-}})$ described in Proposition 1.3.

Lemma 1.4**.**

One has $\eta_{\bar{\mathfrak{p}}}(\operatorname{Frob}_{\mathfrak{p}})=\eta_{\mathfrak{p}}(\operatorname{Frob}_{\bar{\mathfrak{p}}})=-\mathscr{L}_{\mathfrak{p}}$.

Proof.

As $u_{\mathfrak{p}}\bar{u}_{\mathfrak{p}}\in\mathbb{Z}_{p}^{\times}\cdot p^{\mathbb{Z}}$, using the exact sequence from Global Class Field Theory

[TABLE]

one finds $\operatorname{ord}_{\mathfrak{p}}(u_{\mathfrak{p}})\cdot\eta_{\bar{\mathfrak{p}}}(\operatorname{Frob}_{\mathfrak{p}})=\operatorname{res}_{\mathfrak{p}}(\eta_{\bar{\mathfrak{p}}})(u_{\mathfrak{p}})=-\operatorname{res}_{\bar{\mathfrak{p}}}(\eta_{\bar{\mathfrak{p}}})(u_{\mathfrak{p}})=-\log_{p}(\bar{u}_{\mathfrak{p}})=\log_{p}(u_{\mathfrak{p}})$.∎

As $[\eta]\in\operatorname{H}^{1}(K,\psi_{\mbox{-}})$ and $\eta_{\mathfrak{p}}\in\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})$ both restrict to the same element of $\operatorname{H}^{1}(\mathrm{I}_{\mathfrak{p}},\overline{\mathbb{Q}}_{p})$, their difference (considered as element of $\operatorname{H}^{1}(K_{\mathfrak{p}},\overline{\mathbb{Q}}_{p})$) is unramified at $\mathfrak{p}$, and the following proposition computes its value on an arithmetic Frobenius element.

Proposition 1.5**.**

One has $(\eta-\eta_{\mathfrak{p}})(\operatorname{Frob}_{\mathfrak{p}})=\mathscr{L}(\psi_{\mbox{-}}^{\tau})-\mathscr{L}_{\mathfrak{p}}$.

Proof.

Restricting to $\Gamma_{H}$ allows us to see $(\eta-\eta_{\mathfrak{p}})$ in $\operatorname{H}^{1}(H,\overline{\mathbb{Q}}_{p})$ and one has to compute its value at $\operatorname{Frob}_{v_{0}}$. By the exact sequence from Global Class Field Theory

[TABLE]

and by Proposition 1.3 and its proof, the restriction of $\eta$ to $(\mathcal{O}_{H}\otimes\mathbb{Z}_{p})^{\times}$ is given by

[TABLE]

Therefore, for $y_{0}\in\mathcal{O}_{H}[\tfrac{1}{v_{0}}]^{\times}$ such that $\operatorname{ord}_{v_{0}}(y_{0})\neq 0$, one has

[TABLE]

The desired formula then follows from the definition of $\mathscr{L}(\psi_{\mbox{-}}^{\tau})$ in (6), using the computation

[TABLE]

the last equality following from the fact that $p$ splits completely in $H$.∎

1.5. An interlude on the Six Exponentials Theorem

We first recall some standard results and conjectures in $p$-adic transcendence. As in the previous section $\log_{p}:\overline{\mathbb{Q}}{}_{p}^{\times}\to\overline{\mathbb{Q}}_{p}$ is the standard $p$-adic logarithm sending $p$ to [math] and we use the same notation for its pre-composition with $\iota_{p}:\overline{\mathbb{Q}}^{\times}\hookrightarrow\overline{\mathbb{Q}}_{p}^{\times}$.

Denote by $\mathcal{L}\subset\overline{\mathbb{Q}}_{p}$ the $\overline{\mathbb{Q}}$-vector space generated by $1$ and elements of $\log_{p}(\overline{\mathbb{Q}}{}^{\times})$.

Theorem 1.6** (Baker–Brumer [7]).**

If $\lambda_{1},.....,\lambda_{n}\in\mathcal{L}$ are linearly independent over $\mathbb{Q}$, then they are linearly independent over $\overline{\mathbb{Q}}$.

Conjecture 1.7** (Strong Four Exponentials Conjecture).**

A matrix $\left(\begin{smallmatrix}\lambda_{11}&\lambda_{12}\\ \lambda_{21}&\lambda_{22}\end{smallmatrix}\right)$ with entries in $\mathcal{L}$ whose lines and whose columns are linearly independent over $\mathbb{Q}$ has rank $2$.

We will need the following Strong version of the Six Exponentials Theorem, proved by Roy [36] (see also [39]).

Theorem 1.8** (Strong Six Exponentials Theorem).**

A matrix $L=\left(\begin{smallmatrix}\lambda_{11}&\lambda_{12}&\lambda_{13}\\ \lambda_{21}&\lambda_{22}&\lambda_{23}\end{smallmatrix}\right)$ with entries in $\mathcal{L}$ whose lines and whose columns are linearly independent over $\mathbb{Q}$ has rank $2$.

1.6. Non-vanishing of $\mathscr{L}$-invariants

The following $\mathscr{L}$-invariant will play a prominent role in the paper, as will be intimately related to the local geometry of the eigencurve at $f$.

Definition 1.9**.**

The anti-cyclotomic $\mathscr{L}$-invariant of $\psi_{\mbox{-}}$ is $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})=\mathscr{L}(\psi_{\mbox{-}})-2\mathscr{L}_{\mathfrak{p}}$.

Remark 1.10*.*

One can give an intrinsic definition of $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})$. Consider the ‘anti-cyclotomic’ basis $\{\operatorname{ord}_{\mathfrak{p}},\eta_{\mathfrak{p}}-\eta_{\bar{\mathfrak{p}}}\}$ of $\operatorname{Hom}(\Gamma_{K_{\mathfrak{p}}},\overline{\mathbb{Q}}_{p})$. According to Lemma 1.4 and Proposition 1.5 one has

[TABLE]

hence, analogously to [13, §1.3], the anti-cyclotomic $\mathscr{L}$-invariant $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})$ is (the opposite of) the slope of the line generated by $\eta$ is this local basis.

Proposition 1.11**.**

At least one amongst $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})$ and $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})$ is non-zero. If either $\psi_{\mbox{-}}$ is quadratic, or the Four Exponentials Conjecture 1.7 holds, then both are non-zero.

Proof.

Recall that $C=\operatorname{Gal}(H/K)$ and that $y_{0}\in\mathcal{O}_{H}[\tfrac{1}{v_{0}}]^{\times}$ is such that $\operatorname{ord}_{v_{0}}(y_{0})\neq 0$. One has

[TABLE]

Suppose first that $\psi_{\mbox{-}}$ is quadratic and denote by $g$ the non-trivial element of $C$ so that $G=\operatorname{Gal}(H/\mathbb{Q})=\{1,g,\tau,g\tau\}$. To see that $\operatorname{ord}_{v_{0}}(y_{0})\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})=2\log_{p}\left(\tfrac{g(y_{0})}{g\tau(y_{0})}\right)$ does not vanish, it suffices to show that $\iota_{p}\left(\tfrac{g(y_{0})}{g\tau(y_{0})}\right)\in\mathcal{O}_{H,v_{0}}^{\times}$ is a non-torsion element. This is true as $\tfrac{g(y_{0})}{g\tau(y_{0})}\in H^{\times}$ has a non-zero valuation at $g(v_{0})$.

Consider next the case $\psi_{\mbox{-}}\neq\psi_{\mbox{-}}^{\tau}$. A computation similar to the one above shows that

[TABLE]

It suffices to show that Roy’s Six Exponentials Theorem 1.8 applies to the matrix

[TABLE]

where $u_{\psi_{\mbox{-}}^{\tau}}=\tau(u_{\psi_{\mbox{-}}})$, $u_{\bar{\mathfrak{p}},\psi_{\mbox{-}}^{\tau}}=\tau(u_{\mathfrak{p},\psi_{\mbox{-}}})$ and $u_{\bar{\mathfrak{p}},\psi_{\mbox{-}}}=\tau(u_{\mathfrak{p},\psi_{\mbox{-}}^{\tau}})$. By looking at the first column entries, Lemma 1.2 implies that the rows are linearly independent over $\mathbb{Q}$. It remains to show that the columns are linearly independent over $\mathbb{Q}$ as well, for which we will only look at the first row entries. Using the facts that $\log_{p}(\mathrm{N}_{H/K}(u))=0$ and $\log_{p}(\mathrm{N}_{H/\mathbb{Q}}(y_{0}))=0$, a linear relation over $\mathbb{Q}$ between those $3$ entries would give integers $a$, $b$ and $c$, not all zero, such that

[TABLE]

The numbers $\{\tau(y_{0})\}\cup\{g(u),g(y_{0}),g\tau(y_{0})|1\neq g\in C\}$ are (multiplicatively) linearly independent over $\mathbb{Q}$ and are all sent by $\iota_{p}$ to $\mathcal{O}_{H,v_{0}}^{\times}\simeq\mathbb{Z}_{p}^{\times}$, hence their $p$-adic logarithms are linearly independent over $\mathbb{Q}$. This contradicts Baker–Brumer’s Theorem 1.6.

The last claim follows by applying the Four Exponentials Conjecture 1.7 to the minors obtained by removing the second (or the third) column of the above matrix.∎

2. Galois deformations

Recall from §1.2 the unique cohomology class $[\eta]\in\operatorname{H}^{1}(K,\psi_{\mbox{-}})$ whose restriction to $\mathrm{I}_{\mathfrak{p}}$ is given by the $p$-adic logarithm. As $\psi_{\mbox{-}}-\mathbf{1}$ is a basis of the coboundaries, fixing, once and for all, an element $\gamma_{0}\in\Gamma_{K}$ such that $\psi_{\mbox{-}}(\gamma_{0})\neq 1$, we let $\eta:\Gamma_{K}\to\overline{\mathbb{Q}}_{p}$ be the unique representative of $[\eta]$ such that $\eta(\gamma_{0})=0$.

We let $\rho=\left(\begin{smallmatrix}\psi&\eta\psi^{\tau}\\ 0&\psi^{\tau}\end{smallmatrix}\right):\Gamma_{K}\to\operatorname{GL}_{2}(\overline{\mathbb{Q}}_{p})$ in the canonical basis $(e_{1},e_{2})$ of $\overline{\mathbb{Q}}{}_{p}^{2}$. The unique $\Gamma_{K_{\mathfrak{p}}}$-stable filtration of $V_{\rho}=\overline{\mathbb{Q}}_{p}^{2}$ with unramified quotient is given by

[TABLE]

2.1. Ordinary deformations of $\rho$

Let $\mathscr{C}$ be the category of complete Noetherian local $\overline{\mathbb{Q}}_{p}$-algebras $A$ with maximal ideal $\mathfrak{m}_{A}$ and residue field $\overline{\mathbb{Q}}{}_{p}$, where the morphisms are local homomorphisms of $\overline{\mathbb{Q}}_{p}$-algebras.

Consider the functor $\mathcal{D}^{\operatorname{univ}}_{\rho}$ associating to $A$ in $\mathscr{C}$ the set of lifts $\rho_{A}:\Gamma_{K}\to\operatorname{GL}_{2}(A)$ of $\rho$ modulo strict equivalence and satisfying

[TABLE]

As $\psi_{\mbox{-}}$ and $[\eta]$ are both non-trivial, the centralizer of the image of $\rho$ consists only of scalar matrices, hence $\mathcal{D}^{\operatorname{univ}}_{\rho}$ is representable by a universal deformation ring $\mathcal{R}^{\operatorname{univ}}_{\rho}$ (see [33]).

Let $\mathcal{D}_{\operatorname{Fil}}$ be the functor assigning to $A$ in $\mathscr{C}$ the set of free $A$-submodules $F_{A}\subset A^{2}$ such that $F_{A}\otimes_{A}\overline{\mathbb{Q}}_{p}=F_{\rho}$ (such $F_{A}$ is necessarily a direct rank $1$ summand of $A^{2}$). It is representable by the strict completed local ring $\mathcal{R}_{\operatorname{Fil}}\simeq\overline{\mathbb{Q}}_{p}\lsem U\rsem$ of the projective space $\mathbb{P}^{1}$ at the point corresponding to the line $F_{\rho}\subset\overline{\mathbb{Q}}{}_{p}^{2}$. The universal submodule $F_{\mathcal{R}_{\operatorname{Fil}}}\subset\mathcal{R}_{\operatorname{Fil}}^{2}$ has basis given by $e_{1}+Ue_{2}$.

Definition 2.1**.**

The functor $\mathcal{D}^{\operatorname{ord}}_{\rho}$ assigns to $A\in\mathscr{C}$ the set of tuples $(\rho_{A},F_{A})$ such that

1. (i)

$\rho_{A}:\Gamma_{K}\to\operatorname{GL}_{2}(A)$* is a continuous representation such that $\rho_{A}\mod\mathfrak{m}_{A}=\rho$, and $\operatorname{tr}\rho_{A}(\tau g\tau)=\operatorname{tr}\rho_{A}(g)$ for all $g\in\Gamma_{K}$, and* 2. (ii)

$F_{A}\subset A^{2}$* is a free direct factor over $A$ of rank $1$ which is $\Gamma_{K_{\mathfrak{p}}}$-stable and such that $\Gamma_{K_{\mathfrak{p}}}$ acts on $A^{2}/F_{A}$ by an unramified character, denoted $\chi_{A}$,*

modulo the strict equivalence relation $[(\rho_{A},F_{A})]=[(P\rho_{A}P^{-1},P\cdot F_{A})]$ with $P\in 1+M_{2}(\mathfrak{m}_{A})$.

One has $F_{A}\otimes_{A}\overline{\mathbb{Q}}_{p}=F_{\rho}$ as the latter is the unique $\Gamma_{K_{\mathfrak{p}}}$-stable line in $\overline{\mathbb{Q}}_{p}^{2}$ with unramified quotient. As in [6, §1.2], given a representative of the universal deformation of $\rho$ to $\mathcal{R}^{\operatorname{univ}}_{\rho}$, there exists a monic natural transformation $\mathcal{D}^{\operatorname{ord}}_{\rho}\hookrightarrow\mathcal{D}^{\operatorname{univ}}_{\rho}\times\mathcal{D}_{\operatorname{Fil}}$. Moreover, it has been shown in loc. cit. that the composition $\mathcal{D}^{\operatorname{ord}}_{\rho}\hookrightarrow\mathcal{D}^{\operatorname{univ}}_{\rho}\times\mathcal{D}_{\operatorname{Fil}}\to\mathcal{D}^{\operatorname{univ}}_{\rho}$ is again monic and is independent of the above choice of representative. Therefore $\mathcal{D}^{\operatorname{ord}}_{\rho}$ is a sub-functor of $\mathcal{D}^{\operatorname{univ}}_{\rho}$ defined by a closed condition guaranteeing its representability. In particular, $(\rho_{A},F_{A})\in\mathcal{D}^{\operatorname{ord}}_{\rho}(A)$ is characterised uniquely by $\rho_{A}$ alone, i.e. when the ordinary filtration exists, then it is unique. For this reason, and because $\chi_{A}$ plays a major role in this paper, we will denote the point $(\rho_{A},F_{A})$ of $\mathcal{D}^{\operatorname{ord}}_{\rho}(A)$ by $(\rho_{A},\chi_{A})$.

Lemma 2.2**.**

1. (i)

The functor $\mathcal{D}^{\operatorname{ord}}_{\rho}$ is representable by a quotient $\mathcal{R}^{\operatorname{ord}}_{\rho}$ of $\mathcal{R}^{\operatorname{univ}}_{\rho}$ and a universal ordinary deformation $\rho_{\mathcal{R}}:\Gamma_{K}\to\operatorname{GL}_{2}(\mathcal{R}^{\operatorname{ord}}_{\rho})$. 2. (ii)

The determinant of $\rho_{\mathcal{R}}$ extends in two different ways to a character $\Gamma_{\mathbb{Q}}\to(\mathcal{R}^{\operatorname{ord}}_{\rho})^{\times}$.

Proof.

(i) The argument is exactly the same as in [6, §1.2].

(ii) As $2\det(\rho_{\mathcal{R}}(g))=\operatorname{tr}(\rho_{\mathcal{R}}(g))^{2}-\operatorname{tr}(\rho_{\mathcal{R}}(g^{2}))$, we deduce that $\det(\rho_{\mathcal{R}})$ is a $\tau$-invariant character of $\Gamma_{K}$, hence can be extended to a character of $\Gamma_{\mathbb{Q}}$ by sending the order $2$ element $\tau$ either to $1$ or to $-1$. Alternatively, one can observe that the character $\det(\rho_{\mathcal{R}})\det(\rho)^{-1}:\Gamma_{K}\to 1+\mathfrak{m}_{\mathcal{R}^{\operatorname{ord}}_{\rho}}$ extends uniquely, because $\mathcal{R}^{\operatorname{ord}}_{\rho}$ is complete and $K$ has a unique $\mathbb{Z}_{p}$-extension on which $\tau$ acts trivially (this argument is more general as it does not use the existence of a section of $\Gamma_{\mathbb{Q}}\to\Gamma_{K/\mathbb{Q}}$ given by the complex conjugation $\tau$).∎

As the two extensions in Lemma 2.2(ii) differ by the quadratic character $\epsilon_{K}$ of $\Gamma_{K/\mathbb{Q}}$, exactly one of them is odd, and we denote it $\det(\rho_{\mathcal{R}}):\Gamma_{\mathbb{Q}}\to\mathcal{R}^{\operatorname{ord}}_{\rho}$. It reduces modulo the maximal ideal to $\det(\operatorname{Ind}^{\mathbb{Q}}_{K}\psi)=(\psi\circ\mathrm{Ver})\cdot\epsilon_{K}:\Gamma_{\mathbb{Q}}\to\overline{\mathbb{Q}}{}_{p}^{\times}$, where $\mathrm{Ver}:\Gamma_{\mathbb{Q}}^{\mathrm{ab}}\to\Gamma_{K}^{\mathrm{ab}}$ denotes the transfer homomorphism. As $\varLambda\in\mathscr{C}$ is the universal deformation ring of that character (see [3, §6]) the natural transformation $\rho_{A}\mapsto\det(\rho_{A})$, endows $\mathcal{R}^{\operatorname{ord}}_{\rho}$ with a $\varLambda$-algebra structure.

2.2. Reducible and CM deformations of $\rho$

Definition 2.3**.**

1. (i)

Let $\mathcal{D}^{\operatorname{red}}_{\rho}$ be the subfunctor of $\mathcal{D}^{\operatorname{ord}}_{\rho}$ consisting of $\Gamma_{K}$-reducible deformations. 2. (ii)

Let $\mathcal{D}^{K}_{\rho}$ be the subfunctor of $\mathcal{D}^{\operatorname{ord}}_{\rho}$ consisting of deformations of $\rho$ having a free $\Gamma_{K}$-quotient of rank $1$ which is unramified at $\mathfrak{p}$.

Lemma 2.4**.**

1. (i)

The functor $\mathcal{D}^{\operatorname{red}}_{\rho}$ is representable by a quotient $\mathcal{R}^{\operatorname{red}}_{\rho}$ of $\mathcal{R}^{\operatorname{ord}}_{\rho}$. 2. (ii)

The functor $\mathcal{D}^{K}_{\rho}$ is representable by a quotient $\mathcal{R}^{K}_{\rho}=\mathcal{R}^{\operatorname{ord}}_{\rho}/(c(g);g\in\Gamma_{K})$, where $\rho_{\mathcal{R}}(g)=\left(\begin{smallmatrix}a(g)&b(g)\\ c(g)&d(g)\end{smallmatrix}\right)$ in an ordinary basis.

Proof.

(i) As $\rho(\gamma_{0})$ is a diagonal matrix with distinct eigenvalues, Hensel’s lemma implies that $\rho_{\mathcal{R}}(\gamma_{0})$ can be strictly conjugated to a diagonal matrix as well. The assertion then follows from exactly the same arguments as those in the proof of [6, Lem. 1.3].

