Big Image of Galois Representations Associated with Finite Slope p-adic Families of Modular Forms

Abstract

We prove that the Lie algebra of the image of the Galois representation associated with a finite slope family of modular forms contains a congruence subalgebra of a certain level. We interpret this level in terms of congruences with CM forms.

The work of Conti and Tilouine is supported by the Programs ArShiFo ANR-10-BLAN-0114 and PerColaTor ANR-14-CE25-0002-01.

1 Introduction

Let f be a non-CM cuspidal eigenform and let $\ell$ be a prime integer. By the work of Ribet [15, 17] and Momose [13], it is known that the $\ell$-adic Galois representation $\rho _{f,\ell }$ associated with f has large image for every $\ell$ and that for almost every $\ell$ it satisfies

(cong${}_\ell$) $\mathrm {Im}\,\rho _{f,\ell }$ contains the conjugate of a principal congruence subgroup $\Gamma (\ell ^m)$ of $\mathrm {SL}_2(\mathbb Z_\ell )$.

For instance if $\mathrm {Im}\,\rho _{f,\ell }$ contains an element with eigenvalues in $\mathbb Z_\ell ^\times$ distinct modulo $\ell$ then (cong${}_\ell$) holds.

In [9], Hida proved an analogous statement for p-adic families of non-CM ordinary cuspidal eigenforms, where p is any odd prime integer. We fix once and for all an embedding $\overline{\mathbb Q}\hookrightarrow \overline{\mathbb Q}_p$, identifying $\mathrm {Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)$ with a decomposition subgroup $G_p$ of $\mathrm {Gal}(\overline{\mathbb Q}/\mathbb Q)$. We also choose a topological generator u of $\mathbb Z_p^\times$. Let $\Lambda =\mathbb Z_p[[T]]$ be the Iwasawa algebra and let $\mathfrak {m}=(p,T)$ be its maximal ideal. A special case of Hida’s first main theorem ([9, Theorem I]) is the following.

Theorem 1.1

Let $\mathbf{f}$ be a non-CM Hida family of ordinary cuspidal eigenforms defined over a finite extension $\mathbb I$ of $\Lambda$ and let $\rho _\mathbf{f}:\mathrm {Gal}(\overline{\mathbb Q}/\mathbb Q)\rightarrow \mathrm {GL}_2(\mathbb I)$ be the associated Galois representation. Assume that $\rho _\mathbf{f}$ is residually irreducible and that there exists an element d in its image with eigenvalues $\alpha ,\beta \in \mathbb Z_p^\times$ such that $\alpha ^2\not \equiv \beta ^2\pmod {p}$. Then there exists a nonzero ideal ${\mathfrak l}\subset \Lambda$ and an element $g\in \mathrm {GL}_2(\mathbb I)$ such that

$$ g\Gamma ({\mathfrak l})g^{-1}\subset \mathrm {Im}\,\rho _\mathbf{f}, $$

where $\Gamma ({\mathfrak l})$ denotes the principal congruence subgroup of $\mathrm {SL}_2(\Lambda )$ of level ${\mathfrak l}$.

Under mild technical assumptions it is also shown in [9, Theorem II] that if the image of the residual representation of $\rho _\mathbf{f}$ contains a conjugate of $\mathrm {SL}_2(\mathbb F_p)$ then ${\mathfrak l}$ is trivial or $\mathfrak {m}$-primary, and if the residual representation is dihedral “of CM type” the height one prime factors P of ${\mathfrak l}$ are exactly those of the g.c.d. of the adjoint p-adic L function of $\mathbf{f}$ and the anticyclotomic specializations of Katz’s p-adic L functions associated with certain Hecke characters of an imaginary quadratic field. This set of primes is precisely the set of congruence primes between the given non-CM family and the CM families.

In her Ph.D. dissertation (see [12]), J. Lang improved on Hida’s Theorem I. Let $\mathbb T$ be Hida’s big ordinary cuspidal Hecke algebra; it is finite and flat over $\Lambda$. Let $\mathrm {Spec}\,\mathbb I$ be an irreducible component of $\mathbb T$. It corresponds to a surjective $\Lambda$-algebra homomorphism $\theta :\mathbb T\rightarrow \mathbb I$ (a $\Lambda$-adic Hecke eigensystem). We also call $\theta$ a Hida family. Assume that it is not residually Eisenstein. It gives rise to a residually irreducible continuous Galois representation $\rho _\theta :G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I)$ that is p-ordinary. We suppose for simplicity that $\mathbb I$ is normal. Consider the $\Lambda$-algebra automorphisms $\sigma$ of $\mathbb I$ for which there exists a finite order character $\eta _\sigma :G_\mathbb Q\rightarrow \mathbb I^\times$ such that for every prime $\ell$ not dividing the level, (see [12, 17]). These automorphisms form a finite abelian 2-group $\Gamma$. Let $\mathbb I_0$ be the subring of $\mathbb I$ fixed by $\Gamma$. Let $H_0=\bigcap _{\sigma \in \Gamma }\ker \,\eta _\sigma$; it is a normal open subgroup of $G_\mathbb Q$. One may assume, up to conjugation by an element of $\mathrm {GL}_2(\mathbb I)$, that $\rho _\theta \vert _{H_0}$ takes values in $\mathrm {GL}_2(\mathbb I_0)$.

Theorem 1.2

[12, Theorem 2.4] Let $\theta :\mathbb T\rightarrow \mathbb I$ be a non-CM Hida family such that $\overline{\rho }_\theta$ is absolutely irreducible. Assume that $\overline{\rho }_\theta \vert _{H_0}$ is an extension of two distinct characters. Then there exists a nonzero ideal ${\mathfrak l}\subset \mathbb I_0$ and an element $g\in \mathrm {GL}_2(\mathbb I)$ such that

$$g\Gamma ({\mathfrak l})g^{-1}\subset \mathrm {Im}\,\rho _{\theta },$$

where $\Gamma ({\mathfrak l})$ denotes the principal congruence subgroup of $\mathrm {SL}_2(\mathbb I_0)$ of level ${\mathfrak l}$.

For all of these results it is important to assume the ordinarity of the family, as it implies the ordinarity of the Galois representation and in particular that some element of the image of inertia at p is conjugate to the matrix

$$C_T=\begin{pmatrix} u^{-1}(1+T) &{} *\\ 0 &{} 1 \end{pmatrix}.$$

Conjugation by the element above defines a $\Lambda$-module structure on the Lie algebra of a pro-p subgroup of $\mathrm {Im}\,\rho _{\theta }$ and this is used to produce the desired ideal ${\mathfrak l}$. Hida and Lang use Pink’s theory of Lie algebras of pro-p subgroups of $\mathrm {SL}_2(\mathbb I)$.

In this paper we propose a generalization of Hida’s work to the finite slope case. We establish analogues of Hida’s Theorems I and II. These are Theorems 6.2, 7.1 and 7.4 in the text. Moreover, we put ourselves in the more general setting considered in Lang’s work. In the positive slope case the existence of a normalizing matrix analogous to $C_T$ above is obtained by applying relative Sen theory ([19, 21]) to the expense of extending scalars to the completion $\mathbb C_p$ of an algebraic closure of $\mathbb Q_p$.

More precisely, for every $h\in (0, \infty )$, we define an Iwasawa algebra $\Lambda _h=\mathcal {O}_h[[t]]$ (where $t=p^{-s_h}T$ for some $s_h\in \mathbb Q\cap ] \frac{1}{p-1}, \infty [$ and $\mathcal {O}_h$ is a finite extension of $\mathbb Z_p$ containing $p^{s_h}$ such that its fraction field is Galois over $\mathbb Q_p$) and a finite torsion free $\Lambda _h$-algebra $\mathbb T_h$ (see Sect. 3.1), called an adapted slope $\leqslant h$ Hecke algebra. Let $\theta :\mathbb T_h\rightarrow \mathbb I^\circ$ be an irreducible component; it is finite and torsion-free over $\Lambda _h$. The notation $\mathbb I^\circ$ is borrowed from the theory of Tate algebras, but $\mathbb I^\circ$ is not a Tate or an affinoid algebra. We write $\mathbb I=\mathbb I^\circ [p^{-1}]$. We assume for simplicity that $\mathbb I^\circ$ is normal. The finite slope family $\theta$ gives rise to a continuous Galois representation $\rho _\theta :G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I^\circ )$. We assume that the residual representation $\overline{\rho _\theta }$ is absolutely irreducible. We introduce the finite abelian 2-group $\Gamma$ as above, together with its fixed ring $\mathbb I_0$ and the open normal subgroup $H_0\subset G_\mathbb Q$. In Sect. 5.1 we define a ring $\mathbb B_{r}$ (with an inclusion $\mathbb I_0\hookrightarrow \mathbb B_r$) and a Lie algebra ${\mathfrak {H}}_{r}\subset {\mathfrak {sl}}_2(\mathbb B_{r})$ attached to the image of $\rho _\theta$. In the positive slope case CM families do not exist (see Sect. 3.3) hence no “non-CM” assumption is needed in the following. As before we can assume, after conjugation by an element of $\mathrm {GL}_2(\mathbb I^\circ )$, that $\rho _\theta (H_0)\subset \mathrm {GL}_2(\mathbb I_0^\circ )$. Let $P_1\subset \Lambda _h$ be the prime $(u^{-1}(1+T)-1)$.

Theorem 1.3

(Theorem 6.2) Let $\theta :\mathbb T_h\rightarrow \mathbb I^\circ$ be a positive slope family such that $\overline{\rho }_\theta \vert _{H_0}$ is absolutely irreducible. Assume that there exists $d\in \rho _\theta (H_0)$ with eigenvalues $\alpha ,\beta \in \mathbb Z_p^\times$ such that $\alpha ^2\not \equiv \beta ^2\pmod {p}$. Then there exists a nonzero ideal ${\mathfrak l}\subset \mathbb I_0[P_1^{-1}]$ such that

$$ {\mathfrak {l}}\cdot {\mathfrak {sl}}_2(\mathbb B_{r})\subset {\mathfrak {H}}_r. $$

The largest such ideal ${\mathfrak {l}}$ is called the Galois level of $\theta$.

We also introduce the notion of fortuitous CM congruence ideal for $\theta$ (see Sect. 3.4). It is the ideal ${\mathfrak {c}}\subset \mathbb I$ given by the product of the primary ideals modulo which a congruence between $\theta$ and a slope $\leqslant h$ CM form occurs. Following the proof of Hida’s Theorem II we are able to show (Theorem 7.1) that the set of primes of $\mathbb I_0=\mathbb I_0^\circ [p^{-1}]$ containing ${\mathfrak {l}}$ coincides with the set of primes containing ${\mathfrak {c}}\cap \mathbb I_0$, except possibly for the primes of $\mathbb I_0$ above $P_1$ (the weight 1 primes).

Several generalizations of the present work are currently being studied by one of the authors.¹ They include a generalization of [10], where the authors treated the ordinary case for $\mathrm {GSp}_4$ with a residual representation induced from the one associated with a Hilbert modular form, to the finite slope case and to bigger groups and more types of residual representations.

Acknowledgements. This paper owes much to Hida’s recent paper [9]. We also thank Jaclyn Lang for making her dissertation [12] available to us and for some very useful remarks pertaining to Sect. 4. We thank the referee of this article for the careful reading of the manuscript and for useful suggestions which hopefully led to improvements.

2 The Eigencurve

2.1 The Weight Space

Fix a prime integer $p>2$. We call weight space the rigid analytic space over $\mathbb Q_p$, $\mathcal {W}$, canonically associated with the formal scheme over $\mathbb Z_p$, $\mathrm{Spf}(\mathbb Z_p[[\mathbb Z_p^\times ]])$. The $\mathbb C_p$-points of $\mathcal {W}$ parametrize continuous homomorphisms $\mathbb Z_p^\times \rightarrow \mathbb C_p^\times$.

Let X be a rigid analytic space defined over some finite extension $L/\mathbb Q_p$. We say that a subset S of $X(\mathbb C_p)$ is Zariski-dense if the only closed analytic subvariety Y of X satisfying $S\subset Y(\mathbb C_p)$ is X itself.

For every $r>0$, we denote by $\mathcal {B}(0,r)$, respectively $\mathcal {B}(0,r^-)$, the closed, respectively open, disc in $\mathbb C_p$ of centre 0 and radius r. The space $\mathcal {W}$ is isomorphic to a disjoint union of $p-1$ copies of the open unit disc $\mathcal {B}(0,1^-)$ centre in 0 and indexed by the group $\mathbb Z/(p-1)\mathbb Z=\widehat{\mu }_{p-1}$. If u denotes a topological generator of $1+p\mathbb Z_p$, then an isomorphism is given by

$$ \mathbb Z/(p-1)\mathbb Z\times \mathcal {B}(0,1^-)\rightarrow \mathcal {W},\quad (i,v)\mapsto \chi _{i,v}, $$

where $\chi _{i,v}((\zeta ,u^x))=\zeta ^i (1+v)^x$. Here we wrote an element of $\mathbb Z_p^\times$ uniquely as a pair $(\zeta , u^x)$ with $\zeta \in \mu _{p-1}$ and $x\in \mathbb Z_p$. We make once and for all the choice $u=1+p$.

We say that a point $\chi \in \mathcal {W}(\mathbb C_p)$ is classical if there exists $k\in \mathbb N$ and a finite order character $\psi :\mathbb Z_p^\times \rightarrow \mathbb C_p^\times$ such that $\chi$ is the character $z\mapsto z^k\psi (z)$. The set of classical points is Zariski-dense in $\mathcal {W}(\mathbb C_p)$.

If $\mathrm {Spm}\, R\subset \mathcal {W}$ is an affinoid open subset, we denote by $\kappa =\kappa _R:\mathbb Z_p^\times \rightarrow R^\times$ its tautological character given by $\kappa (t)(\chi )=\chi (t)$ for every $\chi \in \mathrm {Spm}\, R$. Recall ([3, Proposition 8.3]) that $\kappa _R$ is r-analytic for every sufficiently small radius $r>0$ (by which we mean that it extends to a rigid analytic function on $\mathbb Z_p^\times \mathcal {B}(1,r)$).

2.2 Adapted Pairs and the Eigencurve

Let N be a positive integer prime to p. We recall the definition of the spectral curve $Z^N$ and of the cuspidal eigencurve $C^N$ of tame level $\Gamma _1(N)$. These objects were constructed in [6] for $p>2$ and $N=1$ and in [3] in general. We follow the presentation of [3, Part II]. Let $\mathrm {Spm}\,R \subset \mathcal {W}$ be an affinoid domain and let $r=p^{-s}$ for $s\in \mathbb Q$ be a radius smaller than the radius of analyticity of $\kappa _R$. We denote by $M_{R,r}$ the R-module of r-overconvergent modular forms of weight $\kappa _R$. It is endowed it with a continuous action of the Hecke operators $T_\ell$, $\ell \not \mid Np$, and $U_p$. The action of $U_p$ on $M_{R,r}$ is completely continuous, so we can consider its associated Fredholm series $F_{R,r}(T)=\det (1-U_pT\vert M_{R,r})\in R\{\{T\}\}$. These series are compatible when R and r vary, in the sense that there exists $F\in \Lambda \{\{T\}\}$ that restricts to $F_{R,r}(T)$ for every R and r.

The series $F_{R,r}(T)$ converges everywhere on the R-affine line $\mathrm {Spm}\,R\times \mathbb A^{1,{\mathrm an}}$, so it defines a rigid curve $Z^{N}_{R,r}=\{F_{R,r}(T)=0\}$ in $\mathrm {Spm}\,R\times \mathbb A^{1,{\mathrm an}}$. When R and r vary, these curves glue into a rigid space $Z^N$ endowed with a quasi-finite and flat morphism $w_Z:Z^N\rightarrow \mathcal {W}$. The curve $Z^N$ is called the spectral curve associated with the $U_p$-operator. For every $h\geqslant 0$, let us consider

$$ Z_R^{N,\leqslant h}=Z^N_R\cap \left( \mathrm {Spm}\,R\times B(0,p^h)\right) . $$

By [3, Lemma 4.1] $Z_R^{N,\leqslant h}$ is quasi-finite and flat over $\mathrm {Spm}\,R$.

We now recall how to construct an admissible covering of $Z^N$.

Definition 2.1

We denote by $\mathcal {C}$ the set of affinoid domains $Y\subset Z$ such that:

• there exists an affinoid domain $\mathrm {Spm}\,R\subset \mathcal {W}$ such that Y is a union of connected components of $w_Z^{-1}(\mathrm {Spm}\,R)$;

• the map $w_Z\vert _Y:Y\rightarrow \mathrm {Spm}\,R$ is finite.

Proposition 2.2

[3, Theorem 4.6] The covering $\mathcal {C}$ is admissible.

Note in particular that an element $Y\in \mathcal {C}$ must be contained in $Z_R^{N,\leqslant h}$ for some h.

For every R and r as above and every $Y\in \mathcal {C}$ such that $w_Z(Y)=\mathrm {Spm}\,R$, we can associate with Y a direct factor $M_Y$ of $M_{R,r}$ by the construction in [3, Sect. I.5]. The abstract Hecke algebra $\mathcal {H}=\mathbb Z[T_\ell ]_{\ell \not \mid Np}$ acts on $M_{R,r}$ and $M_Y$ is stable with respect to this action. Let $\mathbb T_Y$ be the R-algebra generated by the image of $\mathcal {H}$ in $\mathrm {End}_R(M_Y)$ and let $C_Y^N=\mathrm {Spm}\,\mathbb T_Y$. Note that it is reduced as all Hecke operators are self-adjoint for a certain pairing and mutually commute.

For every Y the finite covering $C_Y^N\rightarrow \mathrm {Spm}\,R$ factors through $Y\rightarrow \mathrm {Spm}\,R$. The eigencurve $C^N$ is defined by gluing the affinoids $C_Y^N$ into a rigid curve, endowed with a finite morphism $C^N\rightarrow Z^N$. The curve $C^N$ is reduced and flat over $\mathcal {W}$ since it is so locally.

We borrow the following terminology from Bellaïche.

Definition 2.3

[1, Definition II.1.8] Let $\mathrm {Spm}\,R\subset \mathcal {W}$ be an affinoid open subset and $h>0$ be a rational number. The couple (R, h) is called adapted if $Z_R^{N,\leqslant h}$ is an element of $\mathcal {C}$.

By [1, Corollary II.1.13] the sets of the form $Z_R^{N,\leqslant h}$ are sufficient to admissibly cover the spectral curve.

Now we fix a finite slope h. We want to work with families of slope $\leqslant h$ which are finite over a wide open subset of the weight space. In order to do this it will be useful to know which pairs (R, h) in a connected component of $\mathcal {W}$ are adapted. If $\mathrm {Spm}\,R^\prime \subset \mathrm {Spm}\,R$ are affinoid subdomains of $\mathcal {W}$ and (R, h) is adapted then $(R^\prime ,h)$ is also adapted by [1, Proposition II.1.10]. By [3, Lemma 4.3], the affinoid $\mathrm {Spm}\,R$ is adapted to h if and only if the weight map $Z_R^{N,\leqslant h}\rightarrow \mathrm {Spm}\, R$ has fibres of constant degree.

Remark 2.4

Given a slope h and a classical weight k, it would be interesting to have a lower bound for the radius of a disc of centre k adapted to h. A result of Wan ([24, Theorem 2.5]) asserts that for a certain radius $r_h$ depending only on h, N and p, the degree of the fibres of $Z_{\mathcal {B}(k,r_h)}^{N,\leqslant h}\rightarrow \mathrm {Spm}\, \mathcal {B}(k,r_h)$ at classical weights is constant. Unfortunately we do not know whether the degree is constant at all weights of $\mathcal {B}(k,r_h)$, so this is not sufficient to answer our question. Estimates for the radii of adapted discs exist in the case of eigenvarieties for groups different than $\mathrm {GL}_2$; see for example the results of Chenevier on definite unitary groups ([4, Sect. 5]).

2.3 Pseudo-characters and Galois Representations

Let K be a finite extension of $\mathbb Q_p$ with valuation ring $\mathcal {O}_K$. Let X be a rigid analytic variety defined over K. We denote by $\mathcal {O}(X)$ the ring of global analytic functions on X equipped with the coarsest locally convex topology making the restriction map $\mathcal {O}(X)\rightarrow \mathcal {O}(U)$ continuous for every affinoid $U\subset X$. It is a Fréchet space isomorphic to the inverse limit over all affinoid domains U of the K-Banach spaces $\mathcal {O}(U)$. We denote by $\mathcal {O}(X)^\circ$ the $\mathcal {O}_K$-algebra of functions bounded by 1 on X, equipped with the topology induced by that on $\mathcal {O}(X)$. The question of the compactness of this ring is related to the following property of X.

Definition 2.5

[2, Definition 7.2.10] We say that a rigid analytic variety X defined over K is nested if there is an admissible covering $X=\bigcup X_i$ by open affinoids $X_i$ defined over K such that the maps $\mathcal {O}(X_{i+1})\rightarrow \mathcal {O}(X_i)$ induced by the inclusions are compact.

We equip the ring $\mathcal {O}(X)^\circ$ with the topology induced by that on $\mathcal {O}(X)={\varprojlim }_{i} \mathcal {O}(X_i)$.

Lemma 2.6

[2, Lemma 7.2.11(ii)] If X is reduced and nested, then $\mathcal {O}(X)^\circ$ is a compact (hence profinite) $\mathcal {O}_K$-algebra.

We will be able to apply Lemma 2.6 to the eigenvariety thanks to the following.

Proposition 2.7

[2, Corollary 7.2.12] The eigenvariety $C^N$ is nested for $K=\mathbb Q_p$.

Given a reduced nested subvariety X of $C^N$ defined over a finite extension K of $\mathbb Q_p$ there is a pseudo-character on X obtained by interpolating the classical ones. Let $\mathbb Q^{N_p}$ be the maixmal extension of $\mathbb Q$ uniamified outside $N_p$ and let $G_\mathbb Q, N_p = Gal (\mathbb Q^{N_p}/\mathbb Q)$.

Proposition 2.8

[1, Theorem IV.4.1] There exists a unique pseudo-character

$$ \tau :G_{\mathbb Q,Np}\rightarrow \mathcal {O}(X)^\circ $$

of dimension 2 such that for every $\ell$ prime to Np, $\tau (\mathrm {Frob}_\ell )=\psi _X(T_\ell )$, where $\psi _X$ is the composition of $\psi :\mathcal {H}\rightarrow \mathcal {O}(C^N)^\circ$ with the restriction map $\mathcal {O}(C^N)^\circ \rightarrow \mathcal {O}(X)^\circ$.

Remark 2.9

One can take as an example of X a union of irreducible components of $C^N$ in which case $K=\mathbb Q_p$. Later we will consider other examples where $K\ne \mathbb Q_p$.

3 The Fortuitous Congruence Ideal

In this section we will define families with slope bounded by a finite constant and coefficients in a suitable profinite ring. We will show that any such family admits at most a finite number of classical specializations which are CM modular forms. Later we will define what it means for a point (not necessarily classical) to be CM and we will associate with a family a congruence ideal describing its CM points. Contrary to the ordinary case, the non-ordinary CM points do not come in families so the points detected by the congruence ideal do not correspond to a crossing between a CM and a non-CM family. For this reason we call our ideal the “fortuitous congruence ideal”.

