THE EISENSTEIN IDEAL OF WEIGHT k AND RANKS OF HECKE ALGEBRAS

Abstract

Let p and $\ell $ be primes such that $p> 3$ and $p \mid \ell -1$ and k be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight k and level $\Gamma _0(\ell )$ at the maximal Eisenstein ideal containing p. We give a necessary and sufficient condition for the $\mathbb {Z}_p$-rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for $k=2$ using our methods. In addition, we prove some $R=\mathbb {T}$ theorems under certain hypotheses.

1 Introduction

Let p and $\ell $ be primes such that $p> 3$ and $p \mid \ell -1$ .

Let $\mathbb {T}$ be the Hecke algebra over $\mathbb {Z}_p$ , acting faithfully on the space of modular forms of level $\Gamma _0(\ell )$ and weight k, and $\mathfrak {m}$ be its Eisenstein maximal ideal containing p (i.e. the maximal ideal of $\mathbb {T}$ generated by p and the prime ideal corresponding to the classical Eisenstein series of level $\Gamma _0(\ell )$ and weight k having Atkin-Lehner eigenvalue $-1$ ). Let $\mathbb {T}_{\mathfrak {m}}$ be the completion of $\mathbb {T}$ at $\mathfrak {m}$ and let $\mathbb {T}^0_{\mathfrak {m}}$ be its cuspidal quotient.

In the setting given above, Mazur, in his landmark work on Eisenstein ideal ([19]), studied the cuspidal Hecke algebra $\mathbb {T}^0_{\mathfrak {m}}$ in the case of $k=2$ . He proved (among many other things) that $\mathbb {T}^0_{\mathfrak {m}} \neq 0$ and also asked whether one can say anything about the $\mathbb {Z}_p$ -rank of $\mathbb {T}^0_{\mathfrak {m}}$ . Since then this question has been studied in detail by various authors using different approaches. We will now give a brief summary of their works on the $\mathbb {Z}_p$ -rank of $\mathbb {T}^0_{\mathfrak {m}}$ when $k=2$ .

1.1 History

In [21], Merel proved that the $\mathbb {Z}_p$ -rank of $\mathbb {T}^0_{\mathfrak {m}}$ is greater than $1$ if and only if the image of $\prod_{i=1}^{\frac {\ell -1}{2}}i^i$ in $(\mathbb {Z}/\ell \mathbb {Z})^{\times }$ is a p-th power. His method was mainly based on computation of some Eisenstein elements in the first homology group of a modular curve. In [18], Lecouturier extended Merel’s result by combining the same circle of ideas with new methods. In particular, he gave a necessary and sufficient condition for the $\mathbb {Z}_p$ -rank of $\mathbb {T}^0_{\mathfrak {m}}$ to be greater than $2$ in terms of a numerical invariant which is similar to the Merel’s invariant mentioned above (see [18, Theorem $1.2$ ]). He also gave an alternate proof of Merel’s result in [18].

On the other hand, in [6], Calegari and Emerton studied this question using deformation theory of Galois representations. They proved that $\mathbb {T}^0_{\mathfrak {m}} = \mathbb {Z}_p$ if the p-part of the class group of $\mathbb {Q}(\ell ^{1/p})$ is cyclic. In [28], Wake and Wang-Erickson used techniques from deformation theory of Galois pseudo-representations to prove that the $\mathbb {Z}_p$ -rank of $\mathbb {T}^0_{\mathfrak {m}}$ is greater than $1$ if and only if the cup product of certain global Galois cohomology classes vanishes. They also recovered many results of Calegari–Emerton. The key step in both these works is a suitable $R=\mathbb {T}$ theorem. In [29], Wake and Wang-Erickson studied this question in the case of squarefree level. We refer the reader to the well-written introductions of [21], [6], [28], [29] and [18] for a summary of the known results, nice exposition of various approaches to the problem and their comparison.

One can say that the approach of Merel and Lecouturier is on the “analytic side” and the approach of Calegari–Emerton and Wake–Wang-Erickson is on the “algebraic side”. In [26], Wake studied the Hecke algebras $\mathbb {T}_{\mathfrak {m}}$ and $\mathbb {T}^0_{\mathfrak {m}}$ and their Eisenstein ideals for weights $k> 2$ by unifying the two approaches mentioned above. In particular, he gave a necessary and sufficient condition (in terms of the Eisenstein ideal and the derivative of Mazur–Tate $\zeta $ -function that he defines) for the $\mathbb {Z}_p$ -rank of $\mathbb {T}^0_{\mathfrak {m}}$ to be $1$ . This is an analogue of Merel’s result ([21, Théoreme $2$ ]) for higher weights.

1.2 Aim and Setup

The main aim of this article is to obtain necessary and sufficient conditions for $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}}) \geq 2$ when $k> 2$ in terms of vanishing of cup products of certain global Galois cohomology classes and class groups. In particular, we prove analogues of [28, Theorem $\mathrm{1.2.1}$ ] and [28, Corollary $\mathrm{1.2.2}$ ] for $k> 2$ , and we recover these results when $k=2$ .

Our approach is based on deformation theory of Galois representations and pseudo-representations, so it is similar to the approach of [6] and [28]. However, our methods are different. To be precise, even though our main tool is comparison between deformation rings (of either representations or pseudo-representations) and Hecke algebras, our main results are not based on $R=\mathbb {T}$ theorems. We instead use the description of $\dfrac {\mathbb {F}_p[\epsilon ]}{(\epsilon ^2)}$ -valued ordinary pseudo-representations, analysis of pseudo-representations arising from actual representations and results from [26] and [19]. The results from [26] that we use are about the reducibility properties of the $\mathbb {T}_{\mathfrak {m}}$ -valued pseudo-representation ([26, Theorem $\mathrm{5.1.1}$ ]) and the index of Eisenstein ideal in $\mathbb {T}^0_{\mathfrak {m}}$ ([26, Theorem $\mathrm{5.1.2}$ ]).

Note that, in [28], Wake and Wang-Erickson work with pseudo-representations which are finite flat at p (a notion that they define and study in [27]). But since this condition is not present in weight $k> 2$ , we work with pseudo-representations that are ordinary at p and recover the results of Wake and Wang-Erickson mentioned above using them. In addition, we also prove some $R=\mathbb {T}$ theorems in certain cases.

Note that in the case of $k=2$ , we need [19, Proposition II. $9.6$ ], but it is not needed in the works of Wake–Wang-Erickson ([28]) and Calegari–Emerton ([6]). Moreover, both sets of authors recover [19, Proposition II. $9.6$ ] using their methods.

Before stating our main results, we describe the setup with which we will be working.

Setup 1.1. Let $p> 3$ be a prime and $\ell $ be a prime such that $p \mid \ell -1$ . Let $G_{\mathbb {Q},p\ell }$ be the Galois group of the maximal extension of $\mathbb {Q}$ unramified outside p, $\ell $ and $\infty $ over $\mathbb {Q}$ , and let $G_{\mathbb {Q},p}$ be the Galois group of the maximal extension of $\mathbb {Q}$ unramified outside p and $\infty $ over $\mathbb {Q}$ . Denote the mod p cyclotomic character of $G_{\mathbb {Q},p\ell }$ by $\omega _p$ and the p-adic cyclotomic character by $\chi _p$ . By abuse of notation, we will also denote the mod p cyclotomic character of $G_{\mathbb {Q},p}$ by $\omega _p$ . Let $k \geq 2$ be an even integer and $\bar \rho _0 : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {F}_p)$ be the continuous, odd representation given by $\bar \rho _0 = 1 \oplus \omega _p^{k-1}$ . Let $\zeta _p$ denote a primitive p-th root of unity. Suppose the following hypotheses hold:

1. (1) $p-1 \nmid k$ ,

2. (2) the $\omega _p^{1-k}$ -component of the p-part of the class group of $\mathbb {Q}(\zeta _p)$ is trivial,

3. (3) $\dim _{\mathbb {F}_p}(H^1(G_{\mathbb {Q},p},\omega _p^{k-1})) =1$ .

For a positive integer n, let $B_n$ be the n-th Bernoulli number.

Remark 1.2. Using the Herbrand–Ribet theorem and Kummer’s congruences, we conclude that Condition (2) of Setup 1.1 holds if and only if p does not divide $B_k$ . Combining this with Kummer’s congruences, we get that $\zeta (1-k) \in \mathbb {Z}_{(p)}^{\times }$ . Hence, the hypotheses of [26] are satisfied in our setup.

Remark 1.3. From [5, Lemma 21], we know that Condition (3) of Setup 1.1 holds if and only if the $\omega _p^{p+1-k}$ -component of the p-part of the class group of $\mathbb {Q}(\zeta _p)$ is trivial. Let $0 < k_0 <p-1$ be the integer such that $k \equiv k_0 \pmod {p-1}$ . Hence, by combining the reflection principle ([30, Theorem 10.9]) and the Herbrand–Ribet theorem, Condition (3) of Setup 1.1 holds if $p \nmid B_{p+1-k_0}$ .

Remark 1.4. Combining Remarks 1.2 and 1.3, we get that Conditions (2) and (3) of Setup 1.1 hold if one of the following conditions hold:

• • p is a regular prime.

• • Vandiver’s conjecture holds for p.

• • $p \nmid B_kB_{p+1-k_0}$ , where $0 < k_0 <p-1$ is the integer such that $k \equiv k_0 \pmod {p-1}$ .

• • $p> 7$ and $k=4,6$ .

• • $p \equiv 3 \pmod {4}$ and $k= \dfrac {p+1}{2}$ .

Note that we get Condition (3) of Setup 1.1 for $k=4$ from [16, Corollary 3.8] and for $k=6$ from [12, Corollary 7.1]. On the other hand, Conditions (2) and (3) for $k = \dfrac {p+1}{2}$ follow from the Herbrand–Ribet Theorem and [23, Theorem 1.1].

In the rest of the article, we assume that we are in Setup 1.1 unless mentioned otherwise.

Let $\mathbb {T}_{\mathfrak {m}}$ be the Hecke algebra of level $\Gamma _0(\ell )$ and weight k as defined in §4 and $\mathbb {T}^0_{\mathfrak {m}}$ be its cuspidal quotient.

Denote the absolute Galois group of $\mathbb {Q}_p$ and $\mathbb {Q}_{\ell }$ by $G_{\mathbb {Q}_p}$ and $G_{\mathbb {Q}_{\ell }}$ , respectively, and denote their inertia subgroups by $I_p$ and $I_{\ell }$ , respectively. Now our assumptions imply that $ \dim _{\mathbb {F}_p}(\ker (H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k}) \to H^1(I_p,\omega _p^{1-k})))=1$ (see Lemma 2.4). Choose a generator $c_0 \in \ker (H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k}) \to H^1(I_p,\omega _p^{1-k}))$ . Let $\bar \rho _{c_0} : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {F}_p)$ be the representation given by $\bar \rho _{c_0} = \begin {pmatrix} 1 & *\\ 0 & \omega _p^{k-1} \end {pmatrix}$ where $*$ corresponds to $c_0$ .

Note that both $\ker (H^1(G_{\mathbb {Q},p\ell },1) \to H^1(I_p,1))$ and $H^1(G_{\mathbb {Q},p},\omega _p^{k-1})$ are also one-dimensional. Choose generators $a_0 \in \ker (H^1(G_{\mathbb {Q},p\ell },1) \to H^1(I_p,1))$ and $b_0 \in H^1(G_{\mathbb {Q},p},\omega _p^{k-1})$ . Denote the cup product of $c_0$ and $b_0$ by $c_0 \cup b_0$ and the cup product of $c_0$ and $a_0$ by $c_0 \cup a_0$ . So in particular, $c_0 \cup b_0 \in H^2(G_{\mathbb {Q},p\ell },1)$ and $c_0 \cup a_0 \in H^2(G_{\mathbb {Q},p\ell }, \omega _p^{1-k})$ .

1.3 Main Results

We are now ready to state the main results.

Theorem A (see Corollary 5.2, Corollary 5.3, Theorem 5.5)

Suppose we are in Setup 1.1. Then:

1. (1) If $k=2$ , then $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}}) = 1$ if and only if $c_0 \cup a_0 \neq 0$ .

2. (2) If $k> 2$ , then $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}}) = 1$ if and only if $c_0 \cup b_0 \neq 0$ and $c_0 \cup a_0 \neq 0$ .

Note that part (1) of Theorem A has already been proved by Wake and Wang-Erickson in [28] using a similar approach but different methods.

In [26], Wake has proved that when $k> 2$ , $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}}) = 1$ if and only if the Eisenstein ideal of $\mathbb {T}^0_{\mathfrak {m}}$ is principal and a certain element $\xi ^{\prime}_{\text {MT}} \in \mathbb {F}_p$ (that he defines in [26, Section $\mathrm{1.2.2}$ ]) is nonzero. See [26, Theorem $\mathrm{1.2.4}$ ] for more details. We don’t use this result to prove part (2) of Theorem A, but we do need some other results from [26].

To be precise, when $c_0 \cup b_0 =0$ , we prove part (2) of Theorem A by proving that the Eisenstein ideal of $\mathbb {T}^0_{\mathfrak {m}}$ is not principal (see Theorem 5.5 and Theorem 5.6). As a consequence of our analysis, we get the following result regarding the principality of the Eisenstein ideal of $\mathbb {T}^0_{\mathfrak {m}}$ :

Corollary A. Suppose we are in Setup 1.1 and $k>2$ . Then the Eisenstein ideal of $\mathbb {T}^0_{\mathfrak {m}}$ is principal if and only if $c_0 \cup b_0 \neq 0$ . Moreover, if Vandiver’s conjecture holds for p, then these assertions hold if and only if $\prod _{j=1}^{p-1}(1-\zeta _p^j)^{j^{k-2}} \in (\mathbb {Z}/\ell \mathbb {Z})^{\times }$ (where $\zeta _p \in \mathbb {Z}/\ell \mathbb {Z}$ is a primitive p-th root of unity) is not a p-th power.

Remark 1.5. Note that Corollary A matches with the prediction made by Wake in [26, Section $\mathrm{1.2.3}$ , Remark 3.2.1].

Remark 1.6. If p is a regular prime, then Vandiver’s conjecture holds for p.

When $c_0 \cup b_0 \neq 0$ , the Eisenstein ideal is principal. In this case, we prove that $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}}) = 1$ if and only if $c_0 \cup a_0 \neq 0$ by using an analysis of pseudo-representations arising from representations.

Let $\zeta _{\ell }$ be a primitive $\ell $ -th root of unity, and let ${\zeta _{\ell }}^{(p)} \in \mathbb {Q}(\zeta _{\ell })$ be an element such that $[\mathbb {Q}(\zeta _{\ell }^{(p)}) : \mathbb {Q}] = p$ . Denote by $\text {Cl}(\mathbb {Q}(\zeta _{\ell }^{(p)},\zeta _p))$ the class group of $\mathbb {Q}(\zeta _{\ell }^{(p)},\zeta _p)$ , and let $(\text {Cl}(\mathbb {Q}(\zeta _{\ell }^{(p)},\zeta _p))/\text {Cl}(\mathbb {Q}(\zeta _{\ell }^{(p)},\zeta _p))^p)[\omega _p^{1-k}]$ be the subspace of $\text {Cl}(\mathbb {Q}(\zeta _{\ell }^{(p)},\zeta _p))/\text {Cl}(\mathbb {Q}(\zeta _{\ell }^{(p)},\zeta _p))^p$ on which $\text {Gal}(\mathbb {Q}(\zeta _p)/\mathbb {Q})$ acts by $\omega _p^{1-k}$ . Now we get the following corollaries (see Proposition 5.7):

Corollary B. Suppose we are in Setup 1.1 and $k=2$ . Then the following are equivalent:

1. (1) $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}}) = 1$ .