(ii) The statement is clear as the $\Gamma_{K}$-filtration necessarily coincides with the ordinary filtration. ∎

Finally, for $?\in\{\operatorname{univ},\operatorname{ord},\operatorname{red},K\}$ we let $\mathcal{D}^{?}_{\rho,0}$ denote the sub-functor of $\mathcal{D}^{?}_{\rho}$ classifying deformations with fixed determinant, represented by $\mathcal{R}^{?}_{\rho,0}=\mathcal{R}^{?}_{\rho}/\mathfrak{m}_{\varLambda}\mathcal{R}^{?}_{\rho}$ (see §2.1).

Remark 2.5*.*

While $\rho$ is reducible and ordinary for the same filtration (8), this is not necessarily true for deformations in $\mathcal{D}^{\operatorname{red}}_{\rho}$. In fact, we show in §2.4 that

[TABLE]

2.3. Non-CM deformations of $\rho$

We recall the indecomposable representation $\rho=\left(\begin{smallmatrix}\psi&\eta\psi^{\tau}\\ 0&\psi^{\tau}\end{smallmatrix}\right)$ from §2.1 and its universal ordinary deformation $\rho_{\mathcal{R}}$.

We can do a similar construction with $\psi_{\mbox{-}}^{\tau}$ instead of $\psi_{\mbox{-}}$. Namely, we let $\eta^{\prime}:\Gamma_{K}\to\overline{\mathbb{Q}}_{p}$ be the unique representative of $[\eta^{\prime}]\in\operatorname{H}^{1}(K,\psi_{\mbox{-}}^{\tau})$ whose restriction to $\mathrm{I}_{\mathfrak{p}}$ is given by the $p$-adic logarithm and such that $\eta^{\prime}(\gamma_{0})=0$. Then we consider the indecomposable representation

[TABLE]

and denote by $\mathcal{D}^{\operatorname{ord}}_{\rho^{\prime}}$ its ordinary deformation functor, representable by a universal deformation ring $\mathcal{R}^{\operatorname{ord}}_{\rho^{\prime}}$. By definition the universal deformation $\rho^{\prime}_{\mathcal{R}}$ has a $\operatorname{Gal}(K/\mathbb{Q})$-invariant trace and admits a rank $1$ unramified $\Gamma_{K_{\mathfrak{p}}}$-quotient, denoted $\chi^{\prime}_{\mathcal{R}}$.

We highlight that as $[\eta^{\tau}]$ and $[\eta^{\prime}]$ are proportional (in fact $\eta^{\tau}=\mathscr{S}(\psi_{\mbox{-}})\eta^{\prime}+\eta^{\tau}(\gamma_{0})\tfrac{\psi_{\mbox{-}}-1}{\psi_{\mbox{-}}(\gamma_{0})-1}$), it follows that $\rho^{\prime}\simeq\rho^{\tau}$. However $\rho^{\prime}_{\mathcal{R}}$ need not be isomorphic to $\rho_{\mathcal{R}}^{\tau}$ as the former admits a $\Gamma_{K_{\mathfrak{p}}}$-unramified quotient, while the latter admits a $\Gamma_{K_{\bar{\mathfrak{p}}}}$-unramified quotient, and not necessarily vice versa.

The following definition (similar to [6, Def.1.6]) is an attempt to describe Galois theoretically the local ring at $f$ of the closed analytic subspace $\mathscr{E}^{\perp}$ of $\mathscr{E}^{\operatorname{ord}}$ defined by the union of irreducible components without CM by $K$. As we will later show in Theorem 3.3 one indeed has $f\in\mathscr{E}^{\perp}$.

Definition 2.6**.**

Let $\mathcal{D}^{\perp}$ be the functor assigning to $A\in\mathscr{C}$ the set of strict equivalence classes of pairs $((\rho_{A},\chi_{A}),(\rho^{\prime}_{A},\chi^{\prime}_{A}))$ in $\mathcal{D}^{\operatorname{ord}}_{\rho}(A)\times\mathcal{D}^{\operatorname{ord}}_{\rho^{\prime}}(A)$ such that $\operatorname{tr}(\rho_{A})=\operatorname{tr}(\rho^{\prime}_{A})$ and $\chi_{A}(\operatorname{Frob}_{\mathfrak{p}})=\chi^{\prime}_{A}(\operatorname{Frob}_{\mathfrak{p}})$.

The definition of $\mathcal{D}^{\perp}$ is meant to exclude $\varLambda$-adic deformations of $\rho$ having CM by $K$. Note that the $\tau$-conjugate of any $\mathfrak{p}$-ordinary deformation of $\rho$ is a $\bar{\mathfrak{p}}$-ordinary deformation of $\rho^{\prime}$ which is not necessarily $\mathfrak{p}$-ordinary, so it does not in general define a point of our functor.

The functor $\mathcal{D}^{\perp}$ is representable by a $\varLambda$-algebra $\mathcal{R}^{\perp}$, quotient of $\mathcal{R}^{\operatorname{ord}}_{\rho}\widehat{\otimes}\mathcal{R}^{\operatorname{ord}}_{\rho^{\prime}}$ by the ideal

[TABLE]

Let $\rho_{\mathcal{R}^{\perp}}$ (resp. $\rho^{\prime}_{\mathcal{R}^{\perp}}$) be the $\mathfrak{p}$-ordinary deformation of $\rho$ (resp. $\rho^{\prime}$) obtained by functoriality from the natural homomorphism $\mathcal{R}_{\rho}^{\operatorname{ord}}\rightarrow\mathcal{R}^{\perp}$ (resp. $\mathcal{R}_{\rho^{\prime}}^{\operatorname{ord}}\rightarrow\mathcal{R}^{\perp}$).

Lemma 2.7**.**

The natural homomorphisms $\mathcal{R}^{\operatorname{ord}}_{\rho}\to\mathcal{R}^{\perp}$ and $\mathcal{R}^{\operatorname{ord}}_{\rho^{\prime}}\to\mathcal{R}^{\perp}$ are surjective. Moreover $\rho^{\prime}_{\mathcal{R}^{\perp}}$ and $\rho_{\mathcal{R}^{\perp}}^{\tau}$ are conjugated by an element of $\operatorname{GL}_{2}(\mathcal{R}^{\perp})$, in particular $\rho_{\mathcal{R}^{\perp}}^{\tau}$ is $\mathfrak{p}$-ordinary.

Proof.

The first claim follows from similar arguments that have already been used in [6, Lem. 1.7]. In fact, as $\dim\operatorname{H}^{1}(K,\psi_{\mbox{-}})=\dim\operatorname{H}^{1}(K,\psi_{\mbox{-}}^{\tau})=1$ by Proposition 1.3, one can apply [30, Cor. 1.1.4(ii)] to prove that the algebras $\mathcal{R}^{\operatorname{univ}}_{\rho}$ and $\mathcal{R}^{\operatorname{univ}}_{\rho^{\prime}}$ (and hence any of their quotients) are generated over $\bar{\mathbb{Q}}_{p}$ by the traces of their universal deformations.

As highlighted before, the representations $\left(\begin{smallmatrix}\mathscr{S}(\psi_{\mbox{-}})&\tfrac{\eta^{\tau}(\gamma_{0})}{1-\psi_{\mbox{-}}(\gamma_{0})}\\ 0&1\end{smallmatrix}\right)^{-1}\rho_{\mathcal{R}^{\perp}}^{\tau}\left(\begin{smallmatrix}\mathscr{S}(\psi_{\mbox{-}})&\tfrac{\eta^{\tau}(\gamma_{0})}{1-\psi_{\mbox{-}}(\gamma_{0})}\\ 0&1\end{smallmatrix}\right)$ and $\rho^{\prime}_{\mathcal{R}^{\perp}}$ both reduce to $\rho^{\prime}$ and share the same trace, hence by loc. cit. they correspond to the same point of $\mathcal{D}^{\operatorname{univ}}_{\rho^{\prime}}(\mathcal{R}^{\perp})$ and thus they are conjugated by an element of $1+M_{2}(\mathfrak{m}_{\mathcal{R}^{\perp}})$.∎

The local ring $\mathcal{R}^{\perp}_{0}=\mathcal{R}^{\perp}/\mathfrak{m}_{\varLambda}\mathcal{R}^{\perp}$ represents the subfunctor $\mathcal{D}^{\perp}_{0}$ of $\mathcal{D}^{\perp}$ consisting of deformations with fixed determinant.

2.4. Tangent spaces

Using results from Galois Cohomology, Class Field Theory and Transcendence Theory we will now describe the tangent spaces of the functors introduced in the previous subsections and compute their dimensions.

Let $\overline{\mathbb{Q}}_{p}[\epsilon]=\overline{\mathbb{Q}}_{p}\lsem X\rsem/(X^{2})$ be the $\overline{\mathbb{Q}}_{p}$-algebra of dual numbers. There is a natural injection

[TABLE]

inducing an injection $t^{\operatorname{univ}}_{\rho,0}=\mathcal{D}^{\operatorname{univ}}_{\rho,0}(\overline{\mathbb{Q}}_{p}[\epsilon])\hookrightarrow\operatorname{H}^{1}(K,\operatorname{ad}^{0}(\rho))$, where $\operatorname{ad}(\rho)$, resp. $\operatorname{ad}^{0}(\rho)$, is the adjoint representation on $\operatorname{End}_{\overline{\mathbb{Q}}_{p}}(V_{\rho})$, resp. on its trace zero elements $\operatorname{End}^{0}_{\overline{\mathbb{Q}}_{p}}(V_{\rho})$. We have a natural decomposition of $\Gamma_{K}$-modules $\operatorname{ad}\rho\simeq\operatorname{ad}\rho^{0}\oplus\mathbf{1}$. For $?\in\{\operatorname{ord},\operatorname{red},K\}$, the tangent space $t^{?}_{\rho}=\mathcal{D}^{?}_{\rho}(\overline{\mathbb{Q}}_{p}[\epsilon])$, resp. $t^{?}_{\rho,0}=\mathcal{D}^{?}_{\rho,0}(\overline{\mathbb{Q}}_{p}[\epsilon])$, is thus naturally a subspace of $\operatorname{H}^{1}(K,\operatorname{ad}(\rho))$, resp. $\operatorname{H}^{1}(K,\operatorname{ad}^{0}(\rho))$. Let

[TABLE]

be the short exact sequence of $\overline{\mathbb{Q}}_{p}[\Gamma_{K}]$-modules arising from the fact that $\rho$ is reducible and let $W_{\rho}^{0}=W_{\rho}\cap\operatorname{End}^{0}_{\overline{\mathbb{Q}}_{p}}(V_{\rho})$. As $\rho=\left(\begin{smallmatrix}\psi&\eta\psi^{\tau}\\ 0&\psi^{\tau}\end{smallmatrix}\right)$, the subspace $W_{\rho}$ of $\operatorname{End}_{\overline{\mathbb{Q}}_{p}}(V_{\rho})=\mathrm{M}_{2}(\overline{\mathbb{Q}}_{p})$ consists of upper triangular matrices $\left(\begin{smallmatrix}a&b\\ 0&d\end{smallmatrix}\right)$. Consider the short exact sequences of $\overline{\mathbb{Q}}_{p}[\Gamma_{K}]$-modules

[TABLE]

Lemma 2.8**.**

Let $a,d\in\operatorname{Hom}(\Gamma_{K},\overline{\mathbb{Q}}_{p})$ and $s\in\{\pm 1\}$. If $\psi a+s\psi^{\tau}d=s\psi^{\tau}a^{\tau}+\psi d^{\tau}$ as functions on $\Gamma_{K}$, then $d=a^{\tau}$.

Proof.

The relation can be rewritten as $\psi_{\mbox{-}}(a-d^{\tau})=s(a^{\tau}-d)$. As $\psi_{\mbox{-}}$ is unramified at $p$, it follows that the elements $(a-d^{\tau})$ and $s(a^{\tau}-d)$ of $\operatorname{Hom}(\Gamma_{K},\overline{\mathbb{Q}}_{p})$ coincide on both inertia subgroups $\mathrm{I}_{\mathfrak{p}}$ and $\mathrm{I}_{\bar{\mathfrak{p}}}$, hence they are equal. However $\psi_{\mbox{-}}\neq\mathbf{1}$, hence $a-d^{\tau}=0$. ∎

Definition 2.9**.**

Let $\operatorname{Hom}(\Gamma_{K},\overline{\mathbb{Q}}{}_{p}^{2})^{\tau}$ be the subspace of $\operatorname{Hom}(\Gamma_{K},\overline{\mathbb{Q}}{}_{p}^{2})$ consisting of elements $(a,d)$ such that $d=a^{\tau}$, and let $\operatorname{H}^{1}(K,W_{\rho})^{\tau}$ be the inverse image of $\operatorname{Hom}(\Gamma_{K},\overline{\mathbb{Q}}{}_{p}^{2})^{\tau}$ under the morphism $(a_{*},d_{*}):\operatorname{H}^{1}(K,W_{\rho})\to\operatorname{Hom}(\Gamma_{K},\overline{\mathbb{Q}}{}_{p}^{2})$ coming by functoriality from (11).

Put $\operatorname{H}^{1}(K,W_{\rho}^{0})^{\tau}=\operatorname{H}^{1}(K,W_{\rho})^{\tau}\cap\operatorname{H}^{1}(K,W_{\rho}^{0})$.

Proposition 2.10**.**

We have $t^{\operatorname{red}}_{\rho}=t^{\operatorname{ord}}_{\rho}\simeq\operatorname{H}^{1}(K,{W_{\rho}})^{\tau}$ and $t^{\operatorname{red}}_{\rho,0}=t^{\operatorname{ord}}_{\rho,0}\simeq\operatorname{H}^{1}(K,{W^{0}_{\rho}})^{\tau}$.

Proof.

By Definition 2.3 we have $t^{\operatorname{red}}_{\rho}\subset t^{\operatorname{ord}}_{\rho}$ . To prove the opposite inclusion, we represent an infinitesimal ordinary deformation $[\rho_{\epsilon}]\in t^{\operatorname{ord}}_{\rho}$ by a cocycle

[TABLE]

As $\operatorname{H}^{0}(K,\psi_{\mbox{-}}^{\tau})=\{0\}$, we obtain from (10) the following long exact sequence

[TABLE]

where the map $\operatorname{ad}\rho\to\psi_{\mbox{-}}^{\tau}$ is given by $\left[\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\right]\mapsto[c]$. By ordinarity (see [6, Proof of Prop. 2.1]), the image of $t^{\operatorname{ord}}_{\rho}\subset\operatorname{H}^{1}(K,\operatorname{ad}\rho)$ under the morphism $c_{*}$ lands in

[TABLE]

which is trivial by Proposition 1.3. Thus $t^{\operatorname{ord}}_{\rho}=t^{\operatorname{red}}_{\rho}\subset\operatorname{H}^{1}(K,{W_{\rho}})$.

By (13) any element of $\operatorname{H}^{1}(K,{W_{\rho}})$ can be represented by an infinitesimal reducible lift

[TABLE]

If the lift comes from an element of $t^{\operatorname{ord}}_{\rho}$, because of the relation $\operatorname{tr}\rho_{\epsilon}=\operatorname{tr}\rho^{\tau}_{\epsilon}$ (see (9)), Lemma 2.8 implies that $a=d^{\tau}$, hence $t^{\operatorname{ord}}_{\rho}\subset\operatorname{H}^{1}(K,{W_{\rho}})^{\tau}$ (see Definition 2.9). The same argument also shows that $t^{\operatorname{red}}_{\rho,0}=t^{\operatorname{ord}}_{\rho,0}\subset\operatorname{H}^{1}(K,{W^{0}_{\rho}})^{\tau}$.

It remains to show the inclusion $\operatorname{H}^{1}(K,{W_{\rho}})^{\tau}\subset t^{\operatorname{ord}}_{\rho}$. As observed above, any element of $\operatorname{H}^{1}(K,{W_{\rho}})^{\tau}$ can be represented by an infinitesimal reducible lift

[TABLE]

where $d\in\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})$. It is sufficient to show that $\rho_{\epsilon}$ is $\mathfrak{p}$-ordinary, i.e., find $\alpha\in\overline{\mathbb{Q}}_{p}$ such that the line of $\overline{\mathbb{Q}}_{p}[\epsilon]^{2}$ generated by $e_{1}+\epsilon\alpha e_{2}$ is $\Gamma_{K_{\mathfrak{p}}}$-stable with unramified quotient. A direct computation shows that the restriction of $\rho_{\epsilon}$ to $\Gamma_{K_{\mathfrak{p}}}$ in the ordinary basis $(e_{1}+\epsilon\alpha e_{2},e_{2})$ equals

[TABLE]

As $\eta_{|\mathrm{I}_{\mathfrak{p}}}=\log_{\mathfrak{p}}$ generates the image of the restriction map $\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})\to\operatorname{H}^{1}(\mathrm{I}_{\mathfrak{p}},\overline{\mathbb{Q}}_{p})$, there exists a unique $\alpha\in\overline{\mathbb{Q}}_{p}$ such that $d_{|\mathrm{I}_{\mathfrak{p}}}=\alpha\eta_{|\mathrm{I}_{\mathfrak{p}}}$, for which the character $\chi_{\epsilon}=\psi(1+\epsilon(d-\alpha\eta))$ will be unramified at $\mathfrak{p}$. This completes the proof.∎

Let $\operatorname{H}^{1}(K,W^{0}_{\rho})\xrightarrow{d_{*}}\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})$ be morphism induced by (11) sending $\left[\left(\begin{smallmatrix}-d&b\\ 0&d\end{smallmatrix}\right)\right]$ to $[d]$.

Lemma 2.11**.**

1. (i)

One has $\dim_{\overline{\mathbb{Q}}_{p}}\operatorname{H}^{2}(K,\psi_{\mbox{-}})=0$. 2. (ii)

The map $\operatorname{H}^{1}(K,W^{0}_{\rho})\xrightarrow{d_{*}}\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})$ is an isomorphism.

Proof.

(i) By the global Euler characteristic formula the dimension of $\operatorname{H}^{2}(K,\psi_{\mbox{-}})$ equals

[TABLE]

(ii) As $\operatorname{H}^{2}(K,\psi_{\mbox{-}})=\{0\}$ and $\dim\operatorname{H}^{1}(K,\psi_{\mbox{-}})=1$, the map $d_{*}$ in the long exact sequence

[TABLE]

coming from (11) is necessarily an isomorphism.∎

Proposition 2.12**.**

One has $\dim t^{\operatorname{ord}}_{\rho}=2$, $\dim t^{\operatorname{ord}}_{\rho,0}=\dim t^{K}_{\rho}=1$ and $t^{K}_{\rho,0}=\{0\}$.

Proof.

Using the direct sum decomposition $W_{\rho}=W_{\rho}^{0}\oplus\overline{\mathbb{Q}}_{p}$, Lemma 2.11(ii) implies that

[TABLE]

is an isomorphism, from which one deduces the isomorphism

[TABLE]

In view of Proposition 2.10, we deduce $\dim t^{\operatorname{red}}_{\rho}=\dim t^{\operatorname{ord}}_{\rho}=\dim\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})=2$. Moreover, as $d_{*}$ induces an isomorphism between $\operatorname{H}^{1}(K,W_{\rho}^{0})^{\tau}$ and the anti-cyclotomic line $\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})^{\tau=-1}$, one deduces that $\dim t^{\operatorname{red}}_{\rho,0}=\dim t^{\operatorname{ord}}_{\rho,0}=1$.

By definition $t^{K}_{\rho}=\ker(t^{\operatorname{red}}_{\rho}\simeq\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})\xrightarrow{\operatorname{res}_{\mathfrak{p}}}\operatorname{Hom}(\mathrm{I}_{\mathfrak{p}},\overline{\mathbb{Q}}_{p}))$, hence $\dim t^{K}_{\rho}=1$. Finally

[TABLE]

∎

Let $t^{\perp}=\mathcal{D}^{\perp}(\overline{\mathbb{Q}}_{p}[\epsilon])$, resp. $t^{\perp}_{0}=\mathcal{D}^{\perp}_{0}(\overline{\mathbb{Q}}_{p}[\epsilon])$, be the tangent, resp. relative tangent, space of the functor $\mathcal{D}^{\perp}$. Recall the $\mathscr{L}$-invariants introduced in Definition 1.9

[TABLE]

Theorem 2.13**.**

1. We have $\dim t^{\perp}=1$. Moreover, $t^{\perp}_{0}=\{0\}$ if and only if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})+\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})\neq 0$. In particular, $t^{\perp}_{0}=\{0\}$ if $\psi_{\mbox{-}}$ is quadratic.