3.1 The Adapted Slope $\leqslant h$ Hecke Algebra

Throughout this section we fix a slope $h>0$. Let $C^{N,\leqslant h}$ be the subvariety of $C^N$ whose points have slope $\leqslant h$. Unlike the ordinary case treated in [9] the weight map $w^{\leqslant h}:C^{N,\leqslant h}\rightarrow \mathcal {W}$ is not finite which means that a family of slope $\leqslant h$ is not in general defined by a finite map over the entire weight space. The best we can do in the finite slope situation is to place ourselves over the largest possible wide open subdomain U of $\mathcal {W}$ such that the restricted weight map $w^{\leqslant h}\vert _U:C^{N,\leqslant h}\times _{\mathcal {W}}U\rightarrow U$ is finite. This is a domain “adapted to h” in the sense of Definition 2.3 where only affinoid domains were considered. The finiteness property will be necessary in order to apply going-up and going-down theorems.

Let us fix a rational number $s_h$ such that for $r_h=p^{-s_h}$ the closed disc $B(0,r_h)$ is adapted for h. We assume that $s_h>\frac{1}{p-1}$ (this will be needed later to assure the convergence of the exponential map). Let $\eta _h\in \overline{\mathbb Q}_p$ be an element of p-adic valuation $s_h$. Let $K_h$ be the Galois closure (in $\mathbb C_p$) of $\mathbb Q_p(\eta _h)$ and let $\mathcal {O}_h$ be its valuation ring. Recall that T is the variable on the open disc of radius 1. Let $t=\eta _h^{-1}T$ and $\Lambda _h=\mathcal {O}_h[[t]]$. This is the ring of analytic functions, with $\mathcal {O}_h$-coefficients and bounded by one, on the wide open disc $\mathcal {B}_h$ of radius $p^{-s_h}$. There is a natural map $\Lambda \rightarrow \Lambda _h$ corresponding to the restriction of analytic functions on the open disc of radius 1, with $\mathbb Z_p$ coefficients and bounded by 1, to the open disc of radius $r_h$. The image of this map is the ring $\mathbb Z_p[[\eta t]]\subset \mathcal {O}_h[[t]]$.

For $i\geqslant 1$, let $s_i=s_h+1/i$ and $\mathcal {B}_i=\mathcal {B}(0,p^{-s_i})$. The open disc $\mathcal {B}_h$ is the increasing union of the affinoid discs $\mathcal {B}_i$. For each i a model for $\mathcal {B}_i$ over $K_h$ is given by Berthelot’s construction of $\mathcal {B}_h$ as the rigid space associated with the $\mathcal {O}_h$-formal scheme $\mathrm {Spf}\,\Lambda _h$. We recall it briefly following [7, Sect. 7]. Let

$$ A_{r_i}^\circ =\mathcal {O}_h\langle t,X_i\rangle /(pX_i-t^i). $$

We have $\mathcal {B}_i=\mathrm {Spm}\,A_{r_i}^\circ [p^{-1}]$ as rigid space over $K_h$. For every i we have a morphism $A_{r_{i+1}}^\circ \rightarrow A_{r_i}^\circ$ given by

$$ X_{i+1}\mapsto X_i t $$

$$ t\mapsto t $$

We have induced compact morphisms $A_{r_{i+1}}^\circ [p^{-1}]\rightarrow A_{r_i}^\circ [p^{-1}]$, hence open immersions $\mathcal {B}_i\rightarrow \mathcal {B}_{i+1}$ defined over $K_h$. The wide open disc $\mathcal {B}_h$ is defined as the inductive limit of the affinoids $\mathcal {B}_i$ with these transition maps. We have $\Lambda _h={\varprojlim }_{i} A_{r_i}^\circ$.

Since the $s_i$ are strictly bigger than $s_h$ for each i, $\mathcal {B}(0,p^{-s_i})=\mathrm {Spm}\, A_{r_i}^\circ [p^{-1}]$ is adapted to h. Therefore for every $r>0$ sufficiently small and for every $i\geqslant 1$ the image of the abstract Hecke algebra acting on $M_{A_{r_i},r}$ provides a finite affinoid $A_{r_i}^\circ$-algebra $\mathbb T_{A_{r_i}^\circ ,r}^{\leqslant h}$. The morphism $w_{A_{r_i}^\circ ,r}:\mathrm {Spm}\,\mathbb T_{A_{r_i}^\circ ,r}^{\leqslant h}\rightarrow \mathrm {Spm}\, A_{r_i}^\circ$ is finite. For $i<j$ we have natural open immersions $\mathrm {Spm}\,\mathbb T_{A_{r_j}^\circ ,r}^{\leqslant h}\rightarrow \mathrm {Spm}\,\mathbb T_{A_{r_i}^\circ ,r}^{\leqslant h}$ and corresponding restriction maps $\mathbb T_{A_{r_i}^\circ ,r}^{\leqslant h}\rightarrow \mathbb T_{A_{r_j}^\circ ,r}^{\leqslant h}$. We call $C_h$ the increasing union $\bigcup _{i\in \mathbb N,r>0}\mathrm {Spm}\,\mathbb T_{A_{r_i}^\circ ,r}^{\leqslant h}$; it is a wide open subvariety of $C^{N}$. We denote by $\mathbb T_h$ the ring of rigid analytic functions bounded by 1 on $C_h$. We have $\mathbb T_h=\mathcal {O}(C_h)^\circ ={\varprojlim }_{i,r}\mathbb T_{A_{r_i}^\circ ,r}^{\leqslant h}$. There is a natural weight map $w_h:C_h\rightarrow \mathcal {B}_h$ that restricts to the maps $w_{A_{r_i}^\circ ,r}$. It is finite because the closed ball of radius $r_h$ is adapted to h.

3.2 The Galois Representation Associated with a Family of Finite Slope

Since $\mathcal {O}(B_h)^\circ =\Lambda _h$, the map $w_h$ gives $\mathbb T_h$ the structure of a finite $\Lambda _h$-algebra; in particular $\mathbb T_h$ is profinite.

Let ${\mathfrak {m}}$ be a maximal ideal of $\mathbb T_h$. The residue field $k=\mathbb T_h/{\mathfrak {m}}$ is finite. Let $\mathbb T_{\mathfrak {m}}$ denote the localization of $\mathbb T_h$ at ${\mathfrak {m}}$. Since $\Lambda _h$ is henselian, $\mathbb T_{\mathfrak {m}}$ is a direct factor of $\mathbb T_h$, hence it is finite over $\Lambda _h$; it is also local noetherian and profinite. It is the ring of functions bounded by 1 on a connected component of $C_h$. Let $W=W(k)$ be the ring of Witt vectors of k. By the universal property of W, $\mathbb T_{\mathfrak {m}}$ is a W-algebra. The affinoid domain $\mathrm {Spm}\,\mathbb T_{\mathfrak {m}}$ contains a zarisiki-dense set of points x corresponding to cuspidal eigenforms $f_x$ of weight $w(x)=k_x\geqslant 2$ and level Np. The Galois representations $\rho _{f_x}$ associated with the $f_x$ give rise to a residual representation $\overline{\rho }:G_{\mathbb Q,Np}\rightarrow \mathrm {GL}_2(k)$ that is independent of $f_x$. By Proposition 2.8, we have a pseudo-character

$$ \tau _{\mathbb T_{\mathfrak {m}}}:G_{\mathbb Q,Np}\rightarrow \mathbb T_{\mathfrak {m}}$$

such that for every classical point $x:\mathbb T_{\mathfrak {m}}\rightarrow L$, defined over some finite extension $L/\mathbb Q_p$, the specialization of $\tau _{\mathbb T_{\mathfrak {m}}}$ at x is the trace of $Lf_x$.

Proposition 3.1

If $\overline{\rho }$ is absolutely irreducible there exists a unique continuous irreducible Galois representation

$$ \rho _{\mathbb T_{\mathfrak {m}}}:G_{\mathbb Q,Np}\rightarrow \mathrm {GL}_2(\mathbb T_{\mathfrak {m}}), $$

lifting $\overline{\rho }$ and whose trace is $\tau _{\mathbb T_{\mathfrak {m}}}$.

This follows from a result of Nyssen and Rouquier ([14], [18, Corollary 5.2]), since $\mathbb T_{\mathfrak {m}}$ is local henselian.

Let $\mathbb I^\circ$ be a finite torsion-free $\Lambda _h$-algebra. We call family an irreducible component of $\mathrm {Spec}\,\mathbb T_h$ defined by a surjective morphism $\theta :\mathbb T_h\rightarrow \mathbb I^\circ$ of $\Lambda _h$-algebras. Since such a map factors via $\mathbb T_{\mathfrak {m}}\rightarrow \mathbb I^\circ$ for some maximal ideal ${\mathfrak {m}}$ of $\mathbb T_h$, we can define a residual representation $\overline{\rho }$ associated with $\theta$. Suppose that $\overline{\rho }$ is irreducible. By Proposition 3.1 we obtain a Galois representation $\rho :G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I^\circ )$ associated with $\theta$.

Remark 3.2

If $\eta _h\notin \mathbb Q_p$, $\Lambda _h$ is not a power series ring over $\mathbb Z_p$.

3.3 Finite Slope CM Modular Forms

In this section we study non-ordinary finite slope CM modular forms. We say that a family is CM if all its classical points are CM. We prove that for every $h>0$ there are no CM families with positive slope $\leqslant h$. However, contrary to the ordinary case, every family of finite positive slope may contain classical CM points of weight $k\geqslant 2$. Let F be an imaginary quadratic field, ${\mathfrak {f}}$ an integral ideal in F, $I_{{\mathfrak {f}}}$ the group of fractional ideals prime to ${{\mathfrak {f}}}$. Let $\sigma _1,\sigma _2$ be the embeddings of F into $\mathbb C$ (say that $\sigma _1=\mathrm {Id}_F$) and let $(k_1,k_2)\in \mathbb Z^2$. A Grössencharacter $\psi$ of infinity type $(k_1,k_2)$ defined modulo ${\mathfrak {f}}$ is a homomorphism $\psi :I_{{\mathfrak {f}}}\rightarrow \mathbb C^*$ such that $\psi ((\alpha ))=\sigma _1(\alpha )^{k_1}\sigma _2(\alpha )^{k_2}$ for all $\alpha \equiv 1$ $(\mathrm {mod}^\times {\mathfrak {f}})$. Consider the q-expansion

$$ \sum _{{\mathfrak {a}}\subset \mathcal {O}_F, ({\mathfrak {a}},{\mathfrak {f}})=1}\psi ({\mathfrak {a}})q^{N({\mathfrak {a}})}, $$

where the sum is over ideals ${\mathfrak {a}}\subset \mathcal {O}_F$ and $N({\mathfrak {a}})$ denotes the norm of ${\mathfrak {a}}$. Let $F/\mathbb Q$ be an imaginary quadratic field of discriminant D and let $\psi$ be a Grössencharacter of exact conductor ${\mathfrak {f}}$ and infinity type $(k-1,0)$. By [22, Lemma 3] the expansion displayed above defines a cuspidal newform $f(F,\psi )$ of level $N({\mathfrak {f}})D$.

Ribet proved in [16, Theorem 4.5] that if a newform g of weight $k\geqslant 2$ and level N has CM by an imaginary quadratic field F, one has $g=f(F,\psi )$ for some Grössencharacter $\psi$ of F of infinity type $(k-1,0)$.

Definition 3.3

We say that a classical modular eigenform g of weight k and level Np has CM by an imaginary quadratic field F if its Hecke eigenvalues for the operators $T_\ell$, $\ell \not \mid Np$, coincide with those of $f(F,\psi )$ for some Grössencharacter $\psi$ of F of infinity type $(k-1,0)$. We also say that g is CM without specifying the field.

Remark 3.4

For g as in the definition the Galois representations $\rho _g,\rho _{f(F,\psi )}:G_\mathbb Q\rightarrow \mathrm {GL}_2(\overline{\mathbb Q}_p)$ associated with g and $f(F,\psi )$ are isomorphic, hence the image of the representation $\rho _g$ is contained in the normalizer of a torus in $\mathrm {GL}_2$.

Proposition 3.5

Let g be a CM modular eigenform of weight k and level $Np^m$ with N prime to p and $m\geqslant 0$. Then its p-slope is either 0, $\frac{k-1}{2}$, $k-1$ or infinite.

Proof

Let F be the quadratic imaginary field and $\psi$ the Grössencharacter of F associated with the CM form g by Definition 3.3. Let ${\mathfrak {f}}$ be the conductor of $\psi$.

We assume first that g is p-new, so that $g=f(F,\psi )$. Let $a_p$ be the $U_p$-eigenvalue of g. If p is inert in F we have $a_p=0$, so the p-slope of g is infinite. If p splits in F as ${\mathfrak {p}}\bar{{\mathfrak {p}}}$, then $a_p=\psi ({\mathfrak {p}})+\psi (\bar{{\mathfrak {p}}})$. We can find an integer n such that ${\mathfrak {p}}^n$ is a principal ideal $(\alpha )$ with $\alpha \equiv 1\,(\mathrm {mod}^\times {\mathfrak {f}})$. Hence $\psi ((\alpha ))=\alpha ^{k-1}$. Since $\alpha$ is a generator of ${\mathfrak {p}}^n$ we have $\alpha \in {\mathfrak {p}}$ and $\alpha \notin \bar{{\mathfrak {p}}}$; moreover $\alpha ^{k-1}=\psi ((\alpha ))=\psi ({\mathfrak {p}})^n$, so we also have $\psi ({\mathfrak {p}})\in {\mathfrak {p}}-\bar{{\mathfrak {p}}}$. In the same way we find $\psi (\bar{{\mathfrak {p}}})\in \bar{{\mathfrak {p}}}-{\mathfrak {p}}$. We conclude that $\psi ({\mathfrak {p}})+\psi (\bar{{\mathfrak {p}}})$ does not belong to ${\mathfrak {p}}$, so its p-adic valuation is 0.

If p ramifies as ${\mathfrak {p}}^2$ in F, then $a_p=\psi ({\mathfrak {p}})$. As before we find n such that ${\mathfrak {p}}^n=(\alpha )$ with $\alpha \equiv 1\,(\mathrm {mod}^\times {\mathfrak {f}})$. Then $(\psi ({\mathfrak {p}}))^n\psi ({\mathfrak {p}}^n)=\psi ((\alpha ))=\alpha ^{k-1}={\mathfrak {p}}^{n(k-1)}$. By looking at p-adic valuations we find that the slope is $\frac{k-1}{2}$.

If g is not p-new, it is the p-stabilization of a CM form $f(F,\psi )$ of level prime to p. If $a_p$ is the $T_p$-eigenvalue of $f(F,\psi )$, the $U_p$-eigenvalue of g is a root of the Hecke polynomial $X^2-a_pX+\zeta p^{k-1}$ for some root of unity $\zeta$. By our discussion of the p-new case, the valuation of $a_p$ belongs to the set $\left\{ 0,\frac{k-1}{2},k-1\right\}$. Then it is easy to see that the valuations of the roots of the Hecke polynomial belong to the same set.$\square$

We state a useful corollary.

Corollary 3.6

There are no CM families of strictly positive slope.

Proof

We show that the eigencurve $C_h$ contains only a finite number of points corresponding to classical CM forms. It will follow that almost all classical points of a family in $C_h$ are non-CM. Let f be a classical CM form of weight k and positive slope. By Proposition 3.5 its slope is at least $\frac{k-1}{2}$. If f corresponds to a point of $C_h$ its slope must be $\leqslant h$, so we obtain an inequality $\frac{k-1}{2}\leqslant h$. The set of weights $\mathcal {K}$ satisfying this condition is finite. Since the weight map $C_h\rightarrow B_h$ is finite, the set of points of $C_h$ whose weight lies in $\mathcal {K}$ is finite. Hence the number of CM forms in $C_h$ is also finite.$\square$

We conclude that, in the finite positive slope case, classical CM forms can appear only as isolated points in an irreducible component of the eigencurve $C_h$. In the ordinary case, the congruence ideal of a non-CM irreducible component is defined as the intersection ideal of the CM irreducible components with the given non-CM component. In the case of a positive slope family $\theta :\mathbb T_h\rightarrow \mathbb I^\circ$, we need to define the congruence ideal in a different way.

3.4 Construction of the Congruence Ideal

Let $\theta :\mathbb T_h\rightarrow \mathbb I^\circ$ be a family. We write $\mathbb I=\mathbb I^\circ [p^{-1}]$.

Fix an imaginary quadratic field F where p is inert or ramified; let $-D$ be its discriminant. Let ${\mathfrak {Q}}$ be a primary ideal of $\mathbb I$; then ${\mathfrak {q}}={\mathfrak {Q}}\cap \Lambda _h$ is a primary ideal of $\Lambda _h$. The projection $\Lambda _h\rightarrow \Lambda _h/{\mathfrak {q}}$ defines a point of $\mathcal {B}_h$ (possibly non-reduced) corresponding to a weight $\kappa _{\mathfrak {Q}}:\mathbb Z_p^*\rightarrow (\Lambda _h/{\mathfrak {q}})^*$. For $r>0$ we denote by $\mathcal {B}_r$ the ball of centre 1 and radius r in $\mathbb C_p$. By [3, Proposition 8.3] there exists $r>0$ and a character $\kappa _{{\mathfrak {Q}},r}:\mathbb Z_p^\times \cdot \mathcal {B}_r\rightarrow (\Lambda _h/{\mathfrak {q}})^\times$ extending $\kappa _{\mathfrak {Q}}$.

Let $\sigma$ be an embedding $F\hookrightarrow \mathbb C_p$. Let r and $\kappa _{{\mathfrak {Q}},r}$ be as above. For m sufficiently large $\sigma (1+p^m\mathcal {O}_F)$ is contained in $\mathbb Z_p^\times \cdot \mathcal {B}_r$, the domain of definition of $\kappa _{{\mathfrak {Q}},r}$.

For an ideal ${\mathfrak {f}}\subset \mathcal {O}_F$ let $I_{{\mathfrak {f}}}$ be the group of fractional ideals prime to ${{\mathfrak {f}}}$. For every prime $\ell$ not dividing Np we denote by $a_{\ell ,{\mathfrak {Q}}}$ the image of the Hecke operator $T_\ell$ in $\mathbb I^\circ /{\mathfrak {Q}}$. We define here a notion of non-classical CM point of $\theta$ (hence of the eigencurve $C_h$) as follows.

Definition 3.7

Let $F,\sigma ,{\mathfrak {Q}},r,\kappa _{{\mathfrak {Q}},r}$ be as above. We say that ${\mathfrak {Q}}$ defines a CM point of weight $\kappa _{{\mathfrak {Q}},r}$ if there exist an integer $m>0$, an ideal ${\mathfrak {f}}\subset \mathcal {O}_F$ with norm $N({\mathfrak {f}})$ such that $DN({\mathfrak {f}})$ divides N, a quadratic extension $(\mathbb I/{\mathfrak {Q}})^\prime$ of $\mathbb I/{\mathfrak {Q}}$ and a homomorphism $\psi :I_{{\mathfrak {f}}p^m}\rightarrow (\mathbb I/{\mathfrak {Q}})^{\prime \times }$ such that:

• 1. $\sigma (1+p^m\mathcal {O}_F)\subset \mathbb Z_p^\times \cdot \mathcal {B}_r$;

• 2. for every $\alpha \in \mathcal {O}_F$ with $\alpha \equiv 1\, (\mathrm {mod}^\times {\mathfrak {f}}p^m)$, $\psi ((\alpha ))=\kappa _{{\mathfrak {Q}},r}(\alpha )\alpha ^{-1}$;

• 3. $a_{\ell ,{\mathfrak {Q}}}=0$ if L is a prime inert in F and not dividing Np;

• 4. $a_{\ell ,{\mathfrak {Q}}}=\psi ({\mathfrak {l}})+\psi (\bar{{\mathfrak {l}}})$ if $\ell$ is a prime splitting as ${\mathfrak {l}}\bar{{\mathfrak {l}}}$ in F and not dividing Np.

Note that $\kappa _{{\mathfrak {Q}},r}(\alpha )$ is well defined thanks to condition 1.

Remark 3.8

If ${\mathfrak {P}}$ is a prime of $\mathbb I$ corresponding to a classical form f then ${\mathfrak {P}}$ is a CM point if and only if f is a CM form in the sense of Sect. 3.3.

Proposition 3.9

The set of CM points in $\mathrm {Spec}\,\mathbb I$ is finite.

Proof

By contradiction assume it is infinite. Then we have an injection $\mathbb I\hookrightarrow \prod _{\mathfrak {P}}\mathbb I/{\mathfrak {P}}$ where ${\mathfrak {P}}$ runs over the set of CM prime ideals of $\mathbb I$. One can assume that the imaginary quadratic field of complex multiplication is constant along $\mathbb I$. We can also assume that the ramification of the associated Galois characters $\lambda _{{\mathfrak {P}}}:G_F\rightarrow (\mathbb I/{\mathfrak {P}})^\times$ is bounded (in support and in exponents). On the density one set of primes of F prime to ${\mathfrak {f}}p$ and of degree one, they take values in the image of $\mathbb I^\times$ hence they define a continuous Galois character $\lambda :G_F\rightarrow \mathbb I^\times$ such that $\rho _\theta =\mathrm {Ind}^{G_\mathbb Q}_{G_F}\lambda$, which is absurd (by Corallary 3.6 and specialization at non-CM classical points which do exist).$\square$

Definition 3.10

The (fortuitous) congruence ideal ${\mathfrak {c}}_\theta$ associated with the family $\theta$ is defined as the intersection of all the primary ideals of $\mathbb I$ corresponding to CM points.

Remark 3.11

(Characterizations of the CM locus)

• 1. Assume that $\overline{\rho }_\theta =\mathrm {Ind}^{G_\mathbb Q}_{G_K}\overline{\lambda }$ for a unique imaginary quadratic field K. Then the closed subscheme $V({\mathfrak {c}}_\theta )=\mathrm {Spec}\,\mathbb I/{\mathfrak {c}}_\theta \subset \mathrm {Spec}\,\mathbb I$ is the largest subscheme on which there is an isomorphism of Galois representations $\rho _\theta \cong \rho _\theta \otimes \left( \frac{K/\mathbb Q}{\bullet }\right)$. Indeed, for every artinian $\mathbb Q_p$-algebra A, a CM point $x :\mathbb I\rightarrow A$ is characterized by the conditions $x(T_\ell )=x(T_\ell )\left( \frac{K/\mathbb Q}{\ell }\right)$ for all primes $\ell$ not dividing Np.

• 2. Note that N is divisible by the discriminant D of K. Assume that $\mathbb I$ is N-new and that D is prime to N / D. Let $W_D$ be the Atkin-Lehner involution associated with D. Conjugation by $W_D$ defines an automorphism $\iota _D$ of $\mathbb T_h$ and of $\mathbb I$. Then $V({\mathfrak {c}}_\theta )$ coincides with the (schematic) invariant locus $(\mathrm {Spec}\,\mathbb I)^{\iota _D=1}$.

4 The Image of the Representation Associated with a Finite Slope Family

It is shown by Lang in [12, Theorem 2.4] that, under some technical hypotheses, the image of the Galois representation $\rho :G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I^\circ )$ associated with a non-CM ordinary family $\theta :\mathbb T\rightarrow \mathbb I^\circ$ contains a congruence subgroup of $\mathrm {SL}_2(\mathbb I^\circ _0)$, where $\mathbb I^\circ _0$ is the subring of $\mathbb I^\circ$ fixed by certain “symmetries” of the representation $\rho$. In order to study the Galois representation associated with a non-ordinary family we will adapt some of the results in [12] to this situation. Since the crucial step ([12, Theorem 4.3]) requires the Galois ordinarity of the representation (as in [9, Lemma 2.9]), the results of this section will not imply the existence of a congruence subgroup of $\mathrm {SL}_2(\mathbb I^\circ _0)$ contained in the image of $\rho$. However, we will prove in later sections the existence of a “congruence Lie subalgebra” of ${\mathfrak {sl}}_2(\mathbb I_0^\circ )$ contained in a suitably defined Lie algebra of the image of $\rho$ by means of relative Sen theory.