2. (2) $\dim _{\mathbb {F}_p}((\text {Cl}(\mathbb {Q}(\zeta _{\ell }^{(p)},\zeta _p))/\text {Cl}(\mathbb {Q}(\zeta _{\ell }^{(p)},\zeta _p))^p)[\omega _p^{1-k}]) = 1$ .

3. (3) $\prod _{i=1}^{\frac {\ell -1}{2}}i^{i}$ in $(\mathbb {Z}/\ell \mathbb {Z})^{\times }$ is not a p-th power.

Note that Corollary B has already been proved by Wake and Wang-Erickson in [28] and by Lecouturier in [18] using different methods. However, Wake and Wang-Erickson use results of [17] to prove that the second part of Corollary B implies the first part. We give a slightly different proof of the same (see Proposition 5.7 and its proof).

Corollary C. Suppose we are in Setup 1.1 and $k>2$ . Then the following are equivalent:

1. (1) $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}}) = 1$ .

2. (2) $\dim _{\mathbb {F}_p}((\text {Cl}(\mathbb {Q}(\zeta _{\ell }^{(p)},\zeta _p))/\text {Cl}(\mathbb {Q}(\zeta _{\ell }^{(p)},\zeta _p))^p)[\omega _p^{1-k}]) = 1$ , and the restriction map $H^1(G_{\mathbb {Q},p},\omega _p^{k-1}) \to H^1(G_{\mathbb {Q}_{\ell }},\omega _p^{k-1})$ is not the zero map.

3. (3) $\prod _{i=1}^{\ell -1}i^{(\sum _{j=1}^{i-1}j^{k-1})} \in (\mathbb {Z}/\ell \mathbb {Z})^{\times }$ is not a p-th power, and the restriction map $H^1(G_{\mathbb {Q},p},\omega _p^{k-1}) \to H^1(G_{\mathbb {Q}_{\ell }},\omega _p^{k-1})$ is not the zero map.

Moreover, if Vandiver’s conjecture holds for p, then the above assertions hold if and only if $\prod _{i=1}^{\ell -1}i^{(\sum _{j=1}^{i-1}j^{k-1})} \in (\mathbb {Z}/\ell \mathbb {Z})^{\times }$ is not a p-th power and $\prod _{i=1}^{p-1}(1-\zeta _p^i)^{i^{k-2}} \in (\mathbb {Z}/\ell \mathbb {Z})^{\times }$ (where $\zeta _p \in \mathbb {Z}/\ell \mathbb {Z}$ is a primitive p-th root of unity) is not a p-th power.

Let $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ be the universal p-ordinary, $\ell $ -unipotent deformation ring of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ with determinant $\chi _p^{k-1}$ (see Definition 2.1 and the paragraph after it). Let $R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell )$ be the universal p-ordinary, Steinberg-or-unramified at $\ell $ deformation ring of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ with determinant $\chi _p^{k-1}$ (see Definition 2.3 and the paragraph after it). Let $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ be the quotient of the universal ordinary deformation ring of $\bar \rho _{c_0}$ with determinant $\chi _p^{k-1}$ defined in §2.

Note that there is a surjective map $\phi _{\mathbb {T}} : R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to \mathbb {T}_{\mathfrak {m}}$ such that $\phi _{\mathbb {T}}$ factors through $R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell )$ , giving the map $\psi _{\mathbb {T}} : R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell ) \to \mathbb {T}_{\mathfrak {m}}$ (see Lemma 4.1). Moreover, the map $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to \mathbb {T}^0_{\mathfrak {m}}$ obtained by composing $\phi _{\mathbb {T}}$ with the surjective map $\mathbb {T}_{\mathfrak {m}} \to \mathbb {T}^0_{\mathfrak {m}}$ factors through $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell ) $ , giving the map $\phi _{\mathbb {T}^0} : R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell ) \to \mathbb {T}^0_{\mathfrak {m}}$ (see Lemma 4.2).

We are now ready to state the $R=\mathbb {T}$ theorems that we prove. Let $(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ))^{\text {red}}$ be the maximal reduced quotient of $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) $ . Recall, from Corollary A, that $c_0 \cup b_0 \neq 0$ if and only if the Eisenstein ideal of $\mathbb {T}^0_{\mathfrak {m}}$ is principal.

Theorem B. Suppose we are in Setup 1.1 and $c_0 \cup b_0 \neq 0$ . Then

1. (1) $\phi _{\mathbb {T}} : R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to \mathbb {T}_{\mathfrak {m}}$ induces an isomorphism $(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ))^{\text {red}} \simeq \mathbb {T}_{\mathfrak {m}}$ of local complete intersection rings.

2. (2) $\psi _{\mathbb {T}} : R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell ) \to \mathbb {T}_{\mathfrak {m}}$ is an isomorphism of local complete intersection rings.

3. (3) $\phi _{\mathbb {T}^0} : R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell ) \to \mathbb {T}^0_{\mathfrak {m}}$ is an isomorphism of local complete intersection rings.

From part (3) of Theorem B, we get the following analogue of [6, Corollary $1.6$ ]:

Corollary D. Suppose we are in Setup 1.1 and $c_0 \cup b_0 \neq 0$ . Then the $\mathbb {Z}_p$ -rank of $\mathbb {T}^0_{\mathfrak {m}}$ is the largest integer n for which there exists an ordinary deformation $\rho : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {F}_p[\epsilon ]/(\epsilon ^n))$ of $\bar \rho _{c_0}$ such that $\det (\rho ) = \omega _p^{k-1}$ , $\operatorname {tr}(\rho (g))=2$ for all $g \in I_{\ell }$ , and the set $\{\operatorname {tr}(\rho (g)) \mid g \in G_{\mathbb {Q},p\ell }\}$ generates $\mathbb {F}_p[\epsilon ]/(\epsilon ^n)$ as an $\mathbb {F}_p$ -algebra.

1.4 Sketch of the proofs of main results

We will now give a brief outline of the proof of Theorem A. We first analyze the space of deformations $(t,d) : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ which are p-ordinary and $\ell $ -unipotent with determinant $\chi _p^{k-1}$ to obtain its properties. To be precise, we prove that the space of such deformations has dimension either $1$ or $2$ (see Lemma 3.4) and this space is one-dimensional if $c_0 \cup b_0 \neq 0$ (see Lemma 3.6).

So we split the proof of Theorem A in two cases. In the first case, we assume either $k=2$ or $c_0 \cup b_0 \neq 0$ , and in the second case, we assume $k> 2$ and $c_0 \cup b_0 = 0$ . In the first case, we know that the tangent space of $\mathbb {T}_{\mathfrak {m}}/(p)$ has dimension $1$ , and hence its Eisenstein ideal is principal. We then prove that in this case, all the first order deformations of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ arising from $\mathbb {T}_{\mathfrak {m}}$ are reducible. From Lemma 3.12, we know that these reducible pseudo-representations arise from actual representations if and only if ${c_0 \cup a_0 = 0}$ . On the Hecke side, we know, from Lemma 4.2, that the pseudo-representation $(\tau ^0_{\ell },\delta ^0_{\ell }) : G_{\mathbb {Q},p\ell } \to \mathbb {T}^0_{\mathfrak {m}}$ arises from an ordinary deformation $\rho _{\mathbb {T}^0} : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {T}^0_{\mathfrak {m}})$ of $\bar \rho _{c_0}$ . Therefore, combining these two facts, we see that if $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}})> 1$ then $c_0 \cup a_0 =0$ .

On the other hand, suppose $c_0 \cup a_0 =0$ . Let $\phi : R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ be the map induced by the universal deformation taking values in $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ and $F : \mathbb {T}_{\mathfrak {m}} \to \mathbb {T}^0_{\mathfrak {m}}$ be the natural surjective map. Then we prove, using Lemma 4.2, that $\phi _{\mathbb {T}}(\ker (\phi )) \subset \ker (F)$ , and Lemma 3.12 implies that $\phi _{\mathbb {T}}(\ker (\phi )) \subset (p,\mathfrak {m}^2)$ . Combining these facts along with the principality of the Eisenstein ideal, [26, Theorem $\mathrm{5.1.2}$ ] and [19, Proposition II. $9.6$ ] (which give the index of Eisenstein ideal in $\mathbb {T}^0_{\mathfrak {m}}$ ), we show that $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}})> 1$ , which proves Theorem A in the first case.

In the second case, we split the proof of Theorem A in two steps. In the first step we prove that $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}})> 1$ if $c_0 \cup b_0 =0$ and $p \mid k$ (Theorem 5.5). To prove this, we use the relation between the tame inertia group and the Frobenius, techniques from Generalized Matrix Algebras (GMAs) along with [26, Theorem $\mathrm{5.1.2}$ ] and [26, Theorem $\mathrm{5.1.1}$ ] (which describes the biggest quotient of $\mathbb {T}_{\mathfrak {m}}$ in which $(\tau _{\ell },\delta _{\ell })$ is reducible) to prove that the Eisenstein ideal is not principal. To prove $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}})> 1$ when $c_0 \cup b_0 =0$ and $p \nmid k$ , we combine Theorem 5.5 and a result of Jochnowitz ([15]) about finiteness of the space of p-ordinary modular forms modulo p. Indeed, the result of Jochnowitz, along with some standard duality results, implies that the $\mathbb {Z}_p$ -rank of $\mathbb {T}^0_{\mathfrak {m}}$ is same as the $\mathbb {Z}_p$ -rank of the corresponding Hecke algebra of weight $k'$ for any $k'> k$ such that $k' \equiv k \pmod {p-1}$ . After taking such a $k'$ with $p \mid k'$ , we use Theorem 5.5 to prove the result.

1.5 Structure of the paper

In §2, we define various deformation rings that we will be working with throughout the article. In §3, we gather several preliminary results from deformation theory which will be used crucially in the proofs of main theorems. In §4, we define the Hecke algebras that we will be working with and gather their properties. In §5, we state and prove the main theorems of this article, as well as their corollaries.

1.6 Notation

In this subsection, we will develop some notation that will be used in the rest of the article. Recall that we denoted the absolute Galois groups of $\mathbb {Q}_p$ and $\mathbb {Q}_{\ell }$ by $G_{\mathbb {Q}_p}$ and $G_{\mathbb {Q}_{\ell }}$ , respectively, and their inertia groups by $I_p$ and $I_{\ell }$ , respectively. Denote the Frobenius at $\ell $ by $\text {Frob}_{\ell }$ . Fix embeddings $i_{\ell } : G_{\mathbb {Q}_{\ell }} \to G_{\mathbb {Q},p\ell }$ and $i_{p} : G_{\mathbb {Q}_{p}} \to G_{\mathbb {Q},p\ell }$ . Note that such embeddings are well-defined up to conjugacy. For a representation $\rho $ of $G_{\mathbb {Q},p\ell }$ , we denote the representation $\rho \circ i_{\ell }$ (resp. $\rho \circ i_{p}$ ) by $\rho |_{G_{\mathbb {Q}_{\ell }}}$ (resp. by $\rho |_{G_{\mathbb {Q}_{p}}}$ ) and denote the restriction of $\rho |_{G_{\mathbb {Q}_{\ell }}}$ (resp. of $\rho |_{G_{\mathbb {Q}_{p}}}$ ) to $I_{\ell }$ (resp. to $I_p$ ) by $\rho |_{I_{\ell }}$ (resp. by $\rho |_{I_p}$ ). By abuse of notation, we also denote $\omega _p|_{G_{\mathbb {Q}_{p}}}$ and $\chi _p|_{G_{\mathbb {Q}_{p}}}$ by $\omega _p$ and $\chi _p$ , respectively.

Now $(\operatorname {tr}(\bar \rho _0), \det (\bar \rho _0)) : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p$ is a $2$ -dimensional pseudo-representation of $G_{\mathbb {Q},p\ell }$ (in the sense of Chenevier ([8])). See [5, Section $1.4$ ] for the definition and properties of $2$ -dimensional pseudo-representations. In this article, we will only consider $2$ -dimensional pseudo-representations. If $(t,d) : G \to R$ is a pseudo-representation and I is an ideal of R, then we denote by $(t \pmod {I} , d \pmod {I})$ the pseudo-representation $G \to R/I$ obtained by composing $(t,d)$ with the quotient map $R \to R/I$ . All the representations, pseudo-representations and cohomology groups that we consider are assumed to be continuous unless mentioned otherwise.

If $(t,d) : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ is a pseudo-representation deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ , then we call it a first order deformation of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ . If $\rho : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {F}_p[\epsilon ]/(\epsilon ^2))$ is a deformation of $\bar \rho _{c_0}$ , then we call it a first order deformation of $\bar \rho _{c_0}$ .

If G is either a quotient of a class group of exponent p or a Galois cohomology group of an $\mathbb {F}_p$ -representation of either a local or a global Galois group, then we denote by $\dim (G)$ the $\mathbb {F}_p$ -dimension of G. If $\rho $ is a representation of $G_{\mathbb {Q},p\ell }$ and $c \in H^i(G_{\mathbb {Q},p\ell },\rho )$ , then we denote by $c|_{G_{\mathbb {Q}_\ell }}$ the image of c under the restriction map $H^i(G_{\mathbb {Q},p\ell },\rho ) \to H^1(G_{\mathbb {Q}_{\ell }},\rho )$ . If c and $c'$ are two Galois cohomology classes (either local or global), then denote by $c \cup c'$ their cup product.

Let $\mathcal {C}$ be the category of local complete noetherian rings with residue field $\mathbb {F}_p$ . If R is an object of $\mathcal {C}$ , then denote its maximal ideal by $m_R$ , denote its tangent space by $\text {tan}(R)$ and denote the $\mathbb {F}_p$ -dimension of $\text {tan}(R)$ by $\dim (\text {tan}(R))$ . By abuse of notation, we denote the character $G_{\mathbb {Q},p\ell } \to R^{\times }$ obtained by composing $\chi _p$ with the natural map $\mathbb {Z}_p^{\times } \to R^{\times }$ by $\chi _p$ . If $p = 0$ in R, then sometimes we will denote it by $\omega _p$ .

Let $\nu $ be the highest power of p dividing $\ell -1$ and $v_p(k)$ be the highest power of p dividing k (i.e., the p-valuation of k).

2 Deformation rings

Let $R^{\text {pd}}_{\bar \rho _0}$ be the universal deformation ring of the pseudo-representation $(\operatorname {tr}(\bar \rho _0), \det (\bar \rho _0)) : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p$ in $\mathcal {C}$ . Note that the existence of $R^{\text {pd}}_{\bar \rho _0}$ is proved in [8]. Let $(T^{\text {univ}}, D^{\text {univ}}) : G_{\mathbb {Q},p\ell } \to R^{\text {pd}}_{\bar \rho _0}$ be the universal pseudo-representation deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ . We will now define the deformation problems (and their deformation rings) that we will be working with.