Proof.

By definition

[TABLE]

By (15), $d_{*}$ identifies $t^{\operatorname{ord}}_{\rho}$ with $\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})$, hence $[\rho_{\epsilon}]$ is uniquely determined by an element $d\in\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})$ as follows (see (14))

[TABLE]

Write $d=\alpha\eta_{\mathfrak{p}}+\beta\eta_{\bar{\mathfrak{p}}}$ with $\alpha,\beta\in\overline{\mathbb{Q}}_{p}$ in the basis $\{\eta_{\mathfrak{p}},\eta_{\bar{\mathfrak{p}}}\}$ of $\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})$ (see §1.2). Note that

[TABLE]

Analogously, an element of $t^{\operatorname{ord}}_{\rho^{\prime}}$ can be represented by an infinitesimal reducible lift

[TABLE]

uniquely determined by $d^{\prime}\in\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})$. As $\operatorname{tr}\rho_{\epsilon}^{\prime}=\operatorname{tr}\rho_{\epsilon}$, Lemma 2.8 implies that $d^{\prime}=d^{\tau}$.

By the end proof of Proposition 2.10, one has $\chi_{\epsilon}=\psi(1+\epsilon(d-\alpha\eta))$, hence

[TABLE]

Analogously, from $d^{\prime}=d^{\tau}=\beta\eta_{\mathfrak{p}}+\alpha\eta_{\bar{\mathfrak{p}}}$, one deduces $\chi^{\prime}_{\epsilon}=\psi(1+\epsilon(d^{\tau}-\beta\eta^{\prime}))$ hence

[TABLE]

Comparing with (19) shows that

[TABLE]

Using Lemma 1.4 and Proposition 1.5 the tangent space can be rewritten in terms of $\mathscr{L}$-invariants:

[TABLE]

By Proposition 1.11, $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})$ and $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})$ cannot be both zero. Hence $\dim t^{\perp}=1$.

(ii) It follows from (18) that the tangent space $t^{\perp}_{0}$ is parametrized by $(\alpha,\beta)\in t^{\perp}$ such that $\alpha+\beta=0$. Hence $t^{\perp}_{0}=\{0\}$ if and only if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})+\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})\neq 0$.∎

Taking $(\alpha,\beta)$ from the above computations as coordinates of the plane $t^{\operatorname{ord}}_{\rho}$, the lines $t^{K}_{\rho}$ , $t^{\perp}$ and $t^{\operatorname{ord}}_{\rho,0}$ can be drawn as follows

$\beta$-line$\alpha$-linet^{\perp}:\alpha\mathscr{L}_{-}(\psi_{\mbox{-}}^{\tau})=\beta\mathscr{L}_{-}(\psi_{\mbox{-}})$$t_{\rho}^{K}:\alpha=0 (CM line)$t^{\operatorname{ord}}_{\rho,0}:\alpha+\beta=0$

Corollary 2.14**.**

We have $t^{\operatorname{ord}}_{\rho}=t^{\perp}\oplus t^{K}_{\rho}$ if and only if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\neq 0$.

Proof.

As in the proof of Theorem 2.13 one can use $(\alpha,\beta)\in\overline{\mathbb{Q}}{}_{p}^{2}$ as coordinates on $t^{\operatorname{ord}}_{\rho}$. By (20) the equation defining $t^{\perp}$ is $\alpha\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})=\beta\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})$, while by (16) the equation defining $t^{K}_{\rho}$ is $\alpha=0$.∎

3. Components of the eigencurve containing $f$

The weight $1$ theta series $\theta_{\psi}$ has level $N=D\cdot\mathrm{N_{K/\mathbb{Q}}}(\mathfrak{c}_{\psi})$, where $-D$ is the fundamental discriminant of $K$ and $\mathfrak{c}_{\psi}$ is the conductor of $\psi$.

The $p$-adic cuspidal eigencurve $\mathscr{E}$ of tame level $N$ is endowed with a flat and locally finite morphism $\kappa:\mathscr{E}\rightarrow\mathcal{W}$ of reduced rigid analytic spaces over $\mathbb{Q}_{p}$, where the weight space $\mathcal{W}$ represents continuous homomorphisms from $\mathbb{Z}_{p}^{\times}\times(\mathbb{Z}/N\mathbb{Z})^{\times}$ to $\mathbb{G}_{m}$. Locally $\mathcal{O}_{\mathscr{E}}$ is generated as $\mathcal{O}_{\mathcal{W}}$-algebra by the Hecke operators $U_{p}$ and $T_{\ell}$ for $\ell\nmid Np$. There exists a universal $2$-dimensional pseudo-character $\tau_{\mathscr{E}}:\Gamma_{\mathbb{Q}}\rightarrow\mathcal{O}(\mathscr{E})$ unramified at $\ell\nmid Np$ and sending $\operatorname{Frob}_{\ell}$ on $T_{\ell}$ (see [3, (15)]). It is defined by interpolation, so that its specialization at any classical point of $\mathscr{E}(\overline{\mathbb{Q}}_{p})$ equals the trace of the semi-simple $p$-adic Galois representation $\Gamma_{\mathbb{Q}}\to\operatorname{GL}_{2}(\overline{\mathbb{Q}}_{p})$ attached to the corresponding eigenform.

If $\psi_{\mbox{-}}$ is trivial then the corresponding theta series $\theta_{\psi}$ coincides with the Eisenstein series attached to $\psi$ and $\psi\epsilon_{K}$, and the local structure of the eigencurve at such points has been extensively studied in [6]. If $\psi_{\mbox{-}}(\mathfrak{p})\neq 1$ then $\theta_{\psi}$ is a cuspform whose $p$-stabilizations

[TABLE]

belong each to a unique Hida family necessarily having CM by $K$ (see [3, Cor. 1.2]).

We henceforth assume that $\psi_{\mbox{-}}$ is non-trivial but $\psi_{\mbox{-}}(\mathfrak{p})=1$, and we let $f$ denote the unique $p$-stabilization of the cuspform $\theta_{\psi}$.

3.1. Components with CM by K containing $f$

The reciprocity map in Class Field Theory, denoted by $\mathrm{Art}$, is normalized so that it sends uniformizers to arithmetic Frobenii. We let $\varepsilon_{\mathrm{cyc}}:\Gamma_{\mathbb{Q}}\twoheadrightarrow\mathbb{Z}_{p}^{\times}$ be the $p$-adic cyclotomic character, $\omega_{p}$ be the Teichmüller character and $\varepsilon_{p}=\varepsilon_{\mathrm{cyc}}\omega_{p}^{-1}:\Gamma_{\mathbb{Q}}\twoheadrightarrow 1+p^{\nu}\mathbb{Z}_{p}$, where $\nu=2$ if $p=2$, and $\nu=1$ otherwise. We recall the universal cyclotomic character $\chi_{p}:\Gamma_{\mathbb{Q}}\to\bar{\mathbb{Q}}_{p}\lsem 1+p^{\nu}\mathbb{Z}_{p}\rsem^{\times}\simeq\varLambda^{\times}$ as well as the character $\eta_{\mathbf{1}}\in\operatorname{H}^{1}(\mathbb{Q},\bar{\mathbb{Q}}_{p})$ from [6, §2.4].

As $(p)=\mathfrak{p}\bar{\mathfrak{p}}$ splits in $K$, we have a short exact sequence of groups

[TABLE]

where the exponent $(\cdot)^{(p)}$ denotes the $p$-primary part of an abelian group. Letting $\mathbb{W}_{\mathfrak{p}}$ denote the maximal torsion-free quotient of $\mathscr{C}\ell_{K}(\mathfrak{p}^{\infty})$, seen as the Galois group of the unique $\mathfrak{p}$-ramified $\mathbb{Z}_{p}$-extension of $K$, for any prime to $p$ ideal $\mathfrak{c}$ of $K$, there exists an isomorphism

[TABLE]

sending $1+p^{\nu}\mathbb{Z}_{p}\subset\mathcal{O}_{K,\mathfrak{p}}^{\times}=\mathbb{Z}_{p}^{\times}$ to the inertia subgroup $\mathrm{I}_{\mathfrak{p}}\subset\mathbb{W}_{\mathfrak{p}}$ at $\mathfrak{p}$. The quotient $\mathbb{W}_{\mathfrak{p}}/\mathrm{I}_{\mathfrak{p}}$ is cyclic of order $p^{h}$, for some $h\in\mathbb{Z}_{\geqslant 1}$, and we let $w_{\mathfrak{p}}\in\mathbb{W}_{\mathfrak{p}}$ be the topological generator such that $w_{\mathfrak{p}}^{p^{h}}=\mathrm{Art}(1+p^{\nu})\in\mathrm{I}_{\mathfrak{p}}$. A $p$-adic avatar of a Hecke character of infinity type $(-1,0)$ is given by the continuous character valued in the ring of integers $\mathcal{O}$ of a sufficiently large $p$-adic field

[TABLE]

where the second map sends $w_{\mathfrak{p}}$ to $z_{\mathfrak{p}}\in 1+\mathfrak{m}_{\mathcal{O}}$ such that $z_{\mathfrak{p}}^{p^{h}}=(1+p^{\nu})^{-1}$.

The homomorphism $\eta_{\mathfrak{p}}:\Gamma_{K}\to\overline{\mathbb{Q}}_{p}$ from §1.2 factors through $\mathbb{W}_{\mathfrak{p}}$ and its restriction to $\mathrm{I}_{\mathfrak{p}}$ is given by $\log_{p}\circ\mathrm{Art}^{-1}$, hence $\eta_{\mathfrak{p}}=-\log_{p}\circ\varepsilon_{\mathfrak{p}}$. One can check that $\varepsilon_{\mathfrak{p}}\circ\mathrm{Ver}=\varepsilon_{p}$ and $\eta_{\mathfrak{p}}\circ\mathrm{Ver}=\eta_{\mathbf{1}}$, where $\mathrm{Ver}:\Gamma_{\mathbb{Q}}^{\mathrm{ab}}\to\Gamma_{K}^{\mathrm{ab}}$ is the transfer homomorphism.

The completed strict local ring $\varLambda_{\mathfrak{p}}$ of $\mathcal{O}\lsem\mathbb{W}_{\mathfrak{p}}\rsem$ at the augmentation ideal $\ker\left(\mathcal{O}\lsem\mathbb{W}_{\mathfrak{p}}\rsem\to\mathcal{O}\right)$ is naturally isomorphic to $\varLambda$. Indeed, the inclusion $\mathcal{O}\lsem\mathrm{I}_{\mathfrak{p}}\rsem\hookrightarrow\mathcal{O}\lsem\mathbb{W}_{\mathfrak{p}}\rsem$ can be identified to the generically étale map $\mathcal{O}\lsem X\rsem\hookrightarrow\mathcal{O}\lsem W\rsem$ sending $X$ to $(1+W)^{p^{h}}-1$. The resulting map

[TABLE]

is an isomorphism allowing to consider the universal character $\chi_{\mathfrak{p}}:\Gamma_{K}\twoheadrightarrow\mathbb{W}_{\mathfrak{p}}\to\mathcal{O}\lsem\mathbb{W}_{\mathfrak{p}}\rsem^{\times}\hookrightarrow\varLambda^{\times}_{\mathfrak{p}}$ as $\varLambda$-valued. It follows that $\chi_{\mathfrak{p}}\circ\mathrm{Ver}=\chi_{p}$ and moreover (compare with [6, (25)]) one has

[TABLE]

Similarly, we define the universal character $\chi_{\bar{\mathfrak{p}}}=\chi_{\mathfrak{p}}^{\tau}:\Gamma_{K}\to\varLambda_{\mathfrak{p}}^{\times}$ unramified away from $\bar{\mathfrak{p}}$.

Recall the CM Hida family $\Theta_{\psi}$ whose specialization in weight $k\in\mathbb{Z}_{\geq 1}$ is given by the classical theta series $\theta_{\psi,k}$ corresponding to the Hecke character $\lambda_{\psi,k}:\mathscr{C}\ell_{K}(\mathfrak{p}^{\infty}\mathfrak{c}_{\psi})\to\mathcal{O}\lsem\mathbb{W}_{\mathfrak{p}}\rsem^{\times}\to\overline{\mathbb{Q}}{}_{p}^{\times}$ of infinity type $(1-k,0)$ obtained as the composed map of the algebra homomorphism attached to the group homomorphism $\varepsilon_{\mathfrak{p}}^{k-1}:\mathbb{W}_{\mathfrak{p}}\rightarrow\overline{\mathbb{Q}}{}_{p}^{\times}$ and the $\psi$-projection $\mathscr{C}\ell_{K}(\mathfrak{p}^{\infty}\mathfrak{c}_{\psi})\to\mathcal{O}\lsem\mathbb{W}_{\mathfrak{p}}\rsem^{\times}$ sending $(z,w)\in\mathscr{C}\ell_{K}(\mathfrak{p}^{\infty}\mathfrak{c}_{\psi})_{\operatorname{tor}}\times\mathbb{W}_{\mathfrak{p}}=\mathscr{C}\ell_{K}(\mathfrak{p}^{\infty}\mathfrak{c}_{\psi})$ to $\psi(z,w)[w]$.

Definition 3.1**.**

The CM by $K$ part $\mathscr{E}^{K}$ of $\mathscr{E}$ is the closed locus where $\tau_{\mathscr{E}}(g)$ vanishes for all $g\in\Gamma_{\mathbb{Q}}\backslash\Gamma_{K}$. We will say that an irreducible component $\mathcal{Z}$ of $\mathscr{E}$ has CM by $K$, if $\mathcal{Z}\subset\mathscr{E}^{K}$.

As $\rho_{f}=\operatorname{Ind}^{\mathbb{Q}}_{K}\psi$ is irreducible, the localization of the pseudo-character $\tau_{\mathscr{E}}$ at $f$ yields

[TABLE]

deforming $\rho_{f}$ and such that $\operatorname{tr}(\rho_{\mathcal{T}})(\operatorname{Frob}_{\ell})=T_{\ell}$ for all primes $\ell\nmid Np$ (see [35]).

Proposition 3.2**.**

Let $\mathcal{T}^{K}$ be the completed local ring of $\mathscr{E}^{K}$ at $f$. There is an isomorphism of local $\varLambda$-algebras $\mathcal{T}^{K}\xrightarrow{\sim}\varLambda_{\mathfrak{p}}\times_{\overline{\mathbb{Q}}_{p}}\varLambda_{\mathfrak{p}}$. Moreover, the Galois representations of the two components $\Theta_{\psi}$ and $\Theta_{\psi^{\tau}}$ of $\mathscr{E}^{K}$ containing $f$ are $\operatorname{Ind}^{\mathbb{Q}}_{K}(\psi\chi_{\mathfrak{p}})$ and $\operatorname{Ind}^{\mathbb{Q}}_{K}(\psi^{\tau}\chi_{\mathfrak{p}})$, respectively.

Proof.

Let $\mathcal{Z}$ be an irreducible component of $\mathscr{E}$ having CM by $K$ and containing $f$, and denote by $A=\mathcal{O}_{\mathcal{Z},f}^{\wedge}$ its completed local $\varLambda$-algebra. As $\rho_{f}$ is irreducible, the pseudo-character $\tau_{A}:\Gamma_{\mathbb{Q}}\to A$ comes from a representation $\rho_{A}:\Gamma_{\mathbb{Q}}\to\operatorname{GL}_{2}(A)$, which in view of Definition 3.1 (and the uniqueness of the lifting) is such that $\rho_{A}\simeq\rho_{A}\otimes\epsilon_{K}$. By slightly adapting the arguments of [16, Lem. 3.2] to local rings in the category $\mathscr{C}$ we obtain that $\rho_{A}=\operatorname{Ind}^{\mathbb{Q}}_{K}\psi_{A}$, where $\psi_{A}:\Gamma_{K}\to A^{\times}$ is a character lifting, say, $\psi$ (otherwise it lifts $\psi^{\tau}$ and the argument is the same). By $\mathfrak{p}$-ordinarity either the character $\psi_{A}\psi^{-1}:\Gamma_{K}\to 1+\mathfrak{m}_{A}$ or its $\tau$-conjugate factors through $\mathbb{W}_{\mathfrak{p}}$. Moreover one has $\chi_{p}=\det(\operatorname{Ind}^{\mathbb{Q}}_{K}(\psi_{A}\psi^{-1}))\cdot\epsilon_{K}=(\psi_{A}\psi^{-1})\circ\mathrm{Ver}$, hence $\psi_{A}\psi^{-1}=\chi_{\mathfrak{p}}$ or $\chi_{\bar{\mathfrak{p}}}$. So far we have shown that there is an injection

[TABLE]

where $\ell\nmid Np$ is a prime splitting in $K$ as $(\ell)=\lambda\bar{\lambda}$. To show that it is an isomorphism, it suffices to see that $\chi_{\mathfrak{p}}\psi+\chi_{\bar{\mathfrak{p}}}\psi^{\tau}$ and $\chi_{\mathfrak{p}}\psi^{\tau}+\chi_{\bar{\mathfrak{p}}}\psi$ are not congruent mod $X^{2}$, which in view of (23) amounts to show that $\eta_{\mathfrak{p}}\psi+\eta_{\bar{\mathfrak{p}}}\psi^{\tau}\neq\eta_{\mathfrak{p}}\psi^{\tau}+\eta_{\bar{\mathfrak{p}}}\psi$. This is clear as $\psi_{\mbox{-}}\neq\mathbf{1}$.∎

3.2. Existence of a component without CM by K containing $f$

The full eigencurve $\mathscr{E}_{\operatorname{full}}$ is locally generated over $\mathscr{E}$ by the Hecke operators $U_{q}$ for $q\mid N$. The resulting morphism $\mathscr{E}_{\operatorname{full}}\to\mathscr{E}$ is locally finite, surjective, compatible with all the structures, and not an isomorphism when $N>1$. Hida’s $p$-ordinary Hecke algebra of tame level $N$ is a canonical integral model for the ordinary locus of $\mathscr{E}_{\operatorname{full}}$ which is the admissible open of $\mathscr{E}_{\operatorname{full}}$ defined by $|U_{p}|_{p}=1$, and whose irreducible components correspond to Hida families of tame level $N$. In this language Proposition 3.2 asserts that $\Theta_{\psi}$ and $\Theta_{\psi^{\tau}}$ are the only Hida families containing $f$ and having CM by $K$.

Let $\mathcal{U}$ be an affinoid subdomain of $\mathcal{W}$ containing $\kappa(f)$ and small enough to ensure the existence of the finite type, projective $\mathcal{O}(\mathcal{U})$-module $S^{\dagger,0}_{\mathcal{U}}(N)$ of $p$-ordinary families of cuspidal overconvergent modular forms of tame level $N$. By construction, there exists an affinoid neighbourhood $\mathcal{V}$ of $f$ in $\mathscr{E}_{\operatorname{full}}$ such that $\mathcal{U}=\kappa(\mathcal{V})$, and $\mathcal{O}(\mathcal{V})$ is the finite $\mathcal{O}(\mathcal{U})$-subalgebra of $\mathrm{End}_{\mathcal{O}(\mathcal{U})}(S^{\dagger,0}_{\mathcal{U}}(N))$ generated by $T_{\ell}$ for $\ell\nmid Np$ and $U_{q}$ for $q\mid Np$ (see [9, §7.1]). Taking the first coefficient in $q$-expansions at the cusp $\infty$ yields a natural isomorphism of $\mathcal{O}(\mathcal{U})$-modules (see Hida [25, §2] and Coleman [10, Prop. B.5.6]):

[TABLE]

By [3, Prop. 7.1], $\mathscr{E}$ and $\mathscr{E}_{\operatorname{full}}$ are locally isomorphic at $f$. In particular, $U_{q}\in\mathcal{T}$ for all $q\mid N$. It follows then from (25) (see also [11, Proof of Prop. 1.1]) that

[TABLE]

where $\mathcal{T}/\mathfrak{m}_{\varLambda}\cdot\mathcal{T}$ is the local ring of the fiber $\kappa^{-1}(\kappa(f))$ at $f$, and $S_{1}^{\dagger}(N)\lsem f\rsem$ is the generalized eigenspace attached to $f$ inside the space of weight $1$, level $N$, ordinary $p$-adic modular forms.