For every ring R we denote by Q(R) its total ring of fractions.

4.1 The Group of Self-twists of a Family

We follow [12, Sect. 2] in this subsection. Let $h\geqslant 0$ and $\theta :\mathbb T_h\rightarrow \mathbb I^\circ$ be a family of slope $\leqslant h$ defined over a finite torsion free $\Lambda _h$-algebra $\mathbb I^\circ$. Recall that there is a natural map $\Lambda \rightarrow \Lambda _h$ with image $\mathbb Z_p[[\eta t]]$.

Definition 4.1

We say that $\sigma \in \mathrm {Aut}_{Q(\mathbb Z_p[[\eta t]])}(Q(\mathbb I^\circ ))$ is a conjugate self-twist for $\theta$ if there exists a Dirichlet character $\eta _\sigma :G_\mathbb Q\rightarrow \mathbb I^{\circ ,\times }$ such that

$$ \sigma (\theta (T_\ell ))=\eta _\sigma (\ell )\theta (T_\ell ) $$

for all but finitely many primes $\ell$.

Any such $\sigma$ acts on $\Lambda _h=\mathcal {O}_h[[t]]$ by restriction, trivially on t and by a Galois automorphism on $\mathcal {O}_h$. The conjugates self-twists for $\theta$ form a subgroup of $\mathrm {Aut}_{Q(\mathbb Z_p[[\eta t]])}(Q(\mathbb I^\circ ))$. We recall the following result which holds without assuming the ordinarity of $\theta$.

Lemma 4.2

[12, Lemma 7.1] $\Gamma$ is a finite abelian 2-group.

We suppose from now on that $\mathbb I^\circ$ is normal. The only reason for this hypothesis is that in this case $\mathbb I^\circ$ is stable under the action of $\Gamma$ on $Q(\mathbb I^\circ )$, which is not true in general. This makes it possible to define the subring $\mathbb I^\circ _0$ of elements of $\mathbb I^\circ$ fixed by $\Gamma$.

Remark 4.3

The hypothesis of normality of $\mathbb I^\circ$ is just a simplifying one. We could work without it by introducing the $\Lambda _h$-order $\mathbb I^{\circ ,\prime }=\Lambda _h[\theta (T_\ell ),\ell \not \mid Np]\subset \mathbb I^\circ$: this is an analogue of the $\Lambda$-order $\mathbb I^\prime$ defined in [12, Sect. 2] and it is stable under the action of $\Gamma$. We would define $\mathbb I^\circ _0$ as the fixed subring of $\mathbb I^{\circ ,\prime }$ and the arguments in the rest of the article could be adapted to this setting.

The subring of $\Lambda _h$ fixed by $\Gamma$ is an $\mathcal {O}_{h,0}$ form of $\Lambda _h$ for some subring $\mathcal {O}_{h,0}$ of $\mathcal {O}_{h}$. We denote it by $\Lambda _{h,0}$ the field of fractions of $\mathcal {O}_{h,0}$.

Remark 4.4

By definition $\Gamma$ fixes $\mathbb Z_p[[\eta t]]$, so we have $\mathbb Z_p[[\eta t]]\subset \Lambda _{h,0}$. In particular it makes sense to speak about the ideal $P_k\Lambda _{h,0}$ for every arithmetic prime $P_k=(1+\eta t-u^k)\subset \mathbb Z_p[[\eta t]]$. Note that $P_k\Lambda _h$ defines a prime ideal of $\Lambda _h$ if and only if the weight k belongs to the open disc $B_h$, otherwise $P_k\Lambda _h=\Lambda _h$. We see immediately that the same statement is true if we replace $\Lambda _h$ by $\Lambda _{h,0}$.

Note that $\mathbb I^\circ _0$ is a finite extension of $\Lambda _{h,0}$ because $\mathbb I^\circ$ is a finite $\Lambda _h$-algebra. Moreover, we have $K_h^\Gamma =K_{h,0}$ (although the inclusion $\Lambda _h\cdot \mathbb I_0^\circ \subset \mathbb I^\circ$ may not be an equality).

We define two open normal subgroups of $G_\mathbb Q$ by:

• $H_0=\bigcap _{\sigma \in \Gamma }\ker \eta _\sigma$;

• $H=H_0\cap \ker (\det \overline{\rho })$.

Note that $H_0$ is an open normal subgroup of $G_\mathbb Q$ and that H is a n open normal subgroup of $H_0$ and $G_\mathbb Q$.

4.2 The Level of a General Ordinary Family

We recall the main result of [12]. Denote by $\mathbb T$ the big ordinary Hecke algebra, which is finite over $\Lambda =\mathbb Z_p[[T]]$. Let $\theta :\mathbb T\rightarrow \mathbb I^\circ$ be an ordinary family with associated Galois representation $\rho :G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I^\circ )$. The representation $\rho$ is p-ordinary, which means that its restriction $\rho \vert _{D_p}$ to a decomposition subgroup $D_p\subset G_\mathbb Q$ is reducible. There exist two characters $\varepsilon ,\delta :D_p\rightarrow \mathbb I^{\circ ,\times }$, with $\delta$ unramified, such that $\rho \vert _{D_p}$ is an extension of $\varepsilon$ by $\delta$.

Denote by $\mathbb F$ the residue field of $\mathbb I^\circ$ and by $\overline{\rho }$ the representation $G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb F)$ obtained by reducing $\rho$ modulo the maximal ideal of $\mathbb I^\circ$. Lang introduces the following technical condition.

Definition 4.5

The p-ordinary representation $\overline{\rho }$ is called $H_0$-regular if $\overline{\varepsilon }\vert _{D_p\cap H_0}\ne \overline{\delta }\vert _{D_p\cap H_0}$.

The following result states the existence of a Galois level for $\rho$.

Theorem 4.6

[12, Theorem 2.4] Let $\rho :G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I^\circ )$ be the representation associated with an ordinary, non-CM family $\theta :\mathbb T\rightarrow \mathbb I^\circ$. Assume that $p>2$, the cardinality of $\mathbb F$ is not 3 and the residual representation $\overline{\rho }$ is absolutely irreducible and $H_0$-regular. Then there exists $\gamma \in \mathrm {GL}_2(\mathbb I^\circ )$ such that $\gamma \cdot \mathrm {Im}\,\rho \cdot \gamma ^{-1}$ contains a congruence subgroup of $\mathrm {SL}_2(\mathbb I^\circ _0)$.

The proof relies on the analogous result proved by Ribet [15] and Momose [13] for the p-adic representation associated with a classical modular form.

4.3 An Approximation Lemma

In this subsection we prove an analogue of [10, Lemma 4.5]. It replaces in our approach the use of Pink’s Lie algebra theory, which is relied upon in the case of ordinary representations in [9, 12]. Let $\mathbb I_0^\circ$ be a local domain that is finite torsion free over $\Lambda _h$. It does not need to be related to a Hecke algebra for the moment.

Let N be an open normal subgroup of $G_\mathbb Q$ and let $\rho :N\rightarrow \mathrm {GL}_2(\mathbb I_0^\circ )$ be an arbitrary continuous representation. We denote by ${\mathfrak {m}}_{\mathbb I_0^\circ }$ the maximal ideal of $\mathbb I_0^\circ$ and by $\mathbb F=\mathbb I_0^\circ /{\mathfrak {m}}_{\mathbb I_0^\circ }$ its residue field of cardinality q. In the lemma we do not suppose that $\rho$ comes from a family of modular forms. We will only assume that it satisfies the following technical condition:

Definition 4.7

Keep notations as above. We say that the representation $\rho :N\rightarrow \mathrm {GL}_2(\mathbb I_0^\circ )$ is $\mathbb Z_p$-regular if there exists $d\in \mathrm {Im}\,\rho$ with eigenvalues $d_1,d_2\in \mathbb Z_p$ such that $d_1^2\not \equiv d_2^2\pmod {p}$. We call d a $\mathbb Z_p$-regular element. If $N^\prime$ is an open normal subgroup of N then we say that $\rho$ is $(N^\prime ,\mathbb Z_p)$-regular if $\rho \vert _{N^\prime }$ is $\mathbb Z_p$-regular.

Let $B^{\pm }$ denote the Borel subgroups consisting of upper, respectively lower, triangular matrices in $\mathrm {GL}_2$. Let $U^{\pm }$ be the unipotent radical of $B^{\pm }$.

Proposition 4.8

Let $\mathbb I_0^\circ$ be a finite torsion free $\Lambda _{h,0}$-algebra, N an open normal subgroup of $G_\mathbb Q$ and $\rho$: $N\rightarrow \mathrm {GL}_2(\mathbb I_0^\circ )$ a continuous representation that is $\mathbb Z_p$-regular. Suppose (upon replacing $\rho$ by a conjugate) that a $\mathbb Z_p$-regular element is diagonal. Let $\mathbf{P}$ be an ideal of $\mathbb I_0^\circ$ and $\rho _\mathbf{P}:N\rightarrow \mathrm {GL}_2(\mathbb I_0^\circ /\mathbf{P})$ be the representation given by the reduction of $\rho$ modulo $\mathbf{P}$. Let $U^\pm (\rho )$, and $U^\pm (\rho _\mathbf{P})$ be the upper and lower unipotent subgroups of $\mathrm {Im}\,\rho$, and $\mathrm {Im}\,\rho _\mathbf{P}$, respectively. Then the natural maps $U^+(\rho )\rightarrow U^+(\rho _\mathbf{P})$ and $U^-(\rho )\rightarrow U^-(\rho _\mathbf{P})$ are surjective.

Remark 4.9

The ideal $\mathbf{P}$ in the proposition is not necessarily prime. At a certain point we will need to take $\mathbf{P}=P\mathbb I_0^\circ$ for a prime ideal P of $\Lambda _{h,0}$.

As in [10, Lemma 4.5] we need two lemmas. Since the argument is the same for $U^+$ and $U^-$, we will only treat here the upper triangular case $U=U^+$ and $B=B^+$.

For $*=U, B$ and every $j\geqslant 1$ we define the groups

$$ \Gamma _{*}(\mathbf{P}^j)=\{x\in \mathrm {SL}_2(\mathbb I_0^\circ )\, |\, x\,\, (\mathrm {mod}\,\,\mathbf{P}^j)\in *(\mathbb I_0^\circ /\mathbf{P}^j)\}.$$

Let $\Gamma _{\mathbb I_0^\circ }(\mathbf{P}^j)$ be the kernel of the reduction morphism $\pi _j:\mathrm {SL}_2(\mathbb I_0^\circ )\rightarrow \mathrm {SL}_2(\mathbb I_0^\circ /\mathbf{P}^j)$. Note that $\Gamma _{U}(\mathbf{P}^j)=\Gamma _{\mathbb I_0^\circ }(\mathbf{P}^j)U(\mathbb I_0^\circ )$ consists of matrices $\begin{pmatrix} a &{} b \\ c &{} d\end{pmatrix}$ such that $a,d\equiv 1 \pmod {\mathbf{P}^j}$, $c\equiv 0\pmod {\mathbf{P}^j}$. Let $K=\mathrm {Im}\,\rho$ and

$$ K_{U}(\mathbf{P}^j)=K\cap \Gamma _{U}(\mathbf{P}^j),\quad K_{B}(\mathbf{P}^j)=K\cap \Gamma _{B}(\mathbf{P}^j). $$

Since $U(\mathbb I_0^\circ )$ and $\Gamma _{\mathbb I_0^\circ }(\mathbf{P})$ are p-profinite, the groups $\Gamma _{U}(\mathbf{P}^j)$ and $K_{U}(\mathbf{P}^j)$ for all $j\geqslant 1$ are also p-profinite. Note that

$$ \left[ \left( {\begin{matrix}a&{}b\\ c&{}-a\end{matrix}}\right) ,\left( {\begin{matrix}e&{}f\\ g&{}-e\end{matrix}}\right) \right] =\left( {\begin{matrix}bg-cf&{}2(af-be)\\ 2(ce-ag)&{}cf-bg\end{matrix}}\right) . $$

From this we obtain the following.

Lemma 4.10

If $X,Y\in {{\mathfrak {sl}}}_2(\mathbb I_0^\circ )\cap \left( {\begin{matrix}{\mathbf{P}}^j&{}{\mathbf{P}}^k\\ {\mathbf{P}}^i&{}{\mathbf{P}}^j\end{matrix}}\right)$ with $i\geqslant j\geqslant k$, then ${[}X,Y{]}\in \left( {\begin{matrix}{\mathbf{P}}^{i+k}&{}{\mathbf{P}}^{j+k}\\ {\mathbf{P}}^{i+j}&{}{\mathbf{P}}^{i+k}\end{matrix}}\right)$.

We denote by $\mathrm {D}\Gamma _{U}(\mathbf{P}^j)$ the topological commutator subgroup $(\Gamma _{U}(\mathbf{P}^j),\Gamma _{U}(\mathbf{P}^j))$. Lemma 4.10 tells us that

$$\begin{aligned} \mathrm {D}\Gamma _{U}(\mathbf{P}^j)\subset \Gamma _{B}(\mathbf{P}^{2j})\cap \Gamma _{U}(\mathbf{P}^j). \end{aligned}$$

(1)

By the $\mathbb Z_p$-regularity assumption, there exists a diagonal element $d\in K$ with eigenvalues in $\mathbb Z_p$ and distinct modulo p. Consider the element $\delta =\lim _{n\rightarrow \infty }d^{p^n}$, which belongs to K since this is p-adically complete. In particular $\delta$ normalizes K. It is also diagonal with coefficients in $\mathbb Z_p$, so it normalizes $K_{U}(\mathbf{P}^j)$ and $\Gamma _{B}(\mathbf{P}^j)$. Since $\delta ^p=\delta$, the eigenvalues $\delta _1$ and $\delta _2$ of $\delta$ are roots of unity of order dividing $p-1$. They still satisfy $\delta _1^2\ne \delta _2^2$ as $p\ne 2$.

Set $\alpha =\delta _1/\delta _2\in \mathbb F_p^\times$ and let a be the order of $\alpha$ as a root of unity. We see $\alpha$ as an element of $\mathbb Z_p^\times$ via the Teichmüller lift. Let H be a p-profinite group normalized by $\delta$. Since H is p-profinite, every $x\in H$ has a unique a-th root. We define a map $\Delta :H\rightarrow H$ given by

$$ \Delta (x)=[x\cdot \mathrm {ad}(\delta ) (x)^{\alpha ^{-1}}\cdot \mathrm {ad}(\delta ^2)(x)^{\alpha ^{-2}} \cdots \mathrm {ad}(\delta ^{a-1})(x)^{\alpha ^{1-a}}]^{1/a} $$

Lemma 4.11

If $u\,{\in } \Gamma _{U}(\mathbf{P}^j)$ for some $j\geqslant 1$, then $\Delta ^2(u)\,{\in } \Gamma _{U}(\mathbf{P}^{2j})$ and $\pi _j(\Delta (u))=\pi _j(u)$.

Proof

If $u\in \Gamma _{U}(\mathbf{P}^j)$, we have $\pi _j(\Delta (u))=\pi _j(u)$ as $\Delta$ is the identity map on $U(\mathbb I_0^\circ /\mathbf{P}^j)$. Let $\mathrm {D}\Gamma _{U}(\mathbf{P}^j)$ be the topological commutator subgroup of $\Gamma _{U}(\mathbf{P}^j)$. Since $\Delta$ induces the projection of the $\mathbb Z_p$-module $\Gamma _{U}(\mathbf{P}^j)/\mathrm {D}\Gamma _{U}(\mathbf{P}^j)$ onto its $\alpha$-eigenspace for $\mathrm {ad}(d)$, it is a projection onto $U(\mathbb I_0^\circ ) \mathrm {D}\Gamma _{U}(\mathbf{P}^j)/\mathrm {D}\Gamma _{U}(\mathbf{P}^j)$. The fact that this is exactly the $\alpha$-eigenspace comes from the Iwahori decomposition of $\Gamma _{U}(\mathbf{P}^j)$, hence a similar direct sum decomposition holds in the abelianization $\Gamma _{U}(\mathbf{P}^j)/\mathrm {D}\Gamma _{U}(\mathbf{P}^j)$.

By (1), we have $\mathrm {D}\Gamma _{U}(\mathbf{P}^j)\subset \Gamma _{B}(\mathbf{P}^{2j})\cap \Gamma _{U}(\mathbf{P}^j)$. Since the $\alpha$-eigenspace of $\Gamma _{U}(\mathbf{P}^j)/\mathrm {D}\Gamma _{U}(\mathbf{P}^j)$ is inside $\Gamma _{B}(\mathbf{P}^{2j})$, $\Delta$ projects $u\Gamma _{U}(\mathbf{P}^j)$ to

$$ \overline{\Delta }(u)\in (\Gamma _{B}(\mathbf{P}^{2j})\cap \Gamma _{U}(\mathbf{P}^j))/\mathrm {D}\Gamma _{U}(\mathbf{P}^j). $$

In particular, $\Delta (u)\in \Gamma _{B}(\mathbf{P}^{2j})\cap \Gamma _{U}(\mathbf{P}^j)$. Again apply $\Delta$. Since $\Gamma _{B}(\mathbf{P}^{2j})/\Gamma _{\mathbb I_0^\circ }(\mathbf{P}^{2j})$ is sent to $\Gamma _{U}(\mathbf{P}^{2j})/\Gamma _{\mathbb I_0^\circ }(\mathbf{P}^{2j})$ by $\Delta$, we get $\Delta ^2(u)\in \Gamma _{U}(\mathbf{P}^{2j})$ as desired.$\square$

Proof

We can now prove Proposition 4.8. Let $\overline{u}\in U(\mathbb I_0^\circ /\mathbf{P})\cap \mathrm {Im}(\rho _\mathbf{P})$. Since the reduction map $\mathrm {Im}(\rho )\rightarrow \mathrm {Im}(\rho _\mathbf{P})$ induced by $\pi _1$ is surjective, there exists $v \in \mathrm {Im}(\rho )$ such that $\pi _1(v)=\overline{u}$. Take $u_1\in U(\mathbb I_0^\circ )$ such that $\pi _1(u_1)=\overline{u}$ (this is possible since $\pi _1:U(\Lambda _h)\rightarrow U(\Lambda _h/P)$ is surjective). Then $v u_1^{-1}\in \Gamma _{\mathbb I_0^\circ }(\mathbf{P})$, so $v\in K_{U}(\mathbf{P})$.

By compactness of $K_{U}(\mathbf{P})$ and by Lemma 4.11, starting with v as above, we see that $\lim _{m\rightarrow \infty }\Delta ^m(v)$ converges $\mathbf{P}$-adically to $\Delta ^\infty (v)\in U(\mathbb I_0^\circ )\cap K$ with $\pi _1(\Delta ^\infty (v))=\overline{u}$.$\square$

Remark 4.12

Proposition 4.8 is true with the same proof if we replace $\Lambda _{h,0}$ by $\Lambda _h$ and $\mathbb I_0^\circ$ by a finite torsion free $\Lambda _h$-algebra.

As a first application of Proposition 4.8 we give a result that we will need in the next subsection. Given a representation $\rho :G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I^\circ )$ and every ideal $\mathbf{P}$ of $\mathbb I^\circ$ we define $\rho _\mathbf{P}$, $U^{\pm }(\rho )$ and $U^{\pm }(\rho _\mathbf{P})$ as above, by replacing $\mathbb I_0^\circ$ by $\mathbb I^\circ$.

Proposition 4.13

Let $\theta :\mathbb T_h\rightarrow \mathbb I^\circ$ be a family of slope $\leqslant h$ and $\rho _\theta :G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I^\circ )$ be the representation associated with $\theta$. Suppose that $\rho _\theta$ is $(H_0,\mathbb Z_p)$-regular and let $\rho$ be a conjugate of $\rho _\theta$ such that $\mathrm {Im}\,\rho \vert _{H_0}$ contains a diagonal $\mathbb Z_p$-regular element. Then $U^+(\rho )$ and $U^-(\rho )$ are both nontrivial.

Proof

By density of classical points in $\mathbb T_h$ we can choose a prime ideal $\mathbf{P}\subset \mathbb I^\circ$ corresponding to a classical modular form f. The modulo $\mathbf{P}$ representation $\rho _\mathbf{P}$ is the p-adic representation classically associated with f. By the results of [13, 15] and the hypothesis of $(H_0,\mathbb Z_p)$-regularity of L, there exists an ideal ${\mathfrak {l}}_\mathbf{P}$ of $\mathbb Z_p$ such that $\mathrm {Im}\,\rho _\mathbf{P}$ contains the congruence subgroup $\Gamma _{\mathbb Z_p}({\mathfrak {l}}_\mathbf{P})$. In particular $U^+(\rho _\mathbf{P})$ and $U^-(\rho _\mathbf{P})$ are both nontrivial. Since the maps $U^+(\rho )\rightarrow U^+(\rho _\mathbf{P})$ and $U^-(\rho )\rightarrow U^-(\rho _\mathbf{P})$ are surjective we find nontrivial elements in $U^+(\rho )$ and $U^-(\rho )$.$\square$

We adapt the work in [12, Sect. 7] to show the following.

Proposition 4.14

Suppose that the representation $\rho :G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I^\circ )$ is $(H_0,\mathbb Z_p)$-regular. Then there exists $g\in \mathrm {GL}_2(\mathbb I^\circ )$ such that the conjugate representation $g\rho g^{-1}$ satisfies the following two properties:

• 1. the image of $g\rho g^{-1}\vert _{H_0}$ is contained in $\mathrm {GL}_2(\mathbb I^\circ _0)$;

• 2. the image of $g\rho g^{-1}\vert _{H_0}$ contains a diagonal $\mathbb Z_p$-regular element.

Proof

As usual we choose a $\mathrm {GL}_2(\mathbb I^\circ )$-conjugate of $\rho$ such that a $\mathbb Z_p$-regular element d is diagonal. We still write $\rho$ for this conjugate representation and we show that it also has property (1).

Recall that for every $\sigma \in \Gamma$ there is a character $\eta _\sigma :G_\mathbb Q\rightarrow (\mathbb I^\circ )^\times$ and an equivalence $\rho ^\sigma \cong \rho \otimes \eta _\sigma$. Then for every $\sigma \in \Gamma$ there exists $\mathbf{t}_\sigma \in \mathrm {GL}_2(\mathbb I^\circ )$ such that, for all $g\in G_\mathbb Q$,

$$\begin{aligned} \rho ^\sigma (g)=\mathbf{t}_\sigma \eta _\sigma (g)\rho (g)\mathbf{t}_\sigma ^{-1}. \end{aligned}$$

(2)

We prove that the matrices $\mathbf{t}_\sigma$ are diagonal. Let $\rho (t)$ be a non-scalar diagonal element in $\mathrm {Im}\,\rho$ (for example d). Evaluating (2) at $g=t$ we find that $\mathbf{t}_\sigma$ must be either a diagonal or an antidiagonal matrix. Now by Proposition 4.13 there exists a nontrivial element $\rho (u^+)\in \mathrm {Im}\,\rho \cap U^+(\mathbb I^\circ )$. Evaluating (2) at $g=u^+$ we find that $\mathbf{t}_\sigma$ cannot be antidiagonal.

It is shown in [12, Lemma 7.3] that there exists an extension A of $\mathbb I^\circ$, at most quadratic, and a function $\zeta :\Gamma \rightarrow A^\times$ such that $\sigma \rightarrow \mathbf{t}_\sigma \zeta (\sigma )^{-1}$ defines a cocycle with values in $\mathrm {GL}_2(A)$. The proof of this result does not require the ordinarity of $\rho$. Equation (2) remains true if we replace $\mathbf{t}_\sigma$ with $\mathbf{t}_\sigma \zeta (\sigma )^{-1}$, so we can and do suppose from now on that $\mathbf{t}_\sigma$ is a cocycle with values in $\mathrm {GL}_2(A)$. In the rest of the proof we assume for simplicity that $A=\mathbb I^\circ$, but everything works in the same way if A is a quadratic extension of $\mathbb I^\circ$ and $\mathbb F$ is the residue field of A.