Definition 2.1. Given an object R of $\mathcal {C}$ , a pseudo-representation $(t,d) : G_{\mathbb {Q},p\ell } \to R$ is called a p-ordinary, $\ell $ -unipotent deformation of $(\operatorname {tr}(\bar \rho _0), \det (\bar \rho _0))$ with determinant $\chi _p^{k-1}$ if the following conditions hold:

1. (1) $(t,d) \pmod {m_R} = (\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ ,

2. (2) $d : G_{\mathbb {Q},p\ell } \to R^{\times }$ is the character $\chi _{p}^{k-1}$ ,

3. (3) $t(g)=2$ for all $g \in I_{\ell }$ ,

4. (4) For all $g' \in G_{\mathbb {Q},p\ell }$ and $g, h \in I_p$ , $t(g'(g-\chi _{p}^{k-1}(g))(h-1)) = 0.$

Let $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ be the object of $\mathcal {C}$ representing the functor from $\mathcal {C}$ to the category of sets which sends an object R of $\mathcal {C}$ to the set of p-ordinary, $\ell $ -unipotent pseudo-representations $(t,d) : G_{\mathbb {Q},p\ell } \to R$ with determinant $\chi _p^{k-1}$ deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ .

It is easy to verify that $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ exists and it is given by the quotient of $R^{\text {pd}}_{\bar \rho _0}$ by the ideal $I_k$ generated by the set

(1) $$ \begin{align} \{D^{\text{univ}}(g) - \chi_{p}^{k-1}(g) \mid g \in G_{\mathbb{Q},p\ell}\} \cup \{T^{\text{univ}}(g)-2 \mid g \in I_{\ell}\} \nonumber\\ \cup \{T^{\text{univ}}(g'(g-\chi_{p}^{k-1}(g))(h-1)) \mid g' \in G_{\mathbb{Q},p\ell}, g,h \in I_p\}. \end{align} $$

Note that our notion of p-ordinariness, given by point (4) of Definition 2.1, is inspired from the notion of ordinary pseudo-representations defined by Calegari and Specter ([7, Definition $2.5$ ]). But we have slightly changed their notion to make it suitable for our purpose.

Remark 2.2. The auxiliary parameter $\alpha $ appearing in the definition of the p-ordinary pseudo-representations in [7, Definition $2.5$ ] is required to account for the presence of the Hecke operator $T_p$ in the Hecke algebra, especially in the non-p-distinguished case. But we are assuming that k is even, which means that $1 \neq \omega _p^{k-1}$ (i.e., we are in the p-distinguished case), so we do not need this auxiliary parameter.

Definition 2.3. Given an object R of $\mathcal {C}$ , a p-ordinary, $\ell $ -unipotent deformation $(t,d) : G_{\mathbb {Q},p\ell } \to R$ of $(\operatorname {tr}(\bar \rho _0), \det (\bar \rho _0))$ with determinant $\chi _p^{k-1}$ is called Steinberg-or-unramified at $\ell $ if for any lift $g_{\ell } \in G_{\mathbb {Q}_{\ell }}$ of $\text {Frob}_{\ell }$ , we have

$$\begin{align*}t(g(g_{\ell}-\ell^{\frac{k}{2}})(h-1))=0,\end{align*}$$

for every $h \in I_{\ell }$ and $g \in G_{\mathbb {Q},p\ell }$ .

Let $R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell )$ be the object of $\mathcal {C}$ representing the functor from $\mathcal {C}$ to the category of sets which sends an object R of $\mathcal {C}$ to the set of p-ordinary, Steinberg-or-unramified at $\ell $ pseudo-representations $(t,d) : G_{\mathbb {Q},p\ell } \to R$ with determinant $\chi _p^{k-1}$ deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ . It is easy to verify that $R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell )$ exists and it is given by the quotient of $R^{\text {pd}}_{\bar \rho _0}$ by the ideal $J_k$ generated by the ideal $I_k$ along with the set

$$\begin{align*}\{T^{\text{univ}}(g(g_{\ell}-\ell^{\frac{k}{2}})(h-1)) \mid g_{\ell} \in G_{\mathbb{Q}_{\ell}} \text{ is a lift of } \text{Frob}_{\ell}, h \in I_{\ell}, g \in G_{\mathbb{Q},p\ell}\}.\end{align*}$$

Note that our notion of a Steinberg-or-unramified at $\ell $ pseudo-representation is inspired from the unramified-or-Steinberg at $\ell $ condition defined and studied by Wake and Wang-Erickson in [29, Section $\mathrm{3.4}$ ]. In an unpublished version of [7], Calegari and Specter also define a similar notion (which they call ordinary at $\ell $ pseudo-representation).

Lemma 2.4. Suppose $p-1 \nmid k$ , k is even and the $\omega _p^{1-k}$ -component of the p-part of the class group of $\mathbb {Q}(\zeta _p)$ is trivial. Then $\dim (H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k}))=2$ and $\dim (\ker (H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k}) \to H^1(I_p,\omega _p^{1-k})))=1$ .

Proof. As we are assuming that the $\omega _p^{1-k}$ -component of the p-part of the class group of $\mathbb {Q}(\zeta _p)$ is trivial, it follows that $\ker (H^1(G_{\mathbb {Q},p},\omega _p^{1-k}) \to H^1(G_{\mathbb {Q}_p},\omega _p^{1-k}))$ is trivial. As $p-1 \nmid k$ , it follows, from local Euler characteristic formula, that $\dim (H^1(G_{\mathbb {Q}_p},\omega _p^{1-k}))=1$ and hence, $\dim (H^1(G_{\mathbb {Q},p},\omega _p^{1-k})) \leq 1$ . As $\omega _p^{1-k}$ is odd, global Euler characteristic formula implies that $\dim (H^1(G_{\mathbb {Q},p},\omega _p^{1-k})) \geq 1$ . So, we have $\dim (H^1(G_{\mathbb {Q},p},\omega _p^{1-k})) = 1$ .

Thus, by the Greenberg–Wiles formula ([31, Theorem $2$ ]), we get that $\ker (H^1(G_{\mathbb {Q},p}, \omega _p^k) \to H^1(G_{\mathbb {Q}_p},\omega _p^k))$ is trivial. Therefore, we conclude that

$$\begin{align*}\ker(H^1(G_{\mathbb{Q},p\ell},\omega_p^k) \to H^1(G_{\mathbb{Q}_p},\omega_p^k) \times H^1(G_{\mathbb{Q}_{\ell}},\omega_p^k)) = 0.\end{align*}$$

Hence, we get that

$$\begin{align*}\dim(H^1(G_{\mathbb{Q},p\ell},\omega_p^{1-k})) = 1+\dim(H^0(G_{\mathbb{Q}_p},\omega_p^k))+\dim(H^0(G_{\mathbb{Q}_{\ell}},\omega_p^k))=1+0+1=2.\end{align*}$$

So $1 \leq \dim (\ker (H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k}) \to H^1(G_{\mathbb {Q}_p},\omega _p^{1-k}))) \leq 2$ . Now we can view $H^1(G_{\mathbb {Q},p}, \omega _p^{1-k})$ as a subgroup of $H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k})$ , and we have seen that $\ker (H^1(G_{\mathbb {Q},p},\omega _p^{1-k}) \to H^1(G_{\mathbb {Q}_p},\omega _p^{1-k}))$ is trivial. Hence, it follows that $\dim (\ker (H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k}) \to H^1(G_{\mathbb {Q}_p}, \omega _p^{1-k})))=1$ . As $\omega _p^{1-k}|_{G_{\mathbb {Q}_p}} \neq 1$ , we see that

$$\begin{align*}\ker(H^1(G_{\mathbb{Q},p\ell},\omega_p^{1-k}) \to H^1(G_{\mathbb{Q}_p},\omega_p^{1-k})) = \ker(H^1(G_{\mathbb{Q},p\ell},\omega_p^{1-k}) \to H^1(I_p,\omega_p^{1-k})).\end{align*}$$

This proves the lemma.

Recall that we have fixed a generator $c_0 \in \ker (H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k}) \to H^1(I_p,\omega _p^{1-k}))$ . Note that there exists a $g_0 \in I_p$ such that $\omega _p^{k-1}(g_0) \neq 1$ . Fix such a $g_0 \in I_p$ . Let $\bar \rho _{c_0} : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {F}_p)$ be the representation such that $\bar \rho _{c_0}(g_0) = \begin {pmatrix} 1 & 0\\ 0 & \omega _p^{k-1}(g_0) \end {pmatrix}$ and $\bar \rho _{c_0}(g) = \begin {pmatrix} 1 & *\\ 0 & \omega _p^{k-1}(g) \end {pmatrix}$ for all $g \in G_{\mathbb {Q},p\ell }$ , where $*$ corresponds to $c_0$ .

Definition 2.5. Let $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}$ be the universal ordinary deformation ring (in the sense of Mazur) of $\bar \rho _{c_0}$ with constant determinant $\chi _p^{k-1}$ in $\mathcal {C}$ . So it represents the functor from $\mathcal {C}$ to the category of sets which sends an object R of $\mathcal {C}$ to the set of equivalence classes of representations $\rho : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(R)$ such that

1. (1) $\rho\ \pmod {m_R} = \bar \rho _{c_0}$ ,

2. (2) There exists an isomorphism $\rho |_{G_{\mathbb {Q}_p}} \simeq \begin {pmatrix} \eta _1 & *\\ 0 & \eta _2\end {pmatrix},$ where $\eta _2$ is an unramified character of $G_{\mathbb {Q}_p}$ lifting the trivial character $1$ ,

3. (3) $\det (\rho )=\chi _{p}^{k-1}$ .

As $c_0 \neq 0$ , the existence of $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}$ follows from [20] and [24].

Let $\rho ^{\text {univ}} : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k})$ be the universal ordinary deformation of $\bar \rho _{c_0}$ with constant determinant $\chi _p^{k-1}$ . Let $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell ) := R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}/I$ , where I is the ideal of $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}$ generated by the set $\{\operatorname {tr}(\rho ^{univ}(g)) - 2 \mid g \in I_{\ell }\}$ . Let $\rho ^{\text {univ},\ell } : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell ))$ be the representation obtained by composing $\rho ^{\text {univ}}$ with the natural surjective map $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k} \to R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ .

3 Preliminary results

In this section, we gather various preliminary results which will be crucially used in the proofs of the main theorems. We begin by recalling the notion of Generalized Matrix Algebras from [4].

3.1 Generalized Matrix Algebras

Let R be a complete noetherian ring with residue field $\mathbb {F}_p$ , and let A be a topological Generalized Matrix Algebra (GMA) over R of type $(1,1)$ as defined in [2, Section $2.2$ , Section $2.3$ ]. This means that there exist topological R-modules B and C such that $A=\begin {pmatrix} R & B\\ C & R\end {pmatrix}$ (i.e., every element of A can be written as $\begin {pmatrix} a & b\\c & d\end {pmatrix}$ with $a,d \in R$ , $b \in B$ and $c \in C$ ), and there exists a continuous morphism $m : B \otimes _R C \to R$ of R-modules such that A becomes a (not necessarily commutative) topological R-algebra under the addition and multiplication given by:

1. (1) Addition:

   $$\begin{align*}\begin{pmatrix} a_1 & b_1\\ c_1 & d_1\end{pmatrix} + \begin{pmatrix} a_2 & b_2\\ c_2 & d_2\end{pmatrix} = \begin{pmatrix} a_1+a_2 & b_1+b_2\\ c_1+c_2 & d_1+d_2\end{pmatrix},\end{align*}$$

2. (2) Multiplication:

   $$\begin{align*}\begin{pmatrix} a_1 & b_1\\ c_1 & d_1\end{pmatrix}. \begin{pmatrix} a_2 & b_2\\ c_2 & d_2\end{pmatrix} = \begin{pmatrix} a_1a_2+m(b_1 \otimes c_2) & a_1b_2+d_2b_1\\ d_1c_2+a_2c_1 & d_1d_2+m(b_2 \otimes c_1)\end{pmatrix}.\end{align*}$$

We refer the reader to [2, Section $2.2$ , Section $2.3$ ] for more details.

From now on, we assume that all the GMAs that we consider are topological GMAs of type $(1,1)$ unless mentioned otherwise.

Definition 3.1. Let $A=\begin {pmatrix} R & B\\ C & R\end {pmatrix}$ be a GMA over R as above. Keeping the notation developed above, we say that the GMA A is faithful if $m(b \otimes c)=0$ for all $c \in C$ implies that $b=0$ and $m(b \otimes c)=0$ for all $b \in B$ implies that $c=0$ .

Given a GMA A over R with R-modules B and C and the multiplication map $m : B \otimes _R C \to R$ as given above, we denote $m(b \otimes c)$ by $bc$ for all $b \in B, c \in C$ , and we denote by $BC$ the image of the map $m : B \otimes _R C \to R$ .

3.2 General results

If R is a complete noetherian local ring with residue field $\mathbb {F}_p$ and $(t,d) : G_{\mathbb {Q},p\ell } \to R$ is a pseudo-representation deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ , then [8, Lemma $3.8$ ] implies that the pseudo-representation $(t|_{I_{\ell }},d|_{I_{\ell }}) : I_{\ell } \to R$ (i.e., the restriction of the pseudo-representation $(t,d)$ to $I_{\ell }$ ) factors through the $\mathbb {Z}_p$ -quotient of the tame inertia group at $\ell $ . Fix a lift $i_{\ell } \in I_{\ell }$ of a topological generator of this $\mathbb {Z}_p$ -quotient of $I_{\ell }$ .

Recall that in §2, we have fixed a $g_0 \in I_p$ such that $\omega _p^{k-1}(g_0) \neq 1$ . We will now prove a result relating pseudo-representations with GMAs and establishing various properties of these GMAs. It will be extensively used throughout the article.

Lemma 3.2. Suppose k is even, the $\omega _p^{1-k}$ -component of the p-part of the class group of $\mathbb {Q}(\zeta _p)$ is trivial and $\dim _{\mathbb {F}_p}(H^1(G_{\mathbb {Q},p},\omega _p^{k-1})) =1$ . Let R be a complete noetherian ring with residue field $\mathbb {F}_p$ and $(t,d) : G_{\mathbb {Q},p\ell } \to R$ be a p-ordinary, $\ell $ -unipotent pseudo-representation with determinant $\chi _p^{k-1}$ deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ . Then there exists a faithful GMA $A = \begin {pmatrix} R & B \\ C & R\end {pmatrix}$ over R (in the sense of Definition 3.1 above) and a representation $\rho : G_{\mathbb {Q},p\ell } \to A^{\times }$ such that

1. (1) $\operatorname {tr}(\rho ) = t$ , $\det (\rho ) = d$ and $BC \subset m_R$ ,

2. (2) If $\rho (g) = \begin {pmatrix} a_g & b_g \\ c_g & d_g\end {pmatrix}$ , then $a_g \equiv 1 \pmod {m_R}$ and $d_g \equiv \omega _p^{k-1}(g) \pmod {m_R}$ for all $g \in G_{\mathbb {Q},p\ell }$ ,

3. (3) $\rho (g_0) = \begin {pmatrix} a_0 & 0\\ 0 & d_0\end {pmatrix}$ and $R[\rho (G_{\mathbb {Q},p\ell })] =A$ ,

4. (4) $\rho |_{I_{\ell }}$ factors through the tame $\mathbb {Z}_p$ -quotient of $I_{\ell }$ , $\rho (i_{\ell }) = \begin {pmatrix} 1+x & b_{\ell } \\ c_{\ell } & 1-x \end {pmatrix}$ with $x \in m_R$ and $B=Rb_{\ell }$ (i.e., B is generated by $b_{\ell }$ as an R-module).