Let $\mathcal{T}_{\operatorname{split}}$ be the sub-algebra of $\mathcal{T}$ generated over $\varLambda$ by $T_{\ell}$ for $\ell\nmid Np$ split in $K$.

Theorem 3.3**.**

1. (i)

The operator $U_{p}$ does not belong to $\mathcal{T}_{\operatorname{split}}$. 2. (ii)

There exists an irreducible component $\mathcal{Z}$ of $\mathscr{E}^{\operatorname{ord}}$ containing $f$ and without CM by $K$.

Proof.

(i) A direct computation shows that $(U_{p}-\psi(\mathfrak{p}))(\theta_{\psi})=\psi(\mathfrak{p})f$. Moreover, $\theta_{\psi}$ and $f$ sharing the same Hecke eigenvalues away from $U_{p}$, the space spanned by those two forms is contained in $S_{1}^{\dagger}(N)\lsem f\rsem$. The above computation also shows that the action of $U_{p}$ on that space is not semi-simple, while $\mathcal{T}_{\operatorname{split}}$ acts on it semi-simply (diagonally).

(ii) Assume that any component of $\mathscr{E}$ containing $f$ has CM by $K$, i.e., $\mathcal{T}=\mathcal{T}^{K}$. Then, by Proposition 3.2, $\mathcal{T}=\varLambda_{\mathfrak{p}}\times_{\overline{\mathbb{Q}}_{p}}\varLambda_{\mathfrak{p}}$ and $\mathcal{T}/\mathfrak{m}_{\varLambda}\cdot\mathcal{T}\simeq\overline{\mathbb{Q}}_{p}[X]/(X^{2})$, hence by (26) we have

[TABLE]

in particular $\{f,\theta_{\psi}\}$ is a basis of $S_{1}^{\dagger}(N)\lsem f\rsem$. On the other hand, Proposition 3.2 also implies (under the identification $\varLambda_{\mathfrak{p}}=\varLambda$) that the image of $U_{p}$ in $\mathcal{T}^{K}$ equals $((\psi\chi_{\bar{\mathfrak{p}}})(\mathfrak{p}),(\psi^{\tau}\chi_{\bar{\mathfrak{p}}})(\mathfrak{p}))$ which belongs to $\varLambda^{\times}$ embedded diagonally in $\mathcal{T}^{K}$ as $\psi_{\mbox{-}}(\mathfrak{p})=1$. Hence $(U_{p}-\psi(\mathfrak{p}))\in\mathfrak{m}_{\varLambda}$ and therefore $U_{p}$ acts as $\psi(\mathfrak{p})$ on $S_{1}^{\dagger}(N)\lsem f\rsem$, yielding a contradiction with the fact, established in (i), that its action on that space is not semi-simple.∎

Remark 3.4*.*

Theorem 3.3(ii) not only answers a question raised in [15, §7.4(1)] about the existence of a Hida family without CM by $K$ containing a weight $1$ form having CM by $K$, but it further asserts that this is systematically the case: any CM weight $1$ cuspform irregular at $p$ belongs to such a family.

Definition 3.5**.**

Let $\mathscr{E}^{\perp}$ be the closed analytic subspace of $\mathscr{E}$, union of irreducible components having no CM by $K$.

Theorem 3.3 implies that $f\in\mathscr{E}^{\perp}$. As well known from Hida theory, $\mathcal{T}$ is equidimensional of dimension $1$, hence so is the completed local ring $\mathcal{T}^{\perp}$ of $\mathscr{E}^{\perp}$ at $f$, which can also be seen as the largest non-CM by $K$ quotient of $\mathcal{T}$. We will use this modular input to prove that the Krull dimension of $\mathcal{R}^{\perp}$ is at least $1$ (recall that by Theorem 2.13 its dimension is at most $1$).

The specialization of (24) by the surjective homomorphism $\pi^{\perp}:\mathcal{T}\to\mathcal{T}^{\perp}$, yields a deformation $\rho_{\mathcal{T}}^{\perp}:\Gamma_{\mathbb{Q}}\to\operatorname{GL}_{2}(\mathcal{T}^{\perp})$ of $\rho_{f}$. Next, we show that after conjugating $\rho_{\mathcal{T}^{\perp}|\Gamma_{K}}$ by a matrix in the total field of fractions $Q(\mathcal{T}^{\perp})$, we obtain $\mathcal{T}^{\perp}$-valued deformations of both $\rho$ and $\rho^{\prime}$.

Proposition 3.6**.**

Denote $\chi_{\mathcal{T}}^{\perp}$ the unramified character of $\Gamma_{K_{\mathfrak{p}}}\subset\Gamma_{K}$ sending $\operatorname{Frob}_{\mathfrak{p}}$ to $U_{p}$. There exists a deformation $\rho_{\mathcal{T}^{\perp}}:\Gamma_{K}\to\operatorname{GL}_{2}(\mathcal{T}^{\perp})$ of $\rho$ such that $\rho_{\mathcal{T}^{\perp}\mid\Gamma_{K_{\mathfrak{p}}}}=\left(\begin{smallmatrix}*&*\\ 0&\chi_{\mathcal{T}}^{\perp}\end{smallmatrix}\right)$, and $\operatorname{tr}\rho_{\mathcal{T}^{\perp}}=\operatorname{tr}(\rho^{\perp}_{\mathcal{T}\mid\Gamma_{K}})$. Similarly, there exists a deformation $\rho^{\prime}_{\mathcal{T}^{\perp}}:\Gamma_{K}\to\operatorname{GL}_{2}(\mathcal{T}^{\perp})$ of ${\rho^{\prime}}$ such that $\operatorname{tr}(\rho^{\prime}_{\mathcal{T}^{\perp}})=\operatorname{tr}(\rho_{\mathcal{T}^{\perp}})$ and $\rho^{\prime}{}_{\!\!\mathcal{T}^{\perp}\mid\Gamma_{K_{\mathfrak{p}}}}=\left(\begin{smallmatrix}*&*\\ 0&\chi_{\mathcal{T}}^{\perp}\end{smallmatrix}\right)$, hence there is a $\varLambda$-algebra homomorphism $\varphi^{\perp}:\mathcal{R}^{\perp}\to\mathcal{T}^{\perp}$.

Proof.

We first observe that for any localization $L$ of $\mathcal{T}^{\perp}$ at a minimal prime, the resulting representation $\rho_{L}:\Gamma_{\mathbb{Q}}\to\operatorname{GL}_{2}(L)$ remains irreducible when restricted to $\Gamma_{K}$. In fact, if $\operatorname{Hom}_{\Gamma_{K}}(\rho_{L},\psi_{L})\neq\{0\}$ for some character $\psi_{L}:\Gamma_{K}\to L^{\times}$, the Frobenius Reciprocity Theorem would imply that $\operatorname{Hom}_{\Gamma_{\mathbb{Q}}}(\rho_{L},\operatorname{Ind}^{\mathbb{Q}}_{K}\psi_{L})\neq\{0\}$, thus contradicting the assumption that the above minimal prime does not correspond to a component having CM by $K$. Proposition 1.3 then allows us to apply [1, Cor. 2] to find an adapted basis $(e_{1}^{\perp},e_{2}^{\perp})$ (resp. $(e_{1}^{\perp^{\prime}},e_{2}^{\perp^{\prime}})$) of $Q(\mathcal{T}^{\perp})^{2}$ such that the representation $\rho_{Q(\mathcal{T}^{\perp})}:\Gamma_{K}\to\operatorname{GL}_{2}(Q(\mathcal{T}^{\perp}))$ takes values in $\operatorname{GL}_{2}(\mathcal{T}^{\perp})$ and reduces to $\rho$ (resp. $\rho^{\prime}$) modulo the maximal ideal of $\mathcal{T}^{\perp}$. The resulting representations, denoted $\rho_{\mathcal{T}^{\perp}}$ and $\rho^{\prime}_{\mathcal{T}^{\perp}}$, share the same trace and determinant.

For the ordinariness statement at $\mathfrak{p}$, we write an exact sequence of $\mathcal{T}^{\perp}[\Gamma_{K_{\mathfrak{p}}}]$-modules as in [3, (16)] and adapt the argument as follows: the fact that $\rho_{\mid\mathrm{I}_{\mathfrak{p}}}$ (resp. $\rho^{\prime}_{\mid\mathrm{I}_{\mathfrak{p}}}$) has an infinite image and admits a unique $\mathrm{I}_{\mathfrak{p}}$-stable line, shows that the last term of that exact sequence is a monogenic $\mathcal{T}^{\perp}$-module, hence it is free, as it is generically free of rank $1$ and $\mathcal{T}^{\perp}$ is reduced.∎

As $\mathcal{R}^{\perp}$ is generated over $\varLambda$ by the trace of its universal deformation (see Lemma 2.7), the Chebotarev Density Theorem and Proposition 3.6 imply that the image of the $\varLambda$-algebra homomorphism $\varphi^{\perp}:\mathcal{R}^{\perp}\to\mathcal{T}^{\perp}$ equals the image $\mathcal{T}^{\perp}_{\operatorname{split}}$ of $\mathcal{T}_{\operatorname{split}}$ in $\mathcal{T}^{\perp}$.

Theorem 3.7**.**

The $\varLambda$-algebra $\mathcal{R}^{\perp}$ is a discrete valuation ring isomorphic to $\mathcal{T}^{\perp}_{\operatorname{split}}$ and $\rho_{\mathcal{R}^{\perp}}$ is irreducible. The structural homomorphism $\varLambda\to\mathcal{R}^{\perp}$ is étale if and only if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})+\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})\neq 0$.

Proof.

By Theorem 2.13 the Zariski tangent space $t^{\perp}$ of $\mathcal{R}^{\perp}$ has dimension $1$. As $\mathcal{T}^{\perp}_{\operatorname{split}}$ is finite and flat over $\varLambda$, it follows that its Krull dimension is $1$ and that the surjective $\varLambda$-algebra homomorphism $\mathcal{R}^{\perp}\twoheadrightarrow\mathcal{T}^{\perp}_{\operatorname{split}}$ is an isomorphism of regular local rings of dimension $1$, i.e. discrete valuation rings. The irreducibility of $\rho_{\mathcal{R}^{\perp}}$ then follows from the proof of Proposition 3.6. The last claim follows directly from Theorem 2.13 where one gives a necessary and sufficient condition for the vanishing of the relative tangent space $t^{\perp}_{0}$ of $\mathcal{R}^{\perp}$ over $\varLambda$.∎

3.3. The reducibility ideal of $\mathcal{R}^{\perp}$

We recall that $\rho_{\mathcal{R}^{\perp}}$ (resp. $\rho^{\prime}_{\mathcal{R}^{\perp}}$) is the $\mathfrak{p}$-ordinary deformation of $\rho$ (resp. $\rho^{\prime}$) obtained by functoriality from the natural surjection $\mathcal{R}_{\rho}^{\operatorname{ord}}\twoheadrightarrow\mathcal{R}^{\perp}$ (resp. $\mathcal{R}_{\rho^{\prime}}^{\operatorname{ord}}\twoheadrightarrow\mathcal{R}^{\perp}$) from Lemma 2.7. In order to determine $\mathcal{T}^{\perp}$ we need to study the possible extensions of $\operatorname{tr}(\rho_{\mathcal{R}^{\perp}})=\operatorname{tr}(\rho^{\prime}_{\mathcal{R}^{\perp}})$ to $\Gamma_{\mathbb{Q}}$. For this we first observe that the reducibility ideal (see [2, §1.5]) of the pseudo-character $\operatorname{tr}(\rho_{\mathcal{R}^{\perp}})=\operatorname{tr}(\rho^{\prime}_{\mathcal{R}^{\perp}})$ is non-zero, as $\rho_{\mathcal{R}^{\perp}}$ is irreducible by Theorem 3.7. We recall that $\mathcal{R}^{\perp}$ is a discrete valuation ring and is flat over $\varLambda$.

Definition 3.8**.**

1. (i)

Let $e\geqslant 1$ be the ramification index of $\mathcal{R}^{\perp}$ over $\varLambda$, and let $Y$ denote a uniformizer of $\mathcal{R}^{\perp}$, so that $\mathcal{R}^{\perp}=\overline{\mathbb{Q}}_{p}\lsem Y\rsem\simeq\overline{\mathbb{Q}}_{p}\lsem X^{1/e}\rsem$. 2. (ii)

Let $r\geqslant 1$ be the valuation of the reducibility ideal of $\operatorname{tr}(\rho_{\mathcal{R}^{\perp}})$.

We state for later use a lemma whose elementary proof uses the fact that $\rho$ has non-isomorphic Jordan-Hölder factors, and that $\mathcal{R}^{\perp}/(Y^{r})$ is a complete local ring, hence Henselian.

Lemma 3.9**.**

For $m\leqslant r$, $\operatorname{tr}(\rho_{\mathcal{R}^{\perp}})\mod{Y^{m}}$ can be uniquely written as sum of two characters.

We recall from §2 the element $\gamma_{0}\in\Gamma_{K}$ such that $\psi_{\mbox{-}}(\gamma_{0})\neq 1$ and $\eta(\gamma_{0})=0$ and we write $\rho_{\mathcal{R}^{\perp}}=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)$ in a basis $(e_{1},e_{2})$ where $\rho_{\mathcal{R}^{\perp}}(\gamma_{0})=\left(\begin{smallmatrix}*&0\\ 0&*\end{smallmatrix}\right)$.

It follows from Proposition 2.10 that $r\geqslant 2$. Much of what follows will depend on $r$, so we first give a numerical criterion for its lowest (conjecturally only possible) value.

Lemma 3.10**.**

One has $r=2$ if and only if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\cdot\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})\neq 0$.

Proof.

By (17) one has

[TABLE]

where $c\in\mathrm{Z}^{1}(\Gamma_{K},\psi_{\mbox{-}}^{\tau})$ is such that $c(\gamma_{0})=0$. By (20), after rescaling the uniformizer $Y$ by an element of $\overline{\mathbb{Q}}{}_{p}^{\times}$, we may and do assume that $\alpha=\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})$ and $\beta=\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})$. By Proposition 3.11 one has $r\geqslant 3$ if and only if $c=0$ which in view of the injectivity of the restriction $\operatorname{H}^{1}(K,\psi_{\mbox{-}}^{\tau})\rightarrow\operatorname{H}^{1}(K_{\mathfrak{p}},\overline{\mathbb{Q}}_{p})$ (see Proposition 1.3) is also equivalent to $c_{|\Gamma_{K_{\mathfrak{p}}}}=0$. We will now use the $\mathfrak{p}$-ordinarity of $\rho_{\mathcal{R}^{\perp}}$ and show that the vanishing of $c_{|\Gamma_{K_{\mathfrak{p}}}}$ is equivalent to $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\cdot\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})=0$.

By the proof of Proposition 2.10 the $\mathfrak{p}$-ordinary filtration modulo $(Y^{3})$ is generated by a vector of the form $e_{1}+(\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})Y+Y^{2}\cdot*)e_{2}$, and a direct computation using (27) shows that

[TABLE]

for all $g\in\Gamma_{K_{\mathfrak{p}}}$, which completes the proof, as $\eta^{\prime}_{|\Gamma_{K_{\mathfrak{p}}}}\neq 0$.∎

Let $\iota$ be the automorphism sending $Z$ to $-Z$ of the local $\varLambda$-algebra

[TABLE]

Note that the latter ring $\widetilde{\mathcal{R}}^{\perp}\simeq\overline{\mathbb{Q}}_{p}\lsem X\rsem[X^{\frac{r}{2}}]$ is not normal.

We write $\rho^{\prime}_{\mathcal{R}^{\perp}}=\left(\begin{smallmatrix}a^{\prime}&b^{\prime}\\ c^{\prime}&d^{\prime}\end{smallmatrix}\right)$ in a basis where $\rho^{\prime}_{\mathcal{R}^{\perp}}(\gamma_{0})=\left(\begin{smallmatrix}*&0\\ 0&*\end{smallmatrix}\right)$.

Proposition 3.11**.**

The reducibility ideal equals $(c(g);g\in\Gamma_{K})=(c^{\prime}(g);g\in\Gamma_{K})=(Y^{r})$. Moreover $\widetilde{\rho}^{\perp}_{\mathcal{R}}=\left(\begin{smallmatrix}Z&0\\ 0&1\end{smallmatrix}\right)\rho_{\mathcal{R}^{\perp}}\left(\begin{smallmatrix}Z^{-1}&0\\ 0&1\end{smallmatrix}\right)$ extends to a representation $\widetilde{\rho}^{\perp}_{\mathcal{R}}:\Gamma_{\mathbb{Q}}\to\operatorname{GL}_{2}(\widetilde{\mathcal{R}}^{\perp})$ reducing to $\rho_{f}$ modulo the maximal ideal of $\widetilde{\mathcal{R}}^{\perp}$. Finally $\iota(\widetilde{\rho}^{\perp}_{\mathcal{R}})=\left(\begin{smallmatrix}-1&0\\ 0&1\end{smallmatrix}\right)(\widetilde{\rho}^{\perp}_{\mathcal{R}}\otimes\epsilon_{K})\left(\begin{smallmatrix}-1&0\\ 0&1\end{smallmatrix}\right)$.

Proof.

Recall that $(Y^{r})$ is the smallest ideal modulo which $\operatorname{tr}(\rho_{\mathcal{R}^{\perp}})=\operatorname{tr}(\rho^{\prime}_{\mathcal{R}^{\perp}})$ can be written as a sum of two characters. A direct computation shows that $(Y^{r})=(c(g)b(g^{\prime});g,g^{\prime}\in\Gamma_{K})=(c^{\prime}(g)b^{\prime}(g^{\prime});g,g^{\prime}\in\Gamma_{K})$ implying the first claim, as $(b(g^{\prime});g^{\prime}\in\Gamma_{K})=(b^{\prime}(g^{\prime});g^{\prime}\in\Gamma_{K})=\mathcal{R}^{\perp}$.