Let $V=(\mathbb I^\circ )^2$ be the space on which $G_\mathbb Q$ acts via $\rho$. As in [12, Sect. 7] we use the cocycle $\mathbf{t}_\sigma$ to define a twisted action of $\Gamma$ on $(\mathbb I^\circ )^2$. For $v=(v_1,v_2)\in V$ we denote by $v^\sigma$ the vector $(v_1^\sigma ,v_2^\sigma )$. We write $v^{[\sigma ]}$ for the vector $\mathbf{t}_\sigma ^{-1}v^\sigma$. Then $v\rightarrow v^{[\sigma ]}$ gives an action of $\Gamma$ since $\sigma \mapsto \mathbf{t}_\sigma$ is a cocycle. Note that this action is $\mathbb I^\circ _0$-linear.

Since $\mathbf{t}_\sigma$ is diagonal for every $\sigma \in \Gamma$, the submodules $V_1=\mathbb I^\circ (1,0)$ and $V_2=\mathbb I^\circ (0,1)$ are stable under the action of $\Gamma$. We show that each $V_i$ contains an element fixed by $\Gamma$. We denote by $\mathbb F$ the residue field $\mathbb I^\circ /{\mathfrak {m}}_\mathbb I^\circ$. Note that the action of $\Gamma$ on $V_i$ induces an action of $\Gamma$ on the one-dimensional $\mathbb F$-vector space $V_i\otimes \mathbb I^\circ /{\mathfrak {m}}_{\mathbb I^\circ }$. We show that for each i the space $V_i\otimes \mathbb I^\circ /{\mathfrak {m}}_{\mathbb I^\circ }$ contains a nonzero element $\overline{v}_i$ fixed by $\Gamma$. This is a consequence of the following argument, a form of which appeared in an early preprint of [12]. Let w be any nonzero element of $V_i\otimes \mathbb I^\circ /{\mathfrak {m}}_{\mathbb I^\circ }$ and let a be a variable in $\mathbb F$. The sum

$$ S_{aw}=\sum _{\sigma \in \Gamma }(aw)^{[\sigma ]} $$

is clearly $\Gamma$-invariant. We show that we can choose a such that $S_{aw}\ne 0$. Since $V_i\otimes \mathbb I^\circ /{\mathfrak {m}}_{\mathbb I^\circ }$ is one-dimensional, for every $\sigma \in \Gamma$ there exists $\alpha _\sigma \in \mathbb F$ such that $w^{[\sigma ]}=\alpha _\sigma w$. Then

$$ S_{aw}=\sum _{\sigma \in \Gamma }(aw)^{[\sigma ]}=\sum _{\sigma \in \Gamma }a^\sigma w^{[\sigma ]}=\sum _{\sigma \in \Gamma }a^\sigma \alpha _\sigma w=\left( \sum _{\sigma \in \Gamma }a^\sigma \alpha _\sigma a^{-1}\right) aw. $$

By Artin’s lemma on the independence of characters, the function $f(a)=\sum _{\sigma \in \Gamma }a^\sigma \alpha _\sigma a^{-1}$ cannot be identically zero on $\mathbb F$. By choosing a value of a such that $f(a)\ne 0$ we obtain a nonzero element $\overline{v}_i=S_{aw}$ fixed by $\Gamma$.

We show that $\overline{v}_i$ lifts to an element $v_i\in V_i$ fixed by $\Gamma$. Let $\sigma _0\in \Gamma$. By Lemma 4.2 $\Gamma$ is a finite abelian 2-group, so the minimal polynomial $P_m(X)$ of $[\sigma _0]$ acting on $V_i$ divides $X^{2^k}-1$ for some integer k. In particular the factor $X-1$ appears with multiplicity at most 1. We show that its multiplicity is exactly 1. If $\overline{P_m}$ is the reduction of $P_m$ modulo ${\mathfrak {m}}_{\mathbb I^\circ }$ then $\overline{P_m}([\sigma _0])=0$ on $V_i\otimes \mathbb I^\circ /{\mathfrak {m}}_{\mathbb I^\circ }$. By our previous argument there is an element of $V_i\otimes \mathbb I^\circ /{\mathfrak {m}}_{\mathbb I^\circ }$ fixed by $\Gamma$ (hence by $[\sigma _0]$) so we have $(X-1)\mid \overline{P_m(X)}$. Since $p>2$ the polynomial $X^{2^k}-1$ has no double roots modulo ${\mathfrak {m}}_{\mathbb I^\circ }$, so neither does $\overline{P_m}$. By Hensel’s lemma the factor $X-1$ lifts to a factor $X-1$ in $P_m$ and $\overline{v}_i$ lifts to an element $v_i\in V_i$ fixed by $[\sigma _0]$. Note that $\mathbb I^\circ \cdot v_i=V_i$ by Nakayama’s lemma since $\overline{v}_i\ne 0$.

We show that $v_i$ is fixed by all of $\Gamma$. Let $W_{[\sigma _0]}=\mathbb I^\circ v_i$ be the one-dimensional eigenspace for $[\sigma _0]$ in $V_i$. Since $\Gamma$ is abelian $W_{[\sigma _0]}$ is stable under $\Gamma$. Let $\sigma \in \Gamma$. Since $\sigma$ has order $2^k$ in $\Gamma$ for some $k\geqslant 0$ and $v_i^{[\sigma ]}\in W_{[\sigma _0]}$, there exists a root of unity $\zeta _\sigma$ of order $2^k$ satisfying $v_i^{[\sigma ]}=\zeta _\sigma v_i$. Since $\overline{v}_i^{[\sigma ]}=\overline{v}_i$, the reduction of $\zeta _\sigma$ modulo ${\mathfrak {m}}_{\mathbb I^\circ }$ must be 1. As before we conclude that $\zeta _\sigma =1$ since $p\ne 2$.

We found two elements $v_1\in V_1$, $v_2\in V_2$ fixed by $\Gamma$. We show that every element of $v\in V$ fixed by $\Gamma$ must belong to the $\mathbb I^\circ _0$-submodule generated by $v_1$ and $v_2$. We proceed as in the end of the proof of [12, Theorem 7.5]. Since $V_1$ and $V_2$ are $\Gamma$-stable we must have $v\in V_1$ or $v\in V_2$. Suppose without loss of generality that $v\in V_1$. Then $v=\alpha v_1$ for some $\alpha \in \mathbb I^\circ$. If $\alpha \in \mathbb I^\circ _0$ then $v\in \mathbb I^\circ _0 v_1$, as desired. If $\alpha \notin \mathbb I^\circ _0$ then there exists $\sigma \in \Gamma$ such that $\alpha ^\sigma \ne \alpha$. Since $v_1$ is $[\sigma ]$-invariant we obtain $(\alpha v_1)^{[\sigma ]}=\alpha ^\sigma v_1^{[\sigma ]}=\alpha ^\sigma v_1\ne \alpha v_1$, so $\alpha v_1$ is not fixed by $[\sigma ]$, a contradiction.

Now $(v_1,v_2)$ is a basis for V over $\mathbb I^\circ$, so the $\mathbb I^\circ _0$ submodule $V_0=\mathbb I^\circ _0v_1+\mathbb I^\circ _0v_2$ is an $\mathbb I^\circ _0$-lattice in V. Recall that $H_0=\bigcap _{\sigma \in \Gamma }\ker \eta _\sigma$. We show that $V_0$ is stable under the action of $H_0$ via $\rho \vert _{H_0}$, i.e. that if $v\in V$ is fixed by $\Gamma$, so is $\rho (h)v$ for every $h\in H_0$. This is a consequence of the following computation, where v and h are as before and $\sigma \in \Gamma$:

$$ (\rho (h)v)^{[\sigma ]}=\mathbf{t}_\sigma ^{-1}\rho (h)^\sigma v^\sigma =\mathbf{t}_\sigma ^{-1}\eta _\sigma (h)\rho (h)^\sigma v^\sigma =\mathbf{t}_\sigma ^{-1}\mathbf{t}_\sigma \rho (h)\mathbf{t}_\sigma ^{-1}v^\sigma =\rho (h)v^{[\sigma ]}. $$

Since $V_0$ is an $\mathbb I^\circ _0$-lattice in V stable under $\rho \vert _{H_0}$, we conclude that $\mathrm {Im}\,\rho \vert _{H_0}\subset \mathrm {GL}_2(\mathbb I^\circ _0)$.$\square$

4.4 Fullness of the Unipotent Subgroups

From now on we write $\rho$ for the element in its $\mathrm {GL}_2(\mathbb I^\circ )$ conjugacy class such that $\rho \vert _{H_0}\in \mathrm {GL}_2(\mathbb I^\circ _0)$. Recall that H is the open subgroup of $H_0$ defined by the condition $\det \overline{\rho }(h)=1$ for every $h\in H$. As in [12, Sect. 4] we define a representation $H\rightarrow \mathrm {SL}_2(\mathbb I^\circ _0)$ by

$$ \rho _0=\rho \vert _H\otimes \left( \det \rho \vert _H\right) ^{-\frac{1}{2}}. $$

We can take the square root of the determinant thanks to the definition of H. We will use the results of [12] to deduce that the $\Lambda _{h,0}$-module generated by the unipotent subgroups of the image of $\rho _0$ is big. We will later deduce the same for $\rho$.

We fix from now on a height one prime $P\subset \Lambda _{h,0}$ with the following properties:

• 1. there is an arithmetic prime $P_k\subset \mathbb Z_p[[\eta t]]$ satisfying $k>h+1$ and $P=P_k\Lambda _{h,0}$;

• 2. every prime ${\mathfrak {P}}\subset \mathbb I^\circ$ lying above P corresponds to a non-CM point.

Such a prime always exists. Indeed, by Remark 4.4 every classical weight $k>h+1$ contained in the disc $B_h$ defines a prime $P=P_k\Lambda _{h,0}$ satisfying (1), so such primes are Zariski-dense in $\Lambda _{h,0}$, while the set of CM primes in $\mathbb I^\circ$ is finite by Proposition 3.9.

Remark 4.15

Since $k>h+1$, every point of $\mathrm {Spec}\,\mathbb T_h$ above $P_k$ is classical by [5, Theorem 6.1]. Moreover the weight map is étale at every such point by [11, Theorem 11.10]. In particular the prime $P\mathbb I_0^\circ =P_k\mathbb I_0^\circ$ splits as a product of distinct primes of $\mathbb I_0^\circ$.

Make the technical assumption that the order of the residue field $\mathbb F$ of $\mathbb I^\circ$ is not 3. For every ideal $\mathbf{P}$ of $\mathbb I^\circ _0$ over P we let $\pi _\mathbf{P}$ be the projection $\mathrm {SL}_2(\mathbb I^\circ _0)\rightarrow \mathrm {SL}_2(\mathbb I^\circ _0/\mathbf{P})$. We still denote by $\pi _\mathbf{P}$ the restricted maps $U^\pm (\mathbb I^\circ _0)\rightarrow U^\pm (\mathbb I^\circ _0/\mathbf{P})$.

Let $G=\mathrm {Im}\,\rho _0$. For every ideal $\mathbf{P}$ of $\mathbb I^\circ _0$ we denote by $\rho _{0,\mathbf{P}}$ the representation $\pi _\mathbf{P}(\rho _0)$ and by $G_\mathbf{P}$ the image of $\rho _\mathbf{P}$, so that $G_\mathbf{P}=\pi _\mathbf{P}(G)$. We state two results from Lang’s work that come over unchanged to the non-ordinary setting.

Proposition 4.16

[12, Corollary 6.3] Let ${\mathfrak {P}}$ be a prime of $\mathbb I^\circ _0$ over P. Then $G_{{\mathfrak {P}}}$ contains a congruence subgroup $\Gamma _{\mathbb I^\circ _0/{\mathfrak {P}}}({\mathfrak {a}})\subset \mathrm {SL}_2(\mathbb I^\circ _0/{\mathfrak {P}})$. In particular $G_{\mathfrak {P}}$ is open in $\mathrm {SL}_2(\mathbb I^\circ _0/{\mathfrak {P}})$.

Proposition 4.17

[12, Proposition 5.1] Assume that for every prime ${\mathfrak {P}}\subset \mathbb I^\circ _0$ over P the subgroup $G_{\mathfrak {P}}$ is open in $\mathrm {SL}_2(\mathbb I^\circ _0/{\mathfrak {P}})$. Then the image of G in $\prod _{{\mathfrak {P}}|P}\mathrm {SL}_2(\mathbb I^\circ _0/{\mathfrak {P}})$ through the map $\prod _{{\mathfrak {P}}|P}\pi _{\mathfrak {P}}$ contains a product of congruence subgroups $\prod _{{\mathfrak {P}}|P}\Gamma _{\mathbb I^\circ _0/{\mathfrak {P}}}({\mathfrak {a}}_{\mathfrak {P}})$.

Remark 4.18

The proofs of Propositions 4.16 and 4.17 rely on the fact that the big ordinary Hecke algebra is étale over $\Lambda$ at every arithmetic point. In order for these proofs to adapt to the non-ordinary setting it is essential that the prime P satisfies the properties above Remark 4.15.

We let $U^\pm (\rho _0)=G\cap U^{\pm }(\mathbb I^\circ _0)$ and $U^\pm (\rho _\mathbf{P})=G_\mathbf{P}\cap U^{\pm }(\mathbb I^\circ _0/\mathbf{P})$. We denote by $U(\rho _\mathbf{P})$ either the upper or lower unipotent subgroups of $G_{\mathbf{P}}$ (the choice will be fixed throughout the proof). By projecting to the upper right element we identify $U^+(\rho _0)$ with a $\mathbb Z_p$-submodule of $\mathbb I^\circ _0$ and $U^+(\rho _{0,\mathbf{P}})$ with a $\mathbb Z_p$-submodule of $\mathbb I^\circ _0/\mathbf{P}$. We make analogous identifications for the lower unipotent subgroups. We will use Propositions 4.17 and 4.8 to show that, for both signs, $U^{\pm }(\rho )$ spans $\mathbb I^\circ _0$ over $\Lambda _{h,0}$.

First we state a version of [12, Lemma 4.10], with the same proof. Let A and B be Noetherian rings with B integral over A. We call A-lattice an A-submodule of B generated by the elements of a basis of Q(B) over Q(A).

Lemma 4.19

Any A-lattice in B contains a nonzero ideal of B. Conversely, every nonzero ideal of B contains an A-lattice.

We prove the following proposition by means of Proposition 4.8. We could also use Pink theory as in [12, Sect. 4].

Proposition 4.20

Consider $U^\pm (\rho _0)$ as subsets of $Q(\mathbb I^\circ _0)$. For each choice of sign the $Q(\Lambda _{h,0})$-span of $U^\pm (\rho _0)$ is $Q(\mathbb I^\circ _0)$. Equivalently the $\Lambda _{h,0}$-span of $U^\pm (\rho _0)$ contains a $\Lambda _{h,0}$-lattice in $\mathbb I^\circ _0$.

Proof

Keep notations as above. We omit the sign when writing unipotent subgroups and we refer to either the upper or lower ones (the choice is fixed throughout the proof). Let P be the prime of $\Lambda _{h,0}$ chosen above. By Remark 4.15 the ideal $P\mathbb I_0^\circ$ splits as a product of distinct primes in $\mathbb I_0^\circ$. When ${\mathfrak {P}}$ varies among these primes, the map $\bigoplus _{{\mathfrak {P}}|P}\pi _{\mathfrak {P}}$ gives embeddings of $\Lambda _{h,0}/P$-modules $\mathbb I^\circ _0/P\mathbb I^\circ _0\hookrightarrow \bigoplus _{{\mathfrak {P}}|P}\mathbb I^\circ _0/{\mathfrak {P}}$ and $U(\rho _{P\mathbb I_0^\circ })\hookrightarrow \bigoplus _{{\mathfrak {P}}|P}U(\rho _{\mathfrak {P}})$. The following diagram commutes:

By Proposition 4.17 there exist ideals ${\mathfrak {a}}_{\mathfrak {P}}\subset \mathbb I^\circ _0/{\mathfrak {P}}$ such that $(\bigoplus _{{\mathfrak {P}}|P}\pi _{\mathfrak {P}})(G_{P\mathbb I_0^\circ })\supset \bigoplus _{{\mathfrak {P}}|P}\Gamma _{\mathbb I^\circ _0/{\mathfrak {P}}}({\mathfrak {a}}_{\mathfrak {P}})$. In particular $(\bigoplus _{{\mathfrak {P}}|P}\pi _{\mathfrak {P}})(U(\rho _{P\mathbb I_0^\circ }))\supset \bigoplus _{{\mathfrak {P}}|P}({\mathfrak {a}}_{\mathfrak {P}})$. By Lemma 4.19 each ideal ${\mathfrak {a}}_{\mathfrak {P}}$ contains a basis of $Q(\mathbb I^\circ _0/{\mathfrak {P}})$ over $Q(\Lambda _{h,0}/P)$, so that the $Q(\Lambda _{h,0}/P)$-span of $\bigoplus _{{\mathfrak {P}}|P}{\mathfrak {a}}_{\mathfrak {P}}$ is the whole $\bigoplus _{{\mathfrak {P}}|P}Q(\mathbb I^\circ _0/{\mathfrak {P}})$. Then the $Q(\Lambda _{h,0}/P)$-span of $(\bigoplus _{{\mathfrak {P}}|P}\pi _{\mathfrak {P}})(G_{\mathfrak {P}}\cap U(\rho _{{\mathfrak {P}}}))$ is also $\bigoplus _{{\mathfrak {P}}|P}Q(\mathbb I^\circ _0/{\mathfrak {P}})$. By commutativity of diagram (3) we deduce that the $Q(\Lambda _{h,0}/P)$-span of $G_P\cap U(\rho _{P\mathbb I_0^\circ })$ is $Q(\mathbb I^\circ _0/P\mathbb I^\circ _0)$. In particular $G_{P\mathbb I_0^\circ }\cap U(\rho _{P\mathbb I_0^\circ })$ contains a $\Lambda _{h,0}/P$-lattice, hence by Lemma 4.19 a nonzero ideal ${\mathfrak {a}}_P$ of $\mathbb I^\circ _0/P\mathbb I^\circ _0$.

Note that the representation $\rho _0:H\rightarrow \mathrm {SL}_2(\mathbb I_0^\circ )$ satisfies the hypotheses of Proposition 4.8. Indeed we assumed that $\rho :G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I)$ is $(H_0,\mathbb Z_p)$-regular, so the image of $\rho \vert _{H_0}$ contains a diagonal $\mathbb Z_p$-regular element d. Since H is a normal subgroup of $H_0$, $\rho (H)$ is a normal subgroup of $\rho (H_0)$ and it is normalized by d. By a trivial computation we see that the image of $\rho _0=\rho \vert _H\otimes (\det \rho \vert _H)^{-1/2}$ is also normalized by d.

Let ${\mathfrak {a}}$ be an ideal of $\mathbb I^\circ _0$ projecting to ${\mathfrak {a}}_P\subset U(\rho _{0,P\mathbb I_0^\circ })$. By Proposition 4.8 applied to $\rho _0$ we obtain that the map $U(\rho _0)\rightarrow U(\rho _{0,P\mathbb I_0^\circ })$ is surjective, so the $\mathbb Z_p$-module ${\mathfrak {a}}\cap U(\rho _0)$ also surjects to ${\mathfrak {a}}_P$. Since $\Lambda _{h,0}$ is local we can apply Nakayama’s lemma to the $\Lambda _{h,0}$-module $\Lambda _{h,0}({\mathfrak {a}}\cap U(\rho _0)$ to conclude that it coincides with ${\mathfrak {a}}$. Hence ${\mathfrak {a}}\subset \Lambda _{h,0}\cdot U(\rho _0)$, so the $\Lambda _{h,0}$-span of $U(\rho _0)$ contains a $\Lambda _{h,0}$-lattice in $\mathbb I^\circ _0$ by lemma 4.19.$\square$

We show that Proposition 4.20 is true if we replace $\rho _0$ by $\rho \vert _H$. This will be a consequence of the description of the subnormal sugroups of $\mathrm {GL}_2(\mathbb I^\circ )$ presented in [23], but we need a preliminary step because we cannot induce a $\Lambda _{h,0}$-module structure on the unipotent subgroups of G. For a subgroup $\mathcal {G}\subset \mathrm {GL}_2(\mathbb I^\circ _0)$ define $\mathcal {G}^p=\{g^p,\, g\in G\}$ and $\widetilde{\mathcal {G}}=\mathcal {G}^p\cap (1+p\mathrm {M}_2(\mathbb I^\circ _0))$. Let $\widetilde{\mathcal {G}}^{\Lambda _{h,0}}$ be the subgroup of $\mathrm {GL}_2(\mathbb I^\circ )$ generated by the set $\{g^\lambda :g\in \widetilde{\mathcal {G}}, \lambda \in \Lambda _{h,0}\}$ where $g^\lambda =\exp (\lambda \log g)$. We have the following.

Lemma 4.21

The group $\widetilde{\mathcal {G}}^{\Lambda _{h,0}}$ contains a congruence subgroup of $\mathrm {SL}_2(\mathbb I^\circ _0)$ if and only if both of the unipotent subgroups $\mathcal {G}\cap U^+(\mathbb I^\circ _0)$ and $\mathcal {G}\cap U^-(\mathbb I^\circ _0)$ contain a basis of a $\Lambda _{h,0}$-lattice in $\mathbb I^\circ _0$.

Proof

It is easy to see that $\mathcal {G}\cap U^+(\mathbb I^\circ _0)$ contains the basis of a $\Lambda _{h,0}$-lattice in $\mathbb I^\circ _0$ if and only if the same is true for $\widetilde{\mathcal {G}}\cap U^+(\mathbb I^\circ _0)$. The same is true for $U^-$. By a standard argument, used in the proofs of [9, Lemma 2.9] and [12, Proposition 4.2], $\mathcal {G}^{\Lambda _{h,0}}\subset \mathrm {GL}_2(\mathbb I^\circ _0)$ contains a congruence subgroup of $\mathrm {SL}_2(\mathbb I^\circ _0)$ if and only if both its upper and lower unipotent subgroup contain an ideal of $\mathbb I^\circ _0$. We have $U^+(\mathbb I^\circ _0)\cap \mathcal {G}^{\Lambda _{h,0}}=\Lambda _{h,0}(\mathcal {G}\cap U^+(\mathbb I^\circ _0))$, so by Lemma 4.19 $U^+(\mathbb I^\circ _0)\cap \mathcal {G}^{\Lambda _{h,0}}$ contains an ideal of $\mathbb I^\circ _0$ if and only if $\mathcal {G}\cap U^+(\mathbb I^\circ _0)$ contains a basis of a $\Lambda _{h,0}$-lattice in $\mathbb I^\circ _0$. We proceed in the same way for $U^-$.$\square$

Now let $G_0=\mathrm {Im}\,\rho \vert _H$, $G=\mathrm {Im}\,\rho _0$. Note that $G_0\cap \mathrm {SL}_2(\mathbb I^\circ _0)$ is a normal subgroup of G. Let $f:\mathrm {GL}_2(\mathbb I^\circ _0)\rightarrow \mathrm {SL}_2(\mathbb I^\circ _0)$ be the homomorphism sending g to $\det (g)^{-1/2}g$. We have $G=f(G_0)$ by definition of $\rho _0$. We show the following.