5. (5) There exists $c' \in R$ such that C is generated by the set $\{c_{\ell },c'\}$ as an R-module.

6. (6) If $g \in I_p$ , then $\rho (g)=\begin {pmatrix} 1 & 0\\c_g & \chi _{p}^{k-1}(g)\end {pmatrix}$ .

Proof. The existence of $A = \begin {pmatrix} R & B \\ C & R\end {pmatrix}$ and $\rho : G_{\mathbb {Q},p\ell } \to A^{\times }$ satisfying parts (1), (2) and (3) of the lemma follows directly from [2, Proposition $\mathrm{2.4.2}$ ]. Moreover, it also implies that B and C are finitely generated R-modules. The description of $\rho (i_{\ell })$ follows from the assumption that $t(g)=2$ for all $g \in I_{\ell }$ . Since $(t|_{I_{\ell }},d|_{I_{\ell }})$ factors through the $\mathbb {Z}_p$ -quotient of the tame inertia group at $\ell $ and A is a faithful GMA, $\rho |_{I_{\ell }}$ also factors through this $\mathbb {Z}_p$ -quotient of $I_{\ell }$ .

If $B=0$ , then faithfulness implies that $C=0$ and vice versa. All the parts of the lemma are clearly true in this case. So assume $B \neq 0$ and $C \neq 0$ .

Let $h \in I_p$ , and let $\rho (h) =\begin {pmatrix} a_h & b_h \\ c_h & d_h \end {pmatrix}$ . As $(t,d)$ is p-ordinary, we have for all $g \in G_{\mathbb {Q},p\ell }$

(2) $$ \begin{align} \operatorname{tr}(\rho(g)(\rho(g_0)-\chi_{p}^{k-1}(g_0))(\rho(h) -1))=0. \end{align} $$

As $R[\rho (G_{\mathbb {Q},p\ell })] =A$ , we get that, for all $g' \in A$ ,

$$\begin{align*}\operatorname{tr}(g'(\rho(g_0)-\chi_{p}^{k-1}(g_0))(\rho(h) -1))=0.\end{align*}$$

For $c \in C$ , let $g_c = \begin {pmatrix} 0 & 0\\ c &0\end {pmatrix} \in A$ . Then

$$\begin{align*}\operatorname{tr}(g_c(\rho(g_0)-\chi_{p}^{k-1}(g_0))(\rho(h) -1)) = (a_0 - \chi_{p}^{k-1}(g_0))b_hc.\end{align*}$$

As $a_0 \equiv 1 \pmod {m_R}$ and $\omega _p^{k-1}(g_0) \neq 1$ , it follows that $a_0 - \chi _{p}^{k-1}(g_0) \in R^{\times }$ , and hence $b_hc=0$ for all $c \in C$ . As A is faithful, we get that $b_h=0$ for all $h \in I_p$ .

Taking $g'=\begin {pmatrix} 1 & 0\\ 0 & 0\end {pmatrix}$ in (2), we get $(a_0-\chi _{p}^{k-1}(g_0))(a_h-1)=0$ for all $h \in I_p$ . As $a_0 - \chi _{p}^{k-1}(g_0) \in R^{\times }$ , we get that $a_h=1$ for all $h \in I_p$ . As $\det (\rho (h))=\chi _{p}^{k-1}(h)$ , it follows that $d_h = \chi _{p}^{k-1}(h)$ for all $h \in I_p$ . This proves part (6) of the lemma.

Let $B' := B/Rb_{\ell }$ . Suppose $\phi : B'/m_RB' \to \mathbb {F}_p$ is a map of R-modules. Then it induces a map $\phi ^* : A \to M_2(\mathbb {F})$ of R-algebras which sends $\begin {pmatrix} a & b \\ c & d \end {pmatrix}$ to $\begin {pmatrix} a \pmod {m_R} & \phi (b) \\ 0 & d \pmod {m_R} \end {pmatrix}$ . So the image of $\phi ^*$ defines an element of $H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k})$ .

Thus, we get a map $f : \text {Hom}(B'/m_RB',\mathbb {F}_p) \to H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k})$ of $\mathbb {F}_p$ -vector spaces. It is easy to verify, using $R[\rho (G_{\mathbb {Q},p\ell })] =A$ , that this map is injective (see the proofs of [4, Theorem $\mathrm{1.5.5}$ ] and [9, Lemma $2.5$ ] for more details). Note that if $x \in H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k})$ lies in the image of f, then part (6) of the lemma implies that x is unramified at p. As $\rho |_{I_{\ell }}$ factors through the tame $\mathbb {Z}_p$ -quotient of $I_{\ell }$ , the definitions of $i_{\ell }$ and $b_{\ell }$ imply that x is also unramified at $\ell $ . Since we are assuming that the $\omega _p^{1-k}$ -component of the p-part of the class group of $\mathbb {Q}(\zeta _p)$ is trivial, it follows that $x=0$ . As f is injective, we see that $B'=0$ , which means $B=Rb_{\ell }$ . This finishes the proof of part (4) of the lemma.

Repeating the argument of the previous paragraph for $C'=C/Rc_{\ell }$ , we get an injective map $f' : \text {Hom}(C'/m_RC',\mathbb {F}_p) \to H^1(G_{\mathbb {Q},p\ell },\omega _p^{k-1})$ of $\mathbb {F}_p$ -vector spaces. Now $\rho |_{I_{\ell }}$ factors through the tame $\mathbb {Z}_p$ -quotient of $I_{\ell }$ . Therefore, from the definitions of $i_{\ell }$ and $c_{\ell }$ , it follows that if x is in the image of $f'$ , then x is unramified at $\ell $ . Hence, the image of $f'$ lies in $H^1(G_{\mathbb {Q},p}, \omega _p^{k-1}) \subset H^1(G_{\mathbb {Q},p\ell },\omega _p^{k-1})$ . As we are assuming that $\dim (H^1(G_{\mathbb {Q},p}, \omega _p^{k-1}))=1$ and $f'$ is injective, we get that $C'$ is either $0$ or it is generated by one element as an R-module. This gives us part (5) of the lemma.

Reducible pseudo-representations deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ will play an important role in the proofs of our main results. Their importance is already highlighted in Wake’s work ([26]). We will now prove a basic result about reducible pseudo-representations which is an analogue of [3, Lemme $1$ ]. Its proof is also similar to that of [3, Lemme $1$ ]. But we give it here for the benefit of the reader. We will use it extensively while working with reducible pseudo-representations.

Lemma 3.3. Let $(t,d) :G_{\mathbb {Q},p\ell } \to R$ be a pseudo-representation deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ . Suppose $A = \begin {pmatrix} R & B \\ C & R\end {pmatrix}$ is a (not necessarily faithful) GMA over R and $\rho : G_{\mathbb {Q},p\ell } \to A^{\times }$ is a representation such that

1. (1) $t=\operatorname {tr}(\rho )$ and $d=\det (\rho )$ .

2. (2) If $g \in G_{\mathbb {Q},p\ell }$ and $\rho (g) = \begin {pmatrix} a_g & b_g \\ c_g & d_g\end {pmatrix}$ , then $a_g \equiv 1 \pmod {m_R}$ and $d_g \equiv \omega _p^{k-1}(g) \pmod {m_R}$ .

3. (3) $\rho (g_0) = \begin {pmatrix} a_0 & 0\\ 0 & d_0\end {pmatrix}$ and $R[\rho (G_{\mathbb {Q},p\ell })] =A$ .

Let I be an ideal of R. Then $t \pmod {I}=\chi _1+\chi _2$ , for some characters $\chi _1, \chi _2 : G_{\mathbb {Q},p\ell } \to (R/I)^{\times }$ deforming $1$ and $\omega _p^{k-1}$ , if and only if $BC \subset I$ . Moreover, if this condition is satisfied, then $a_g \pmod {I}=\chi _1(g)$ and $d_g \pmod {I}=\chi _2(g)$ for all $g \in G_{\mathbb {Q},p\ell }$ .

Proof. It is easy to see, from the description of $\rho (g)$ , that if $BC \subset I$ , then $\operatorname {tr}(\rho ) \pmod {I}=\chi _1+\chi _2$ for some characters $\chi _1$ and $\chi _2$ deforming $1$ and $\omega _p^{k-1}$ . Indeed, we can take $\chi _1(g) = a_g \pmod {I}$ and $\chi _2(g)=d_g\pmod {I}$ for all $g \in G_{\mathbb {Q},p\ell }$ .

Now suppose $\operatorname {tr}(\rho ) \pmod {I}$ is a sum of two characters lifting $1$ and $\omega _p^{k-1}$ . If $r \in R$ , then denote its image in $R/I$ by $\overline {r}$ . Suppose $g \in G_{\mathbb {Q},p\ell }$ and $\rho (g)=\begin {pmatrix} a_g & b_g \\ c_g & d_g\end {pmatrix}$ . Then $t(gg_0) = a_0a_g+d_0d_g$ . By our assumption on I, we know that

(3) $$ \begin{align}\overline{a_0a_g+d_0d_g} = \chi_1(g_0g)+\chi_2(g_0g).\end{align} $$

Now $d(g_0)=a_0d_0$ , and hence $a_0$ and $d_0$ are roots of the polynomial $f(X) = X^2-t(g_0)X+d(g_0) \in R[X]$ .

Let $\bar f(X) \in R/I[X]$ be the reduction of f modulo I. So $\overline {a_0}$ and $\overline {d_0}$ are the roots of $\bar f(X)$ . Now as $p>2$ , $d(g)=\frac {t(g)^2-t(g^2)}{2}$ (see [5, Section $1.4$ ]). As $t \pmod {I}=\chi _1+\chi _2$ , it follows that $\overline {t(g)}=\chi _1(g)+\chi _2(g)$ and $\overline {d(g)}=\chi _1(g)\chi _2(g)$ . So $\chi _1(g_0)$ and $\chi _2(g_0)$ are also roots of $\bar f(X)$ . Therefore, we get that $\overline {a_0}=\chi _1(g_0)$ and $\overline {d_0}=\chi _2(g_0)$ by matching their reductions modulo the maximal ideal of $R/I$ .

Hence, by (3), we get $\overline {a_0a_g+d_0d_g} = \chi _1(g)\overline {a_0}+\chi _2(g)\overline {d_0}$ . On the other hand, $\overline {a_g+d_g} = \chi _1(g) + \chi _2(g)$ . So, we get $\overline {a_g(d_0-a_0)}=\chi _1(g)\overline {(d_0-a_0)}$ and $\overline {d_g(a_0-d_0)}=\chi _2(g)\overline {(a_0-d_0)}$ . As $a_0-d_0 \in R^{\times }$ , we get that $\overline {a_g}=\chi _1(g)$ and $\overline {d_g}=\chi _2(g)$ . This proves the second part of the lemma.

Now if $g, g' \in G_{\mathbb {Q},p\ell }$ , then $\rho (gg')=\rho (g)\rho (g')$ . So we get $a_{gg'}=a_ga_{g'}+b_gc_{g'}$ and $d_{gg'}=d_gd_{g'}+c_gb_{g'}$ . From the previous paragraph, we know that $a_{gg'} \equiv a_ga_{g'} \pmod {I}$ and $d_{gg'} \equiv d_gd_{g'} \pmod {I}$ . Hence, for all $g, g' \in G_{\mathbb {Q},p\ell }$ , we have $b_gc_{g'} \in I$ and $c_gb_{g'}\in I$ . So we get $BC \subset I$ which proves the lemma.

3.3 First order deformations

We will now focus on the p-ordinary, $\ell $ -unipotent pseudo-representations $(t,d) : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ with determinant $\omega _p^{k-1}$ deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ . Note that such pseudo-representations arise from the tangent space of $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/(p)$ . We start with determining the possible dimensions of the space of such deformations. This will be useful in studying the structures of the Hecke algebras of interest and their Eisenstein ideals.

Let $(T,D) : G_{\mathbb {Q},p\ell } \to R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ be the universal p-ordinary, $\ell $ -unipotent deformation of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ with determinant $\chi _p^{k-1}$ . Recall from §1.6 that we denote the tangent space of $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/(p)$ by $\text {tan}(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/(p))$ .

Lemma 3.4. $1 \leq \dim (\text {tan}(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/(p))) \leq 2$ .

Proof. Suppose $\eta _1, \eta _2 : G_{\mathbb {Q},p\ell } \to (\mathbb {F}_p[\epsilon ]/(\epsilon ^2))^{\times }$ are characters such that $\eta _1(I_p)=1$ , $\eta _1(i_{\ell })=1+\epsilon $ and $\eta _2=\omega _p^{k-1}\eta _1^{-1}$ . By class field theory and the definition of $i_{\ell }$ , it follows that such characters exist and are unique. Note that $\eta _1$ is a deformation of $1$ , $\eta _2$ is a deformation of $\omega _p^{k-1}$ and $\eta _1\eta _2=\omega _p^{k-1}$ . It is easy to verify that the pseudo-representation

$$\begin{align*}(\eta_1+\eta_2, \eta_1\eta_2) : G_{\mathbb{Q},p\ell} \to \mathbb{F}_p[\epsilon]/(\epsilon^2)\end{align*}$$

is a p-ordinary, $\ell $ -unipotent deformation of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ with determinant $\chi _p^{k-1}$ . Hence, we get that $1 \leq \dim (\text {tan}(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/(p)))$ .

On the other hand, suppose $(t,d) : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ is a reducible, p-ordinary, $\ell $ -unipotent deformation of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ with determinant $\chi _p^{k-1}$ . So there exist two characters $\chi _1, \chi _2 : G_{\mathbb {Q},p\ell } \to (\mathbb {F}_p[\epsilon ]/(\epsilon ^2))^{\times }$ such that $\chi _1$ lifts $1$ , $\chi _2$ lifts $\omega _p^{k-1}$ and $\chi _1\chi _2=\omega _p^{k-1}$ . From part (6) of Lemma 3.2, we get that $\chi _1|_{I_p}=1$ and $\chi _2|_{I_p}=\omega _p^{k-1}$ .