As $\dim\operatorname{H}^{1}(K,\psi_{\mbox{-}})=\dim\operatorname{H}^{1}(K,\psi_{\mbox{-}}^{\tau})=1$, there exist upper triangular matrices $P_{0}$ and $P_{1}$ in $\operatorname{GL}_{2}(\overline{\mathbb{Q}}_{p})$ such that the pair of $\mathcal{R}^{\perp}$-valued $\Gamma_{K}$-representations

[TABLE]

reduces to $(\rho,\rho^{\prime})$ modulo $\mathfrak{m}_{\mathcal{R}^{\perp}}$. We will now check that it defines the same $\mathcal{R}^{\perp}$-point of $\mathcal{D}^{\perp}$ as $(\rho_{\mathcal{R}^{\perp}},\rho^{\prime}_{\mathcal{R}^{\perp}})$. In fact, all representations are $\mathcal{R}^{\perp}$-valued and share the same $\tau$-invariant trace and determinant, and by Lemma 2.7 they are all $\mathfrak{p}$-ordinary (with same unramified character). Hence the two pairs are strictly conjugated, in particular there exists $P\in\operatorname{GL}_{2}(\mathcal{R}^{\perp})$ such that for all $g\in\Gamma_{K}$ we have

[TABLE]

As $d^{\tau}\not\equiv d\pmod{Y}$, reducing the above equation modulo $Y^{r}$ yields $P\in\left(\begin{smallmatrix}*&*\\ 0&*\end{smallmatrix}\right)\pmod{Y^{r}}$. Letting $M=\left(\begin{smallmatrix}0&1\\ Z&0\end{smallmatrix}\right)P\left(\begin{smallmatrix}Z^{-1}&0\\ 0&1\end{smallmatrix}\right)\in\operatorname{GL}_{2}(\widetilde{\mathcal{R}}^{\perp})$ the equation (29) can be rewritten as

[TABLE]

where the representation $\widetilde{\rho}^{\perp}_{\mathcal{R}}=\left(\begin{smallmatrix}Z&0\\ 0&1\end{smallmatrix}\right)\rho_{\mathcal{R}^{\perp}}\left(\begin{smallmatrix}Z^{-1}&0\\ 0&1\end{smallmatrix}\right):\Gamma_{K}\to\operatorname{GL}_{2}(\widetilde{\mathcal{R}}^{\perp})$ is generically absolutely irreducible (as $\rho_{\mathcal{R}^{\perp}}$ is), while reduces to $\psi\oplus\psi^{\tau}$ modulo the maximal ideal. As $\tau^{2}=1$, Schur’s lemma applied to a generic point of $\widetilde{\mathcal{R}}^{\perp}$ implies that $M^{2}$ is scalar. Reducing (30) modulo the maximal ideal of $\widetilde{\mathcal{R}}^{\perp}$ shows that $M$ is not scalar, hence necessarily $\operatorname{tr}(M)=0$. After rescaling $M$ one can assume that $\det(M)=-1$. Then $M^{2}=1$ and letting $\widetilde{\rho}^{\perp}_{\mathcal{R}}(\tau)=M$ (or $-M$) yields the desired extension. Note that $\rho_{f}$ is the unique extension of $\psi\oplus\psi^{\tau}$ to $\Gamma_{\mathbb{Q}}$.

The last point is a direct computation.∎

Denote by $\rho^{\perp}_{\mathcal{R}}$ the push-forward of $\widetilde{\rho}^{\perp}_{\mathcal{R}}$ along the projection $\widetilde{\mathcal{R}}^{\perp}\twoheadrightarrow\mathcal{R}^{\perp}$ sending $Z$ to $Y^{\frac{r}{2}}$, when $r$ is even, and let $\rho^{\perp}_{\mathcal{R}}=\widetilde{\rho}^{\perp}_{\mathcal{R}}$, when $r$ is odd. By the Frobenius Reciprocity Theorem (applied over a finite extension of the fraction field of the integral domain $\mathcal{R}^{\perp}$), $\rho^{\perp}_{\mathcal{R}}$ and $\rho^{\perp}_{\mathcal{R}}\otimes\epsilon_{K}$ are the only possible extensions of $\rho_{\mathcal{R}^{\perp}}$ to $\Gamma_{\mathbb{Q}}$.

Corollary 3.12**.**

The pseudo-characters $\operatorname{tr}(\rho^{\perp}_{\mathcal{R}})$ and $\operatorname{tr}(\rho^{\perp}_{\mathcal{R}}\cdot\epsilon_{K})$ are the only possible extensions of $\operatorname{tr}(\rho_{\mathcal{R}^{\perp}})$ to $\Gamma_{\mathbb{Q}}$. Moreover $\operatorname{tr}(\widetilde{\rho}^{\perp}_{\mathcal{R}})\cdot\epsilon_{K}=\iota\circ\operatorname{tr}(\widetilde{\rho}^{\perp}_{\mathcal{R}})$.

3.4. Components without CM by K containing $f$

Theorem 3.3 gives a Hida family $\mathcal{F}$ containing $f$ and having no CM by $K$ . As $f\otimes\epsilon_{K}=f$, the twist $\mathcal{F}\otimes\epsilon_{K}$ of $\mathcal{F}$ is another such family.

Proposition 3.13**.**

The local ring $\widetilde{\mathcal{R}}^{\perp}$ is topologically generated over $\varLambda$ by $\widetilde{\tau}_{\mathcal{R}}^{\perp}=\operatorname{tr}(\widetilde{\rho}^{\perp}_{\mathcal{R}}(\Gamma_{\mathbb{Q}}))$.

Proof.

Let $A$ be the subring of $\widetilde{\mathcal{R}}^{\perp}$ topologically generated over $\varLambda$ by $\widetilde{\tau}_{\mathcal{R}}^{\perp}(\Gamma_{\mathbb{Q}})$. As $A\supset\mathcal{R}^{\perp}$ (see Theorem 3.7), it suffices to show that $A$ contains the local parameter $Z$. By [35, Thm. 1] there exists a deformation $\rho_{A}=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right):\Gamma_{\mathbb{Q}}\to\operatorname{GL}_{2}(A)$ of $\rho_{f}$ such that $\operatorname{tr}(\rho_{A})=\widetilde{\tau}_{\mathcal{R}}^{\perp}$ and $\rho_{A}(\gamma_{0})$ is diagonal. Then the reducibility ideal of $\widetilde{\tau}^{\perp}_{\mathcal{R}\mid\Gamma_{K}}=\operatorname{tr}\rho_{\mathcal{R}^{\perp}}$ equals $(Y^{r})=(b(g)c(g^{\prime});g,g^{\prime}\in\Gamma_{K})$. We first prove that the ideals $(b(g);g\in\Gamma_{K})$ and $(c(g);g\in\Gamma_{K})$ of $A$ are equal by showing the double inclusion. To show $(b(g);g\in\Gamma_{K})\subset(c(g);g\in\Gamma_{K})$ one reduces the equality $\rho_{A}(\tau g\tau)=\rho_{A}(\tau)\rho_{A}(g)\rho_{A}(\tau)$ for a given $g\in\Gamma_{K}$ modulo $(c(g);g\in\Gamma_{K})$ and uses the fact that $\rho_{A}(g)$ and $\rho_{A}(\tau g\tau)$ are upper triangular and that $\rho_{A}(\gamma_{0})$ is diagonal.

Assume first that $r$ is odd. In that case $\widetilde{\mathcal{R}}^{\perp}$ is an integral domain, which is finite, flat and ramified over $\mathcal{R}^{\perp}$. It suffices then to show that $A$ contains an element of $Y^{\frac{r}{2}}\cdot\overline{\mathbb{Q}}_{p}\lsem Y^{\frac{1}{2}}\rsem^{\times}$, and any $b(g_{0})\in A$ generating $(b(g);g\in\Gamma_{K})=(Y^{\frac{r}{2}})$ would do.

Assume next that $r$ is even. In that case $\widetilde{\mathcal{R}}^{\perp}\simeq\mathcal{R}^{\perp}\times_{\overline{\mathbb{Q}}_{p}[Y]/(Y^{\frac{r}{2}})}\mathcal{R}^{\perp}$, and $A\subset\widetilde{\mathcal{R}}^{\perp}$ surjects on both components $\mathcal{R}^{\perp}$. Hence $A=\mathcal{R}^{\perp}\times_{\overline{\mathbb{Q}}_{p}[Y]/(Y^{s})}\mathcal{R}^{\perp}$ with $s\geq\frac{r}{2}$, and by Proposition 3.11 one has

[TABLE]

where $\iota$ is the involution of $\widetilde{\mathcal{R}}^{\perp}$ exchanging its two components. Writing $b(g)=(b_{1}(g),b_{2}(g))\in A$, there exists $g_{0}\in\Gamma_{K}$ such that $b_{1}(g_{0})=-b_{2}(g_{0})\in Y^{\frac{r}{2}}\cdot\overline{\mathbb{Q}}_{p}\lsem Y\rsem^{\times}$. The above congruence implies that $-b_{1}(g_{0})\equiv b_{1}(g_{0})\pmod{Y^{s}}$, hence $s=\frac{r}{2}$.∎

As $\mathcal{T}^{\perp}$ carries a pseudo-character of $\Gamma_{\mathbb{Q}}$, extending the $\mathcal{R}^{\perp}\simeq\mathcal{T}^{\perp}_{\operatorname{split}}$-valued pseudo-character of $\Gamma_{K}$, Corollary 3.12 implies that given any minimal prime ideal $\mathfrak{p}$ of $\mathcal{T}^{\perp}$ the resulting pseudo-character $\Gamma_{\mathbb{Q}}\to\mathcal{T}^{\perp}/\mathfrak{p}$ factors through $\operatorname{tr}(\widetilde{\rho}^{\perp}_{\mathcal{R}})$. Together with Proposition 3.13 this implies the existence of a $\varLambda$-algebra homomorphism $\widetilde{\varphi}^{\perp}:\widetilde{\mathcal{R}}^{\perp}\to\mathcal{T}^{\perp}$ fitting in the commutative diagram

[TABLE]

As by definition $\widetilde{\varphi}^{\perp}\circ\widetilde{\tau}_{\mathcal{R}}^{\perp}$ is surjective, so is $\widetilde{\varphi}^{\perp}$. Both algebras being reduced, it suffices to check injectivity on generic points, which holds as $\widetilde{\mathcal{R}}^{\perp}$ is a free $\mathcal{R}^{\perp}$-module of rank $2$, while $\mathcal{T}^{\perp}$ has rank $\geqslant 2$ over $\mathcal{T}^{\perp}_{\operatorname{split}}$. As a consequence we determine the $\varLambda$-algebras structure of $\mathcal{T}^{\perp}$.

Theorem 3.14**.**

There is an isomorphism of $\varLambda$-algebras $\widetilde{\varphi}^{\perp}:\widetilde{\mathcal{R}}^{\perp}\xrightarrow{\sim}\mathcal{T}^{\perp}$ extending $\varphi^{\perp}:\mathcal{R}^{\perp}\xrightarrow{\sim}\mathcal{T}_{\operatorname{split}}^{\perp}$ and under which $\iota$ is sent to the involution of $\mathcal{T}^{\perp}$ mapping $T_{\ell}$ to $\epsilon_{K}(\ell)T_{\ell}$, for $\ell\nmid Np$.

In summary, we have shown that $f$ belongs to exactly $3$ or $4$ irreducible components of $\mathscr{E}$ depending on the parity of $r$, corresponding to the Hida families $\Theta_{\psi}$, $\Theta_{\psi^{\tau}}$, $\mathcal{F}$ and $\mathcal{F}\otimes\epsilon_{K}$ containing $f$ (the last two being Galois conjugates if $r$ is odd).

4. Geometry of the eigencurve and congruence ideals

4.1. Iwasawa cohomology and the CM ideal

Let $K_{\infty}^{-}$ be the anti-cyclotomic $\mathbb{Z}_{p}$-extension of $K$ and

[TABLE]

be the universal $p$-adic anti-cyclotomic character. By (23), it satisfies the congruence

[TABLE]

Put $\Psi=\chi_{\mbox{-}}\psi_{\mbox{-}}$ and for $n\in\mathbb{Z}_{\geqslant 1}$ let $\Psi_{n}=\Psi\mod{X^{n}}$. Finally, let $\Gamma_{K}^{Np}$ be the Galois group of the maximal extension of $K$ unramified outside $Np$.

Proposition 4.1**.**

1. (i)

The $\varLambda_{\mathfrak{p}}$-module $\operatorname{H}^{1}(\Gamma_{K}^{Np},\Psi^{\pm 1})$ is free of rank $1$ and for all $n\in\mathbb{Z}_{\geqslant 1}$ the following natural map is an isomorphism

[TABLE] 2. (ii)

Assume that $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\neq 0$. Then for every $n\geqslant 1$, the natural restriction map is injective

[TABLE]

Proof.

(i) The proof is similar to [6, Prop. 2.8], as $\dim\operatorname{H}^{1}(K,\psi_{\mbox{-}})=\dim\operatorname{H}^{1}(K,\psi_{\mbox{-}}^{\tau})=1$ (by Proposition 1.3) and $\dim\operatorname{H}^{2}(K,\psi_{\mbox{-}})=\dim\operatorname{H}^{2}(K,\psi_{\mbox{-}}^{\tau})=0$ (by Lemma 2.11).

(ii) As in loc. cit. we have to check that $\eta^{\prime}$ and $\frac{\eta_{\mathfrak{p}}-\eta_{\bar{\mathfrak{p}}}}{\log_{p}(1+p^{\nu})}\equiv\frac{\chi_{\mbox{-}}^{\tau}-1}{X}\mod X$ (see (31)) generate distinct lines in $\operatorname{H}^{1}(K_{\mathfrak{p}},\overline{\mathbb{Q}}_{p})$. The claim follows from the fact that $(\eta^{\prime}-\eta_{\mathfrak{p}}+\eta_{\bar{\mathfrak{p}}})_{\mid\mathrm{I}_{\mathfrak{p}}}=0$ while $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})=(\eta^{\prime}-\eta_{\mathfrak{p}}+\eta_{\bar{\mathfrak{p}}})(\operatorname{Frob}_{\mathfrak{p}})\neq 0$.∎

We recall that the completed local ring at $f$ of the irreducible component of $\mathscr{E}$ corresponding to $\Theta_{\psi}$ is naturally isomorphic to $\varLambda_{\mathfrak{p}}$.

Corollary 4.2**.**

There exists a natural isomorphism of complete Noetherian local $\varLambda$-algebras $\mathcal{R}^{K}_{\rho}\xrightarrow{\sim}\varLambda_{\mathfrak{p}}$.

Proof.

Taking a generator of the free rank $1$ $\varLambda_{\mathfrak{p}}$-module $\operatorname{H}^{1}(\Gamma_{K}^{Np},\Psi)$ (see Proposition 4.1) yields a point in $\mathcal{D}^{K}_{\rho}(\varLambda_{\mathfrak{p}})$, hence a $\varLambda$-algebra homomorphism $\mathcal{R}^{K}_{\rho}\to\varLambda_{\mathfrak{p}}$. By Proposition 2.12 the relative tangent space $t^{K}_{\rho,0}$ of the structural homomorphism $\varLambda\rightarrow\mathcal{R}^{K}_{\rho}$ vanishes, and hence $\mathcal{R}^{K}_{\rho}/X\mathcal{R}^{K}_{\rho}=\overline{\mathbb{Q}}_{p}$ (i.e. $\varLambda\rightarrow\mathcal{R}^{K}_{\rho}$ is unramified). As $\mathcal{R}^{K}_{\rho}$ is separated for the $X$-adic topology, Nakayama’s lemma implies that $\varLambda\to\mathcal{R}^{K}_{\rho}$ is surjective. The injectivity follows from the fact that the composed map $\varLambda\twoheadrightarrow\mathcal{R}^{K}_{\rho}\to\varLambda_{\mathfrak{p}}$ is given by the natural isomorphism (22). ∎

Let $J$ be the kernel of the $\varLambda$-algebra homomorphism $\mathcal{R}^{\operatorname{ord}}_{\rho}\twoheadrightarrow\mathcal{R}_{\rho}^{K}\simeq\varLambda_{\mathfrak{p}}$. As $\mathcal{R}^{\operatorname{ord}}_{\rho}$ is Noetherian, we know that $J/J^{2}$ is a module of finite type over $\mathcal{R}^{\operatorname{ord}}_{\rho}/J\simeq\varLambda_{\mathfrak{p}}$.

Proposition 4.3**.**

If $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\neq 0$, then the $\varLambda_{\mathfrak{p}}$-module $J/J^{2}$ is torsion of length $\leqslant 1$.

Proof.

It is enough to show that for $n\geqslant 1$ the length of the $\varLambda_{\mathfrak{p}}$-module $\operatorname{Hom}_{\varLambda_{\mathfrak{p}}}(J/J^{2},\varLambda_{\mathfrak{p}}/(X^{n}))$ is at most $1$. We use a homological machinery as in [6, Prop. 4.10]. Let us fix a realization $\rho_{\mathcal{R}}(g)=\left(\begin{smallmatrix}a(g)&b(g)\\ c(g)&d(g)\end{smallmatrix}\right)$ of the universal representation in an ordinary basis. By Lemma 2.4 the ideal $J$ of $\mathcal{R}^{\operatorname{ord}}_{\rho}$ is generated by the set of $c(g)$ for $g\in\Gamma_{K}$. As

[TABLE]

a direct computation using that the basis is $\Gamma_{K_{\mathfrak{p}}}$-ordinary shows that the function

[TABLE]

belongs to $\ker\left(\operatorname{Z}^{1}(\Gamma_{K}^{Np},(\chi_{\mbox{-}}^{\tau}\psi_{\mbox{-}}^{\tau})\otimes_{\varLambda_{\mathfrak{p}}}(J/J^{2}))\to\operatorname{Z}^{1}(K_{\mathfrak{p}},(\chi_{\mbox{-}}^{\tau}\psi_{\mbox{-}}^{\tau})\otimes_{\varLambda_{\mathfrak{p}}}(J/J^{2}))\right)$. As the image of $\bar{c}$ contains a set of generators of $J/J^{2}$ as $\mathcal{R}^{\operatorname{ord}}_{\rho}/J\simeq\varLambda_{\mathfrak{p}}$-module, the natural map

[TABLE]

is injective for all $n\geqslant 1$. Moreover by Proposition 4.1(ii) one has

[TABLE]

As $\Psi_{1}|_{\Gamma_{K_{\mathfrak{p}}}}=\psi_{\mbox{-}}|_{\Gamma_{K_{\mathfrak{p}}}}=\mathbf{1}$, while $\Psi_{2}|_{\Gamma_{K_{\mathfrak{p}}}}\neq\mathbf{1}$, the coboundary in (33) is given by the length $1$ $\varLambda_{\mathfrak{p}}$-module $(X^{n-1})/(X^{n})\simeq\overline{\mathbb{Q}}_{p}$, proving the desired assertion.∎

We begin the study of the congruences between $\Theta_{\psi}$ and a family $\mathcal{F}$ without CM by $K$, by determining the exact congruence ideal at split primes (note that the eigenvalues of $\mathcal{F}$ and $\mathcal{F}\otimes\epsilon_{K}$ coincide at such primes). This amounts to the study of the image $\mathcal{T}_{\rho}^{\operatorname{split}}$ of $\mathcal{T}_{\operatorname{split}}$ in $\mathcal{T}_{\operatorname{split}}^{\perp}\times\varLambda_{\mathfrak{p}}$, which we will now relate to the universal deformation ring $\mathcal{R}_{\rho}^{\operatorname{ord}}$ from §2.

Recall the functorial homomorphism $\mathcal{R}^{\operatorname{ord}}_{\rho}\to\mathcal{R}^{\perp}\times_{\overline{\mathbb{Q}}_{p}}\varLambda_{\mathfrak{p}}$ which surjects on both components. As observed in the proof of Lemma 2.7, the $\varLambda$-algebra $\mathcal{R}_{\rho}^{\operatorname{ord}}$ is generated by the trace of the universal ordinary deformation of $\rho$, hence there exists an integer $m\geqslant 1$ and a commutative diagram of surjective homomorphism of $\varLambda$-algebras:

[TABLE]

where the natural surjection $\mathcal{R}^{\operatorname{ord}}_{\rho}\twoheadrightarrow\mathcal{T}^{\mathrm{split}}_{\rho}$ sends $\chi_{\mathcal{R}}(\operatorname{Frob}_{\mathfrak{p}})$ to $U_{p}=(U_{p}(\mathcal{F}),U_{p}(\Theta_{\psi}))$.

Proposition 4.4**.**

If $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\neq 0$, there is an isomorphism of local complete intersection rings

[TABLE]

Proof.

Corollary 2.14 implies that $m=1$. The isomorphism is then a consequence of a variant of Wiles’ numerical criterion due to Lenstra [32] in view of Proposition 4.3.∎

As $\mathcal{R}_{\rho}^{K}$ is a quotient of $\mathcal{R}_{\rho}^{\operatorname{red}}$, the CM ideal $(X^{m})$ contains the reducibility ideal $(X^{r})$ and plays a role analogous to that of the Eisenstein ideal.

Proposition 4.5**.**

Assume that $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})=0$. Then $m=r-1$, and $\mathcal{T}^{\operatorname{split}}_{\rho}=\varLambda\times_{\overline{\mathbb{Q}}_{p}[X]/(X^{r-1})}\varLambda_{\mathfrak{p}}$.

Proof.