Proposition 4.22

The subgroups $G_0\cap U^{\pm }(\mathbb I^\circ _0)$ both contain the basis of a $\Lambda _{h,0}$-lattice in $\mathbb I^\circ _0$ if and only if $G\cap U^{\pm }(\mathbb I^\circ _0)$ both contain the basis of a $\Lambda _{h,0}$-lattice in $\mathbb I^\circ _0$.

Proof

Since $G=f(G_0)$ we have $\widetilde{G}=f(\widetilde{G_0})$. This implies that $\widetilde{G}^{\Lambda _{h,0}}=f(\widetilde{G_0}^{\Lambda _{h,0}})$. We remark that $\widetilde{G_0}^{\Lambda _{h,0}}\cap \mathrm {SL}_2(\mathbb I_0^\circ )$ is a normal subgroup of $\widetilde{G}^{\Lambda _{h,0}}$. Indeed $\widetilde{G_0}^{\Lambda _{h,0}}\cap \mathrm {SL}_2(\mathbb I_0^\circ )$ is normal in $\widetilde{G_0}^{\Lambda _{h,0}}$, so its image $f(G_0^{\Lambda _{h,0}}\cap \mathrm {SL}_2(\mathbb I_0^\circ ))=G_0^{\Lambda _{h,0}}\cap \mathrm {SL}_2(\mathbb I_0^\circ )$ is normal in $f(G_0^{\Lambda _{h,0}})=\widetilde{G}^{\Lambda _{h,0}}$.

By [23, Corollary 1] a subgroup of $\mathrm {GL}_2(\mathbb I^\circ _0)$ contains a congruence subgroup of $\mathrm {SL}_2(\mathbb I^\circ _0)$ if and only if it is subnormal in $\mathrm {GL}_2(\mathbb I^\circ _0)$ and it is not contained in the centre. We note that $\widetilde{G_0}^{\Lambda _{h,0}}\cap \mathrm {SL}_2(\mathbb I^\circ _0)=(\widetilde{G_0}\cap \mathrm {SL}_2(\mathbb I^\circ _0))^{\Lambda _{h,0}}$ is not contained in the subgroup $\{\pm 1\}$. Otherwise also $\widetilde{G_0}\cap \mathrm {SL}_2(\mathbb I^\circ _0)$ would be contained in $\{\pm 1\}$ and $\mathrm {Im}\,\rho \cap \mathrm {SL}_2(\mathbb I^\circ _0)$ would be finite, since $\widetilde{G_0}$ is of finite index in $G_0^p$. This would give a contradiction: indeed if ${\mathfrak {P}}$ is an arithmetic prime of $\mathbb I^\circ$ of weight greater than 1 and ${\mathfrak {P}}^\prime ={\mathfrak {P}}\cap \mathbb I^\circ _0$, the image of $\rho$ modulo ${\mathfrak {P}}^\prime$ contains a congruence subgroup of $\mathrm {SL}_2(\mathbb I_0^\circ /{\mathfrak {P}}^\prime )$ by the result of [15].

Since $\widetilde{G_0}^{\Lambda _{h,0}}\cap \mathrm {SL}_2(\mathbb I^\circ _0)$ is a normal subgroup of $\widetilde{G}^{\Lambda _{h,0}}$, we deduce by [23, Corollary 1] that $\widetilde{G_0}^{\Lambda _{h,0}}\cap \mathrm {SL}_2(\mathbb I^\circ _0)$ (hence $\widetilde{G_0}^{\Lambda _{h,0}}$) contains a congruence subgroup of $\mathrm {SL}_2(\mathbb I^\circ _0)$ if and only if $\widetilde{G}^{\Lambda _{h,0}}$ does. By applying Lemma 4.21 to $\mathcal {G}=G_0$ and $\mathcal {G}=G$ we obtain the desired equivalence.$\square$

By combining Propositions 4.20 and 4.22 we obtain the following.

Corollary 4.23

The $\Lambda _{h,0}$-span of each of the unipotent subgroups $\mathrm {Im}\,\rho \cap U^{\pm }$ contains a $\Lambda _{h,0}$-lattice in $\mathbb I^\circ _0$.

Unlike in the ordinary case we cannot deduce from the corollary that $\mathrm {Im}\,\rho$ contains a congruence subgroup of $\mathrm {SL}_2(\mathbb I^\circ _0)$, since we are working over $\Lambda _h\ne \Lambda$ and we cannot induce a $\Lambda _h$-module structure (not even a $\Lambda$-module structure) on $\mathrm {Im}\,\rho \cap U^{\pm }$. The proofs of [9, Lemma 2.9] and [12, Proposition 4.3] rely on the existence, in the image of the Galois group, of an element inducing by conjugation a $\Lambda$-module structure on $\mathrm {Im}\,\rho \cap U^{\pm }$. In their situation this is predicted by the condition of Galois ordinarity of $\rho$. In the non-ordinary case we will find an element with a similar property via relative Sen theory. In order to do this we will need to work with a suitably defined Lie algebra rather than with the group itself.

5 Relative Sen Theory

We recall the notations of Sect. 3.1. In particular $r_h=p^{-s_h}$, with $s_h\in \mathbb Q$, is the h-adapted radius (which we also take smaller than $p^{-\frac{1}{p-1}})$, $\eta _h$ is an element in $\mathbb C_p$ of norm $r_h$, $K_h$ is the Galois closure in $\mathbb C_p$ of $\mathbb Q_p(\eta _h)$ and $\mathcal {O}_h$ is the ring of integers in $K_h$. The ring $\Lambda _h$ of analytic functions bounded by 1 on the open disc $\mathcal {B}_h=\mathcal {B}(0,r_h^-)$ is identified to $\mathcal {O}_h[[t]]$. We take a sequence of radii $r_i=p^{-s_h-1/i}$ converging to $r_h$ and denote by $A_{r_i}=K_h\langle t,X_i\rangle /(pX_i-t^i)$ the $K_h$-algebra defined in Sect. 3.1 which is a form over $K_h$ of the $\mathbb C_p$-algebra of analytic functions on the closed ball $\mathcal {B}(0,r_i)$ (its Berthelot model). We denote by $A_{r_i}^\circ$ the $\mathcal {O}_h$-subalgebra of functions bounded by 1. Then $\Lambda _h={\varprojlim }_{i} A_{r_i}^\circ$ where $A_{r_j}^\circ \rightarrow A_{r_i}^\circ$ for $i<j$ is the restriction of analytic functions.

We defined in Sect. 4.1 a subring $\mathbb I_0^\circ \subset \mathbb I^\circ$, finite over $\Lambda _{h,0}\subset \Lambda _h$. For $r_i$ as above, we write $A_{0,r_i}^\circ =\mathcal {O}_{h,0}\langle t,X_i\rangle /(pX_i-t^i)$ with maps $A_{0,r_j}^\circ \rightarrow A_{0,r_i}^\circ$ for $i<j$, so that $\Lambda _{h,0}={\varprojlim }_{i}A_{0,r_i}^\circ$. Let $\mathbb I_{r_i}^\circ =\mathbb I^\circ \widehat{\otimes }_{\Lambda _h} A_{r_i}^\circ$ and $\mathbb I_{0,r_i}^\circ =\mathbb I_0^\circ \widehat{\otimes }_{\Lambda _{h,0}} A_{0,r_i}^\circ$, both endowed with their p-adic topology. Note that $(\mathbb I_{r_i}^\circ )^\Gamma =\mathbb I_{r_i,0}^\circ$.

Consider the representation $\rho :G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I^\circ )$ associated with a family $\theta :\mathbb T_h\rightarrow \mathbb I^\circ$. We observe that $\rho$ is continuous with respect to the profinite topology of $\mathbb I^\circ$ but not with respect to the p-adic topology. For this reason we fix an arbitrary radius r among the $r_i$ defined above and consider the representation $\rho _r:G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I_{r}^\circ )$ obtained by composing $\rho$ with the inclusion $\mathrm {GL}_2(\mathbb I^\circ )\hookrightarrow \mathrm {GL}_2(\mathbb I_{r}^\circ )$. This inclusion is continuous, hence the representation $\rho _r$ is continuous with respect to the p-adic topology on $\mathrm {GL}_2(\mathbb I_{0,r}^\circ )$.

Recall from Proposition 4.14 that, after replacing $\rho$ by a conjugate, there is an open normal subgroup $H_0\subset G_\mathbb Q$ such that the restriction $\rho \vert _{H_0}$ takes values in $\mathrm {GL}_2(\mathbb I_0^\circ )$ and is $(H_0,\mathbb Z_p)$-regular. Then the restriction $\rho _r\vert _{H_0}$ gives a representation $H_0\rightarrow \mathrm {GL}_2(\mathbb I_{0,r}^\circ )$ which is continuous with respect to the p-adic topology on $\mathrm {GL}_2(\mathbb I_{0,r}^\circ )$.

5.1 Big Lie Algebras

Recall that $G_p\subset G_\mathbb Q$ denotes our chosen decomposition group at p. Let $G_r$ and $G_r^\mathrm {loc}$ be the images respectively of $H_0$ and $G_p\cap H_0$ under the representation $\rho _r\vert _{H_0}:H_0\rightarrow \mathrm {GL}_2(\mathbb I_{0,r}^\circ )$. Note that they are actually independent of r since they coincide with the images of $H_0$ and $G_p\cap H_0$ under $\rho$.

For every ring R and ideal $I\subset R$ we denote by $\Gamma _{\mathrm {GL}_2(R)}(I)$ the $\mathrm {GL}_2$-congruence subgroup consisting of elements $g\in \mathrm {GL}_2(R)$ such that $g\equiv \mathrm {Id}_2\pmod {I}$. Let $G_r^{\prime }=G_r\cap \Gamma _{\mathrm {GL}_2(\mathbb I_{0,r}^\circ )}(p)$ and $G_r^{\prime ,\mathrm {loc}}=G_r^\mathrm {loc}\cap \Gamma _{\mathrm {GL}_2(\mathbb I_{0,r}^\circ )}(p)$, so that $G_r^{\prime }$ and $G_{r}^{\prime ,\mathrm {loc}}$ are pro-p groups. Note that the congruence subgroups $\Gamma _{\mathrm {GL}_2(\mathbb I_{0,r})}(p^m)$ are open in $\mathrm {GL}_2(\mathbb I_{0,r})$ for the p-adic topology. In particular $G_r^\prime$ and $G_{r}^{\prime ,\mathrm {loc}}$ can be identified with the images under $\rho$ of the absolute Galois groups of finite extensions of $\mathbb Q$ and respectively $\mathbb Q_p$.

Remark 5.1

We remark that we can choose an arbitrary $r_0$ and set, for every r, $G_r^{\prime }=G_r\cap \Gamma _{\mathrm {GL}_2(\mathbb I_{0,r_0}^\circ )}(p)$. Then $G_r^\prime$ is a pro-p subgroup of $G_r$ for every r and it is independent of r since $G_r$ is. This will be important in Theorem 7.1 where we will take projective limits over r of various objects.

We set $A_{0,r}=A_{0,r}^\circ [p^{-1}]$ and $\mathbb I_{0,r}=\mathbb I_{0,r}^\circ [p^{-1}]$. We consider from now on $G_r^{\prime }$ and $G_r^{\prime ,\mathrm {loc}}$ as subgroups of $\mathrm {GL}_2(\mathbb I_{0,r})$ through the inclusion $\mathrm {GL}_2(\mathbb I_{0,r}^\circ )\hookrightarrow \mathrm {GL}_2(\mathbb I_{0,r})$.

We want to define big Lie algebras associated with the groups $G_r^{\prime }$ and $G_r^{\prime ,\mathrm {loc}}$. For every nonzero ideal ${\mathfrak {a}}$ of the principal ideal domain $A_{0,r}$, we denote by $G_{r,{\mathfrak {a}}}^\prime$ and $G_{r,{\mathfrak {a}}}^{\prime ,\mathrm {loc}}$ the images respectively of $G_r^{\prime }$ and $G_r^{\prime ,\mathrm {loc}}$ under the natural projection $\mathrm {GL}_2(\mathbb I_{0,r})\rightarrow \mathrm {GL}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})$. The pro-p groups $G_{r,{\mathfrak {a}}}^\prime$ and $G_{r,{\mathfrak {a}}}^{\prime ,\mathrm {loc}}$ are topologically of finite type so we can define the corresponding $\mathbb Q_p$-Lie algebras ${\mathfrak {H}}_{r,{\mathfrak {a}}}$ and ${\mathfrak {H}}_{r,{\mathfrak {a}}}^\mathrm {loc}$ using the p-adic logarithm map: ${\mathfrak {H}}_{r,{\mathfrak {a}}}=\mathbb Q_p\cdot \mathrm {Log}\, G_{r,{\mathfrak {a}}}^{\prime }$ and ${\mathfrak {H}}_{r,{\mathfrak {a}}}^\mathrm {loc}=\mathbb Q_p\cdot \mathrm {Log}\, G_{r,{\mathfrak {a}}}^{\prime ,\mathrm {loc}}$. They are closed Lie subalgebras of the finite dimensional $\mathbb Q_p$-Lie algebra $\mathrm {M}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})$.

Let $B_r={\varprojlim }_{({\mathfrak {a}},P_1)=1}A_{0,r}/ {\mathfrak {a}}A_{0,r}$ where the inverse limit is taken over nonzero ideals ${\mathfrak {a}}\subset A_{0,r}$ prime to $P_1=(u^{-1}(1+T)-1)$ (the reason for excluding $P_1$ will become clear later). We endow $B_r$ with the projective limit topology coming from the p-adic topology on each quotient. We have a topological isomorphism of $K_{h,0}$-algebras

$$ B_r\cong \prod _{P\ne P_1} \widehat{(A_{0,r})}_{P}, $$

where the product is over primes P and $\widehat{(A_{0,r})}_P={\varprojlim }_{m\geqslant 1}A_{0,r}/P^mA_{0,r}$ denotes the $K_{h,0}$-Fréchet space inverse limit of the finite dimensional $K_{h,0}$-vector spaces $A_{0,r}/P^mA_{0,r}$. Similarly, let $\mathbb B_r={\varprojlim }_{({\mathfrak {a}},P_1)=1}\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r}$, where as before ${\mathfrak {a}}$ varies over all nonzero ideals of $A_{0,r}$ prime to $P_1$. We have

$$ \mathbb B_r\cong \prod _{P\ne P_1} \widehat{(\mathbb I_{0,r})}_{P\mathbb I_{0,r}}\cong \prod _{{\mathfrak {P}}\not \mid P_1} \widehat{(\mathbb I_{0,r})}_{{\mathfrak {P}}}\cong \varprojlim _{({\mathfrak {Q}},P_1)=1}\mathbb I_{0,r}/{\mathfrak {Q}}, $$

where the second product is over primes ${\mathfrak {P}}$ of $\mathbb I_{0,r}$ and the projective limit is over primary ideals ${\mathfrak {Q}}$ of $\mathbb I_{0,r}$. Here $\widehat{(\mathbb I_{0,r})}_{\mathfrak {P}}$ denotes the projective limit of finite dimensional $K_{h,0}$-algebras (endowed with the p-adic topology). The last isomorphism follows from the fact that $\mathbb I_{0,r}$ is finite over $A_{0,r}$, so that there is an isomorphism $\mathbb I_{0,r}\otimes \widehat{(A_{0,r})}_P=\prod _{\mathfrak {P}}\widehat{(\mathbb I_{0,r})}_{\mathfrak {P}}$ where P is a prime of $A_{0,r}$ and ${\mathfrak {P}}$ varies among the primes of $\mathbb I_{0,r}$ above P. We have natural continuous inclusions $A_{0,r}\hookrightarrow B_r$ and $\mathbb I_{0,r}\hookrightarrow \mathbb B_r$, both with dense image. The map $A_{0,r}\hookrightarrow \mathbb I_{0,r}$ induces an inclusion $B_r\hookrightarrow \mathbb B_r$ with closed image. Note however that $\mathbb B_r$ is not finite over $B_r$. We will work with $\mathbb B_r$ for the rest of this section, but we will need $B_r$ later.

For every ${\mathfrak {a}}$ we have defined Lie algebras ${\mathfrak {H}}_{r,{\mathfrak {a}}}$ and ${\mathfrak {H}}_{r,{\mathfrak {a}}}^\mathrm {loc}$ associated with the finite type Lie groups $G_{r,{\mathfrak {a}}}^\prime$ and $G_{r,{\mathfrak {a}}}^{\prime ,\mathrm {loc}}$. We take the projective limit of these algebras to obtain Lie subalgebras of $\mathrm {M}_2(\mathbb B_r)$.

Definition 5.2

The Lie algebras associated with $G_{r}^\prime$ and $G_{r}^{\prime ,\mathrm {loc}}$ are the closed $\mathbb Q_p$-Lie subalgebras of $\mathrm {M}_2(\mathbb B_r)$ given respectively by

$$ {\mathfrak {H}}_{r}=\varprojlim _{({\mathfrak {a}},P_1)=1}{\mathfrak {H}}_{r,{\mathfrak {a}}} $$

and

$$ {\mathfrak {H}}_{r}^\mathrm {loc}=\varprojlim _{({\mathfrak {a}},P_1)=1}{\mathfrak {H}}_{r,{\mathfrak {a}}}^\mathrm {loc}, $$

where as usual the products are taken over nonzero ideals ${\mathfrak {a}}\subset A_{0,r}$ prime to $P_1$.

For every ideal ${\mathfrak {a}}$ prime to $P_1$, we have continuous homomorphisms ${\mathfrak {H}}_r\rightarrow {\mathfrak {H}}_{r,{\mathfrak {a}}}$ and ${\mathfrak {H}}_r^\mathrm {loc}\rightarrow {\mathfrak {H}}_{r,{\mathfrak {a}}}^\mathrm {loc}$. Since the transition maps are surjective these homomorphisms are surjective.

Remark 5.3

The limits in Definition 5.2 can be replaced by limits over primary ideals of $\mathbb I_{0,r}$. Explicitly, let ${\mathfrak {Q}}$ be a primary ideal of $\mathbb I_{0,r}$. Let $G_{r,{\mathfrak {Q}}}^\prime$ be the image of $G_r^\prime$ via the natural projection $\mathrm {GL}_2(\mathbb I_{0,r})\rightarrow \mathrm {GL}_2(\mathbb I_{0,r}/{\mathfrak {Q}})$ and let ${\mathfrak {H}}_{r,{\mathfrak {Q}}}$ be the Lie algebra associated with $G_{r,{\mathfrak {Q}}}^\prime$ (which is a finite type Lie group). We have an isomorphism of topological Lie algebras

$$ {\mathfrak {H}}_{r}=\varprojlim _{({\mathfrak {Q}},P_1)=1}{\mathfrak {H}}_{r,{\mathfrak {Q}}}, $$

where the limit is taken over primary ideals ${\mathfrak {Q}}$ of $\mathbb I_{0,r}$. This is naturally a subalgebra of $\mathrm {M}_2(\mathbb B_r)$ since $\mathbb B_r\cong {\varprojlim }_{({\mathfrak {Q}},P_1)=1}\mathbb I_{0,r}/{\mathfrak {Q}}$. The same goes for the local algebras.

5.2 The Sen Operator Associated with a Galois Representation

Recall that there is a finite extension $K/\mathbb Q_p$ such that $G_r^{\prime ,\mathrm {loc}}$ is the image of $\rho \vert _{\mathrm {Gal}(\overline{K}/K)}$ and, for an ideal $P\subset A_{0,r}$ and $m\geqslant 1$, $G_{r,P^m}^{\prime ,\mathrm {loc}}$ is the image of $\rho _{r,P^m}\vert _{\mathrm {Gal}(\overline{K}/K)}$. Following [19, 21] we can define a Sen operator associated with $\rho _r\vert _{\mathrm {Gal}(\overline{K}/K)}$ and $\rho _{r,P^m}\vert _{\mathrm {Gal}(\overline{K}/K)}$ for every ideal $P\subset A_{0,r}$ and every $m\geqslant 1$. We will see that these operators satisfy a compatibility property. We write for the rest of the section $\rho _r$ and $\rho _{r,P^m}$ while implicitly taking the domain to be $\mathrm {Gal}(\overline{K}/K)$.

We begin by recalling the definition of the Sen operator associated with a representation $\tau :\mathrm {Gal}(\overline{K}/K)\rightarrow \mathrm {GL}_m(\mathcal {R})$ where $\mathcal {R}$ is a Banach algebra over a p-adic field L. We follow [21]. We can suppose $L\subset K$; if not we just restrict $\tau$ to the open subgroup $\mathrm {Gal}(\overline{K}/KL)\subset \mathrm {Gal}(\overline{K}/K)$.

Let $L_{\infty }$ be a totally ramified $\mathbb Z_p$-extension of L. Let $\gamma$ be a topological generator of $\Gamma =\mathrm {Gal}(L_{\infty }/L)$, $\Gamma _n\subset \Gamma$ the subgroup generated by $\gamma ^{p^n}$ and $L_{n}=L_{\infty }^{\gamma ^{p^n}}$, so that $L_{\infty }=\cup _n L_{n}$. Let $L_n^\prime =L_{n}K$ and $G_n^\prime =\mathrm {Gal}(\overline{L}/L_n^\prime )$. If $\mathcal {R}^m$ is the $\mathcal {R}$-module over which $\mathrm {Gal}(\overline{K}/K)$ acts via $\tau$, define an action of $\mathrm {Gal}(\overline{K}/K)$ on $\mathcal {R}\widehat{\otimes }_L \mathbb C_p$ by letting $\sigma \in \mathrm {Gal}(\overline{K}/K)$ map $x\otimes y$ to $\tau (\sigma )(x)\otimes \sigma (y)$. Then by the results of [19, 21] there is a matrix $M\in \mathrm {GL}_m\left( \mathcal {R}\widehat{\otimes }_L \mathbb C_p\right)$, an integer $n\geqslant 0$ and a representation $\delta :\Gamma _n\rightarrow \mathrm {GL}_m(\mathcal {R}\otimes _L L_{n}^\prime )$ such that for all $\sigma \in G_n^\prime$

$$ M^{-1}\tau (\sigma )\sigma (M)=\delta (\sigma ). $$

Definition 5.4

The Sen operator associated with $\tau$ is

$$ \phi =\lim _{\sigma \rightarrow 1}\frac{\log \bigl (\delta (\sigma )\bigr )}{\log (\chi (\sigma ))}\in \mathrm {M}_m(\mathcal {R}\widehat{\otimes }_L \mathbb C_p). $$

The limit exists as for $\sigma$ close to 1 the map $\displaystyle \sigma \mapsto \frac{\log \bigl (\delta (\sigma )\bigr )}{\log (\chi (\sigma ))}$ is constant. It is proved in [21, Sect. 2.4] that $\phi$ does not depend on the choice of $\delta$ and M.

If $L=\mathcal {R}=\mathbb Q_p$, we define the Lie algebra ${\mathfrak {g}}$ associated with $\tau (\mathrm {Gal}(\overline{K}/K))$ as the $\mathbb Q_p$-vector space generated by the image of the logarithm map in $\mathrm {M}_m(\mathbb Q_p)$. In this situation the Sen operator $\phi$ associated with $\tau$ has the following property.

Theorem 5.5

[19, Theorem 1] For a continuous representation $\tau :G_K\,{\rightarrow }\,\mathrm {GL}_m(\mathbb Q_p)$, the Lie algebra ${\mathfrak {g}}$ of the group $\tau (\mathrm {Gal}(\overline{K}/K))$ is the smallest $\mathbb Q_p$-subspace of $\mathrm {M}_m(\mathbb Q_p)$ such that ${\mathfrak {g}}{\otimes {\mathbb Q_p}} \mathbb C_p$ contains $\phi$.