As $\chi _1$ is unramified at p, it follows, from class field theory and the definition of $i_{\ell }$ , that $\chi _1 = \eta _1^m$ and $\chi _2 =\omega _p^{k-1}\eta _1^{-m}$ for some integer $m \geq 0$ . So in the space of first order deformations of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ , the deformation $(t,d) = (\eta _1^m + \omega _p^{k-1}\eta _1^{-m}, \omega _p^{k-1})$ lies in the subspace generated by the deformation $(\eta _1+\eta _2,\eta _1\eta _2)$ found above. So the space of reducible, p-ordinary, $\ell $ -unipotent, first order deformations of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ with determinant $\chi _p^{k-1}$ has dimension $1$ .

Let $\mathfrak {m}_0$ be the maximal ideal of $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ , and let $R:=R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/(p,\mathfrak {m}_0^2)$ . So $\dim (\text {tan}(R)) = \dim (\text {tan}(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/(p)))$ . Denote the pseudo-representation $G_{\mathbb {Q},p\ell } \to R$ obtained by composing $(T,D)$ with the natural surjective map $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to R$ by $(t,d)$ . Let $A=\begin {pmatrix} R & B\\ C & R\end {pmatrix}$ be the faithful GMA over R and $\rho : G_{\mathbb {Q},p\ell } \to A^{\times }$ be the representation associated to $(t,d)$ by Lemma 3.2. If $BC=0$ , then $(t,d)$ is reducible. Hence, we conclude from the discussion above that $\dim (\text {tan}(R))=1$ . Therefore, we get that $\dim (\text {tan}(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/(p))) =1$ .

Suppose $B \neq 0$ and $C \neq 0$ . Now $\rho (i_{\ell }) = \begin {pmatrix}1+x & b_{\ell } \\ c_{\ell } & 1-x\end {pmatrix}$ . As $d=\omega _p^{k-1}$ , we see that $d(i_{\ell })=1$ . This means $\det (\rho (i_{\ell }))=1$ , which implies that $b_{\ell }c_{\ell }= -x^2$ . As $x \in m_R$ and $m_R^2=0$ , we get $b_{\ell }c_{\ell }=0$ . As A is faithful, part (4) of Lemma 3.2 implies that $c_{\ell }=0$ . Hence, by part (5) of Lemma 3.2, it follows that C is generated by $1$ element over R. Let c be a generator of C as R-module and let $x'=b_{\ell }c \in R$ . So $BC=(b_{\ell }c)=(x')$ .

By Lemma 3.3, we get that $(t \pmod {(x')}, d \pmod {(x')}) : G_{\mathbb {Q},p\ell } \to R/(x')$ is a reducible, p-ordinary, $\ell $ -unipotent deformation of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ with determinant $\chi _p^{k-1}$ . So any first order deformation of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ arising from $R/(x')$ is reducible. Hence, from above, we conclude that $\dim (\text {tan}(R/(x'))) \leq 1$ . This implies that $\dim (\text {tan}(R)) \leq 2$ . Therefore, we get $\dim (\text {tan}(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/(p))) \leq 2$ and the lemma follows.

The next result will be used in determining the generators of the cotangent spaces of the Hecke algebras.

Lemma 3.5. Let R be a quotient of $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ and $(t,d) : G_{\mathbb {Q},p\ell } \to R$ be the pseudo-representation obtained by composing $(T,D)$ with the quotient map $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to R$ . Let A be the faithful GMA over R and $\rho : G_{\mathbb {Q},p\ell } \to A^{\times }$ be the representation attached to $(t,d)$ by Lemma 3.2. Let $\rho (i_{\ell }) = \begin {pmatrix} 1 + x & b_{\ell } \\ c_{\ell } & 1-x \end {pmatrix}$ . If $\dim (\text {tan}(R/(p)))=1$ and the deformation $(t',d') : G_{\mathbb {Q},p\ell } \to R/(p,m_R^2)$ of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ obtained by composing $(t,d)$ with the quotient map $R \to R/(p,m_R^2)$ is reducible, then $m_R$ is generated by p and x.

Proof. As $\dim (\text {tan}(R/(p)))=1$ , we see that $R/(p,m_R^2) \simeq \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ . So $(t',d')$ gives us a nontrivial $\mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ -valued pseudo-representation. We will now use this identification. Since $(t',d')$ is reducible, $t' = \chi _1 + \chi _2$ , where $\chi _1, \chi _2 : G_{\mathbb {Q},p\ell } \to (\mathbb {F}_p[\epsilon ]/(\epsilon ^2))^{\times }$ are characters deforming $1$ and $\omega _p^{k-1}$ , respectively. From the proof of Lemma 3.4, we know that $\chi _1|_{I_p} = 1$ , $\chi _2|_{I_p}=\omega _p^{k-1}$ , $\chi _1(i_{\ell }) = 1 + a\epsilon $ and $\chi _2(i_{\ell }) = 1 - a\epsilon $ for some nonzero $a \in \mathbb {F}_p$ . From Lemma 3.3, we get that the image of $1+x$ in $R/(p,m_R^2) \simeq \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ is $1+a\epsilon $ . Therefore, the image of x in $R/(p,m_R^2) \simeq \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ generates the ideal $(\epsilon )$ . Hence, we conclude that $m_R$ is generated by $(p,x)$ .

Recall that we have chosen a nonzero generator $b_0$ of $H^1(G_{\mathbb {Q},p},\omega _p^{k-1})$ and have denoted the cup product of $c_0$ and $b_0$ by $c_0 \cup b_0$ . So $c_0 \cup b_0 \in H^2(G_{\mathbb {Q},p\ell },1)$ . We now give a necessary condition, in terms of this cup product, for the existence of a first order deformation of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ which is p-ordinary, $\ell $ -unipotent with determinant $\omega _p^{k-1}$ and is not reducible. This lemma is the first step towards establishing the link between the principality of the Eisenstein ideal and the nonvanishing of the cup product $c_0 \cup b_0$ . The proof uses techniques similar to the ones used in [1].

Lemma 3.6. If $\dim (\text {tan}(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/(p))) = 2$ , then $c_0 \cup b_0 =0$ .

Proof. If $\dim (\text {tan}(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/(p))) = 2$ , then the proof of Lemma 3.4 implies that there exists a pseudo-representation $(t,d) : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ such that $(t,d)$ is a p-ordinary, $\ell $ -unipotent deformation of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ with determinant $\omega _p^{k-1}$ and $(t,d)$ is not reducible. Let $(t_0,d_0)$ be such a deformation.

Let $A= \begin {pmatrix} \mathbb {F}_p[\epsilon ]/(\epsilon ^2) & B \\ C & \mathbb {F}_p[\epsilon ]/(\epsilon ^2) \end {pmatrix}$ be the faithful GMA over $\mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ and $\rho : G_{\mathbb {Q},p\ell } \to A^{\times }$ be the representation attached to $(t_0,d_0)$ by Lemma 3.2. So both $B \neq 0$ and $C \neq 0$ . Now we know, from part (4) of Lemma 3.2, that $\rho (i_{\ell }) = \begin {pmatrix} 1+x & b_{\ell } \\ c_{\ell } & 1-x\end {pmatrix}$ with $x \in (\epsilon )$ . As $d_0(i_{\ell })=1$ , we get that $b_{\ell }c_{\ell }=-x^2=0$ . We also know, from part (4) of Lemma 3.2, that $B = \mathbb {F}_p[\epsilon ]/(\epsilon ^2).b_{\ell }$ . Since $B \neq 0$ , we get that $b_{\ell } \neq 0$ , and hence $c_{\ell }=0$ . So we conclude, from part (5) of Lemma 3.2, that C is also generated by one element over $\mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ .

Let c be a generator of C. As $BC \subset (\epsilon )$ , we get $\epsilon BC=0$ . As A is faithful, we get that $\epsilon B =0$ and $\epsilon C=0$ . Hence, both B and C are isomorphic to $\mathbb {F}_p$ as $\mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ -modules. The choice of generators $b_{\ell }$ for B and c for C identifies both B and C with $\mathbb {F}_p$ . In particular, if $g \in G_{\mathbb {Q},p\ell }$ , then $\rho (g) = \begin {pmatrix} 1+a_g\epsilon & b_gb_{\ell } \\ c_gc &\omega _p^{k-1}(g) + d_g\epsilon \end {pmatrix}$ with $a_g, b_g, c_g, d_g \in \mathbb {F}_p$ .

Since $\rho $ is a representation and $\epsilon b_{\ell } = \epsilon c =0$ , we get that the map $G_{\mathbb {Q},p\ell } \to \mathbb {F}_p$ sending g to $c_g$ defines an element of $H^1(G_{\mathbb {Q},p\ell },\omega _p^{k-1})$ and the map $G_{\mathbb {Q},p\ell } \to \mathbb {F}_p$ sending g to $\omega _p^{1-k}(g)b_g$ defines an element of $H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k})$ . As $\mathbb {F}_p[\epsilon ]/(\epsilon ^2)[\rho (G_{\mathbb {Q},p\ell })] = A$ , it is easy to verify (using proof of Lemma 3.2) that both these elements are nonzero.

Note that part (6) of Lemma 3.2 implies that $b_h=0$ for all $h \in I_p$ . Hence, the cohomology class defined by $b_g$ lies in $\ker (H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k}) \to H^1(I_p,\omega _p^{1-k}))$ . Since $c_{\ell }=0$ and $\rho |_{I_{\ell }}$ factors through the tame $\mathbb {Z}_p$ -quotient of $I_{\ell }$ (part (4) of Lemma 3.2), the definition of $i_{\ell }$ implies that the cohomology class defined by $c_g$ lies in $H^1(G_{\mathbb {Q},p},\omega _p^{k-1})$ .

Since both these spaces are one-dimensional and generated by $c_0$ and $b_0$ , respectively, it follows that there exist nonzero elements $\alpha , \beta \in \mathbb {F}_p$ such that $c_g = \alpha b_0(g)$ and $b_g = \beta \omega _p^{k-1}(g)c_0(g)$ for all $g \in G_{\mathbb {Q},p\ell }$ . Let $\gamma \in \mathbb {F}_p$ such that $b_{\ell }c=\gamma \epsilon $ . So $\gamma \neq 0$ . Now we have

$$\begin{align*}1+a_{gh}\epsilon = (1+a_g\epsilon)(1+a_h\epsilon) + b_gc_h\gamma\epsilon = (1+a_g\epsilon)(1+a_h\epsilon) +\alpha\beta\gamma\omega_p^{k-1}(g)c_0(g)b_0(h)\epsilon.\end{align*}$$

So we have $a_{gh} - a_g - a_h = \alpha \beta \gamma \omega _p^{k-1}(g)c_0(g)b_0(h)$ for all $g, h \in G_{\mathbb {Q},p\ell }$ . As $\alpha \beta \gamma \neq 0$ , it follows, from the definition of the cup product, that $c_0 \cup b_0 = 0$ (see [25, Section $2.1$ ]).

We will now give a criteria for the vanishing of the cup product $c_0 \cup b_0$ .

Lemma 3.7. Suppose Vandiver’s conjecture holds for p. Then $c_0 \cup b_0 = 0$ if and only if $\prod _{j=1}^{p-1}(1-\zeta _p^j)^{j^{k-2}} \in (\mathbb {Z}/\ell \mathbb {Z})^{\times }$ is a p-th power in $(\mathbb {Z}/\ell \mathbb {Z})^{\times }$ , where $\zeta _p \in (\mathbb {Z}/\ell \mathbb {Z})^{\times }$ is a primitive p-th root of unity.

Proof. By [25, Proposition $\mathrm{2.4.1}$ ], we know that $c_0 \cup b_0 = 0$ if and only if $c_0|_{G_{\mathbb {Q}_{\ell }}} \cup b_0|_{G_{\mathbb {Q}_{\ell }}} = 0$ . Note that $c_0|_{G_{\mathbb {Q}_{\ell }}}, b_0|_{G_{\mathbb {Q}_{\ell }}} \in H^1(G_{\mathbb {Q}_{\ell }},1)$ . Since $c_0|_{G_{\mathbb {Q}_{\ell }}}$ is ramified at $\ell $ , we see that $c_0|_{G_{\mathbb {Q}_{\ell }}} \neq 0$ . On the other hand, $b_0|_{G_{\mathbb {Q}_{\ell }}}$ is unramified at $\ell $ . So $c_0|_{G_{\mathbb {Q}_{\ell }}} \cup b_0|_{G_{\mathbb {Q}_{\ell }}} = 0$ if and only if $b_0|_{G_{\mathbb {Q}_{\ell }}} = 0$ .

Now let $\bar \rho _{b_0} : G_{\mathbb {Q},p}\to \mathrm {GL}_2(\mathbb {F}_p)$ be the representation given by $\begin {pmatrix} \omega _p^{k-1} & * \\ 0 & 1\end {pmatrix}$ , where $*$ corresponds to $b_0$ , and let $K'$ be the extension of $\mathbb {Q}$ fixed by $\ker (\bar \rho _{b_0})$ . So $b_0|_{G_{\mathbb {Q}_{\ell }}} = 0$ if and only if $\ell $ splits completely in $K'$ . As $p \mid \ell -1$ , $\ell $ splits completely in $K'$ if and only if $\ell $ splits completely in $K'(\zeta _p)$ .

Let $\mathcal {U} = \mathbb {Z}[\zeta _p,p^{-1}]^{\times }$ . So $\dfrac {\mathcal {U}}{\mathcal {U}^p}$ is a $\text {Gal}(\mathbb {Q}(\zeta _p)/\mathbb {Q})$ -module. For any integer i, let $\dfrac {\mathcal {U}}{\mathcal {U}^p}[\omega _p^i]$ be the $\omega _p^i$ -component of $\dfrac {\mathcal {U}}{\mathcal {U}^p}$ . Using the inflation-restriction exact sequence and Kummer theory, we get that $\dfrac {\mathcal {U}}{\mathcal {U}^p}[\omega _p^{2-k}]$ is isomorphic to a subgroup of $H^1(G_{\mathbb {Q},p},\omega _p^{k-1})$ . Since Vandiver’s conjecture holds for p, we know, from Remark 1.3, that $\dim (H^1(G_{\mathbb {Q},p},\omega _p^{k-1}))=1$ (which is consistent with the hypothesis (3) of Setup 1.1). Observe that $\dfrac {\mathcal {U}}{\mathcal {U}^p}[\omega _p^{2-k}]$ is an $\mathbb {F}_p$ -vector space of dimension $1$ , and hence, $H^1(G_{\mathbb {Q},p},\omega _p^{k-1}) \simeq \dfrac {\mathcal {U}}{\mathcal {U}^p}[\omega _p^{2-k}]$ . We refer the reader to the discussion in [25, Section $5.3$ ] appearing just before [25, Theorem $\mathrm{5.3.2}$ ] for more details.

Hence, if $\Xi \in \mathcal {U}$ is an element such that $\Xi \not \in \mathcal {U}^p$ and $g(\Xi ) \equiv \Xi ^{\omega _p^{2-k}(g)} \pmod {\mathcal {U}^p}$ for all $g \in \text {Gal}(\mathbb {Q}(\zeta _p)/\mathbb {Q})$ , then $K'(\zeta _p)$ is obtained by attaching a p-th root of $\Xi $ to $\mathbb {Q}(\zeta _p)$ .