As $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})=0$, the structural homomorphism $\varLambda\xrightarrow{\sim}\mathcal{R}^{\perp}$ is an isomorphism by Theorem 3.7. We recall that $r\geq 3$ by Proposition 3.10, and that

[TABLE]

where $A,B,D$ are polynomials in $X$ of degree $\leqslant r$. As $\mathcal{R}^{\perp}=\overline{\mathbb{Q}}_{p}\lsem X\rsem$ is a discrete valuation ring, the ordinary filtration of $\rho_{\mathcal{R}^{\perp}}$ is generated by $e_{1}+X^{s}e_{2}$ for some $s\geqslant 1$ and, using the corresponding $\mathfrak{p}$-ordinary basis $(e_{1}+X^{s}e_{2},e_{2})$, one has

[TABLE]

It is crucial to first observe that $s\leq r-1$. Let us proceed by absurd, assuming that $s\geq r$. The lower left entry of $\rho_{\mathcal{R}^{\perp}}(g)\mod{X^{r+1}}$ in the ordinary basis $(e_{1}+X^{s}e_{2},e_{2})$ equals

[TABLE]

and it has to be trivial on $\mathrm{I}_{\mathfrak{p}}$, contradicting Proposition 1.3, as $0\neq[\eta^{\prime}]\in\operatorname{H}^{1}(K,\psi_{\mbox{-}}^{\tau})$.

Therefore $s<r$, hence $D\mod X^{r}$ is $\mathfrak{p}$-ramified. The $\mathfrak{p}$-ordinarity of $\rho_{\mathcal{R}^{\perp}}\mod X^{r}$ yields

[TABLE]

As $m\geqslant 2$ (see Cor. 2.14) one has $\operatorname{tr}(\rho_{\mathcal{R}^{\perp}})\equiv\psi\chi_{\mathfrak{p}}+\psi^{\tau}\chi_{\mathfrak{p}}^{\tau}\mod X^{2}$. Lemma 3.9 implies

[TABLE]

which together with (39) implies that $s=r-1$, as $\eta_{\mathfrak{p}|\mathrm{I}_{\mathfrak{p}}}\neq 0$. It then follows from the second relation of (39) that $D\mod X^{r-1}$ is unramified at $\mathfrak{p}$, hence there exists a morphism $\varLambda_{\mathfrak{p}}\rightarrow\mathcal{R}^{\perp}/(X^{r-1})$ sending $\psi^{\tau}\chi_{\mathfrak{p}}^{\tau}$ on $D\mod X^{r-1}$. Using the $\tau$-invariance of $\operatorname{tr}(\rho_{\mathcal{R}^{\perp}})$, a second application of Lemma 3.9 yields that $A\equiv D^{\tau}\pmod{X^{r}}$. Hence $\mathcal{R}^{\perp}/(X^{r-1})$ is topologically generated by $\{D(g),g\in\Gamma_{K}\}$ and therefore the morphism $\varLambda_{\mathfrak{p}}\rightarrow\mathcal{R}^{\perp}/(X^{r-1})$ defined above is in fact a surjection, and it yields that $m\geqslant r-1$.

It remains to show that $m<r$, i.e. $\operatorname{tr}(\rho_{\mathcal{R}^{\perp}})\nequiv\psi\chi_{\mathfrak{p}}+\psi^{\tau}\chi_{\mathfrak{p}}^{\tau}\pmod{X^{r}}$. As $A\equiv D^{\tau}\pmod{X^{r}}$, it suffices (again by Lemma 3.9) to show that $D\nequiv\psi^{\tau}\chi_{\mathfrak{p}}^{\tau}\pmod{X^{r}}$ which follows from the fact that $D\mod X^{r}$ is ramified at $\mathfrak{p}$ as observed earlier. ∎

Proposition 4.5 implies in particular that if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})=0$ then $U_{p}(\mathcal{F})\equiv U_{p}(\Theta_{\psi})\mod X^{r-1}$ and we will now see that this congruence is optimal.

Proposition 4.6**.**

Assume that $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})=0$, then $U_{p}(\mathcal{F})\not\equiv U_{p}(\Theta_{\psi})\mod X^{r}$.

Proof.

We proceed by absurd assuming that $U_{p}(\mathcal{F})\equiv U_{p}(\Theta_{\psi})\mod X^{r}$. As $\mathcal{T}^{\mathrm{split}}_{\rho}$ contains $(X^{r-1},0)$ and $(0,X^{r-1})$, we deduce that $U_{p}-\psi(\mathfrak{p})\in X\cdot\mathcal{T}^{\mathrm{split}}_{\rho}$ hence it is mapped to [math] by

[TABLE]

By Propositions 2.12 and 4.5, (40) gives rise to an isomorphism of tangent spaces

[TABLE]

To obtain a contradiction it then suffices to show that the image of $\chi_{\mathcal{R}}(\operatorname{Frob}_{\mathfrak{p}})-\psi(\mathfrak{p})$ under any non-zero element of the line $t^{\operatorname{ord}}_{\rho,0}$ does not vanish. Recall from §2.4 that $t^{\operatorname{ord}}_{\rho,0}$ is the subspace of $t^{\operatorname{ord}}_{\rho}\simeq\operatorname{Hom}(\Gamma_{K},\overline{\mathbb{Q}}_{p})\simeq\{\alpha\cdot\eta_{\mathfrak{p}}+\beta\cdot\eta_{\bar{\mathfrak{p}}},(\alpha,\beta)\in\overline{\mathbb{Q}}{}_{p}^{2}\}$ given by $\alpha=-\beta$. Hence, by (19)

[TABLE]

As $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})=0$, Proposition 1.11 implies that $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})\neq 0$ proving the claim. ∎

4.2. The congruence ideal

We recall that $C^{0}_{\psi}=\pi_{\psi}(\mathrm{Ann}_{\mathcal{T}}(\ker(\pi_{\psi})))$ is the congruence ideal attached to the $\Theta_{\psi}$-projection $\pi_{\psi}:\mathcal{T}\twoheadrightarrow\varLambda_{\mathfrak{p}}$. By Proposition 4.1 and Corollary 4.2, one may attach to $\Theta_{\psi}$ (resp. $\Theta_{\psi^{\tau}}$) a $\varLambda_{\mathfrak{p}}$-valued $\Gamma_{K}$-reducible deformation of $\rho$ (resp. $\rho^{\prime}$) whose semi-simplification is given by $\psi\chi_{\mathfrak{p}}\oplus\psi^{\tau}\chi_{\mathfrak{p}}^{\tau}$ (resp. $\psi\chi_{\mathfrak{p}}^{\tau}\oplus\psi^{\tau}\chi_{\mathfrak{p}}$). In order to compute $C^{0}_{\psi}$ we will first determine $\mathcal{T}_{\operatorname{split}}$, then $\mathcal{T}$, using the $\varLambda$-algebra homomorphism

[TABLE]

Theorem 4.7**.**

Assume that $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\neq 0$. Then $\mathcal{T}_{\operatorname{split}}[U_{p}]=\varLambda_{\mathfrak{p}}\times_{\overline{\mathbb{Q}}_{p}}\left(\mathcal{R}^{\perp}\times_{\overline{\mathbb{Q}}_{p}[X]/(X^{r-1})}\varLambda_{\mathfrak{p}}\right)$.

1. (i)

If moreover $0\neq\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})\neq-\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})$, then

[TABLE] 2. (ii)

If moreover $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})=0$, then $e=1$, $r\geqslant 3$ and there exists $\xi\in\overline{\mathbb{Q}}{}_{p}^{\times}$ such that

[TABLE] 3. (iii)

If moreover $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})=-\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})$, then $e\geqslant 2=r$ and there exists $\xi\in\overline{\mathbb{Q}}{}_{p}^{\times}$ such that

[TABLE]

Proof.

According to Propositions 3.2, 4.4 and 4.5 one knows that $\mathcal{T}_{\operatorname{split}}$ is a $\varLambda$-sub-algebra of the amalgamated product $A=\varLambda_{\mathfrak{p}}\times_{\overline{\mathbb{Q}}_{p}}\left(\mathcal{R}^{\perp}\times_{\overline{\mathbb{Q}}_{p}[X]/(X^{r-1})}\varLambda_{\mathfrak{p}}\right)$ surjecting to the product of each two amongst the three factors. In particular $\mathcal{T}_{\operatorname{split}}$ contains $(X,b,0)$ and $(X,0,c)$, for certain $b\in\mathcal{R}^{\perp}$ and $c\in\varLambda_{\mathfrak{p}}$, hence their product $(X^{2},0,0)$, from which one deduces that

[TABLE]

As $\mathcal{T}_{\operatorname{split}}\subset\mathcal{T}_{\operatorname{split}}[U_{p}]\subset A$, one deduces that $\mathfrak{m}_{\mathcal{T}_{\operatorname{split}}}$ is given by the kernel of a $\overline{\mathbb{Q}}_{p}$-valued linear form on the maximal ideal $\mathfrak{m}_{A}$, which we will determine as precisely as possible in each case. As $U_{p}\notin\mathcal{T}_{\operatorname{split}}$ by Theorem 3.3(i), we deduce that the above linear form is non-zero and that $\mathcal{T}_{\operatorname{split}}[U_{p}]=A$, proving the first part of the theorem. Let us now write an equation for $\mathfrak{m}_{\mathcal{T}_{\operatorname{split}}}$ in each case.

(i) One sees exactly as in [6] that $\mathfrak{m}_{\mathcal{T}_{\operatorname{split}}}\supset(\mathfrak{m}_{\varLambda}^{2})^{3}$. Moreover the image of $\mathfrak{m}_{\mathcal{T}_{\operatorname{split}}}$ in $(\mathfrak{m}_{\varLambda}/\mathfrak{m}_{\varLambda}^{2})^{3}$ is necessarily a plane (as it surjects onto any two amongst the three factors). The precise equation follows from (18) and Proposition 3.2, in view of (20).

(ii) In this case $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})=0\neq\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})$, hence $e=1$ and $\mathfrak{m}_{\mathcal{T}_{\operatorname{split}}}$ contains the elements $(X,X,X)$, $(X^{2},0,0)$ and $(0,X^{r},0)$, generating an ideal which has co-dimension $2$ in the $\overline{\mathbb{Q}}_{p}$-vector space $\mathfrak{m}_{A}$. Again the surjectivity onto each two amongst the three factors shows that $\mathfrak{m}_{\mathcal{T}_{\operatorname{split}}}$ has co-dimension $1$ in $\mathfrak{m}_{A}$, hence satisfies a linear equation of the desired form.

(iii) We leave its proof, which is very similar to the above case, to the interested reader. ∎

Theorem 4.8**.**

One has $C^{0}_{\psi}=(X)$ if and only if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\neq 0$. More precisely

1. (i)

If $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\cdot\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})\neq 0$, then $\mathcal{T}=\varLambda_{\mathfrak{p}}\times_{\overline{\mathbb{Q}}_{p}}\mathcal{R}^{\perp}\times_{\overline{\mathbb{Q}}_{p}}\mathcal{R}^{\perp}\times_{\overline{\mathbb{Q}}_{p}}\varLambda_{\mathfrak{p}}.$ 2. (ii)

If $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})=0$, then $\mathcal{T}=\varLambda_{\mathfrak{p}}\times_{\overline{\mathbb{Q}}_{p}}\left(\varLambda[Z]/(Z^{2}-X^{r})\times_{\overline{\mathbb{Q}}_{p}[X]/(X^{r-1})}\varLambda_{\mathfrak{p}}\right)$ where the projection to the first component is given by $\pi_{\psi}:\mathcal{T}\twoheadrightarrow\varLambda_{\mathfrak{p}}$, $r\geqslant 3$ and the map $\varLambda[Z]/(Z^{2}-X^{r})\twoheadrightarrow\overline{\mathbb{Q}}_{p}[X]/(X^{r-1})$ is modulo the non-principal ideal $(Z,X^{r-1})$. In particular $C^{0}_{\psi^{\tau}}=(X^{r-1})$.

Proof.

The claim about the congruence ideal would follow from (i) and (ii).

(i) follows directly from Theorems 4.7(i)(iii) and 3.14.

(ii) By Theorem 4.7(ii) it suffices to show that $(0,Z,0)\in\mathcal{T}$. As $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})+\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})\neq 0$ we have $\mathcal{R}^{\perp}=\varLambda$ in this case. Recall the involution $\iota$ of $\widetilde{\mathcal{R}}^{\perp}$ fixing $\mathcal{R}^{\perp}=\varLambda$ and sending $Z$ to $-Z$.

For all $\gamma\in\Gamma_{K}$ one has

[TABLE]

hence $\operatorname{tr}(\rho^{\perp}_{\mathcal{R}})(\gamma\tau)\in Z\cdot\varLambda$. Finally, Proposition 3.13 implies that the $\varLambda$-sub-module of $\widetilde{\mathcal{R}}^{\perp}$ generated by $\operatorname{tr}(\tilde{\rho}^{\perp}_{\mathcal{R}})(\tau\Gamma_{K})$ contains $Z$. ∎

We conclude this subsection by determining the congruences ideal between $\mathcal{F}$ and $\Theta_{\psi}$, as well as the congruence ideal $C^{0}_{\mathcal{F}}$ attached to the projection of $\mathcal{T}$ on its $\mathcal{F}$-component.

Corollary 4.9**.**

1. (i)

Assume that $r\geqslant 3$ is odd.

Then $C^{0}_{\mathcal{F}}=(X^{r-1},Z)\subset\widetilde{\mathcal{R}}^{\perp}=\varLambda[Z]/(Z^{2}-X^{r})$. The congruence ideal between $\mathcal{F}$ and $\Theta_{\psi}$ is given by $(X^{r-1},Z)$, if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})=0$, and by $(X,Z)$, otherwise. 2. (ii)

Assume that $r\geqslant 2$ is even. Then $C^{0}_{\mathcal{F}}=(Y^{r-1})\subset\mathcal{R}^{\perp}=\overline{\mathbb{Q}}_{p}\lsem Y\rsem$. The congruence ideal between $\mathcal{F}$ and $\Theta_{\psi}$ is given by $(Y^{\frac{r}{2}})$, if $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})=0$, and by $(Y)$, otherwise.

4.3. The Katz $p$-adic $L$-function at $s=0$

Given an ideal $\mathfrak{c}$ of $K$ relatively prime to $p$, Katz constructed a measure $\mu_{\mathfrak{c}}$ on the ray class group $\mathscr{C}\ell_{K}(p^{\infty}\mathfrak{c})$ whose value on (the $p$-adic avatar of) a Hecke character $\varphi$ of conductor dividing $\mathfrak{c}$ and infinity type $(k_{1},k_{2})\in\mathbb{Z}_{\leqslant-1}\times\mathbb{Z}_{\geqslant 0}$ is given by (see [5, Thm. 3.1] based on [29] and [14])

[TABLE]

$\Omega_{\infty}$ (resp. $\Omega_{p}$) is a complex (resp. $p$-adic) period of the CM elliptic curve attached to $\bar{\mathfrak{c}}^{-1}$. The above interpolation formula uniquely characterizes $\mu_{\mathfrak{c}}$ as these characters are Zariski dense in $\operatorname{Spec}\mathcal{O}\lsem\mathscr{C}\ell_{K}(p^{\infty}\mathfrak{c})\rsem$. It does not apply however to finite order characters since their infinity type is $(0,0)$.

The Katz $p$-adic $L$-function of a finite order *non-trivial anti-cyclotomic * character $\varphi$ of conductor dividing $\mathfrak{c}$, is defined as the following continuous function on $\mathbb{Z}_{p}^{2}$

[TABLE]

where $\varepsilon_{\mathfrak{p}}$ is the character defined in (21) and $\varepsilon_{\bar{\mathfrak{p}}}=\varepsilon_{\mathfrak{p}}^{\tau}$. In other terms, $L_{p}(\varphi,s_{\mathfrak{p}},s_{\bar{\mathfrak{p}}})$ is the $p$-adic Mellin transform of the push-forward of $\mu_{\mathfrak{c}}$ by $\mathscr{C}\ell_{K}^{(p)}(p^{\infty}\mathfrak{c})_{/\operatorname{tor}}\hookrightarrow\mathbb{W}_{\mathfrak{p}}\times\mathbb{W}_{\bar{\mathfrak{p}}}$. As the function $s\mapsto z_{\mathfrak{p}}^{s}$ equals the analytic function $\exp_{p}(s\cdot p^{-h}\log_{p}(1+p^{\nu}))$ on $p^{h}\mathbb{Z}_{p}$, and as $\mathscr{C}\ell_{K}^{(p)}(p^{\infty})_{/\operatorname{tor}}\hookrightarrow\mathbb{W}_{\mathfrak{p}}\times\mathbb{W}_{\bar{\mathfrak{p}}}\xrightarrow{\sim}p^{-h}\mathbb{Z}_{p}\times p^{-h}\mathbb{Z}_{p}$ consists of pairs with difference in $\mathbb{Z}_{p}$, it follows that $L_{p}(\varphi,s_{\mathfrak{p}},s_{\bar{\mathfrak{p}}})$ is analytic on the set of $(s_{\mathfrak{p}},s_{\bar{\mathfrak{p}}})\in\mathbb{Z}_{p}^{2}$ such that $s_{\mathfrak{p}}+s_{\bar{\mathfrak{p}}}\in p^{h}\mathbb{Z}_{p}$, in particular it is locally analytic at $(0,0)$.

Furthermore Katz constructed a $1$-variable improved $p$-adic $L$-function $L_{p}^{*}(\varphi,s)$ such that

[TABLE]

(see [29, §7.2]) and proved a $p$-adic analogue of the Kronecker’s Second Limit Formula

[TABLE]

(see [29, Cor. 10.2.9], [28, Thm. 1.5.1]), where $\mathfrak{u}_{\varphi}\in(\mathcal{O}_{H}[\mathfrak{c}^{-1}]^{\times}\otimes\overline{\mathbb{Q}})[\varphi]=(\mathcal{O}_{H}^{\times}\otimes\overline{\mathbb{Q}})[\varphi]$ is a specific unit, defined using Robert units (see [28, (1.5.5)]).

Remark 4.10*.*

Assume the Heegner hypothesis for the ideal $\mathfrak{c}$, i.e, $\mathcal{O}_{K}/\mathfrak{c}\mathcal{O}_{K}\simeq\mathbb{Z}/C\mathbb{Z}$ and that $D\neq 3,4$. Let $Y_{1}(C)/\mathbb{Q}$ be the open modular curve of level $C$, $\zeta$ be a $C$-th root of the unity and $g_{\zeta}(q)=q^{\frac{1}{12}}(1-\zeta)\prod_{n>0}(1-q^{n}\zeta)(1-q^{n}\zeta^{-1})\in\mathcal{O}(Y_{1}(C))^{\times}$ be the Siegel unit associated to $\zeta$. The theory of Complex Multiplication implies that the elliptic curve $(\mathbb{C}/\bar{\mathfrak{c}}^{-1},C^{-1})$ defines a point of $Y_{1}(C)$ over the ray class field $K_{\mathfrak{c}}$ of conductor $\mathfrak{c}$ of $K$. The evaluation of the Siegel units $g_{\zeta}(q)$ at this point is an elliptic unit $u_{\zeta,\mathfrak{c}}\in\mathcal{O}_{K_{\mathfrak{c}}}^{\times}$, which can be used to define $\mathfrak{u}_{\varphi}$.

Suppose that $\varphi(\mathfrak{p})=1$. The Euler factor $(1-\varphi(\mathfrak{p}))$ in (43) vanishes yielding a trivial zero

[TABLE]

As $\log_{p}(\mathfrak{u}_{\varphi})\neq 0$ by the Baker–Brumer Theorem, it follows that this is the only case where a trivial zeros of the Katz $p$-adic $L$-function at $(0,0)$ can occur.

Remark 4.11*.*

When $\varphi$ is trivial, a case considered in [6] which we have excluded from the present paper, then one has $L_{p}(\mathbf{1},0,0)=\frac{-1}{2}(1-p^{-1})\log_{p}(u_{\bar{\mathfrak{p}}})\neq 0$, where $u_{\bar{\mathfrak{p}}}$ is a $\bar{\mathfrak{p}}$-unit of $K$ of minimal valuation. Note that there is no improved $p$-adic $L$-function is that case.