This theorem is valid in the absolute case above, but relies heavily on the fact that the image of the Galois group is a finite dimensional Lie group. In the relative case it is doubtful that its proof can be generalized.

5.3 The Sen Operator Associated with $\rho _r$

Set $\mathbb I_{0,r,\mathbb C_p}=\mathbb I_{0,r}\widehat{\otimes }_{K_{h,0}}\mathbb C_p$. It is a Banach space for the natural norm. Let $\mathbb B_{r,\mathbb C_p}=\mathbb B_r\widehat{\otimes }_{K_{h,0}}\mathbb C_p$; it is the topological $\mathbb C_p$-algebra completion of $\mathbb B_r\otimes _{K_{h,0}} \mathbb C_p$ for the (uncountable) set of nuclear seminorms $p_{{\mathfrak {a}}}$ given by the norms on $\mathbb I_{0,r,\mathbb C_p}/{\mathfrak {a}}\mathbb I_{0,r,\mathbb C_p}$ via the specialization morphisms $\pi _{\mathfrak {a}}:\mathbb B_r\otimes _{K_{h,0}}\mathbb C_p\rightarrow \mathbb I_{0,r,\mathbb C_p}/{\mathfrak {a}}\mathbb I_{0,r,\mathbb C_p}$. Let ${\mathfrak {H}}_{r,{\mathfrak {a}},\mathbb C_p}={\mathfrak {H}}_{r,{\mathfrak {a}}}\otimes _{K_{h,0}}\mathbb C_p$ and ${\mathfrak {H}}_{r,{\mathfrak {a}},\mathbb C_p}^\mathrm {loc}={\mathfrak {H}}_{r,{\mathfrak {a}},}^\mathrm {loc}\otimes _{K_{h,0}}\mathbb C_p$. Then we define ${\mathfrak {H}}_{r,\mathbb C_p}={\mathfrak {H}}_r\widehat{\otimes }_{K_{h,0}}\mathbb C_p$ as the topological $\mathbb C_p$-Lie algebra completion of ${\mathfrak {H}}_r\otimes _{K_{0,h}}\mathbb C_p$ for the (uncountable) set of seminorms $p_{\mathfrak {a}}$ given by the norms on ${\mathfrak {H}}_{r,{\mathfrak {a}},\mathbb C_p}$ and similar specialization morphisms $\pi _{\mathfrak {a}}:{\mathfrak {H}}_{r,}\otimes _{K_{h,0}}\mathbb C_p\rightarrow {\mathfrak {H}}_{r,{\mathfrak {a}},\mathbb C_p}$. We define in the same way ${\mathfrak {H}}_{r,\mathbb C_p}^\mathrm {loc}$ in terms of the norms on ${\mathfrak {H}}^{\mathrm {loc}}_{r,{\mathfrak {a}},\mathbb C_p}$. Note that by definition we have

$${\mathfrak {H}}_{r,\mathbb C_p}=\varprojlim _{({\mathfrak {a}},P_1)=1}{\mathfrak {H}}_{r,{\mathfrak {a}},\mathbb C_p},\,\,\mathrm {and}\,\, {\mathfrak {H}}_{r,\mathbb C_p}^\mathrm {loc}=\varprojlim _{({\mathfrak {a}},P_1)=1}{\mathfrak {H}}_{r,{\mathfrak {a}},\mathbb C_p}^\mathrm {loc}.$$

We apply the construction of the previous subsection to $L=K_{h,0}$, $\mathcal {R}=\mathbb I_{0,r}$ which is a Banach L-algebra with the p-adic topology, and $\tau =\rho _r$. We obtain an operator $\phi _r\in \mathrm {M}_2(\mathbb I_{0,r,\mathbb C_p})$. Recall that we have a natural continuous inclusion $\mathbb I_{0,r}\hookrightarrow \mathbb B_r$, inducing inclusions $\mathbb I_{0,r,\mathbb C_p}\hookrightarrow \mathbb B_{r,\mathbb C_p}$ and $\mathrm {M}_2(\mathbb I_{0,r,\mathbb C_p})\hookrightarrow \mathrm {M}_2(\mathbb B_{r,\mathbb C_p})$. We denote all these inclusions by $\iota _{\mathbb B_r}$ since it will be clear each time to which we are referring to. We will prove in this section that $\iota _{\mathbb B_r}(\phi _r$) is an element of ${\mathfrak {H}}_{r,\mathbb C_p}^\mathrm {loc}$.

Let ${\mathfrak {a}}$ be a nonzero ideal of $A_{0,r}$. Let us apply Sen’s construction to $L=K_{h,0}$, $\mathcal {R}=\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r}$ and $\tau =\rho _{r,{\mathfrak {a}}}:\mathrm {Gal}(\overline{K}/K)\rightarrow \mathrm {GL}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})$; we obtain an operator $\phi _{r,{\mathfrak {a}}}\in \mathrm {M}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r}\widehat{\otimes }_{K_{h,0}}\mathbb C_p)$.

Let

$$\pi _{\mathfrak {a}}:\mathrm {M}_2(\mathbb I_{0,r}\widehat{\otimes }_{K_{h,0}}\mathbb C_p)\rightarrow \mathrm {M}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r}\widehat{\otimes }_{K_{h,0}}\mathbb C_p)$$

and

$$\pi _{\mathfrak {a}}^\times :\mathrm {GL}_2(\mathbb I_{0,r}\widehat{\otimes }_{K_{h,0}}\mathbb C_p)\rightarrow \mathrm {GL}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r}\widehat{\otimes }_{K_{h,0}}\mathbb C_p)$$

be the natural projections.

Proposition 5.6

We have $\phi _{r,{\mathfrak {a}}}=\pi _{\mathfrak {a}}(\phi _r)$ for all ${\mathfrak {a}}$.

Proof

Recall from the construction of $\phi _r$ that there exist $M\in \mathrm {GL}_2\left( \mathbb I_{0,r,\mathbb C_p}\right)$, $n\geqslant 0$ and $\delta :\Gamma _n\rightarrow \mathrm {GL}_2(\mathbb I_{0,r}\widehat{\otimes }_{K_{h,0}}K_{h,0,n}^\prime )$ such that for all $\sigma \in G_n^\prime$ we have

$$\begin{aligned} M^{-1}\rho _r(\sigma )\sigma (M)=\delta (\sigma ) \end{aligned}$$

(4)

and

$$\begin{aligned} \phi _r=\lim _{\sigma \rightarrow 1}\frac{\log (\delta \bigl (\sigma )\bigr )}{\log (\chi (\sigma ))}. \end{aligned}$$

(5)

Let $M_{\mathfrak {a}}=\pi ^\times _{\mathfrak {a}}(M)\in \mathrm {GL}_2(\mathbb I_{0,r,\mathbb C_p}/{\mathfrak {a}}\mathbb I_{0,r,\mathbb C_p})$ and

$$\begin{aligned} \delta _{\mathfrak {a}}=\pi _{\mathfrak {a}}^\times \circ \delta :\Gamma _n\rightarrow \mathrm {GL}_2((\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})\widehat{\otimes }_{K_{h,0}}K_{h,0,n}^\prime ). \end{aligned}$$

Denote by $\phi _{r,{\mathfrak {a}}}\in \mathrm {M}_2((\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})\widehat{\otimes }_{K_{h,0}}K_{h,0,n}^\prime )$ the Sen operator associated with $\rho _{r,{\mathfrak {a}}}$. Now (4) gives

$$\begin{aligned} M_{\mathfrak {a}}^{-1}\rho _{r,{\mathfrak {a}}}(\sigma )\sigma (M_{\mathfrak {a}})=\delta _{\mathfrak {a}}(\sigma ) \end{aligned}$$

(6)

so we can calculate $\phi _{r,{\mathfrak {a}}}$ as

$$\begin{aligned} \phi _{r,{\mathfrak {a}}}=\lim _{\sigma \rightarrow 1}\frac{\log (\delta _{\mathfrak {a}}\bigl (\sigma )\bigr )}{\log (\chi (\sigma ))}\in \mathrm {M}_2(\mathcal {R}\widehat{\otimes }_L \mathbb C_p). \end{aligned}$$

(7)

By comparing this with (5) we see that $\phi _{r,{\mathfrak {a}}}=\pi _{\mathfrak {a}}(\phi _r)$.$\square$

Let $\phi _{r,\mathbb B_r}=\iota _{\mathbb B_r}(\phi _r)$. For a nonzero ideal ${\mathfrak {a}}$ of $A_{0,r}$ let $\pi _{\mathbb B_r,{\mathfrak {a}}}$ be the natural projection $\mathbb B_r\rightarrow \mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r}$. Clearly $\pi _{\mathbb B_r,{\mathfrak {a}}}(\phi _{r,\mathbb B_r})=\pi _{{\mathfrak {a}}}(\phi _r)$ and $\phi _{r,{\mathfrak {a}}}=\pi _{{\mathfrak {a}}}(\phi _r)$ by Proposition 5.6, so we have $\phi _{r,\mathbb B_r}={\varprojlim }_{({\mathfrak {a}},P_1)=1}\phi _{r,{\mathfrak {a}}}$.

We apply Theorem 5.5 to show the following.

Proposition 5.7

Let ${\mathfrak {a}}$ be a nonzero ideal of $A_{0,r}$ prime to $P_1$. The operator $\phi _{r,{\mathfrak {a}}}$ belongs to the Lie algebra ${\mathfrak {H}}_{r,{\mathfrak {a}},\mathbb C_p}^\mathrm {loc}$.

Proof

Let n be the dimension over $\mathbb Q_p$ of $\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r}$; by choosing a basis $(\omega _1,\ldots ,\omega _n)$ of this algebra as a $\mathbb Q_p$-vector space, we can define an injective ring homomorphism $\alpha :\mathrm {M}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})\hookrightarrow \mathrm {M}_{2n}(\mathbb Q_p)$ and an injective group homomorphism $\alpha ^\times :\mathrm {GL}_2(\mathbb I_{0,r}/\alpha \mathbb I_{0,r})\hookrightarrow \mathrm {GL}_{2n}(\mathbb Q_p)$. In fact, an endomorphism f of the $(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})$-module $(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})^2=(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})\cdot e_1\oplus (\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})\cdot e_2$ is $\mathbb Q_p$-linear, so it induces an endomorphism $\alpha (f)$ of the $\mathbb Q_p$-vector space $(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})^2=\bigoplus _{i,j} \mathbb Q_p\cdot \omega _i e_j$; furthermore if $\alpha$ is an automorphism then $\alpha (f)$ is one too. In particular $\rho _{r,{\mathfrak {a}}}$ induces a representation $\rho _{r,{\mathfrak {a}}}^\alpha =\alpha ^\times \circ \rho _{r,{\mathfrak {a}}}:\mathrm {Gal}(\overline{K}/K)\rightarrow \mathrm {GL}_{2n}(\mathbb Q_p)$. The image of $\rho _{r,{\mathfrak {a}}}^\alpha$ is the group $G_{r,{\mathfrak {a}}}^{\mathrm {loc},\alpha }=\alpha ^\times (G_{r,{\mathfrak {a}}}^\mathrm {loc})$. We consider its Lie algebra ${\mathfrak {H}}_{r,{\mathfrak {a}}}^{\mathrm {loc},\alpha }=\mathbb Q_p\cdot \mathrm {Log}\,(G_{r,{\mathfrak {a}}}^{\mathrm {loc},\alpha })\subset \mathrm {M}_{2n}(\mathbb Q_p)$. The p-adic logarithm commutes with $\alpha$ in the sense that $\alpha (\mathrm {Log}\, x)=\mathrm {Log}\,(\alpha ^\times (x))$ for every $x\in \Gamma _{\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r}}(p)$, so we have ${\mathfrak {H}}_{r,{\mathfrak {a}}}^{\mathrm {loc},\alpha }=\alpha ({\mathfrak {H}}_{r,{\mathfrak {a}}}^\mathrm {loc})$ (recall that ${\mathfrak {H}}_{r,{\mathfrak {a}}}^\mathrm {loc}=\mathbb Q_p\cdot \mathrm {Log}\, G_{r,{\mathfrak {a}}}^\mathrm {loc})$.

Let $\phi _{r,{\mathfrak {a}}}^\alpha$ be the Sen operator associated with $\rho _{r,{\mathfrak {a}}}^\alpha :\mathrm {Gal}(\overline{K}/K)\rightarrow \mathrm {GL}_{2n}(\mathbb Q_p)$. By Theorem 5.5 we have $\phi _{r,{\mathfrak {a}}}^\alpha \in {\mathfrak {H}}_{r,{\mathfrak {a}},\mathbb C_p}^{\mathrm {loc},\alpha }={\mathfrak {H}}_{r,{\mathfrak {a}}}^{\mathrm {loc},\alpha }\widehat{\otimes }\mathbb C_p$. Denote by $\alpha _{\mathbb C_p}$ the map $\alpha \widehat{\otimes }1:\mathrm {M}_2(\mathbb I_{0,r,\mathbb C_p}/{\mathfrak {a}}\mathbb I_{0,r,\mathbb C_p})\hookrightarrow \mathrm {M}_{2n}(\mathbb C_p)$. We show that $\phi _{r,{\mathfrak {a}}}^{\alpha _{\mathbb C_p}}=\alpha _{\mathbb C_p}(\phi _{r,{\mathfrak {a}}})$, from which it follows that $\phi _{r,{\mathfrak {a}}}\in {\mathfrak {H}}_{r,{\mathfrak {a}},\mathbb C_p}^\mathrm {loc}$ since ${\mathfrak {H}}_{r,{\mathfrak {a}},\mathbb C_p}^{\mathrm {loc},\alpha _{\mathbb C_p}}=\alpha _{\mathbb C_p}({\mathfrak {H}}_{r,{\mathfrak {a}},{\mathbb C_p}}^\mathrm {loc})$ and $\alpha _{\mathbb C_p}$ is injective. Now let $M_{\mathfrak {a}}$, $\delta _{\mathfrak {a}}$ be as in (6) and $M_{\mathfrak {a}}^{\alpha _{\mathbb C_p}}=\alpha _{\mathbb C_p}(M_{\mathfrak {a}})$, $\delta _{\mathfrak {a}}^{\alpha _{\mathbb C_p}}=\alpha _{\mathbb C_p}\circ \delta _{\mathfrak {a}}$. By applying $\alpha _C$ to (4) we obtain $(M_{\mathfrak {a}}^{\alpha _{\mathbb C_p}})^{-1}\rho _{r,{\mathfrak {a}}}^{\alpha _{\mathbb C_p}}(\sigma )\sigma (M_{\mathfrak {a}}^{\alpha _{\mathbb C_p}})=\delta _{\mathfrak {a}}^{\alpha _{\mathbb C_p}}(\sigma )$ for every $\sigma \in G_n^\prime$, so we can calculate

$$ \phi _{r,{\mathfrak {a}}}^{\alpha _{\mathbb C_p}}=\lim _{\sigma \rightarrow 1}\frac{\log (\delta _{\mathfrak {a}}^{\alpha _{\mathbb C_p}}\bigl (\sigma )\bigr )}{\log (\chi (\sigma ))}, $$

which coincides with $\alpha _{\mathbb C_p}(\phi _{r,{\mathfrak {a}}})$.$\square$

Proposition 5.8

The element $\phi _{r,\mathbb B_r}$ belongs to ${\mathfrak {H}}_{r,{\mathbb C_p}}^\mathrm {loc}$, hence to ${\mathfrak {H}}_{r,{\mathbb C_p}}$.

Proof

By definition of the space ${\mathfrak {H}}_{r,{\mathbb C_p}}^\mathrm {loc}$ as completion of the space ${\mathfrak {H}}_{r}^\mathrm {loc}\otimes _{K_{h,0}}\mathbb C_p$ for the seminorms $p_{\mathfrak {a}}$ given by the norms on ${\mathfrak {H}}_{r,{\mathfrak {a}},{\mathbb C_p}}^\mathrm {loc}$, we have ${\mathfrak {H}}_{r,{\mathbb C_p}}^\mathrm {loc}={\varprojlim }_{({\mathfrak {a}},P_1)=1}{\mathfrak {H}}_{r,{\mathfrak {a}},{\mathbb C_p}}^\mathrm {loc}$. By Proposition 5.6, we have $\phi _{r,\mathbb B_r}={\varprojlim }_{{\mathfrak {a}}}\phi _{r,{\mathfrak {a}}}$ and by Proposition 5.7 we have, for every ${\mathfrak {a}}$, $\phi _{r,{\mathfrak {a}}}\in {\mathfrak {H}}_{r,{\mathfrak {a}},{\mathbb C_p}}^\mathrm {loc}$. We conclude that $\phi _{r,\mathbb B_r}\in {\mathfrak {H}}_{r,{\mathbb C_p}}^\mathrm {loc}$.$\square$

Remark 5.9

In order to prove that our Lie algebras are “big” it will be useful to work with primary ideals of $A_r$, as we did in this subsection. However, in light of Remark 5.3, all of the results can be rewritten in terms of primary ideals ${\mathfrak {Q}}$ of $\mathbb I_{0,r}$. This will be useful in the next subsection, when we will interpolate the Sen operators corresponding to the attached to the classical modular forms representations.

From now on we identify $\mathbb I_{0,r,{\mathbb C_p}}$ with a subring of $\mathbb B_{r,{\mathbb C_p}}$ via $\iota _{\mathbb B_r}$, so we also identify $\mathrm {M}_2(\mathbb I_{0,r})$ with a subring of $\mathrm {M}_2(\mathbb B_r)$ and $\mathrm {GL}_2(\mathbb I_{0,r,{\mathbb C_p}})$ with a subgroup of $\mathrm {GL}_2(\mathbb B_{r,{\mathbb C_p}})$. In particular we identify $\phi _r$ with $\phi _{r,\mathbb B_r}$ and we consider $\phi _r$ as an element of ${\mathfrak {H}}_{r,{\mathbb C_p}}\cap \mathrm {M}_2(\mathbb I_{0,r,{\mathbb C_p}})$.

5.4 The Characteristic Polynomial of the Sen Operator

Sen proved the following result.

Theorem 5.10

Let $L_1$ and $L_2$ be two p-adic fields. Assume for simplicity that $L_2$ contains the normal closure of $L_1$. Let $\tau :\mathrm {Gal}(\overline{L}_1/L_1)\rightarrow \mathrm {GL}_m(L_2)$ be a continuous representation. For each embedding $\sigma :L_1\rightarrow L_2$, there is a Sen operator $\phi _{\tau ,\sigma }\in \mathrm {M}_m(\mathbb C_p\otimes _{L_1,\sigma }L_2)$ associated with $\tau$ and $\sigma$. If $\tau$ is Hodge-Tate and its Hodge-Tate weights with respect to $\sigma$ are $h_{1,\sigma },\ldots ,h_{m,\sigma }$ (with multiplicities, if any), then the characteristic polynomial of $\phi _{\tau ,\sigma }$ is $\prod _{i=1}^m(X-h_{i,\sigma })$.

Now let $k\in \mathbb N$ and $P_k=(u^{-k}(1+T)-1)$ be the corresponding arithmetic prime of $A_{0,r}$. Let ${\mathfrak {P}}_f$ be a prime of $\mathbb I_r$ above P, associated with the system of Hecke eigenvalues of a classical modular form f. Let $\rho _r:\mathbb G_Q\rightarrow \mathrm {GL}_2(\mathbb I_r)$ be as usual. The specialization of $\rho _r$ modulo ${\mathfrak {P}}$ is the representation $\rho _f:G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I_r/{\mathfrak {P}})$ classically associated with f, defined over the field $K_f=\mathbb I_r/{\mathfrak {P}}_f\mathbb I_r$. By a theorem of Faltings [8], when the weight of the form f is k, the representation $\rho _f$ is Hodge-Tate of Hodge-Tate weights 0 and $k-1$. Hence by Theorem 5.10 the Sen operator $\phi _f$ associated with $\rho _f$ has characteristic polynomial $X(X-(k-1))$. Let ${\mathfrak {P}}_{f,0}={\mathfrak {P}}_f\cap \mathbb I_{0,r}$. With the notations of the previous subsection, the specialization of $\rho _r$ modulo ${\mathfrak {P}}_{f,0}$ gives a representation $\rho _{r,{\mathfrak {P}}_{f,0}}:\mathrm {Gal}(\overline{K}/K)\rightarrow \mathrm {GL}_2(\mathbb I_{0,r}/{\mathfrak {P}}_{f,0})$, which coincides with $\rho _f\vert _{\mathrm {Gal}(\overline{K}/K)}$. In particular the Sen operator $\phi _{r,{\mathfrak {P}}_{f,0}}$ associated with $\rho _{r,{\mathfrak {P}}_{f,0}}$ is $\phi _f$.

By Proposition 5.6 and Remark 5.9, the Sen operator $\phi _r\in \mathrm {M}_2(\mathbb I_{0,r,\mathbb C_p})$ specializes modulo ${\mathfrak {P}}_{f,0}$ to the Sen operator $\phi _{r,{\mathfrak {P}}_{f,0}}$ associated with $\rho _{r,{\mathfrak {P}}_{f,0}}$, for every f as above. Since the primes of the form ${\mathfrak {P}}_{f,0}$ are dense in $\mathbb I_{0,r,\mathbb C_p}$, the eigenvalues of $\phi _{r,Q}$ are given by the unique interpolation of those of $\rho _{r,{\mathfrak {P}}_{f,0}}$. This way we will recover an element of $\mathrm {GL}_2(\mathbb B_{r,\mathbb C_p})$ with the properties we need.

Given $f\in A_{0,r}$ we define its p-adic valuation by $v_p^\prime (f)=\inf _{x\in \mathcal {B}(0,r)}v_p(f(x))$, where $v_p$ is our chosen valuation on $\mathbb C_p$. Then if $v^\prime (f-1)\leqslant p^{-\frac{1}{p-1}}$ there are well-defined elements $\log (f)$ and $\exp (\log (f))$ in $A_{0,r}$, and $\exp (\log (f))=f$.

Let $\phi ^\prime _r=\log (u)\phi _r$. Note that $\phi ^\prime _r$ is a well-defined element of $\mathrm {M}_2(\mathbb B_{r,\mathbb C_p})$ since $\log (u)\in \mathbb Q_p$. Recall that we denote by $C_T$ the matrix $\mathrm {diag}(u^{-1}(1+T),1)$. We have the following.

Proposition 5.11

• 1. The eigenvalues of $\phi ^\prime _r$ are $\log (u^{-1}(1+T))$ and 0. In particular the exponential $\Phi _r=\exp (\phi ^\prime _r)$ is defined in $\mathrm {GL}_2(\mathbb B_{r,\mathbb C_p})$. Moreover $\Phi ^\prime _r$ is conjugate to $C_T$ in $\mathrm {GL}_2(\mathbb B_{r,\mathbb C_p})$.

• 2. The element $\Phi ^\prime _r$ of part (1) normalizes ${\mathfrak {H}}_{r,{\mathbb C_p}}$.

Proof

For every ${\mathfrak {P}}_{f,0}$ as in the discussion above, the element $\log (u)\phi _r$ specializes to $\log (u)\phi _{r,{\mathfrak {P}}_{f,0}}$ modulo ${\mathfrak {P}}_{f,0}$. If ${\mathfrak {P}}_{f,0}$ is a divisor of $P_k$, the eigenvalues of $\log (u)\phi _{r,{\mathfrak {P}}_{f,0}}$ are $\log (u)(k-1)$ and 0. Since $1+T=u^k$ modulo ${\mathfrak {P}}_{f,0}$ for every prime ${\mathfrak {P}}_{f,0}$ dividing $P_k$, we have $\log (u^{-1}(1+T))=\log (u^{k-1})=(k-1)\log (u)$ modulo ${\mathfrak {P}}_{f,0}$. Hence the eigenvalues of $\log (u)\phi _{r,{\mathfrak {P}}_{f,0}}$ are interpolated by $\log (u^{-1}(1+T))$ and 0.