Now let $\Xi = \prod _{j=1}^{p-1}(1-\zeta _p^j)^{j^{k-2}}$ . Observe that $g(\Xi ) \equiv \Xi ^{\omega _p^{2-k}(g)} \pmod {\mathcal {U}^p}$ for all ${g \in \text {Gal}(\mathbb {Q}(\zeta _p)/\mathbb {Q})}$ . Recall that, by our assumption, Vandiver’s conjecture holds for p. Therefore, by combining [17, Lemma 2.7], [30, Lemma 8.1] and [30, Theorem 8.2], we get that the set $\{1-\zeta _p^a \mid a \in \mathbb {Z}, 0 < a < p/2\}$ is a $\mathbb {Z}$ -basis of the free part of $\mathcal {U}$ . As k is even, this implies that $\Xi \in \mathcal {U}\setminus \mathcal {U}^p$ . Hence, we conclude that the extension $K'(\zeta _p)$ is obtained by attaching a p-th root of $\prod _{j=1}^{p-1}(1-\zeta _p^j)^{j^{k-2}}$ to $\mathbb {Q}(\zeta _p)$ (see also [26, Remark $\mathrm{3.2.1}$ ]).

So $\ell $ splits completely in $K'(\zeta _p)$ if and only if $\prod _{j=1}^{p-1}(1-\zeta _p^j)^{j^{k-2}}$ is a p-th power in $\mathbb {Q}_{\ell }$ . By Hensel’s lemma, $\prod _{j=1}^{p-1}(1-\zeta _p^j)^{j^{k-2}}$ is a p-th power in $\mathbb {Q}_{\ell }$ if and only if $\prod _{j=1}^{p-1}(1-\zeta _p^j)^{j^{k-2}} \in (\mathbb {Z}/\ell \mathbb {Z})^{\times }$ is a p-th power in $(\mathbb {Z}/\ell \mathbb {Z})^{\times }$ which proves the lemma.

3.4 Pseudo-representations and representations

Let $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ be the deformation ring introduced in Definition 2.5 and $\rho ^{\text {univ}, \ell } : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell ))$ be the corresponding deformation. Note that $(\operatorname {tr}(\rho ^{\text {univ},\ell }),\det (\rho ^{\text {univ},\ell })) : G_{\mathbb {Q},p\ell } \to R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ is a deformation of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ . Note that $\det (\rho ^{\text {univ},\ell }) = \chi _{p}^{k-1}$ .

As $\rho ^{\text {univ},\ell }$ is p-ordinary, it follows that, under a suitable basis, $\rho ^{\text {univ},\ell }(g) = \begin {pmatrix} \chi _{p}(g)^{k-1} & b_g \\ 0 & 1 \end {pmatrix}$ for all $g \in I_p$ . So $(\rho ^{\text {univ},\ell }(g) - \chi _{p}(g)^{k-1})(\rho ^{\text {univ},\ell }(h)-1) = \begin {pmatrix} 0 & 0 \\ 0 & 0 \end {pmatrix}$ for all $g,h \in I_p$ . Hence, $(\operatorname {tr}(\rho ^{\text {univ},\ell }),\det (\rho ^{\text {univ},\ell }))$ is p-ordinary. Moreover, it is $\ell $ -unipotent by definition.

Therefore, $(\operatorname {tr}(\rho ^{\text {univ},\ell }),\det (\rho ^{\text {univ},\ell })) $ induces a map $\phi : R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ .

Lemma 3.8. The map $\phi : R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ induced by $(\operatorname {tr}(\rho ^{\text {univ},\ell }),\det (\rho ^{\text {univ},\ell })) $ is surjective.

Proof. To prove the lemma, it suffices to prove that if $\rho : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {F}_p[\epsilon ]/(\epsilon ^2))$ is an ordinary deformation of $\bar \rho _{c_0}$ such that $\det (\rho )=\omega _p^{k-1}$ and $\operatorname {tr}(\rho ) = 1 + \omega _p^{k-1}$ , then $\rho \simeq \bar \rho _{c_0}$ . Denote $\mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ by R and let $\rho $ be such a representation. By [10, Lemma $3.1$ ], after replacing $\rho $ by a suitable element in its equivalence class if necessary, we can assume that $\rho (g_0)=\begin {pmatrix}a_0 & 0\\ 0 & d_0\end {pmatrix}$ . Since , [2, Lemma $\mathrm{2.4.5}$ ] implies that there exist ideals $B, C \subset R$ such that $R[\rho (G_{\mathbb {Q},p\ell })] = \begin {pmatrix}R & B\\ C & R\end {pmatrix}$ .

Note that $R[\rho (G_{\mathbb {Q},p\ell })]$ is a GMA over R with multiplication of B and C given by multiplication in R. As $\rho \pmod {(\epsilon )} = \bar \rho _{c_0}$ , it follows that $B=R$ . Now $(\operatorname {tr}(\rho ),\det (\rho ))$ is reducible. Hence, applying Lemma 3.3 to the GMA $R[\rho (G_{\mathbb {Q},p\ell })]$ , we get that $BC=0$ . As $B=R$ , we have $C=0$ . Moreover, Lemma 3.3 also implies that $\rho (g) = \begin {pmatrix}1 & b_g\\ 0 & \omega _p^{k-1}(g)\end {pmatrix}$ for all $g \in G_{\mathbb {Q},p\ell }$ .

So if $f_1,f_2 : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p$ are functions such that $b_g = f_1(g) + \epsilon (f_2(g))$ , then $\omega _p^{1-k}f_1, \omega _p^{1-k}f_2 \in H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k})$ . As $\rho $ is p-ordinary, it follows that $\rho |_{I_p} \simeq 1 \oplus \omega _p^{k-1}$ . Hence, by changing the basis if necessary, we can assume that $b_h=0$ for all $h \in I_p$ .

Thus, we see that $\omega _p^{1-k}f_1, \omega _p^{1-k}f_2 \in \ker (H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k}) \to H^1(I_p,\omega _p^{1-k}))$ . From Lemma 2.4, we know that $\ker (H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k}) \to H^1(I_p,\omega _p^{1-k}))$ is generated by $c_0$ . Hence, there exist $\alpha , \beta \in \mathbb {F}_p$ such that $\omega _p^{1-k}f_1 = \alpha c_0$ and $\omega _p^{1-k}f_2 = \beta c_0$ . Note that $\alpha = 1$ as $\rho \pmod {(\epsilon )} = \bar \rho _{c_0}$ . So conjugating $\rho $ by $\begin {pmatrix} (1+ \epsilon \beta )^{-1} & 0 \\ 0 & 1\end {pmatrix}$ gives us $\bar \rho _{c_0}$ which proves the lemma.

Definition 3.9.

1. (1) Let $I_0$ be the ideal of $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ generated by the set $\{\operatorname {tr}(\rho ^{\text {univ},\ell }(\text {Frob}_q)) - (1+q^{k-1}) \mid q \text { is a prime } \neq p,\ell \}$ . We call it the Eisenstein ideal of $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ .

2. (2) Let $J_0$ be the ideal of $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ generated by the set $\{T(\text {Frob}_q) - (1+q^{k-1}) \mid q \text { is a prime } \neq p,\ell \}$ . We call it the Eisenstein ideal of $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ .

We will now give an upper bound on the index of $I_0$ in $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ in terms of $\nu $ and $v_p(k)$ introduced in §1.6.

Lemma 3.10. The ideal $I_0$ has finite index in $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ and $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )/I_0$ is cyclic of order at most $ p^{\nu +v_p(k)}$ .

Proof. Note that the ideal generated by $\phi (J_0)$ in $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ is $I_0$ where $\phi : R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ is the surjective map obtained in Lemma 3.8. By definition of $J_0$ , it follows that $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) /J_0 \simeq \mathbb {Z}_p$ . Indeed, it is the kernel of the map $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to \mathbb {Z}_p$ induced by the pseudo-representation $(1+\chi _p^{k-1},\chi _p^{k-1})$ . So $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )/I_0$ is a quotient of $\mathbb {Z}_p$ , and hence it is cyclic.

Let $R=R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )/I_0$ and $\rho : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(R)$ be the representation obtained by composing $\rho ^{\text {univ},\ell }$ with the natural surjective map $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell ) \to R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )/I_0$ . Note that $\operatorname {tr}(\rho ) = 1 + \chi _{p}^{k-1}$ and $\det (\rho )=\chi _{p}^{k-1}$ . Using the arguments of Lemma 3.8, we conclude that (after replacing $\rho $ with a representation in its equivalence class if necessary) $\rho = \begin {pmatrix} 1 & * \\ 0 & \chi _{p}^{k-1}\end {pmatrix}$ , where $*$ is nonzero and is unramified at p.

As $\rho $ lifts $\bar \rho _{c_0}$ , it follows, from the definition of $i_{\ell }$ , that $\rho (i_{\ell }) = \begin {pmatrix} 1 & b_{\ell } \\ 0 & 1\end {pmatrix}$ and $b_{\ell } \in (R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )/I_0)^{\times }$ . Let $g_{\ell } \in G_{\mathbb {Q}_{\ell }}$ be a lift of $\text {Frob}_{\ell }$ . As $\rho (g_{\ell }i_{\ell }g_{\ell }^{-1})=\rho (i_{\ell })^{\ell }$ , we get that $(\ell ^{1-k}-\ell )b_{\ell }=0$ . Since $b_{\ell }$ is a unit, we get that $\ell ^k-1=0$ . Hence, it follows that $R/I_0$ is finite (as it is a quotient of $\mathbb {Z}_p$ ) and $|R/I_0| \leq |\mathbb {Z}_p/(\ell ^k-1)\mathbb {Z}_p|$ . Now the highest power of p dividing $\ell ^k-1$ is $\nu +v_p(k)$ , and the lemma follows from this.

We will now give a necessary and sufficient condition for the existence of a reducible, p-ordinary first order deformation of $\bar \rho _{c_0}$ with determinant $\omega _p^{k-1}$ . Before proceeding further, let us develop some notation.

Let $\text {Ad}(\bar \rho _{c_0})$ be the adjoint representation of $\bar \rho _{c_0}$ . So it is the space of $2 \times 2$ matrices over $\mathbb {F}_p$ on which $g \in G_{\mathbb {Q},p\ell }$ acts by conjugation by $\bar \rho _{c_0}(g)$ . Let $\text {Ad}^0(\bar \rho _{c_0})$ be the subspace of trace $0$ matrices of $\text {Ad}(\bar \rho _{c_0})$ and V be the subspace of $\text {Ad}^0(\bar \rho _{c_0})$ given by upper triangular matrices.

It is easy to verify that V is a $G_{\mathbb {Q},p\ell }$ -subrepresentation of $\text {Ad}^0(\bar \rho _{c_0})$ and it is isomorphic to $\bar \rho ^{\prime}_{c_0} := \bar \rho _{c_0} \otimes \omega _p^{1-k}$ . Note that $\text {Ad}^0(\bar \rho _{c_0})/\bar \rho ^{\prime}_{c_0} \cong \omega _p^{k-1}$ . So the natural map $H^1(G_{\mathbb {Q},p\ell }, \bar \rho ^{\prime}_{c_0}) \to H^1(G_{\mathbb {Q},p\ell }, \text {Ad}^0(\bar \rho _{c_0}))$ is injective.

By class field theory, we know that $\dim (\ker (H^1(G_{\mathbb {Q},p\ell },1) \to H^1(I_p,1))) = 1$ . Recall that we have chosen a generator $a_0$ of $\ker (H^1(G_{\mathbb {Q},p\ell },1) \to H^1(I_p,1))$ and have denoted by $c_0 \cup a_0$ the cup product of $c_0$ and $a_0$ . So $c_0 \cup a_0 \in H^2(G_{\mathbb {Q},p\ell }, \omega _p^{1-k})$ .

Let R be an object of $\mathcal {C}$ . We say that a deformation $\rho : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(R)$ of $\bar \rho _{c_0}$ is reducible if there exist characters $\chi _1, \chi _2 : G_{\mathbb {Q},p\ell } \to R^{\times }$ deforming $1$ and $\omega _p^{k-1}$ , respectively, such that $\operatorname {tr}(\rho ) = \chi _1 + \chi _2$ .

We will now prove one of the key lemmas which link first order reducible deformations of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ with the vanishing of the cup product $c_0 \cup a_0$ .

Lemma 3.11. There exists a p-ordinary deformation $\rho : G_{\mathbb {Q},{p\ell }} \to \mathrm {GL}_2(\mathbb {F}_p[\epsilon ]/(\epsilon ^2))$ of $\bar \rho _{c_0}$ with determinant $\omega _p^{k-1}$ which is reducible and not isomorphic to $\bar \rho _{c_0}$ if and only if ${c_0 \cup a_0 =0}$ .

Proof. Let $\rho : G_{\mathbb {Q},{p\ell }} \to \mathrm {GL}_2(\mathbb {F}_p[\epsilon ]/(\epsilon ^2))$ be a deformation of $\bar \rho _{c_0}$ with determinant $\omega _p^{k-1}$ . Let $R=\mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ . From [10, Lemma $3.1$ ], we can assume, by changing the basis if necessary, that $\rho (g_0)=\begin {pmatrix} a_0 &0\\ 0 & d_0\end {pmatrix}$ with $a_0 \pmod {(\epsilon )} =1 $ and $d_0 \pmod {(\epsilon )} = \omega _p^{k-1}(g_0)$ .

Now [2, Lemma $\mathrm{2.4.5}$ ] implies that there exists an ideal $ C \subset R$ such that $R[\rho (G_{\mathbb {Q},p\ell })] = \begin {pmatrix}R & R\\ C & R\end {pmatrix}$ . By Lemma 3.3, it follows that $\rho $ is reducible if and only if $C=0$ and in this case, $\rho \simeq \begin {pmatrix} \chi _1 & *\\ 0 & \chi _2\end {pmatrix}$ , where $\chi _1$ and $\chi _2$ are characters of $G_{\mathbb {Q},p\ell }$ deforming $1$ and $\chi _p^{k-1}$ , respectively.

Note that $\rho $ corresponds to an element of $x \in H^1(G_{\mathbb {Q},p\ell },\text {Ad}^0(\bar \rho _{c_0}))$ . So from the previous paragraph, it follows that $\rho $ is reducible if and only if $x \in H^1(G_{\mathbb {Q},p\ell },\bar \rho ^{\prime}_{c_0}) \subset H^1(G_{\mathbb {Q},p\ell },\text {Ad}^0(\bar \rho _{c_0}))$ . Since $\chi _1$ is a lift of $1$ and $\chi _2$ is a lift of $\omega _p^{k-1}$ , it follows that if $\rho $ is reducible, then $\rho $ is p-ordinary if and only if $\rho |_{I_p} \simeq 1 \oplus \omega _p^{k-1}$ .

So $\rho $ is reducible and p-ordinary if and only if $x \in \ker (H^1(G_{\mathbb {Q},p\ell },\bar \rho ^{\prime}_{c_0}) \to H^1(I_p,\bar \rho ^{\prime}_{c_0}))$ . Thus, there exists a nontrivial, reducible, p-ordinary deformation $\rho : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {F}_p[\epsilon ]/(\epsilon ^2))$ of $\bar \rho _{c_0}$ with determinant $\omega _p^{k-1}$ if and only if $\ker (H^1(G_{\mathbb {Q},p\ell },\bar \rho ^{\prime}_{c_0}) \to H^1(I_p,\bar \rho ^{\prime}_{c_0})) \neq 0$ .