A first insight into what should be the leading term of $L_{p}(\varphi,s_{\mathfrak{p}},s_{\bar{\mathfrak{p}}})$ at the trivial zero point $L_{p}(\varphi,0,0)=0$, is obtained by differentiating (43)

[TABLE]

To gain a further intuition about the linear term, we consider the Katz anti-cyclotomic $p$-adic $L$-function $L_{p}^{-}(\varphi,s)=L_{p}(\varphi,s,-s)$.

As $\varphi$ is anti-cyclotomic of finite order, there exists a finite order Hecke character $\psi$ which can be chosen to have conductor $\mathfrak{c}_{\psi}$ relatively prime to $p$, such that $\varphi=\psi_{\mbox{-}}=\psi/\psi^{\tau}$.

Let $\zeta_{\psi_{\mbox{-}}}^{-}\in\Lambda_{\mathcal{O}}=\mathcal{O}\lsem\mathbb{W}_{\mathfrak{p}}\rsem$ be the power series corresponding to the push-forward of $\mu_{\mathfrak{c}_{\psi_{\mbox{-}}}}$ by

[TABLE]

where the last map is the $\psi$-projection (see §3.1). By definition $(\varepsilon_{\mathfrak{p}}^{s})(\zeta_{\psi_{\mbox{-}}}^{-})=\mu_{\mathfrak{c}_{\psi_{\mbox{-}}}}(\psi_{\mbox{-}}\varepsilon_{\mathfrak{p}}^{s}\varepsilon_{\bar{\mathfrak{p}}}^{-s})=L_{p}^{-}(\psi_{\mbox{-}},s)$ is analytic in $s\in\mathbb{Z}_{p}$. The vanishing (45) of $L_{p}^{-}(\psi_{\mbox{-}},0)$ implies $\zeta_{\psi_{\mbox{-}}}^{-}\in\ker\left(\mathcal{O}\lsem\mathbb{W}_{\mathfrak{p}}\rsem\to\mathcal{O}\right)$, hence the image of $\zeta_{\psi_{\mbox{-}}}^{-}$ under the localization at $(X)$ morphism $\mathcal{O}\lsem\mathbb{W}_{\mathfrak{p}}\rsem\hookrightarrow\varLambda_{\mathfrak{p}}$ lands in $(X)$.

Theorem 4.12** (Hida–Tilouine).**

The anti-cyclotomic $p$-adic $L$-function $\zeta^{-}_{\psi_{\mbox{-}}}$ divides the generator of the congruence ideal $C^{0}_{\psi}$ in $\varLambda_{\mathfrak{p}}$.

Proof.

Recall from §3.1 the one variable Iwasawa algebra $\Lambda_{\mathcal{O}}=\mathcal{O}\lsem\mathbb{W}_{\mathfrak{p}}\rsem$ and the Hecke character $\lambda_{\psi,k}$ corresponding to the classical theta series $\theta_{\psi,k}$ which is the specialization in weight $k\in\mathbb{Z}_{\geq 1}$ of the CM Hida family $\Theta_{\psi}$.

Let $H_{\psi}\in\Lambda_{\mathcal{O}}$ be the characteristic power series of the congruence module attached to $\Theta_{\psi}$ introduced by Hida and Tilouine in [27, (6.9)]. Given a Hecke finite order character $\phi$ of $K$ having a prime to $p$ conductor $\mathfrak{c}_{\phi}$ and taking values in $\mathcal{O}$, Hida constructed in [26, Thm. I] an element $\mathscr{D}$ of the fraction field of $\Lambda_{\mathcal{O}}\widehat{\otimes}\Lambda_{\mathcal{O}}$ such that $H_{\psi}\cdot\mathscr{D}\in\Lambda_{\mathcal{O}}\widehat{\otimes}\Lambda_{\mathcal{O}}$, satisfying for all $k>l\geqslant 2$ the interpolation

[TABLE]

up to a factor made explicit in [26, Thm. I], where $D_{p}$ is the Rankin-Selberg convolution without Euler factors at $p$, and $(\theta_{\psi,k},\theta_{\psi,k})$ is the Petersson inner product.

Let $L_{1}$ (resp. $L_{2}$, $L^{-}_{\psi}$) be the unique element of $\Lambda_{\mathcal{O}}\widehat{\otimes}\Lambda_{\mathcal{O}}$ whose specialization at $(\varepsilon_{\mathfrak{p}}^{k-1},\varepsilon_{\mathfrak{p}}^{l-1})$ is given by the value $\mu_{\mathfrak{c}_{\psi/\phi}}(\lambda_{\psi,k}\cdot(\lambda_{\phi,l})^{-1})$ (resp. $\mu_{\mathfrak{c}_{\psi/\phi^{\tau}}}(\lambda_{\psi,k}\cdot(\lambda_{\phi,l}^{\tau})^{-1})$, $\mu_{\mathfrak{c}_{\mbox{-}}}(\lambda_{\psi,k}\cdot(\lambda_{\psi,k}^{\tau})^{-1})$) of the Katz $p$-adic measure. Note that the measure used in [27] differs from $\mu_{\mathfrak{c}}$ by the involution $g\mapsto g^{-1}$ of $\mathscr{C}\ell_{K}(p^{\infty}\mathfrak{c})$. Note also that by definition $L^{-}_{\psi}$ and $H_{\psi}$ are elements of the embedding of $\Lambda_{\mathcal{O}}$ in $\Lambda_{\mathcal{O}}\widehat{\otimes}\Lambda_{\mathcal{O}}$ via the first coordinate.

Let $\Psi\in\Lambda_{\mathcal{O}}\widehat{\otimes}\Lambda_{\mathcal{O}}$ be the Euler product defined in [27, p.249 bottom]. By comparing their respective specializations at each $k>l\geqslant 2$, Hida and Tilouine showed in [27, Thm. 8.1] that the two elements $H_{\psi}\cdot\mathscr{D}$ and $\frac{\Psi L_{1}L_{2}H_{\psi}}{L^{-}_{\psi}}$ differ by a unit, i.e. their quotient belongs to $(\Lambda_{\mathcal{O}}\widehat{\otimes}\Lambda_{\mathcal{O}})[1/p]^{\times}$. The divisibility $L^{-}_{\psi}\mid H_{\psi}$ in $(\Lambda_{\mathcal{O}}\widehat{\otimes}\Lambda_{\mathcal{O}})[1/p]$ (hence in $\Lambda_{\mathcal{O}}[1/p]$) follows from [27, Thm. 8.2] where it is shown that $L_{1},L_{2}$ and $\Psi$ are all relatively prime to $L^{-}_{\psi}$.

The ideal $C^{0}_{\psi}$, resp. $(\zeta^{-}_{\psi_{\mbox{-}}})$, of $\varLambda_{\mathfrak{p}}$ is the localization of $(H_{\psi})$, resp. $L^{-}_{\psi}$ at the height one prime ideal $(X)$ of $\Lambda_{\mathcal{O}}$ corresponding to $f$, hence the theorem.

Alternatively, as we are only interested at establishing this divisibility in $\varLambda_{\mathfrak{p}}$ (i.e. after localizing at $(X)$), we can instead argue as follows. Choosing $\phi$ such that $\phi(\bar{\mathfrak{p}})\neq 1\neq\phi(\mathfrak{p})$, the formulas (43) and (44) imply the non-vanishing of $L_{1}$ and of $L_{2}$ when evaluated at $k=l=1$, and the same is true for $\Psi$ by the explicit formula defining it. ∎

The description of the congruence ideal in Theorem 4.8 has the following consequence.

Corollary 4.13**.**

Assume that $\mathscr{L}_{\mbox{-}}(\varphi)\neq 0$. Then $C^{0}_{\psi}=(\zeta^{-}_{\psi_{\mbox{-}}})=(X)$, in particular, the order of vanishing of the anti-cyclotomic Katz $p$-adic $L$-functions $L_{p}^{-}(\varphi,s)$ at $s=0$ is $1$.

This relates a conjecture of Hida on the adjoint $p$-adic $L$-function of the CM family $\Theta_{\psi}$ (see [27, p.192]) to the well-established Four Exponentials Conjecture in Transcendence Theory, and shows that it holds for more than half of the characters, i.e., for at least one character in each couple $\{\psi_{\mbox{-}},\psi_{\mbox{-}}^{\tau}\}$.

Combining Corollary 4.13 with (46) leads us to propose the following formula

[TABLE]

or equivalently, using Katz cyclotomic $p$-adic $L$-function $L_{p}(\varphi,s)=L_{p}(\varphi,s,s)$

[TABLE]

Remark 4.14*.*

In [4] Benois has formulated a precise conjecture on trivial zeros in a generality that covers the case of rank $2$ Artin motives which are critical in the sense of Deligne. His definition of a cyclotomic $\mathscr{L}$-invariant depends on a choice of a regular sub-module which, in the case of a Galois representation which is locally scalar at $p$, can be arbitrarily chosen. We expect however that there should exist a choice for which (49) concords with his conjecture. We should also mention a recent work [8] of Büyükboduk and Sakamoto on the leading term, under the additional hypotheses that $p$ is relatively prime both to the class number of $K$ and to the order of $\psi_{\mbox{-}}$, although their methods are totally different from ours.

For the rest of this section we assume that $\varphi=\psi_{\mbox{-}}$ is quadratic and we will check the compatibility between (49) and the Greenberg–Ferrero formula [17]. Denote by $F=\mathbb{Q}(\sqrt{d})$ the real quadratic subfield of $H$ and by $K^{\prime}\neq K$ its other imaginary quadratic subfield. Recall Gross’s factorization of the Katz cyclotomic $p$-adic $L$-function as product of two Kubota-Leopoldt $p$-adic $L$-functions (see [20], and also [19] for an arbitrary conductor)

[TABLE]

where $\omega_{p}$ is the Teichmüller character. Then $L_{p}(\epsilon_{K^{\prime}}\cdot\omega_{p},s)$ has a trivial zero at $s=0$ and

[TABLE]

where $\mathscr{L}(\epsilon_{K^{\prime}})$ is the cyclotomic $\mathscr{L}$-invariant of the odd Dirichlet character $\epsilon_{K^{\prime}}$ (see [6, (15)]), which turns out to be equal to $\mathscr{L}(\varphi)$. By the Class Number Formula $L(\epsilon_{K^{\prime}},0)=\tfrac{h_{K^{\prime}}}{|\mathcal{O}_{K^{\prime}}^{\times}{}_{/\{\pm 1\}}|}$.

On the other hand, Leopoldt’s formula yields that

[TABLE]

As $\sum_{a=1}^{d}\epsilon_{F}(a)\log_{p}(1-\zeta_{d}^{a})\in\mathcal{O}_{F}^{\times}[\epsilon_{F}]\subset\mathcal{O}_{H}^{\times}[\varphi]$, it follows that (49) holds in this case, up to a multiplication by an element of $\mathbb{Q}^{\times}$.

5. An $R=T$ modularity lifting theorem for non-Gorenstein local rings

The goal of this section is to introduce a universal ring $\mathcal{R}^{\triangle}$ representing deformations of $\rho_{f}$ which are generically ordinary, and show that its nilreduction $\mathcal{R}^{\triangle}_{\operatorname{red}}$ is isomorphic to $\mathcal{T}$.

5.1. Ordinary framed deformations of a $2$-dimensional trivial representation

The deformation functor $\mathcal{D}^{\Box}_{\operatorname{loc}}$ associating to $A$ in $\mathscr{C}$ the set of framed deformations $\rho_{A}:\Gamma_{\mathbb{Q}_{p}}\to\operatorname{GL}_{2}(A)$ of $\mathbf{1}_{2}$ is representable by $\mathcal{R}_{\operatorname{loc}}^{\Box}$ together with $\rho^{\Box}:\Gamma_{\mathbb{Q}_{p}}\to\operatorname{GL}_{2}(\mathcal{R}_{\operatorname{loc}}^{\Box})$.

Recall that $\rho_{A}\in\mathcal{D}^{\Box}_{\operatorname{loc}}(A)$ is ordinary if and only if there exists a $\Gamma_{\mathbb{Q}_{p}}$-stable direct factor $\operatorname{Fil}_{A}\subset A^{2}$ of rank $1$ over $A$ and such that $\Gamma_{\mathbb{Q}_{p}}$ acts on $A^{2}/\operatorname{Fil}_{A}$ by an unramified character $\chi_{A}$. Being ordinary is not a deformation condition on $\mathcal{D}^{\Box}_{\operatorname{loc}}$ as it does not satisfy Schlessinger’s Criterion (see for example [18, §3.1]).

Recall that for any $\overline{\mathbb{Q}}_{p}$-scheme $S$, an $S$-point of $\mathbb{P}^{1}$ is an invertible $\mathcal{O}_{S}$-sub-module $\operatorname{Fil}_{S}\subset\mathcal{O}_{S}^{2}$ which is locally a direct summand. We introduce a variant, when the residual characteristic is [math], of a functor defined in [18, §3.1].

Proposition 5.1**.**

Let $\mathcal{D}^{\triangle}_{\mathrm{loc}}$ be the subfunctor of $\mathbb{P}^{1}\times_{\overline{\mathbb{Q}}_{p}}\mathcal{R}^{\Box}_{\operatorname{loc}}\lsem U\rsem$ whose $A$-points correspond to a filtration $\operatorname{Fil}_{A}\in\mathbb{P}^{1}(A)$ and a morphism $\pi_{A}:\mathcal{R}^{\Box}_{\operatorname{loc}}\lsem U\rsem\to A$, such that $\operatorname{Fil}_{A}$ is $\Gamma_{\mathbb{Q}_{p}}$-stable for the action on $A^{2}$ defined via $\mathcal{R}^{\Box}_{\operatorname{loc}}\to\mathcal{R}^{\Box}_{\operatorname{loc}}\lsem U\rsem\overset{\pi_{A}}{\longrightarrow}A$ and moreover $A^{2}/\operatorname{Fil}_{A}$ is unramified with $\operatorname{Frob}_{p}$ acting by $\phi_{A}=\pi_{A}(1+U)\in A^{\times}$.

The functor $\mathcal{D}^{\triangle}_{\mathrm{loc}}$ is representable by a closed subscheme $\mathscr{X}\subset\mathbb{P}^{1}\times_{\overline{\mathbb{Q}}_{p}}\mathcal{R}^{\Box}_{\operatorname{loc}}\lsem U\rsem$.

Proof.

See [18, Lem. 3.1.1].∎

Let $\operatorname{Spec}\mathcal{R}^{\triangle}_{\operatorname{loc}}\subset\operatorname{Spec}\mathcal{R}^{\Box}_{\operatorname{loc}}\lsem U\rsem$ be the scheme theoretical image of $\mathscr{X}$ under the second projection $\operatorname{pr}_{2}:\mathbb{P}^{1}\times_{\overline{\mathbb{Q}}_{p}}\mathcal{R}^{\Box}_{\operatorname{loc}}\lsem U\rsem\to\operatorname{Spec}\mathcal{R}^{\Box}_{\operatorname{loc}}\lsem U\rsem$, that is to say $\operatorname{Spec}\mathcal{R}^{\triangle}_{\operatorname{loc}}$ is the Zariski closure of $\operatorname{pr}_{2}(\mathscr{X})$. As $\mathbb{P}^{1}$ is proper (hence universally closed) over $\overline{\mathbb{Q}}_{p}$ and as $\mathscr{X}$ is a closed subscheme of $\mathbb{P}^{1}\times_{\overline{\mathbb{Q}}_{p}}\mathcal{R}^{\Box}_{\operatorname{loc}}\lsem U\rsem$, the natural morphism $\operatorname{pr}_{2}:\mathscr{X}\to\operatorname{Spec}\mathcal{R}^{\triangle}_{\operatorname{loc}}$ is surjective.

[TABLE]

We denote by $\phi\in(\mathcal{R}^{\triangle}_{\operatorname{loc}})^{\times}$ the image of $(1+U)$ under the canonical surjection $\mathcal{R}^{\Box}_{\operatorname{loc}}\lsem U\rsem\twoheadrightarrow\mathcal{R}^{\triangle}_{\operatorname{loc}}$.

5.2. Generically ordinary deformations of $\rho_{f}$

Let $(e_{1},e_{2})$ be a basis of $\overline{\mathbb{Q}}{}_{p}^{2}$ in which $\rho_{f\mid\Gamma_{K}}=\left(\begin{smallmatrix}\psi&0\\ 0&\psi^{\tau}\end{smallmatrix}\right)$. Let $\rho^{\operatorname{univ}}:\Gamma_{\mathbb{Q}}\to\operatorname{GL}_{2}(\mathcal{R}_{\rho_{f}}^{\operatorname{univ}})$ be the universal deformation of $\rho_{f}$.

Let $\mathcal{R}^{\triangle}=\mathcal{R}^{\triangle}_{\operatorname{loc}}\widehat{\otimes}_{\mathcal{R}^{\Box}_{\operatorname{loc}}}\mathcal{R}_{\rho_{f}}^{\operatorname{univ}}$ and $\rho^{\triangle}_{\mathcal{R}}:\Gamma_{\mathbb{Q}}\to\operatorname{GL}_{2}(\mathcal{R}^{\triangle})$ be the corresponding universal deformation, where the natural morphism $\mathcal{R}^{\Box}_{\operatorname{loc}}\to\mathcal{R}_{\rho_{f}}^{\operatorname{univ}}$ comes from $\psi^{-1}\otimes\rho^{\operatorname{univ}}_{|\Gamma_{\mathbb{Q}_{p}}}$. We next show that the schematic points of $\operatorname{Spec}\mathcal{R}^{\triangle}$ correspond to ordinary representations.

Lemma 5.2**.**

Let $x:\mathcal{R}^{\triangle}\to L$ be a point over a field $L$. Then the corresponding representation $\rho_{x}:\Gamma_{\mathbb{Q}}\to\operatorname{GL}_{2}(L)$ has an unramified rank $1$ $\Gamma_{\mathbb{Q}_{p}}$-quotient on which $\operatorname{Frob}_{p}$ acts by $x(\phi\otimes 1)\in L^{\times}$.

Proof.

By definition of $\mathcal{R}^{\triangle}$, $x$ yields a point $y:\mathcal{R}^{\triangle}_{\operatorname{loc}}\to L$ such that $y(\phi)=x(\phi\otimes 1)\in L^{\times}$. As $\operatorname{pr}_{2}:\mathscr{X}\to\operatorname{Spec}\mathcal{R}^{\triangle}_{\operatorname{loc}}$ is surjective, we can lift $y$ to $y^{\prime}\in\mathscr{X}(L^{\prime})$, with $L^{\prime}$ a field containing $L$. By Proposition 5.1, $(\rho_{x}\otimes_{L}L^{\prime})_{\mid\Gamma_{\mathbb{Q}_{p}}}$ stabilizes a line $\operatorname{Fil}_{L^{\prime}}\subset(L^{\prime})^{2}$ and acts on the quotient $(L^{\prime})^{2}/\operatorname{Fil}_{L^{\prime}}$ by an unramified character sending $\operatorname{Frob}_{p}$ to $\operatorname{pr}_{2}(y^{\prime})(\phi)\in(L^{\prime})^{\times}$.