Recall that in Sect. 3.1 we chose $r_h$ smaller than $p^{-\frac{1}{p-1}}$. Since $r<r_h$, $v_p^\prime (T)<p^{-\frac{1}{p-1}}$. In particular $\log (u^{-1}(1+T))$ is defined and $\exp (\log (u^{-1}(1+T)))=u^{-1}(1+T)$, so $\Phi _r=\exp (\phi ^\prime _r)$ is also defined and its eigenvalues are $u^{-1}(1+T)$ and 1. The difference between the two is $u^{-1}(1+T)-1$; this elements belongs to $P_1$, hence it is invertible in $\mathbb B_r$. This proves (1).

By Proposition 5.8, $\phi _r\in {\mathfrak {H}}_{r,{\mathbb C_p}}$. Since ${\mathfrak {H}}_{r,{\mathbb C_p}}$ is a $\mathbb Q_p$-Lie algebra, $\log (u)\phi _r$ is also an element of ${\mathfrak {H}}_{r,{\mathbb C_p}}$. Hence its exponential $\Phi ^\prime _r$ normalizes ${\mathfrak {H}}_{r,{\mathbb C_p}}$.$\square$

6 Existence of the Galois Level for a Family with Finite Positive Slope

Let $r_h\in p^\mathbb Q\cap ]0,p^{-\frac{1}{p-1}}[$ be the radius chosen in Sect. 3. As usual we write r for any one of the radii $r_i$ of Sect. 3.1. Recall that ${\mathfrak {H}}_{r}\subset \mathrm {M}_2(\mathbb B_r)$ is the Lie algebra attached to the image of $\rho _{r}$ (see Definition 5.2) and ${\mathfrak {H}}_{r,\mathbb C_p}={\mathfrak {H}}_{r}\widehat{\otimes }\mathbb Q_{p}\mathbb C_p$. Let ${\mathfrak {u}}^\pm$ and ${\mathfrak {u}}^\pm _{\mathbb C_p}$ be the upper and lower nilpotent subalgebras of ${\mathfrak {H}}_{r}$, and ${\mathfrak {H}}_{r,\mathbb C_p}$ respectively.

Remark 6.1

The commutative Lie algebra ${\mathfrak {u}}^\pm$ is independent of r because it is equal to $\mathbb Q_p\cdot \mathrm {Log}(U(\mathbb I_0^\circ )\cap G_r^\prime )$ which is independent of r, provided $r_1\leqslant r<r_h$.

We fix $r_0\in p^\mathbb Q\cap ]0,r_h[$ arbitrarily and we work from now on with radii r satisfying $r_0\leqslant r<r_h$. As in Remark 5.1 this fixes a finite extension of $\mathbb Q$ corresponding to the inclusion $G_r^\prime \subset G_r$. For $r<r^\prime$ we have a natural inclusion $\mathbb I_{0,r^\prime }\hookrightarrow \mathbb I_{0,r}$. Since $\mathbb B_r={\varprojlim }_{({\mathfrak {a}}P_1)=1}\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r}$ this induces an inclusion $\mathbb B_{r^\prime }\hookrightarrow \mathbb B_r$. We will consider from now on $\mathbb B_{r^\prime }$ as a subring of $\mathbb B_r$ for every $r<r^\prime$. We will also consider $\mathrm {M}_2(\mathbb I_{0,r^\prime ,\mathbb C_p})$ and $\mathrm {M}_2(\mathbb B_{r^\prime })$ as subsets of $\mathrm {M}_2(\mathbb I_{0,r,\mathbb C_p})$ and $\mathrm {M}_2(\mathbb B_r)$ respectively. These inclusions still hold after taking completed tensor products with $\mathbb C_p$.

Recall the elements $\phi ^\prime _r\,{=}\,\log (u)\phi _r\,{\in }\, \mathrm {M}_2(\mathbb B_{r,\mathbb C_p})$ and $\Phi ^\prime _r\,{=}\,\exp (\phi ^\prime _r)\in \mathrm {GL}_2(\mathbb B_{r,\mathbb C_p})$ defined at the end of the previous section. The Sen operator $\phi _r$ is independent of r in the following sense: if $r<r^\prime <r_h$ and $\mathbb B_{r^\prime ,\mathbb C_p}\rightarrow \mathbb B_{r,\mathbb C_p}$ is the natural inclusion then the image of $\phi _{r^\prime }$ under the induced map $\mathrm {M}_2(\mathbb B_{r^\prime ,\mathbb C_p})\rightarrow \mathrm {M}_2(\mathbb B_{r,\mathbb C_p})$ is $\phi _r$. We deduce that $\phi ^\prime _r$ and $\Phi ^\prime _r$ are also independent of r (in the same sense).

By Proposition 5.11, for every $r<r_h$ there exists an element $\beta _r\in \mathrm {GL}_2(\mathbb B_{r,\mathbb C_p})$ such that $\beta _r\Phi ^\prime _r\beta _r^{-1}=C_T$. Since $\Phi ^\prime _r$ normalizes ${\mathfrak {H}}_{r,\mathbb C_p}$, $C_T=\beta _r\Phi ^\prime _r\beta _r^{-1}$ normalizes $\beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}$.

We denote by ${\mathfrak {U}}^{\pm }$ the upper and lower nilpotent subalgebras of ${\mathfrak {sl}}_2$. The action of $C_T$ on ${\mathfrak {H}}_{r,\mathbb C_p}$ by conjugation is semisimple, so we can decompose $\beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}$ as a sum of eigenspaces for $C_T$:

$$\begin{aligned}&\beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}\\&\quad = \left( \beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}\right) [1]\oplus \left( \beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}\right) [u^{-1}(1+T)]\oplus \left( \beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}\right) [u(1+T)^{-1}] \end{aligned}$$

with $\left( \beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}\right) [u^{-1}(1+T)]\subset {\mathfrak {U}}^+(\mathbb B_{r,\mathbb C_p})$ and $\left( \beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}\right) [u(1+T)^{-1}]\subset {\mathfrak {U}}^-(\mathbb B_{r,\mathbb C_p})$.

Moreover, the formula

$$\begin{pmatrix} u^{-1}(1+T)&{}0\\ 0&{}1\end{pmatrix} \begin{pmatrix} 1&{}\lambda \\ 0&{}1\end{pmatrix} \begin{pmatrix} u^{-1}(1+T)&{}0\\ 0&{}1\end{pmatrix}^{-1}= \begin{pmatrix} 1&{}u^{-1}(1+T)\lambda \\ 0&{}1\end{pmatrix}$$

shows that the action of $C_T$ by conjugation coincides with multiplication by $u^{-1}(1+T)$. By linearity this gives an action of the polynomial ring $\mathbb C_p[T]$ on $\beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}\cap {\mathfrak {U}}^+(\mathbb B_{r,\mathbb C_p})$, compatible with the action of ${\mathbb C_p}[T]$ on ${\mathfrak {U}}^+(\mathbb B_{r,\mathbb C_p})$ given by the inclusions $\mathbb C_p[T]\subset \Lambda _{h,0,{\mathbb C_p}}\subset B_{r,\mathbb C_p}\subset \mathbb B_{r,\mathbb C_p}$. Since ${\mathbb C_p}[T]$ is dense in $A_{h,0,{\mathbb C_p}}$ for the p-adic topology, it is also dense in $B_{r,\mathbb C_p}$. Since ${\mathfrak {H}}_{r,\mathbb C_p}$ is a closed Lie subalgebra of $\mathrm {M}_2(\mathbb B_{r,\mathbb C_p})$, we can define by continuity a $B_{r,\mathbb C_p}$-module structure on $\beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}\cap {\mathfrak {U}}^+(\mathbb B_{r,\mathbb C_p})$, compatible with that on ${\mathfrak {U}}^+(\mathbb B_{r,\mathbb C_p})$. Similarly we have

$$ \begin{pmatrix} u^{-1}(1+T)&{}0\\ 0&{}1\end{pmatrix} \begin{pmatrix}1&{}0\\ \mu &{}1\end{pmatrix} \begin{pmatrix} u^{-1}(1+T)&{}0\\ 0&{}1\end{pmatrix}^{-1}= \begin{pmatrix} 1&{}0\\ u(1+T)^{-1}\mu &{}1\end{pmatrix}. $$

We note that $1+T$ is invertible in $A_{0,r}$ since $T=p^{s_h}t$ where $r_h=p^{-s_h}$. Therefore $C_T$ is invertible and by twisting by $(1+T)\mapsto (1+T)^{-1}$ we can also give $\beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}\cap {\mathfrak {U}}^-(\mathbb B_{r,\mathbb C_p})$ a structure of $B_{r,\mathbb C_p}$-module compatible with that on ${\mathfrak {U}}^-(\mathbb B_{r,\mathbb C_p})$.

By combining the previous remarks with Corollary 4.23, we prove the following “fullness” result for the big Lie algebra ${\mathfrak {H}}_r$.

Theorem 6.2

Suppose that the representation $\rho$ is $(H_0,\mathbb Z_p)$-regular. Then there exists a nonzero ideal ${\mathfrak {l}}$ of $\mathbb I_0$, independent of $r<r_h$, such that for every such r the Lie algebra ${\mathfrak {H}}_r$ contains ${\mathfrak {l}}\cdot {\mathfrak {sl}}_2(\mathbb B_r)$.

Proof

Since $U^\pm (\mathbb B_r)\cong \mathbb B_r$, we can and shall identify ${\mathfrak {u}}^+=\mathbb Q_p\cdot \mathrm {Log}\, G_r^\prime \cap {\mathfrak {U}}^+(\mathbb B_r)$ with a $\mathbb Q_p$-vector subspace of $\mathbb B_r$ (actually of $\mathbb I_0$), and ${\mathfrak {u}}^+_{\mathbb C_p}$ with a $\mathbb C_p$-vector subspace of $\mathbb B_{r,\mathbb C_p}$. We repeat that these spaces are independent of r since $G_r^\prime$ is, provided that $r_0\leqslant r<r_h$ (see Remark 5.1). By Corollary 4.23, ${\mathfrak {u}}^\pm \cap \mathbb I_0$ contains a basis $\{e_{i,\pm }\}_{i\in I}$ for $Q(\mathbb I_0)$ over $Q(\Lambda _{h,0})$. The set $\{e_{i,+}\}_{i\in I}\subset {\mathfrak {u}}^+$ is a basis for $Q(\mathbb I_{0})$ over $Q(\Lambda _{h,0})$, so ${\mathfrak {u}}^+$ contains the basis of a $\Lambda _{h,0}$-lattice in $\mathbb I_0$. By Lemma 4.19 we deduce that $\Lambda _{h,0}{\mathfrak {u}}^+$ contains a nonzero ideal ${\mathfrak {a}}^+$ of $\mathbb I_0$. Hence we also have $B_{r,\mathbb C_p}{\mathfrak {u}}^+_{\mathbb C_p}\supset B_{r,\mathbb C_p} {\mathfrak {a}}^+$. Now ${\mathfrak {a}}^+$ is an ideal of $\mathbb I_0$ and $B_{r,\mathbb C_p}\mathbb I_{0,\mathbb C_p}=\mathbb B_{r,\mathbb C_p}$, so $B_{r,\mathbb C_p}{\mathfrak {a}}^+={\mathfrak {a}}^+\mathbb B_{r,\mathbb C_p}$ is an ideal in $\mathbb B_{r,\mathbb C_p}$. We conclude that $B_{r,\mathbb C_p}\cdot {\mathfrak {u}}^+\supset {\mathfrak {a}}^+\mathbb B_{r,\mathbb C_p}$ for a nonzero ideal ${\mathfrak {a}}^+$ of $\mathbb I_0$. We proceed in the same way for the lower unipotent subalgebra, obtaining $B_{r,\mathbb C_p}\cdot {\mathfrak {u}}^-\supset {\mathfrak {a}}^-\mathbb B_{r,\mathbb C_p}$ for some nonzero ideal ${\mathfrak {a}}^-$ of $\mathbb I_0$.

Consider now the Lie algebra $B_{r,\mathbb C_p}{\mathfrak {H}}_{\mathbb C_p}\subset \mathrm {M}_2(\mathbb B_{r,\mathbb C_p})$. Its nilpotent subalgebras are $B_{r,\mathbb C_p}{\mathfrak {u}}^+$ and $B_{r,\mathbb C_p}{\mathfrak {u}}^-$, and we showed $B_{r,\mathbb C_p}{\mathfrak {u}}^+\supset {\mathfrak {a}}^+\mathbb B_{r,\mathbb C_p}$ and $B_{r,\mathbb C_p}{\mathfrak {u}}^-\supset {\mathfrak {a}}^-\mathbb B_{r,\mathbb C_p}$. Denote by ${\mathfrak {t}}\subset {\mathfrak {sl}}_2$ the subalgebra of diagonal matrices over $\mathbb Z$. By taking the Lie bracket, we see that $[{\mathfrak {U}}^+({\mathfrak {a}}^+\mathbb B_{r,\mathbb C_p}),{\mathfrak {U}}^-({\mathfrak {a}}^-\mathbb B_{r,\mathbb C_p})]$ spans ${\mathfrak {a}}^+\cdot {\mathfrak {a}}^-\cdot {\mathfrak {t}}(\mathbb B_{r,\mathbb C_p})$ over $B_{r,\mathbb C_p}$. We deduce that $B_{r,\mathbb C_p}{\mathfrak {H}}_{\mathbb C_p}\supset {\mathfrak {a}}^+\cdot {\mathfrak {a}}^-\cdot {\mathfrak {sl}}_2(\mathbb B_{r,\mathbb C_p})$. Let ${\mathfrak {a}}={\mathfrak {a}}^+\cdot {\mathfrak {a}}^-$. Now ${\mathfrak {a}}\cdot {\mathfrak {sl}}_2(\mathbb B_{r,\mathbb C_p})$ is a $\mathbb B_{r,\mathbb C_p}$-Lie subalgebra of ${\mathfrak {sl}}_2(\mathbb B_{r,\mathbb C_p})$. Recall that $\beta _r\in \mathrm {GL}_2(\mathbb B_{r,\mathbb C_p})$; hence by stability by conjugation we have $\beta _r\left( {\mathfrak {a}}\cdot {\mathfrak {sl}}_2(\mathbb B_{r,\mathbb C_p})\right) \beta _r^{-1}={\mathfrak {a}}\cdot {\mathfrak {sl}}_2(\mathbb B_{r,\mathbb C_p})$. Thus, we constructed ${\mathfrak {a}}$ such that $B_{r,\mathbb C_p}\left( \beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}\right) \supset {\mathfrak {a}}\cdot {\mathfrak {sl}}_2(\mathbb B_{r,\mathbb C_p})$. In particular, if ${\mathfrak {u}}_{\mathbb C_p}^{\pm ,\beta _r}$ denote the unipotent subalgebras of $\beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}$, we have $B_{r,\mathbb C_p}{\mathfrak {u}}_{\mathbb C_p}^{\pm ,\beta _r}\supset {\mathfrak {a}}\mathbb B_{r,\mathbb C_p}$ for both signs. By the discussion preceding the proposition the subalgebras ${\mathfrak {u}}_{\mathbb C_p}^{\pm ,\beta _r}$ have a structure of $B_{r,\mathbb C_p}$-modules, which means that ${\mathfrak {u}}_{\mathbb C_p}^{\pm ,\beta _r}=B_{r,\mathbb C_p}{\mathfrak {u}}_{\mathbb C_p}^{\pm ,\beta _r}$. We conclude that ${\mathfrak {u}}_{\mathbb C_p}^{\pm ,\beta _r}\supset \beta _r\left( {\mathfrak {a}}\cdot {\mathfrak {U}}^\pm (\mathbb B_{r,\mathbb C_p})\right) \beta _r^{-1}$ for both signs. By the usual argument of taking the bracket, we obtain $\beta _r{\mathfrak {H}}_{r,\mathbb C_p}\beta _r^{-1}\supset {\mathfrak {a}}^2\cdot {\mathfrak {sl}}_2(\mathbb B_{r,\mathbb C_p})$. We can untwist by the invertible matrix $\beta _r$ to conclude that, for ${\mathfrak {l}}={\mathfrak {a}}^2$, we have ${\mathfrak {H}}_{r,\mathbb C_p}\supset {\mathfrak {l}}\cdot {\mathfrak {sl}}_2(\mathbb B_{r,\mathbb C_p})$.

Let us get rid of the completed extension of scalars to $\mathbb C_p$. For every ideal ${\mathfrak {a}}\subset \mathbb I_{0,r}$ not dividing $P_1$, let ${\mathfrak {H}}_{r,{\mathfrak {a}}}$ be the image of ${\mathfrak {H}}_r$ in $\mathrm {M}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})$. Consider the two finite dimensional $\mathbb Q_p$-vector spaces ${\mathfrak {H}}_{r,{\mathfrak {a}}}$ and ${\mathfrak {l}}\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})$. Note that they are both subspaces of the finite dimensional $\mathbb Q_p$-vector space $\mathrm {M}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})$. After extending scalars to $\mathbb C_p$, we have

$$\begin{aligned} {\mathfrak {l}}\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})\otimes \mathbb C_p\subset {\mathfrak {H}}_{r,{\mathfrak {a}}}\otimes \mathbb C_p. \end{aligned}$$

(8)

Let $\{e_i\}_{i\in I}$ be an orthonormal basis of the Banach space $\mathbb C_p$ over $\mathbb Q_p$, with I some index set, such that $1\in \{e_i\}_{i\in I}$. Let $\{v_j\}_{j=1,...,n}$ be a $\mathbb Q_p$-basis of $\mathrm {M}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})$ such that, for some $d\leqslant n$, $\{v_j\}_{j=1,...,d}$ is a $\mathbb Q_p$-basis of ${\mathfrak {H}}_{r,{\mathfrak {a}}}$.

Let v be an element of ${\mathfrak {l}}\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})$. Then $v\otimes 1\in {\mathfrak {l}}\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})\otimes \mathbb C_p$ and by (8) we have $v\otimes 1\in {\mathfrak {H}}_{r,{\mathfrak {a}}}\otimes \mathbb C_p$. As $\{v_j\otimes e_i\}_{1\leqslant j\leqslant d, i\in I}$, and $\{v_j\otimes e_i \}_{1\leqslant j\leqslant n,i\in I}$ are orthonormal bases of ${\mathfrak {H}}_{r,{\mathfrak {a}}}\otimes \mathbb C_p$, and $\mathrm {M}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})\otimes \mathbb C_p$ over $\mathbb Q_p$, respectively there exist $\lambda _{j,i}\in \mathbb Q_p, (j,i)\in \{1,2,...,d\}\times I$ converging to 0 in the filter of complements of finite subsets of $\{1,2,...,d\}\times I$ such that $v\otimes 1=\sum _{j=1,...,d;\, i\in I}\lambda _{j,i}(v_j\otimes e_i)$.

But $v\otimes 1\in \mathrm {M}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})\otimes 1\subset \mathrm {M}_2(\mathbb I_{0,r}/{\mathfrak {a}}\mathbb I_{0,r})\otimes \mathbb C_p$ and therefore $v\otimes 1=\sum _{1\leqslant j\leqslant n} a_j(v_j\otimes 1)$, for some $a_j\in \mathbb Q_p$, $j=1,...,n$. By the uniqueness of a representation of an element in a $\mathbb Q_p$-Banach space in terms of a given orthonormal basis we have

$$ v\otimes 1=\sum _{j=1}^d a_j(v_j\otimes 1),\quad {\text {i.e.}}\quad v=\sum _{j=1}^da_jv_j\in {\mathfrak {H}}_{r,{\mathfrak {a}}}. $$

By taking the projective limit over ${\mathfrak {a}}$, we conclude that

$${\mathfrak {l}}\cdot {\mathfrak {sl}}_2(\mathbb B_r)\subset {\mathfrak {H}}_r.$$

$\square$

Definition 6.3

The Galois level of the family $\theta :\mathbb T_h\rightarrow \mathbb I^\circ$ is the largest ideal ${\mathfrak {l}}_\theta$ of $\mathbb I_{0}[P_1^{-1}]$ such that ${\mathfrak {H}}_{r}\supset {\mathfrak {l}}_\theta \cdot {\mathfrak {sl}}_2(\mathbb B_r)$ for all $r<r_h$.

It follows by the previous remarks that ${\mathfrak {l}}_\theta$ is nonzero.

7 Comparison Between the Galois Level and the Fortuitous Congruence Ideal

Let $\theta :\mathbb T_h\rightarrow \mathbb I^\circ$ be a slope $\leqslant h$ family. We keep all the notations from the previous sections. In particular $\rho :G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I^\circ )$ is the Galois representation associated with $\theta$. We suppose that the restriction of $\rho$ to $H_0$ takes values in $\mathrm {GL}_2(\mathbb I^\circ _0)$. Recall that $\mathbb I=\mathbb I^\circ [p^{-1}]$ and $\mathbb I_0=\mathbb I_0^\circ [p^{-1}]$. Also recall that $P_1$ is the prime of $\Lambda _{h,0}$ generated by $u^{-1}(1+T)-1$. Let ${\mathfrak {c}}\subset \mathbb I$ be the congruence ideal associated with $\theta$. Set ${\mathfrak {c}}_0={\mathfrak {c}}\cap \mathbb I_0$ and ${\mathfrak {c}}_1={\mathfrak {c}}_0\mathbb I_0[P_1^{-1}]$. Let ${\mathfrak {l}}={\mathfrak {l}}_\theta \subset \mathbb I_0[P_1^{-1}]$ be the Galois level of $\theta$. For an ideal ${\mathfrak {a}}$ of $\mathbb I_0[P_1^{-1}]$ we denote by $V({\mathfrak {a}})$ the set of prime ideals of $\mathbb I_0[P_1^{-1}]$ containing ${\mathfrak {a}}$. We prove the following.

Theorem 7.1

Suppose that

• 1. $\rho$ is $(H_0,\mathbb Z_p)$-regular;

• 2. there exists no pair $(F,\psi )$, where F is a real quadratic field and $\psi :\mathrm {Gal}(\overline{F}/F)\rightarrow \mathbb F^\times$ is a character, such that $\overline{\rho }:G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb F)\cong \mathrm {Ind}_F^\mathbb Q\psi$.

Then we have $V({\mathfrak {l}})=V({\mathfrak {c}}_1)$.

Before giving the proof we make some remarks. Let P be a prime of $\mathbb I_0[P_1^{-1}]$ and Q be a prime factor of $P\mathbb I[P_1^{-1}]$. We consider $\rho$ as a representation $G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I[P_1^{-1}])$ by composing it with the inclusion $\mathrm {GL}_2(\mathbb I)\hookrightarrow \mathrm {GL}_2(\mathbb I[P_1^{-1}])$. We have a representation $\rho _{Q}:G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I[P_1^{-1}]/Q)$ obtained by reducing $\rho$ modulo Q. Its restriction $\rho _{Q}\vert _{H_0}$ takes values in $\mathrm {GL}_2(\mathbb I_0[P_1^{-1}]/(Q\cap \mathbb I_0[P_1^{-1}]))=\mathrm {GL}_2(\mathbb I_0[P_1^{-1}]/P)$ and coincides with the reduction $\rho _P$ of $\rho \vert _{H_0}:H_0\rightarrow \mathrm {GL}_2(\mathbb I_0[P_1^{-1}])$ modulo P. In particular $\rho _{Q}\vert _{H_0}$ is independent of the chosen prime factor Q of $P\mathbb I[P_1^{-1}]$.