Note that an element x of $H^1(G_{\mathbb {Q},p\ell },\bar \rho ^{\prime}_{c_0})$ gives a representation $\rho ^{\prime}_x : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_3(\mathbb {F}_p)$ such that

$$\begin{align*}\rho_x^\prime(g) = \begin{pmatrix} \omega_p^{1-k}(g) & c_0(g) & F(g) \\ 0 & 1 & b(g) \\ 0 & 0 & 1\end{pmatrix} \text{ for all } g \in G_{\mathbb{Q},p\ell}.\end{align*}$$

Note that $b \in H^1(G_{\mathbb {Q},p\ell },1)$ . So $x \in \ker (H^1(G_{\mathbb {Q},p\ell },\bar \rho ^{\prime}_{c_0}) \to H^1(I_p,\bar \rho ^{\prime}_{c_0}))$ if and only if $b(I_p)=0$ and $F(I_p)=0$ in the corresponding $\rho ^{\prime}_x$ (after changing the basis if necessary). Hence, $\ker (H^1(G_{\mathbb {Q},p\ell },\bar \rho ^{\prime}_{c_0}) \to H^1(I_p,\bar \rho ^{\prime}_{c_0})) \neq 0$ if and only if there exists a representation $\rho ' : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_3(\mathbb {F}_p)$ such that

$$\begin{align*}\rho'(g) = \begin{pmatrix} \omega_p^{1-k}(g) & c_0(g) & F(g) \\ 0 & 1 & a_0(g) \\ 0 & 0 & 1\end{pmatrix} \text{ for all } g \in G_{\mathbb{Q},p\ell} \text{ and } F(I_p)=0.\end{align*}$$

Now if such a $\rho '$ exists, then it is easy to verify that the coboundary of $-F : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p$ is $c_0 \cup a_0$ , and hence $c_0 \cup a_0 =0$ .

On the other hand, suppose $c_0 \cup a_0=0$ and let $F : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p$ be the map such that the coboundary of $-F$ is $c_0 \cup a_0$ . Since $c_0$ is unramified at p and $\omega _p^{1-k}|_{G_{\mathbb {Q}_p}} \neq 1$ , it follows that $c_0 |_{G_{\mathbb {Q}_p}} =0$ . Hence, $F|_{G_{\mathbb {Q}_p}} \in H^1(G_{\mathbb {Q}_p},\omega _p^{1-k})$ .

Since we are assuming that the $\omega _p^{1-k}$ -component of the p-part of the class group of $\mathbb {Q}(\zeta _p)$ is trivial, we know that $\ker (H^1(G_{\mathbb {Q},p},\omega _p^{1-k}) \to H^1(G_{\mathbb {Q}_p},\omega _p^{1-k})) =0$ . From the proof of Lemma 2.4, we know that $\dim (H^1(G_{\mathbb {Q},p},\omega _p^{1-k})) = \dim (H^1(G_{\mathbb {Q}_p},\omega _p^{1-k})) =1$ . So the restriction map $H^1(G_{\mathbb {Q},p} , \omega _p^{1-k}) \to H^1(G_{\mathbb {Q}_p},\omega _p^{1-k})$ is an isomorphism. Hence, we can change F by a suitable element of $H^1(G_{\mathbb {Q},p} , \omega _p^{1-k})$ to assume that $F(I_p)=0$ .

This means that there exists a representation $\rho ' : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_3(\mathbb {F}_p)$ such that $\rho '(g) = \begin {pmatrix} \omega _p^{1-k}(g) & c_0(g) & F(g) \\ 0 & 1 & a_0(g) \\ 0 & 0 & 1\end {pmatrix}$ for all $g \in G_{\mathbb {Q},p\ell }$ and $F(I_p)=0$ . This completes the proof of the lemma.

We say that a pseudo-representation $(t,d) : G_{\mathbb {Q},p\ell } \to R$ arises from a representation if there exists a representation $\rho : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(R)$ such that $\operatorname {tr}(\rho ) =t$ and $\det (\rho ) = d$ .

Lemma 3.12. Suppose $(t,d) : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ is a nontrivial, reducible, p-ordinary, $\ell $ -unipotent pseudo-representation with determinant $\omega _p^{k-1}$ deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ . Then $(t,d)$ arises from a p-ordinary representation deforming $\bar \rho _{c_0}$ if and only if $c_0 \cup a_0=0$ .

Proof. Suppose $\rho : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {F}_p[\epsilon ]/(\epsilon ^2))$ is a p-ordinary deformation of $\bar \rho _{c_0}$ such that $t=\operatorname {tr}(\rho )$ and $d=\det (\rho )$ . As $(t,d)$ is nontrivial, reducible and $d=\omega _p^{k-1}$ , Lemma 3.11 implies that $c_0 \cup a_0 =0$ .

Now suppose $c_0 \cup a_0 =0$ . Then Lemma 3.11 implies that there exists a nontrivial, reducible, p-ordinary deformation $\rho : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {F}_p[\epsilon ]/(\epsilon ^2))$ of $\bar \rho _{c_0}$ with determinant $\omega _p^{k-1}$ . So $\operatorname {tr}(\rho ) = \chi _1 + \chi _2$ , where $\chi _1, \chi _2 : G_{\mathbb {Q},p\ell } \to (\mathbb {F}_p[\epsilon ]/(\epsilon ^2))^{\times }$ are characters deforming $1$ and $\omega _p^{k-1}$ , respectively. Since $\rho $ is p-ordinary, $\chi _1$ is unramified at p. If $g \in I_{\ell }$ , then $\chi _1(g) = 1+a_g\epsilon $ for some $a_g \in \mathbb {F}_p$ . As $d(g)=\chi _1(g)\chi _2(g) =\omega _p^{k-1}(g)=1$ , we have $\chi _2(g) = (1+a_g\epsilon )^{-1}=1 - a_g\epsilon $ . So $\operatorname {tr}(\rho (g))=2$ for all $g \in I_{\ell }$ . Therefore, $(\operatorname {tr}(\rho ),\det (\rho )) : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ is a nontrivial, reducible, p-ordinary, $\ell $ -unipotent pseudo-representation with determinant $\omega _p^{k-1}$ deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ .

From the proof of Lemma 3.4, we know that the space of first order deformations of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ which are reducible, p-ordinary, $\ell $ -unipotent with determinant $\omega _p^{k-1}$ has dimension $1$ . Therefore, we can find a deformation $\rho '$ of $\bar \rho _{c_0}$ in the subspace of the first order deformations of $\bar \rho _{c_0}$ generated by $\rho $ such that $\operatorname {tr}(\rho ) =t$ and $\det (\rho )=d$ . This proves the lemma.

4 Hecke algebras

We will now introduce the Hecke algebras that we will be working with and collect their properties. We will mostly follow [26] in this section. Let $M_k(\ell ,\mathbb {Z}_p)$ be the space of classical modular forms of level $\Gamma _0(\ell )$ and weight k with Fourier coefficients in $\mathbb {Z}_p$ , and let $S_k(\ell ,\mathbb {Z}_p)$ be its submodule of cusp forms. Let $\mathbb {T}$ be the $\mathbb {Z}_p$ -subalgebra of $\text {End}_{\mathbb {Z}_p}(M_k(\ell ,\mathbb {Z}_p))$ generated by the Hecke operators $T_q$ for primes $q \neq \ell $ and the Atkin–Lehner operator $w_{\ell }$ at $\ell $ . Let $\mathbb {T}^0$ be the $\mathbb {Z}_p$ -subalgebra of $\text {End}_{\mathbb {Z}_p}(S_k(\ell ,\mathbb {Z}_p))$ generated by the Hecke operators $T_q$ for primes $q \neq \ell $ and the Atkin–Lehner operator $w_{\ell }$ at $\ell $ .

The restriction of the action of Hecke operators from $M_k(\ell ,\mathbb {Z}_p)$ to $S_k(\ell ,\mathbb {Z}_p)$ gives a surjective morphism $\mathbb {T} \to \mathbb {T}^0$ . Let $I^{\text {eis}}$ be the ideal of $\mathbb {T}$ generated by the set $\{w_{\ell }+1, T_q - (1+q^{k-1}) \mid q \neq \ell \text { is a prime}\}$ . It is easy to verify that $I^{\text {eis}}$ is a prime ideal $\mathbb {T}$ and it corresponds to the Eisenstein series of level $\Gamma _0(\ell )$ and weight k having $w_{\ell }$ -eigenvalue $-1$ (see [26, Section $2$ ] for more details). Let $\mathfrak {m}$ be the ideal of $\mathbb {T}$ generated by p and $I^{\text {eis}}$ . So $\mathfrak {m}$ is a maximal ideal of $\mathbb {T}$ . Denote by $\mathbb {T}_{\mathfrak {m}}$ and $\mathbb {T}^0_{\mathfrak {m}}$ the completion of $\mathbb {T}$ and $\mathbb {T}^0$ at $\mathfrak {m}$ , respectively. It follows from [26] that $\mathbb {T}^0_{\mathfrak {m}}$ is nonzero. Note that the surjective map $\mathbb {T} \to \mathbb {T}^0$ induces a surjective map $F : \mathbb {T}_{\mathfrak {m}} \to \mathbb {T}^0_{\mathfrak {m}}$ .

Let $\mathfrak {m}^0$ be the maximal ideal of $\mathbb {T}^0$ . As $\mathbb {T}$ is a finite $\mathbb {Z}_p$ -module, [13, Corollary 7.6] implies that $\mathbb {T}_{\mathfrak {m}}$ is the localization of $\mathbb {T}$ at $\mathfrak {m}$ and it is a finite $\mathbb {Z}_p$ -module. Similarly, $\mathbb {T}^0_{\mathfrak {m}}$ is the localization of $\mathbb {T}^0$ at $\mathfrak {m}^0$ and it is a finite $\mathbb {Z}_p$ -module. Note that both $\mathbb {T}$ and $\mathbb {T}^0$ are reduced. As $\mathbb {T}_{\mathfrak {m}}$ and $\mathbb {T}^0_{\mathfrak {m}}$ are localizations of $\mathbb {T}$ and $\mathbb {T}^0$ , respectively, we get that both $\mathbb {T}_{\mathfrak {m}}$ and $\mathbb {T}^0_{\mathfrak {m}}$ are reduced.

The residue field of both $\mathbb {T}_{\mathfrak {m}}$ and $\mathbb {T}^0_{\mathfrak {m}}$ is $\mathbb {F}_p$ . Denote the maximal ideals of $\mathbb {T}_{\mathfrak {m}}$ and $\mathbb {T}^0_{\mathfrak {m}}$ by $\mathfrak {m}$ and $\mathfrak {m}^0$ , respectively. By abuse of notation, denote by $I^{\text {eis}}$ the ideal of $\mathbb {T}_{\mathfrak {m}}$ generated by the set $\{w_{\ell }+1, T_q - (1+q^{k-1}) \mid q \neq \ell \text { is a prime}\}$ . We call it the Eisenstein ideal of $\mathbb {T}_{\mathfrak {m}}$ . Denote by $I^{\text {eis},0}$ the ideal of $\mathbb {T}^0_{\mathfrak {m}}$ generated by the set $\{w_{\ell }+1, T_q - (1+q^{k-1}) \mid q \neq \ell \text { is a prime}\}$ . We call it the Eisenstein ideal of $\mathbb {T}^0_{\mathfrak {m}}$ .

We will now collect some of the properties of $\mathbb {T}_{\mathfrak {m}}$ and $\mathbb {T}^0_{\mathfrak {m}}$ . We begin by relating $\mathbb {T}_{\mathfrak {m}}$ with the pseudo-deformation ring $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ introduced in §2.

Lemma 4.1. There exists a pseudo-representation $(\tau _{\ell }, \delta _{\ell }) : G_{\mathbb {Q},p\ell } \to \mathbb {T}_{\mathfrak {m}}$ such that $(\tau _{\ell }, \delta _{\ell })$ is a p-ordinary, Steinberg-or-unramified at $\ell $ deformation of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ with determinant $\chi _p^{k-1}$ and $\tau _{\ell }(\text {Frob}_q) = T_q$ for all primes $q \nmid p\ell $ . The morphism $\phi _{\mathbb {T}} : R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to \mathbb {T}_{\mathfrak {m}}$ induced by $(\tau _{\ell },\delta _{\ell })$ is surjective.

Proof. For $k>2$ , the lemma follows from [26, Section $3.2$ ]. For $k=2$ , the lemma follows from [28, Proposition $\mathrm{4.2.4}$ ]. Indeed, the pseudo-representation $(\tau _{\ell },\delta _{\ell })$ is obtained by gluing the pseudo-representations corresponding to the semi-simple p-adic Galois representations attached to the modular eigenforms of level $\Gamma _0(\ell )$ and weight k lifting $\bar \rho _0$ . Here we say that an eigenform f of level $\Gamma _0(\ell )$ and weight k lifts $\bar \rho _0$ if $a_q(f) \equiv 1+q^{k-1} \pmod {\varpi _f}$ for all primes $q \neq \ell $ , where $a_q(f)$ is the $T_q$ -eigenvalue of f and $\varpi _f$ is a uniformizer of the ring of integers of the finite extension of $\mathbb {Q}_p$ obtained by attaching the Hecke eigenvalues of f to $\mathbb {Q}_p$ .

Suppose f is an eigenform of level $\Gamma _0(\ell )$ and weight k lifting $\bar \rho _0$ and $\rho _f$ is the semi-simple p-adic Galois representation attached to f. Then the $T_p$ -eigenvalue of f is a p-adic unit, and hence $\rho _f$ is p-ordinary. This means that the pseudo-representation attached to $\rho _f$ is also p-ordinary (see §3.4). As f is of level $\Gamma _0(\ell )$ , we know that $\rho _f|_{G_{\mathbb {Q}_{\ell }}}$ is either unramified or Steinberg. Hence, the pseudo-representation attached to $\rho _f$ is Steinberg-or-unramified at $\ell $ (see [29, Observation $\mathrm{1.9.2}$ ]). This proves the desired properties of the pseudo-representation $(\tau _{\ell },\delta _{\ell })$ . The surjectivity of $\phi _{\mathbb {T}}$ can be concluded in the same way as in [28, Proposition $\mathrm{4.2.4}$ ].

We now prove that the $\mathbb {T}^0_{\mathfrak {m}}$ -valued pseudo-representation obtained by composing $(\tau _{\ell },\delta _{\ell })$ with the natural surjective map $F : \mathbb {T}_{\mathfrak {m}} \to \mathbb {T}^0_{\mathfrak {m}}$ arises from an actual representation.

Lemma 4.2. There exists a p-ordinary deformation $\rho _{\mathbb {T}^0} : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {T}^0_{\mathfrak {m}})$ of $\bar \rho _{c_0}$ with determinant $\chi _p^{k-1}$ such that $\operatorname {tr}(\rho _{\mathbb {T}^0}(\text {Frob}_q)) = T_q$ for all primes $q \nmid p\ell $ and $\operatorname {tr}(\rho _{\mathbb {T}^0}(g)) =2$ for all $g \in I_{\ell }$ . The morphism $\phi _{\mathbb {T}^0} : R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell ) \to \mathbb {T}^0_{\mathfrak {m}}$ induced by $\rho _{\mathbb {T}^0}$ is surjective.