The commutativity of the diagram (55) implies that $\operatorname{pr}_{2}(y^{\prime})(\phi)=y(\phi)\in L$, and therefore $\operatorname{Fil}_{L^{\prime}}$ has an $L$-rational basis, and $\operatorname{Frob}_{p}$ acts on the quotient by $x(\phi\otimes 1)$ as requested. ∎

5.3. Modularity in the non-Gorenstein case

Recall that $\mathcal{T}$ denotes the completed local $\varLambda$-algebra of $\mathscr{E}$ at $f$. The restriction of the deformation $\rho_{\mathcal{T}}:\Gamma_{\mathbb{Q}}\to\operatorname{GL}_{2}(\mathcal{T})$ to $\Gamma_{\mathbb{Q}_{p}}$ yields a morphism $\mathcal{R}^{\Box}_{\operatorname{loc}}\lsem U\rsem\to\mathcal{T}$ sending $\psi(\mathfrak{p})(1+U)$ on $U_{p}$. Moreover, $\rho_{\mathcal{T}}\otimes Q(\mathcal{T})$ is ordinary at $p$, yielding a $Q(\mathcal{T})$-point of $\mathscr{X}$, hence, in view of (55), we have a following commutative diagram

[TABLE]

Since $\mathcal{T}$ is reduced, the above diagram can be commutatively completed with a morphism $\mathcal{R}^{\triangle}_{\operatorname{loc}}\to\mathcal{T}$. By (24) there exists a homomorphism $\mathcal{R}_{\rho_{f}}^{\operatorname{univ}}\to\mathcal{T}$. As $\mathcal{T}$ is topologically generated over $\varLambda$ by the image of $\operatorname{tr}(\rho^{\operatorname{univ}})$ and by $U_{p}$, we obtain a natural surjective $\varLambda$-algebra homomorphism

[TABLE]

Theorem 5.3**.**

The morphism (56) induces an isomorphism $\mathcal{R}^{\triangle}_{\operatorname{red}}\xrightarrow{\sim}\mathcal{T}$, where $\mathcal{R}^{\triangle}_{\operatorname{red}}$ is the nilreduction of $\mathcal{R}^{\triangle}$. In particular $\mathcal{R}^{\triangle}$ is equidimensional of dimension $1$.

Proof.

It suffices to show that for every minimal prime ideal $\mathfrak{q}$ of $\mathcal{R}^{\triangle}$ there exists a surjective homomorphism $\mathcal{T}\twoheadrightarrow\mathcal{R}^{\triangle}/\mathfrak{q}$ of $\mathcal{R}^{\triangle}$-algebras. In fact, it will then follow that the kernel of $\mathcal{R}^{\triangle}\twoheadrightarrow\mathcal{T}$ is contained in the nilradical of $\mathcal{R}^{\triangle}$ (which is given by the intersection of its primes). As $\mathcal{T}$ is reduced, the desired isomorphism $\mathcal{R}^{\triangle}_{\operatorname{red}}\xrightarrow{\sim}\mathcal{T}$ would follow.

To prove the above claim, we let $\mathcal{A}=\mathcal{R}^{\triangle}/\mathfrak{q}\neq\overline{\mathbb{Q}}_{p}$ and consider the push-forward $\rho_{\mathcal{A}}:\Gamma_{\mathbb{Q}}\to\operatorname{GL}_{2}(\mathcal{A})$ of $\rho_{\mathcal{R}}^{\triangle}$ along $\mathcal{R}^{\triangle}\twoheadrightarrow\mathcal{A}$. As $Q(\mathcal{A})$ is a field, Lemma 5.2 implies that $\rho_{\mathcal{A}}\otimes Q(\mathcal{A})$ has an unramified free rank $1$ $\Gamma_{\mathbb{Q}_{p}}$-quotient on which $\operatorname{Frob}_{p}$ acts by the image of $\phi\otimes 1$ in $\mathcal{A}^{\times}$. Note that $\rho_{\mathcal{A}}$ need not be $p$-ordinary a priori (a posteriori this will be the case when $\mathcal{A}$ is a discrete valuation ring). One distinguishes two cases

• •

$\rho_{\mathcal{A}\mid\Gamma_{K}}$ is reducible. Exactly as in the proof of Proposition 3.2 one shows that $\rho_{\mathcal{A}}\simeq\operatorname{Ind}^{\mathbb{Q}}_{K}\psi_{\mathcal{A}}$, where $\psi_{\mathcal{A}}:\Gamma_{K}\to\mathcal{A}^{\times}$ is a character lifting $\psi$. The $\mathfrak{p}$-ordinariness of $\rho_{\mathcal{A}}\otimes Q(\mathcal{A})$ implies that $\psi^{-1}\psi_{\mathcal{A}}$ is equal either to $\chi_{\mathfrak{p}}$ or to $\chi_{\mathfrak{p}}^{\tau}$. In both cases $\mathcal{T}\twoheadrightarrow\mathcal{A}=\varLambda$.

• •

$\rho_{\mathcal{A}\mid\Gamma_{K}}$ is irreducible. As in Proposition 3.6, one may construct a point of $\mathcal{D}^{\perp}(\mathcal{A})$, i.e. a homomorphism of integral domains $\mathcal{R}^{\perp}\to\mathcal{A}$, which is necessarily injective as $\mathcal{R}^{\perp}$ is a discrete valuation ring. It follows from Theorem 3.14 that either $\mathcal{A}\simeq\mathcal{R}^{\perp}$ when $r$ is even, or $\mathcal{A}\simeq\widetilde{\mathcal{R}}^{\perp}$ when $r$ is odd, hence in all cases $\mathcal{T}\twoheadrightarrow\mathcal{T}^{\perp}\twoheadrightarrow\mathcal{A}$. ∎

The following lemma in commutative algebra justifies the title of the section.

Proposition 5.4**.**

*The dimension of the $\overline{\mathbb{Q}}_{p}$-vector space $\mathcal{T}/(\mathfrak{m}_{\varLambda},\mathfrak{m}_{\mathcal{T}}^{2})$ is $4$.

The eigencurve $\mathscr{E}$ is not Gorenstein at $f$.*

Proof.

The $1$-dimensional reduced local ring $\mathcal{T}$ is Gorenstein if and only if its quotient $\mathcal{T}/a\mathcal{T}$ by a regular element $a\in\mathcal{T}$ is Gorenstein. As $\mathcal{T}/a\mathcal{T}$ is Artinian, the claim is equivalent to showing that its socle $\mathcal{M}=\operatorname{Hom}_{\mathcal{T}}(\overline{\mathbb{Q}}_{p},\mathcal{T}/a\mathcal{T})$ is $1$-dimensional.

Assume first that $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\cdot\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})\neq 0$ and let $a=(X,Y,Y,X)$, where $Y$ denotes a uniformizer of $\mathcal{R}^{\perp}$. By Theorem 4.8 one has $\mathcal{T}=\varLambda\times_{\overline{\mathbb{Q}}_{p}}\mathcal{R}^{\perp}\times_{\overline{\mathbb{Q}}_{p}}\mathcal{R}^{\perp}\times_{\overline{\mathbb{Q}}_{p}}\varLambda$ hence $\mathfrak{m}_{\mathcal{T}}^{2}=a\cdot\mathfrak{m}_{\mathcal{T}}$ and $\mathcal{M}=\mathfrak{m}_{\mathcal{T}}/a\mathcal{T}$. It follows that $\dim(\mathcal{M})=\dim(\mathfrak{m}_{\mathcal{T}}/(a,\mathfrak{m}_{\mathcal{T}}^{2}))=\dim(\mathfrak{m}_{\mathcal{T}}/\mathfrak{m}_{\mathcal{T}}^{2})-1=\dim(\mathfrak{m}_{\mathcal{T}}/(\mathfrak{m}_{\varLambda},\mathfrak{m}_{\mathcal{T}}^{2}))=3>1$, hence $\mathcal{T}$ is not Gorenstein.

Assume next that $\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})\cdot\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}}^{\tau})=0$ and we let $a=(X,X,X)$. Then $\mathcal{T}\simeq\varLambda\times_{\overline{\mathbb{Q}}_{p}}\mathcal{A}$, with

[TABLE]

As the $\overline{\mathbb{Q}}_{p}$-vector space $\mathfrak{m}_{\mathcal{A}}/\mathfrak{m}_{\mathcal{A}}^{2}$ has basis $\{(X,X),(Z,0),(0,X^{r-1})\}$, one computes that $\mathfrak{m}_{\mathcal{A}}^{2}=\mathfrak{m}_{\varLambda}\mathfrak{m}_{\mathcal{A}}$ and $\mathfrak{m}_{\mathcal{T}}^{2}=\mathfrak{m}_{\varLambda}\mathfrak{m}_{\mathcal{T}}$. Again $\mathcal{M}=\mathfrak{m}_{\mathcal{T}}/\mathfrak{m}_{\varLambda}\mathcal{T}$ has dimension $3$, hence $\mathcal{T}$ is not Gorenstein.

In both cases $\mathfrak{m}_{\mathcal{T}}/(\mathfrak{m}_{\varLambda},\mathfrak{m}_{\mathcal{T}}^{2})$ is $3$-dimensional, hence $\mathcal{T}/(\mathfrak{m}_{\varLambda},\mathfrak{m}_{\mathcal{T}}^{2})$ is $4$-dimensional. ∎

We close this paper by establishing Conjecture 4.1 from [12] about the generalized eigenspace $S_{1}^{\dagger}(N)\lsem f\rsem=S_{1}^{\dagger}(N)\lsem\mathfrak{m}_{\mathcal{T}}\rsem$ attached to $f$ inside the space of weight $1$, level $N$, ordinary $p$-adic modular forms. To be more precise, Darmon, Lauder and Rotger only consider the subspace $S_{1}^{\dagger}(N)[\mathfrak{m}_{\mathcal{T}}^{2}]$. The subspace of classical forms $S_{1}(Np)[\mathfrak{m}_{\mathcal{T}}^{2}]$ is two-dimensional having $\{f,\theta_{\psi}\}$ as basis, and a supplement of this space is given by the space $S_{1}^{\dagger}(N)[\mathfrak{m}_{\mathcal{T}}^{2}]_{0}$ of normalized generalized eigenforms, i.e. forms whose first and $p$-th Fourier coefficients both vanish.

Corollary 5.5**.**

*The $\overline{\mathbb{Q}}_{p}$-vector space $S_{1}^{\dagger}(N)[\mathfrak{m}_{\mathcal{T}}^{2}]$ has dimensions $4$.

Moreover Conjecture 4.1 from [12] holds in the CM case, i.e. the two-dimensional vector space $S_{1}^{\dagger}(N)[\mathfrak{m}_{\mathcal{T}}^{2}]_{0}$ is canonically isomorphic to $\operatorname{H}^{1}(\mathbb{Q},\operatorname{ad}^{0}(\rho_{f}))$.*

Proof.

The first claim is merely a translation of the first claim of Proposition 5.4, in view of the duality (26). It follows that $\dim S_{1}^{\dagger}(N)[\mathfrak{m}_{\mathcal{T}}^{2}]_{0}=4-2=2$ and the conjecture then follows from [3, Lem. 3.2] asserting that $\dim\operatorname{H}^{1}(\mathbb{Q},\operatorname{ad}^{0}(\rho_{f}))=2$ as well. ∎

Acknolwedgements: The authors are mostly indebted to H. Hida and J. Tilouine for numerous discussions related to this project. In addition, the second author would like to thank M. Waldschmidt for an extremely helpful correspondence in Transcendence Theory and D. Benois for explaining his conjectures on trivial zeros. The first author acknowledges support from the EPSRC (grant EP/R006563/1) and the START-Prize Y966 of the Austrian Science Fund (FWF). The second author is partially supported by the Agence Nationale de la Recherche (grants ANR-18-CE40-0029 and ANR-16-IDEX-0004). Finally, both authors thank the anonymous referees for their carefully reading of the manuscript.

Glossary of notation

$\mathscr{C}\ell_{K}$ ideal class group of $K$

$C^{0}_{\psi}$ congruence ideal of $\Theta_{\psi}$, §4.2

$C^{0}_{\mathcal{F}}$ congruence ideal of $\mathcal{F}$, §4.2

$C$ Galois group of $H/K$, §1

$-D$ fundamental discriminant of $K$

$e$ ramification index of $\varLambda\to\mathcal{R}^{\perp}$, Def. 3.8

$\mathscr{E}$ eigencurve of tame level $N$, §3

$\mathscr{E}^{\perp}$ union of non-CM (by K) irred. components, §3.2

$f$ the unique $p$-stabilisation of $\theta_{\psi}$

$\operatorname{Frob}_{\mathfrak{p}}$ arithmetic Frobenius in $\Gamma_{K_{\mathfrak{p}}}$

$\mathcal{F}$ eigenfamily without CM by $K$ containing $f$, §3.4

$G$ Galois group of $H/\mathbb{Q}$, §1

$H$ splitting field of $\psi_{\mbox{-}}$, §1.1

$\mathrm{I}_{\mathfrak{p}}$ inertia subgroup of $\Gamma_{K_{\mathfrak{p}}}$

$K$ imaginary quadratic field

$K_{\infty}^{-}$ anti-cyclotomic $\mathbb{Z}_{p}$-extension of $K$, §4.1

$L_{p}^{-}$ anti-cyclotomic Katz $p$-adic $L$-function, §4.3

$\mathscr{L}_{\mathfrak{p}}$ $\mathscr{L}$-invariant (7)

$\mathscr{L}(\psi_{\mbox{-}})$ cyclotomic $\mathscr{L}$-invariant (6)

$\mathscr{L}_{\mbox{-}}(\psi_{\mbox{-}})$ anti-cyclotomic $\mathscr{L}$-invariant, Def. 1.9

$\mathfrak{p}$ place of $K$ above $p$ induced by $\iota_{p}$

$r$ valuation of the reducibility ideal of $\mathcal{R}^{\perp}$

$\mathcal{R}_{\operatorname{loc}}^{\Box}$ framed deformation ring, §5.1

$\mathcal{R}^{\operatorname{univ}}_{\rho}$ deformation ring with $\tau$-invariant trace, §2.1

$\mathcal{R}^{\operatorname{ord}}_{\rho}$ $\mathfrak{p}$-ordinary deformation ring of $\rho$, Def. 2.1

$\mathcal{R}^{\operatorname{ord}}_{\rho^{\prime}}$ $\mathfrak{p}$-ordinary deformation ring of $\rho^{\prime}$, §2.3

$\mathcal{R}^{\operatorname{red}}_{\rho}$ reducible $\mathfrak{p}$-ordinary deformation ring, Def. 2.3

$\mathcal{R}^{K}_{\rho}$ CM $\mathfrak{p}$-ordinary deformation ring, Def. 2.3

$\mathcal{R}^{\perp}$ non-CM $\mathfrak{p}$-ordinary deformation ring, Def. 2.6

$\widetilde{\mathcal{R}}^{\perp}$ $\mathcal{R}^{\perp}[Z]/(Z^{2}-Y^{r})$, §3.3

$\mathcal{R}^{\triangle}$ generically ordinary deformation ring of $\rho_{f}$, §5.1

$\mathscr{S}(\psi_{\mbox{-}})$ slope of $\psi_{\mbox{-}}$, (3)

$S_{1}^{\dagger}(N)\lsem f\rsem$ generalised eigenspace of $f$, §3.2

$\mathcal{T}$ completed local ring of $\mathscr{E}$ at $f$, §3.2

$\mathcal{T}^{\perp}$ completed local ring of $\mathscr{E}^{\perp}$ at $f$, §3.2

$\mathcal{T}_{\operatorname{split}}\subset\mathcal{T}$ generated by $T_{\ell}$ for $\ell\nmid Np$ split in $K$, §3.2

$\mathcal{T}^{\perp}_{\operatorname{split}}$ image of the $\varLambda$-algebra $\mathcal{T}_{\operatorname{split}}$ in $\mathcal{T}^{\perp}$, §3.2

$u_{\psi_{\mbox{-}}}$ $\psi_{\mbox{-}}$–isotypic unit in $\mathcal{O}_{H}^{\times}\otimes\bar{\mathbb{Q}}$, §1.1

$u_{\mathfrak{p},\psi_{\mbox{-}}}$ $\psi_{\mbox{-}}$-isotypic $\mathfrak{p}$-unit in $\mathcal{O}_{H}[1/\mathfrak{p}]^{\times}\otimes\bar{\mathbb{Q}}$ §1.2

$u_{\mathfrak{p}}$ $\mathfrak{p}$-unit of $K$

$v_{0}$ place of $H$ above $\mathfrak{p}$ induced by $\iota_{p}$

$\mathcal{W}$ the weight space, §3

$\mathbb{W}_{\mathfrak{p}}$ Galois group of $\mathfrak{p}$-ramified $\mathbb{Z}_{p}$-extension, §3.1

$X$ uniformizer of $\varLambda$, $\varLambda_{\mathfrak{p}}$ §3.1

$Y$ uniformizer of $\mathcal{R}^{\perp}$, §3

$\gamma_{0}$ fixed element of $\Gamma_{K}$, §2

$\Gamma_{L}$ absolute Galois group of a perfect field $L$

$\Gamma_{K/\mathbb{Q}}$ Galois group of $K/\mathbb{Q}$

$\Gamma_{K_{\mathfrak{p}}}$ decomposition subgroup of $\Gamma_{K}$

$\zeta^{-}_{\psi_{\mbox{-}}}$ $p$-adic inverse Mellin transform of $L_{p}^{-}(\psi_{\mbox{-}},\cdot)$ §4.3

$\eta_{\mathfrak{p}},\eta_{\bar{\mathfrak{p}}}$ basis of $\operatorname{H}^{1}(K,\overline{\mathbb{Q}}_{p})$ §1.2

$[\eta],\eta$ canonical basis of $\operatorname{H}^{1}(K,\psi_{\mbox{-}})$, Prop. 1.3, §2

$[\eta^{\prime}],\eta^{\prime}$ canonical basis of $\operatorname{H}^{1}(K,\psi_{\mbox{-}}^{\tau})$, §2.3

$\theta_{\psi}$ weight one theta-series attached to $\psi$

$\Theta_{\psi},\Theta_{\psi^{\tau}}$ theta families specializing to $f$, §3.1

$\iota_{p}$ embedding of $\bar{\mathbb{Q}}$ in $\bar{\mathbb{Q}}_{p}$

$\kappa$ weight map, §3

$\varLambda$ completed local ring of $\mathcal{W}$ at $\kappa(f)$, §3

$\varLambda_{\mathfrak{p}}$ completed local ring of $\Theta_{\psi}$-component at $f$, §3

$\pi_{\psi}$ $\Theta_{\psi}$-projection, §4.2

$\rho_{f}=\operatorname{Ind}^{\mathbb{Q}}_{K}\psi$ Artin representation associated to $f$

$\rho_{\mathcal{R}}$ ordinary deformation of $\rho$ to $\mathcal{R}^{\operatorname{ord}}_{\rho}$, §2.1

$\rho_{\mathcal{R}^{\perp}}$ $\mathfrak{p}$-ordinary deformations of $\rho$ to $\mathcal{R}^{\perp}$, §2.3

$\rho^{\prime}_{\mathcal{R}^{\perp}}$ $\mathfrak{p}$-ordinary deformations of $\rho^{\prime}$ to $\mathcal{R}^{\perp}$, §2.3

$\widetilde{\rho}^{\perp}_{\mathcal{R}}$ extension of $\rho_{\mathcal{R}^{\perp}}$ to $\Gamma_{\mathbb{Q}}$, Prop. 3.11

$\rho_{\mathcal{T}^{\perp}}$ deformation of $\rho$ to $\mathcal{T}^{\perp}$, §3.2

$\rho^{\prime}_{\mathcal{T}^{\perp}}$ deformation of $\rho^{\prime}$ to $\mathcal{T}^{\perp}$, §3.2

$\rho_{\mathcal{T}}$ deformation of $\rho_{f}$ to $\mathcal{T}$, §3.2

$\rho_{\mathcal{T}}^{\perp}$ deformation of $\rho_{f}$ to $\mathcal{T}^{\perp}$, §3.2

$\tau$ complex conjugation, §1

$\epsilon_{K}$ Dirichlet character of $K/\mathbb{Q}$, §1

$\epsilon_{\mathfrak{p}}$ $\mathfrak{p}$-ramified character of type $(-1,0)$, §3.1

$\chi_{\mathcal{R}}$ ordinary character, §2.3

$\chi_{p}$ universal cyclotomic character, §3.1

$\chi_{\mathfrak{p}}$ universal $\mathfrak{p}$-ramified character, §3.1

$\chi_{\mbox{-}}$ universal anti-cyclotomic character, §4.1

$\psi,\psi^{\tau},\psi_{\mbox{-}}$ finite order Hecke characters of $K$, §1

$\varphi$ anti-cyclotomic Hecke character of $K$, §4.3