We say that a subgroup of $\mathrm {GL}_2(A)$ for some algebra A finite over a p-adic field K is small if it admits a finite index abelian subgroup. Let P, Q be as above, $G_P$ be the image of $\rho _P:H_0\rightarrow \mathrm {GL}_2(\mathbb I_0[P_1^{-1}]/P)$ and $G_{Q}$ be the image of $\rho _{Q}:G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I[P_1^{-1}]/Q)$. By our previous remark $\rho _P$ coincides with the restriction $\rho _{Q}\vert _{H_0}$, so $G_P$ is a finite index subgroup of $G_{Q}$ for every Q. In particular $G_P$ is small if and only if $G_{Q}$ is small for all prime factors Q of $P\mathbb I[P_1^{-1}]$.

Now if Q is a CM point the representation $\rho _{Q}$ is induced by a character of $\mathrm {Gal}(F/\mathbb Q)$ for an imaginary quadratic field F. Hence $G_{Q}$ admits an abelian subgroup of index 2 and $G_P$ is also small.

Conversely, if $G_P$ is small, $G_{Q^\prime }$ is small for every prime $Q^\prime$ above P. Choose any such prime $Q^\prime$; by the argument in [16, Proposition 4.4] $G_{Q^\prime }$ has an abelian subgroup of index 2. It follows that $\rho _{Q^\prime }$ is induced by a character of $\mathrm {Gal}(\overline{F}_{Q^\prime }/F_{Q^\prime })$ for a quadratic field $F_{Q^\prime }$. If $F_{Q^\prime }$ is imaginary then ${Q^\prime }$ is a CM point. In particular, if we suppose that the residual representation $\bar{\rho }:G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb F)$ is not induced by a character of $\mathrm {Gal}(\overline{F}/F)$ for a real quadratic field $F/\mathbb Q$, then $F_{Q^\prime }$ is imaginary and ${Q^\prime }$ is CM. The above argument proves that $G_P$ is small if and only if all points ${Q^\prime }\subset \mathbb I[P_1^{-1}]$ above P are CM.

Proof

We prove first that $V({\mathfrak {c}}_1)\subset V({\mathfrak {l}})$. Fix a radius $r<r_h$. By contradiction, suppose that a prime P of $\mathbb I_0[P_1^{-1}]$ contains ${\mathfrak {c}}_0$ but P does not contain ${\mathfrak {l}}$. Then there exists a prime factor Q of $P\mathbb I[P_1^{-1}]$ such that ${\mathfrak {c}}\subset Q$. By definition of ${\mathfrak {c}}$ we have that Q is a CM point in the sense of Sect. 3.4, hence the representation $\rho _{\mathbb I[P_1^{-1}],Q}$ has small image in $\mathrm {GL}_2(\mathbb I[P_1^{-1}]/Q)$. Then its restriction $\rho _{\mathbb I[P_1^{-1}],Q}\vert _{H_0}=\rho _P$ also has small image in $\mathrm {GL}_2(\mathbb I_0[P_1^{-1}]/P)$. We deduce that there is no nonzero ideal ${\mathfrak {I}}_P$ of $\mathbb I_0[P_1^{-1}]/P$ such that the Lie algebra ${\mathfrak {H}}_{r,P}$ contains ${\mathfrak {I}}_P\cdot {\mathfrak {sl}}_2(\mathbb I_0[P_1^{-1}]/P)$.

Now by definition of ${\mathfrak {l}}$ we have ${\mathfrak {l}}\cdot {\mathfrak {sl}}_2(\mathbb B_r)\subset {\mathfrak {H}}_r$. Since reduction modulo P gives a surjection ${\mathfrak {H}}_r\rightarrow {\mathfrak {H}}_{r,P}$, by looking at the previous inclusion modulo P we find ${\mathfrak {l}}\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r}[P_1^{-1}]/P\mathbb I_{0,r}[P_1^{-1}])\subset {\mathfrak {H}}_{r,P}$. If ${\mathfrak {l}}\not \subset P$ we have ${\mathfrak {l}}/P\ne 0$, which contradicts our earlier statement. We deduce that ${\mathfrak {l}}\subset P$.

We prove now that $V({\mathfrak {l}})\subset V({\mathfrak {c}}_1)$. Let $P\subset \mathbb I_0[P_1^{-1}]$ be a prime containing ${\mathfrak {l}}$. Recall that $\mathbb I_0[P_1^{-1}]$ has Krull dimension one, so $\kappa _{P}=\mathbb I_0[P_1^{-1}]/P$ is a field. Let Q be a prime of $\mathbb I[P_1^{-1}]$ above P. As before $\rho$ reduces to representations $\rho _Q:G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I[P_1^{-1}]/Q)$ and $\rho _P:H_0\rightarrow \mathrm {GL}_2(\mathbb I_0[P_1^{-1}]/P)$. Let ${\mathfrak {P}}\subset \mathbb I_0[P_1^{-1}]$ be the P-primary component of ${\mathfrak {l}}$ and let ${\mathfrak {A}}$ be an ideal of $\mathbb I_0[P_1^{-1}]$ containing ${\mathfrak {P}}$ such that the localization at P of ${\mathfrak {A}}/{\mathfrak {P}}$ is one-dimensional over $\kappa _P$. Choose any $r<r_h$. Let ${\mathfrak {s}}={\mathfrak {A}}/{\mathfrak {P}}\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r}[P_1^{-1}]/{\mathfrak {P}})\cap {\mathfrak {H}}_{r,{\mathfrak {P}}}$, that is a Lie subalgebra of ${\mathfrak {A}}/{\mathfrak {P}}\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r}[P_1^{-1}]/{\mathfrak {P}})$.

We show that ${\mathfrak {s}}$ is stable under the adjoint action $\mathrm {Ad}(\rho _Q)$ of $G_\mathbb Q$. Let ${\mathfrak {Q}}$ be the Q-primary component of ${\mathfrak {l}}\cdot \mathbb I[P_1^{-1}]$. Recall that ${\mathfrak {H}}_{r,{\mathfrak {P}}}$ is the Lie algebra associated with the pro-p group $\mathrm {Im}\,\rho _{r,{\mathfrak {Q}}}\vert _{H_0}\cap \Gamma _{\mathrm {GL}_2(\mathbb I_{0,r_o}[P_1^{-1}]/{\mathfrak {P}})}(p)\subset \mathrm {GL}_2(\mathbb I_{0,r}[P_1^{-1}]/{\mathfrak {P}})$. Since this group is open in $\mathrm {Im}\,\rho _{r,{\mathfrak {Q}}}\subset \mathrm {GL}_2(\mathbb I_r[P_1^{-1}]/{\mathfrak {Q}})$, the Lie algebra associated with $\mathrm {Im}\,\rho _{r,{\mathfrak {Q}}}$ is again ${\mathfrak {H}}_{r,{\mathfrak {P}}}$. In particular ${\mathfrak {H}}_{r,{\mathfrak {P}}}$ is stable under $\mathrm {Ad}(\rho _Q)$. Since ${\mathfrak {H}}_{r,{\mathfrak {P}}}\subset {\mathfrak {sl}}_2(\mathbb I_{0,r}[P_1^{-1}]/{\mathfrak {P}})$ we have ${\mathfrak {A}}/{\mathfrak {P}}\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r}[P_1^{-1}]/{\mathfrak {P}})\cap {\mathfrak {H}}_{r,{\mathfrak {P}}}={\mathfrak {A}}/{\mathfrak {P}}\cdot {\mathfrak {sl}}_2(\mathbb I_r[P_1^{-1}]/{\mathfrak {Q}})\cap {\mathfrak {H}}_{r,{\mathfrak {P}}}$. Now ${\mathfrak {A}}/{\mathfrak {P}}\cdot {\mathfrak {sl}}_2(\mathbb I_r[P_1^{-1}]/{\mathfrak {Q}})$ is clearly stable under $\mathrm {Ad}(\rho _Q)$, so the same is true for ${\mathfrak {A}}/{\mathfrak {P}}\cdot {\mathfrak {sl}}_2(\mathbb I_r[P_1^{-1}]/{\mathfrak {Q}})\cap {\mathfrak {H}}_{r,{\mathfrak {P}}}$, as desired.

We consider from now on ${\mathfrak {s}}$ as a Galois representation via $\mathrm {Ad}(\rho _Q)$. By the proof of Theorem 6.2 we can assume, possibly considering a sub-Galois representation, that ${\mathfrak {H}}_r$ is a $\mathbb B_r$-submodule of ${\mathfrak {sl}}_2(\mathbb B_r)$ containing ${\mathfrak {l}}\cdot {\mathfrak {sl}}_2(\mathbb B_r)$ but not ${\mathfrak {a}}\cdot {\mathfrak {sl}}_2(\mathbb B_r)$ for any ${\mathfrak {a}}$ strictly bigger than ${\mathfrak {l}}$. This allows us to speak of the localization ${\mathfrak {s}}_P$ of ${\mathfrak {s}}$ at P. Note that, since ${\mathfrak {P}}$ is the P-primary component of ${\mathfrak {l}}$ and ${\mathfrak {A}}_P/{\mathfrak {P}}_P\cong \kappa _P$, when P-localizing we find ${\mathfrak {H}}_{r,P}\supset {\mathfrak {P}}_{P}\cdot {\mathfrak {sl}}_2(\mathbb B_{r,P})$ and ${\mathfrak {H}}_{r,P}\not \supset {\mathfrak {A}}_P\cdot {\mathfrak {sl}}_2(\mathbb B_{r,P})$.

The localization at P of ${\mathfrak {a}}/{\mathfrak {P}}\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r}[P_1^{-1}]/{\mathfrak {P}})$ is ${\mathfrak {sl}}_2(\kappa _P)$, so ${\mathfrak {s}}_P$ is contained in ${\mathfrak {sl}}_2(\kappa _P)$. It is a $\kappa _P$-representation of $G_\mathbb Q$ (via $\mathrm {Ad}(\rho _Q)$) of dimension at most 3. We distinguish various cases following its dimension.

We cannot have ${\mathfrak {s}}_P=0$. By exchanging the quotient with the localization we would obtain $({\mathfrak {A}}_{P}\cdot {\mathfrak {sl}}_2(\mathbb B_{r,P})\cap {\mathfrak {H}}_{r,P})/{\mathfrak {P}}_{P}=0$. By Nakayama’s lemma ${\mathfrak {A}}_{P}\cdot {\mathfrak {sl}}_2(\mathbb B_{r,P})\cap {\mathfrak {H}}_{r,P}\,{=}\,0$, which is absurd since ${\mathfrak {A}}_{P}\cdot {\mathfrak {sl}}_2(\mathbb B_{r,P})\cap {\mathfrak {H}}_{r,P}\supset {\mathfrak {P}}_{P}\cdot {\mathfrak {sl}}_2(\mathbb B_{r,P})\,{\ne }~0$.

We also exclude the three-dimensional case. If ${\mathfrak {s}}_{P}={\mathfrak {sl}}_2(\kappa _{P})$, by exchanging the quotient with the localization we obtain $({\mathfrak {A}}_{P}\cdot {\mathfrak {sl}}_2(\mathbb B_{r,P})\cap {\mathfrak {H}}_{r,P})/{\mathfrak {P}}_{P}=({\mathfrak {A}}_P\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r,P}[P_1^{-1}]))/{\mathfrak {P}}_{P}\mathbb I_{0,r,P}[P_1^{-1}]$, because we have ${\mathfrak {A}}_{P}\mathbb I_{0,r,P}[P_1^{-1}]/{\mathfrak {P}}_P\mathbb I_{0,r,P}[P_1^{-1}]=\left( \mathbb I_{0,r,P}[P_1^{-1}]/{\mathfrak {P}}_{P}\mathbb I_{0,r,P}[P_1^{-1}]\right)$ and this is isomorphic to $\kappa _{P}$. By Nakayama’s lemma we would conclude that ${\mathfrak {H}}_{r,P}\supset {\mathfrak {A}}\cdot {\mathfrak {sl}}_2(\mathbb B_{r,P})$, which is absurd.

We are left with the one and two-dimensional cases. If ${\mathfrak {s}}_P$ is two-dimensional we can always replace it by its orthogonal in ${\mathfrak {sl}}_2(\kappa _P)$ which is one-dimensional; indeed the action of $G_\mathbb Q$ via $\mathrm {Ad}(\rho _Q)$ is isometric with respect to the scalar product ${\mathrm {Tr}}(XY)$ on ${\mathfrak {sl}}_2(\kappa _P)$.

Suppose that ${\mathfrak {sl}}_2(\kappa _P)$ contains a one-dimensional stable subspace. Let $\phi$ be a generator of this subspace over $\kappa _P$. Let $\chi :G_\mathbb Q\rightarrow \kappa _P$ denote the character satisfying $\rho _Q(g)\phi \rho _Q(g)^{-1}=\chi (g)\phi$ for all $g\in G_\mathbb Q$. Now $\phi$ induces a nontrivial morphism of representations $\rho _Q\rightarrow \rho _Q\otimes \chi$. Since $\rho _Q$ and $\rho _Q\otimes \chi$ are irreducible, by Schur’s lemma $\phi$ must be invertible. Hence we obtain an isomorphism $\rho _Q\cong \rho _Q\otimes \chi$. By taking determinants we see that $\chi$ must be quadratic. If $F_0/\mathbb Q$ is the quadratic extension fixed by $\ker \chi$, then $\rho _Q$ is induced by a character $\psi$ of $\mathrm {Gal}(\overline{F_0}/F_0)$. By assumption the residual representation $\rho _{{\mathfrak {m}}_\mathbb I}:G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb F)$ is not of the form $\mathrm {Ind}_F^\mathbb Q\psi$ for a real quadratic field F and a character $\mathrm {Gal}(\overline{F}/F)\rightarrow \mathbb F^\times$. We deduce that $F_0$ must be imaginary, so Q is a CM point by Remark 3.11(1). By construction of the congruence ideal ${\mathfrak {c}}\subset Q$ and ${\mathfrak {c}}_0\subset Q\cap \mathbb I_0[P_1^{-1}]=P$.$\square$

We prove a corollary.

Corollary 7.2

If the residual representation $\overline{\rho }:G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb F)$ is not dihedral then ${\mathfrak {l}}=1$.

Proof

Since $\overline{\rho }$ is not dihedral there cannot be any CM point on the family $\theta :\mathbb T_h\rightarrow \mathbb I^\circ$. By Theorem 7.1 we deduce that ${\mathfrak {l}}$ has no nontrivial prime factor, hence it is trivial.$\square$

Remark 7.3

Theorem 7.1 gives another proof of Proposition 3.9. Indeed the CM points of a family $\theta :\mathbb T_h\rightarrow \mathbb I^\circ$ correspond to the prime factors of its Galois level, which are finite in number.

We also give a partial result about the comparison of the exponents of the prime factors in ${\mathfrak {c}}_1$ and ${\mathfrak {l}}$. This is an analogous of what is proved in [9, Theorem 8.6] for the ordinary case; our proof also relies on the strategy there. For every prime P of $\mathbb I_0[P_1^{-1}]$ we denote by ${\mathfrak {c}}_1^P$ and ${\mathfrak {l}}^P$ the P-primary components of ${\mathfrak {c}}_1$ and ${\mathfrak {l}}$ respectively.

Theorem 7.4

Suppose that $\overline{\rho }$ is not induced by a character of $G_F$ for a real quadratic field $F/\mathbb Q$. We have $({\mathfrak {c}}_1^P)^2\subset {\mathfrak {l}}^P\subset {\mathfrak {c}}_1^P$.

Proof

The inclusion ${\mathfrak {l}}^P\subset {\mathfrak {c}}_1^P$ is proved in the same way as the first inclusion of Theorem 7.1.

We show that the inclusion $({\mathfrak {c}}_1^P)^2\subset {\mathfrak {l}}^P$ holds. If ${\mathfrak {c}}_1^P$ is trivial this reduces to Theorem 7.1, so we can suppose that P is a factor of ${\mathfrak {c}}_1$. Let Q denote any prime of $\mathbb I[P_1^{-1}]$ above P. Let ${\mathfrak {c}}_1^Q$ be a Q-primary ideal of $\mathbb I[P_1^{-1}]$ satisfying ${\mathfrak {c}}_1^Q\cap \mathbb I_0[P_1^{-1}]={\mathfrak {c}}_1^P$. Since P divides ${\mathfrak {c}}_1$, Q is a CM point, so we have an isomorphism $\rho _P\cong \mathrm {Ind}_F^\mathbb Q\psi$ for an imaginary quadratic field $F/\mathbb Q$ and a character $\psi :G_F\rightarrow \mathbb C_p^\times$. Choose any $r<r_h$. Consider the $\kappa _P$-vector space ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}={\mathfrak {H}}_r\cap {\mathfrak {c}}_1^P\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r})/{\mathfrak {H}}_r\cap {\mathfrak {c}}_1^PP\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r})$. We see it as a subspace of ${\mathfrak {sl}}_2({\mathfrak {c}}_1^P/{\mathfrak {c}}_1^PP)\cong {\mathfrak {sl}}_2(\kappa _{P})$. By the same argument as in the proof of Theorem 7.1, ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}$ is stable under the adjoint action $\mathrm {Ad}(\rho _{{\mathfrak {c}}_1^QQ}):G_\mathbb Q\rightarrow \mathrm {Aut}({\mathfrak {sl}}_2(\kappa _P))$.

Let $\chi _{F/\mathbb Q}:G_\mathbb Q\rightarrow \mathbb C_p^\times$ be the quadratic character defined by the extension $F/\mathbb Q$. Let $\varepsilon \in G_\mathbb Q$ be an element projecting to the generator of $\mathrm {Gal}(F/\mathbb Q)$. Let $\psi ^\varepsilon :G_F\rightarrow \mathbb C_p^\times$ be given by $\psi ^\varepsilon (\tau )=\psi (\varepsilon \tau \varepsilon ^{-1})$. Set $\psi ^-=\psi /\psi ^\varepsilon$. Since $\rho _Q\cong \mathrm {Ind}_F^\mathbb Q\psi$, we have a decomposition $\mathrm {Ad}(\rho _Q)\cong \chi _{F/\mathbb Q}\oplus \mathrm {Ind}_F^\mathbb Q\psi ^-$, where the two factors are irreducible. Now we have three possibilities for the Galois isomorphism class of ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}$: it is either that of $\mathrm {Ad}(\rho _Q)$ or that of one of the two irreducible factors.

If ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}\cong \mathrm {Ad}(\rho _Q)$, then as $\kappa _P$-vector spaces ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}={\mathfrak {sl}}_2(\kappa _P)$. By Nakayama’s lemma ${\mathfrak {H}}_r\supset {\mathfrak {c}}_1^P\cdot {\mathfrak {sl}}_2(\mathbb B_{r})$. This implies ${\mathfrak {c}}_1^P\subset {\mathfrak {l}}^P$, hence ${\mathfrak {c}}_1^P={\mathfrak {l}}^P$ in this case.

If ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}$ is one-dimensional then we proceed as in the proof of Theorem 7.1 to show that $\rho _{{\mathfrak {c}}_1^QQ}:G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I_r[P_1^{-1}]/{\mathfrak {c}}_1^{Q}Q\mathbb I_r[P_1^{-1}])$ is induced by a character $\psi _{{\mathfrak {c}}_1^QQ}:G_F\rightarrow \mathbb C_p^\times$. In particular the image of $\rho _{{\mathfrak {c}}_1^PP}:H\rightarrow \mathrm {GL}_2(\mathbb I_{0,r}[P_1^{-1}]/{\mathfrak {c}}_1^PP\mathbb I_{0,r})$ is small. This is a contradiction, since ${\mathfrak {c}}_1^P$ is the P-primary component of ${\mathfrak {c}}_1$, hence it is the smallest P-primary ideal ${\mathfrak {A}}$ of $\mathbb I_{0,r}[P_1^{-1}]$ such that the image of $\rho _{\mathfrak {A}}:G_\mathbb Q\rightarrow \mathrm {GL}_2(\mathbb I_r[P_1^{-1}]/{\mathfrak {A}}\mathbb I_r[P_1^{-1}])$ is small.

Finally, suppose that ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}\cong \mathrm {Ind}_F^\mathbb Q\psi ^-$. Let $d=\mathrm {diag}(d_1,d_2)\in \rho (G_\mathbb Q)$ be the image of a $\mathbb Z_p$-regular element. Since $d_1$ and $d_2$ are nontrivial modulo the maximal ideal of $\mathbb I_0^\circ$, the image of d modulo ${\mathfrak {c}}_1^QQ$ is a nontrivial diagonal element $d_{{\mathfrak {c}}_1^QQ}=\mathrm {diag}(d_{1,{\mathfrak {c}}_1^QQ},d_{2,{\mathfrak {c}}_1^QQ})\in \rho _{{\mathfrak {c}}_1^QQ}(G_\mathbb Q)$. We decompose ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}$ in eigenspaces for the adjoint action of $d_{{\mathfrak {c}}_1^QQ}$: we write ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}={\mathfrak {s}}_{{\mathfrak {c}}_1^P}[a]\oplus {\mathfrak {s}}_{{\mathfrak {c}}_1^P}[1]\oplus {\mathfrak {s}}_{{\mathfrak {c}}_1^P}[a^{-1}]$, where $a=d_{1,{\mathfrak {c}}_1^QQ}/d_{2,{\mathfrak {c}}_1^QQ}$. Now ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}[1]$ is contained in the diagonal torus, on which the adjoint action of $G_\mathbb Q$ is given by the character $\chi _{F/\mathbb Q}$. Since $\chi _{F/\mathbb Q}$ does not appear as a factor of ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}$, we must have ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}[1]=0$. This implies that ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}[a]\ne 0$ and ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}[a^{-1}]\ne 0$. Since ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}[a]={\mathfrak {s}}_{{\mathfrak {c}}_1^P}\cap {\mathfrak {u}}^+(\kappa _P)$ and ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}[a^{-1}]={\mathfrak {s}}_{{\mathfrak {c}}_1^P}\cap {\mathfrak {u}}^-(\kappa _P)$, we deduce that ${\mathfrak {s}}_{{\mathfrak {c}}_1^P}$ contains nontrivial upper and lower nilpotent elements $\overline{u^+}$ and $\overline{u^-}$. Then $\overline{u^+}$ and $\overline{u^-}$ are the images of some elements $u^+$ and $u^-$ of ${\mathfrak {H}}_r\cap {\mathfrak {c}}_1^P\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r}[P_1^{-1}])$ nontrivial modulo ${\mathfrak {c}}_1^PP$. The Lie bracket $t=[u^+,u^-]$ is an element of ${\mathfrak {H}}_r\cap {\mathfrak {t}}(\mathbb I_{0,r}[P_1^{-1}])$ (where ${\mathfrak {t}}$ denotes the diagonal torus) and it is nontrivial modulo $({\mathfrak {c}}_1^P)^2P$. Hence the $\kappa _P$-vector space ${\mathfrak {s}}_{({\mathfrak {c}}_1^P)^2}={\mathfrak {H}}_r\cap ({\mathfrak {c}}_1^P)^2\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r,\mathbb C_p}[P_1^{-1}])/{\mathfrak {H}}_r\cap ({\mathfrak {c}}_1^P)^2P\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r,\mathbb C_p}[P_1^{-1}])$ contains nontrivial diagonal, upper nilpotent and lower nilpotent elements, so it is three-dimensional. By Nakayama’s lemma we conclude that ${\mathfrak {H}}_r\supset ({\mathfrak {c}}_1^P)^2\cdot {\mathfrak {sl}}_2(\mathbb I_{0,r}[P_1^{-1}])$, so $({\mathfrak {c}}_1^P)^2\subset {\mathfrak {l}}^P$.$\square$