Proof. Composing $(\tau _{\ell },\delta _{\ell })$ with the surjective map $F : \mathbb {T}_{\mathfrak {m}} \to \mathbb {T}^0_{\mathfrak {m}}$ gives us a pseudo-representation $(\tau ^0_{\ell },\delta ^0_{\ell }) : G_{\mathbb {Q},p\ell } \to \mathbb {T}^0_{\mathfrak {m}}$ which is a p-ordinary, Steinberg-or-unramified at $\ell $ deformation of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ with determinant $\chi _p^{k-1}$ . Let $A=\begin {pmatrix} \mathbb {T}^0_{\mathfrak {m}} & B \\ C & \mathbb {T}^0_{\mathfrak {m}} \end {pmatrix}$ be the GMA over $\mathbb {T}^0_{\mathfrak {m}}$ and $\rho : G_{\mathbb {Q},p\ell } \to A^{\times }$ be the representation associated to $(\tau ^0_{\ell },\delta ^0_{\ell })$ by Lemma 3.2.

Note that $\mathbb {T}^0_{\mathfrak {m}}$ is reduced. Let $K_0$ be the total fraction ring of $\mathbb {T}^0_{\mathfrak {m}}$ . So, by [4, Proposition $\mathrm{1.3.12}$ ], we can assume that B and C are fractional ideals of $K_0$ and the multiplication between B and C is given by the multiplication in $K_0$ . If $I=BC$ , then $\tau ^0_{\ell } \pmod {I}$ is a sum of two characters. But the minimal primes of $\mathbb {T}^0_{\mathfrak {m}}$ correspond to cuspidal eigenforms f of level $\Gamma _0(\ell )$ and weight k lifting $\bar \rho _0$ . Hence, if P is a minimal prime of $\mathbb {T}^0_{\mathfrak {m}}$ , then $(\tau ^0_{\ell } \pmod {P},\delta ^0_{\ell }\pmod {P})$ is not reducible as it is the pseudo-representation corresponding to the p-adic Galois representation attached to the cuspidal eigenform corresponding to P. So I is not contained in any minimal prime of $\mathbb {T}^0_{\mathfrak {m}}$ . So $B \neq 0$ .

If $\alpha B =0$ , then we have $\alpha I=0$ . So $\alpha $ should be in every minimal prime of $\mathbb {T}^0_{\mathfrak {m}}$ . As $\mathbb {T}^0_{\mathfrak {m}}$ is reduced, it means that $\alpha = 0$ . Now if $\rho (i_{\ell }) = \begin {pmatrix} 1+x & b_{\ell } \\ c_{\ell } & 1-x\end {pmatrix}$ , then part (4) of Lemma 3.2 implies that B is generated by $b_{\ell }$ over $\mathbb {T}^0_{\mathfrak {m}}$ . Hence, B is a free $\mathbb {T}^0_{\mathfrak {m}}$ -module of rank $1$ generated by $b_{\ell }$ over $\mathbb {T}^0_{\mathfrak {m}}$ . As I is not contained in any of the minimal primes of $\mathbb {T}^0_{\mathfrak {m}}$ , it follows that $b_{\ell } \in K_0^{\times }$ .

So $\begin {pmatrix} b_{\ell }^{-1} & 0\\ 0 & 1\end {pmatrix} A \begin {pmatrix} b_{\ell } & 0\\ 0 & 1\end {pmatrix} = \begin {pmatrix}\mathbb {T}^0_{\mathfrak {m}} & \mathbb {T}^0_{\mathfrak {m}}\\ I & \mathbb {T}^0_{\mathfrak {m}}\end {pmatrix}$ . Hence, conjugating $\rho $ by $\begin {pmatrix} b_{\ell }^{-1} & 0\\ 0 & 1\end {pmatrix}$ gives us a representation $\rho ' : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {T}^0_{\mathfrak {m}})$ . As $\operatorname {tr}(\rho )=\tau ^0_{\ell }$ , it follows that $\operatorname {tr}(\rho ')=\tau ^0_{\ell }$ . Hence, we get $\operatorname {tr}(\rho '(g))=2$ for all $g \in I_{\ell }$ . From part (6) of Lemma 3.2, it follows that $\rho (h) = \begin {pmatrix} 1 & 0\\ c_h & \chi _p^{k-1}(h)\end {pmatrix}$ for all $h \in I_p$ . Hence, $\rho '$ is p-ordinary as it is a conjugate of $\rho $ by a diagonal matrix.

Therefore, $\rho ' \pmod {\mathfrak {m}^0} = \begin {pmatrix}1 & *\\ 0 & \omega _p^{k-1}\end {pmatrix}$ where $*$ is nonzero and unramified at p (i.e., $\rho '\pmod {\mathfrak {m}^0}$ arises from an element of $\ker (H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k}) \to H^1(G_{\mathbb {Q}_p},\omega _p^{1-k}))$ ). By Lemma 2.4, it follows that $\ker (H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k}) \to H^1(G_{\mathbb {Q}_p},\omega _p^{1-k}))$ is generated by $c_0$ . Hence, conjugating $\rho '$ with a suitable diagonal matrix gives us the representation $\rho _{\mathbb {T}^0} : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(\mathbb {T}^0_{\mathfrak {m}})$ satisfying the statement of the lemma.

The existence of $\rho _{\mathbb {T}^0}$ implies that the map $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to \mathbb {T}^0_{\mathfrak {m}}$ induced by $(\tau ^0_{\ell },\delta ^0_{\ell })$ factors through $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ to give the map $\phi _{\mathbb {T}^0}$ induced by $\rho _{\mathbb {T}^0}$ . Hence, the surjection of $\phi _{\mathbb {T}^0}$ follows from Lemma 4.1.

We now show that the space of first order deformations of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ arising from $\mathbb {T}_{\mathfrak {m}}$ always contains reducible deformations.

Lemma 4.3. If $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p))) = 1$ , then $I^{\text {eis}}$ is principal and the pseudo-representation $(\tau _{\ell } \pmod {(p,\mathfrak {m}^2)},\delta _{\ell } \pmod {(p,\mathfrak {m}^2)}) : G_{\mathbb {Q},p\ell } \to \mathbb {T}_{\mathfrak {m}}/(p,\mathfrak {m}^2)$ is reducible.

Proof. If $\dim (\text {tan}({\mathbb {T}}_{\mathfrak {m}}/(p))) = 1$ , then ${\mathbb {T}}_{\mathfrak {m}}$ is a quotient of . As ${\mathbb {T}}_{\mathfrak {m}}/I^{\text {eis}} \simeq {\mathbb {Z}}_p$ , it follows that $I^{\text {eis}}$ is principal. For $k> 2$ , the reducibility of $(\tau _{\ell } \pmod {(p,{\mathfrak {m}}^2)},\delta _{\ell } \pmod {(p,{\mathfrak {m}}^2)})$ follows from [26, Theorem $\mathrm{5.1.1}$ ].

Suppose $k=2$ . Then the lemma follows from work of Calegari and Emerton (by combining [6, Proposition 3.12] and [6, Proposition 5.5]). However, we will give a different proof here as we are not using the deformation conditions studied by them. It follows from [19] that $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p))) = 1$ and $I^{\text {eis}}$ is principal. So $\mathbb {T}_{\mathfrak {m}}/(p,\mathfrak {m}^2) \simeq \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ . Denote $\mathbb {T}_{\mathfrak {m}}/(p,\mathfrak {m}^2)$ by R and $(\tau _{\ell } \pmod {(p,\mathfrak {m}^2)},\delta _{\ell } \pmod {(p,\mathfrak {m}^2)})$ by $(t,d)$ . Suppose $(t,d)$ is not reducible.

Let A be the faithful GMA over R and $\rho : G_{\mathbb {Q},p\ell } \to A^{\times }$ be the representation associated to $(t,d)$ by Lemma 3.2. From the proof of Lemma 3.6, it follows that there exist nonzero constants $\alpha , \beta , \gamma \in \mathbb {F}_p$ such that if $\rho (g) = \begin {pmatrix} a_g & b_g \\ c_g & d_g \end {pmatrix}$ , then $b_gc_g=\alpha \beta \gamma \omega _p^{k-1}(g)c_0(g)b_0(g)\epsilon $ for all $g \in G_{\mathbb {Q},p\ell }$ . Let $K_{c_0}$ be the extension of $\mathbb {Q}$ fixed by the kernel of the representation $\begin {pmatrix} \omega _p^{1-k} & *\\ 0 & 1\end {pmatrix}$ defined by $c_0$ and $K_{b_0}$ be the extension of $\mathbb {Q}$ fixed by the kernel of the representation $\begin {pmatrix} \omega _p^{k-1} & *\\ 0 & 1\end {pmatrix}$ defined by $b_0$ .

By Chebotarev density theorem, there exists a prime q such that $q \nmid p\ell $ , $p \mid q-1$ and q is not totally split in both $K_{c_0}$ and $K_{b_0}$ . This means that $c_0(\text {Frob}_q) \neq 0$ and $b_0(\text {Frob}_q) \neq 0$ . So if $\rho (\text {Frob}_q) = \begin {pmatrix} a & b\\ c & d\end {pmatrix}$ , then $bc \neq 0$ . Now $a = 1+x\epsilon $ , $d = 1 +y\epsilon $ with $x,y \in \mathbb {F}_p$ and $\det (\rho (\text {Frob}_q))=1$ . Hence, it follows that $x+y \neq 0$ . Therefore, $\operatorname {tr}(\rho (\text {Frob}_q)) - q-1 \neq 0$ , and hence it generates the cotangent space of R. The image of $T_q$ under the surjective map $\mathbb {T}_{\mathfrak {m}} \to R$ is $\operatorname {tr}(\rho (\text {Frob}_q))$ . Hence, it follows that p and $T_q-q-1$ generate the maximal ideal $\mathfrak {m}$ of $\mathbb {T}_{\mathfrak {m}}$ . Note that $T_q-q-1 \in I^{\text {eis}}$ and $\mathbb {T}_{\mathfrak {m}}/(p, T_q-q-1) \simeq \mathbb {F}_p$ . As $\mathbb {T}_{\mathfrak {m}}/I^{\text {eis}} \simeq \mathbb {Z}_p$ , we get that $T_q-q-1$ generates $I^{\text {eis}}$ . Since q is not a nice prime in the sense of Mazur ([19]), [19, Proposition II. $16.1$ ] gives a contradiction. Hence, $(t,d)$ is reducible which implies the lemma.

We will now briefly review modular forms modulo p as they form a crucial ingredient of the proof of Theorem 5.6 and Corollary A.

Let $i> 0$ be an even integer and $M_i(\ell ,\mathbb {Z}_p)$ be the space of classical modular forms of level $\Gamma _0(\ell )$ and weight i with Fourier coefficients in $\mathbb {Z}_p$ . Using the q-expansion principle, we identify $M_i(\ell ,\mathbb {Z}_p)$ with a submodule of . Let $M_i(\ell ,\mathbb {F}_p)$ be the image of $M_i(\ell ,\mathbb {Z}_p)$ under the natural surjective map obtained by reducing the coefficients of power series modulo p. So $M_i(\ell ,\mathbb {F}_p)$ is the space of modular forms modulo p of weight i and level $\Gamma _0(\ell )$ (in the sense of Serre and Swinnerton-Dyer).

Let $\mathbb {T}_i$ be the $\mathbb {Z}_p$ -subalgebra of $\text {End}_{\mathbb {Z}_p}(M_i(\ell ,\mathbb {Z}_p))$ generated by the Hecke operators $T_q$ for primes $q \neq \ell $ and the Atkin–Lehner operator $w_{\ell }$ at $\ell $ . So, under the notation developed above, we have $\mathbb {T}_k = \mathbb {T}$ . Let $\mathfrak {n}$ be a maximal ideal of $\mathbb {T}_i$ , and let $(\mathbb {T}_i)_{\mathfrak {n}}$ be the completion of $\mathbb {T}_i$ at $\mathfrak {n}$ . As $\mathbb {T}_i$ is a finite $\mathbb {Z}_p$ -module, [13, Corollary 7.6] implies that $(\mathbb {T}_i)_{\mathfrak {n}}$ is the localization of $\mathbb {T}_i$ at $\mathfrak {n}$ and $\mathbb {T}_i = (\mathbb {T}_i)_{\mathfrak {n}} \times S$ , where S is the product of localizations of $\mathbb {T}_i$ at maximal ideals other than $\mathfrak {n}$ .

Let $M_i(\ell ,\mathbb {Z}_p)_{\mathfrak {n}}$ be the localization of $M_i(\ell ,\mathbb {Z}_p)$ at $\mathfrak {n}$ . From the product decomposition of $\mathbb {T}_i$ given in the previous paragraph, we conclude that $M_i(\ell ,\mathbb {Z}_p)_{\mathfrak {n}}$ is a submodule of $M_i(\ell ,\mathbb {Z}_p)$ and moreover, it is a direct summand of $M_i(\ell ,\mathbb {Z}_p)$ . Note that $(\mathbb {T}_i)_{\mathfrak {n}}$ is the largest quotient of $\mathbb {T}_i$ acting faithfully on $M_i(\ell ,\mathbb {Z}_p)_{\mathfrak {n}}$ .

The action of $\mathbb {T}_i$ on $M_i(\ell ,\mathbb {Z}_p)$ also gives an action of $\mathbb {T}_i$ on $M_i(\ell ,\mathbb {F}_p)$ and this action factors through $\mathbb {T}_i/(p)$ . Let $M_i(\ell ,\mathbb {F}_p)_{\mathfrak {n}}$ be the localization of $M_i(\ell ,\mathbb {F}_p)$ at the maximal ideal $\mathfrak {n}$ . As $\mathbb {T}_i/(p)$ is Artinian, it follows that $M_i(\ell ,\mathbb {F}_p)_{\mathfrak {n}}$ is the submodule of $M_i(\ell ,\mathbb {F}_p)$ consisting of generalized eigenvectors corresponding to the system of eigenvalues defined by $\mathfrak {n}$ . In other words, $M_i(\ell ,\mathbb {F}_p)_{\mathfrak {n}} = \{ f \in M_i(\ell ,\mathbb {F}_p) \mid \mathfrak {n}^k.f=0 \text { for some } k> 0 \}$ .

Under the mod p reduction map $M_i(\ell ,\mathbb {Z}_p) \to M_i(\ell ,\mathbb {F}_p)$ , $M_i(\ell ,\mathbb {Z}_p)_{\mathfrak {n}}$ gets mapped onto $M_i(\ell ,\mathbb {F}_p)_{\mathfrak {n}}$ . Thus the action of $\mathbb {T}_i$ on $M_i(\ell ,\mathbb {F}_p)_{\mathfrak {n}}$ factors through $(\mathbb {T}_i)_{\mathfrak {n}}/(p)$ .

Lemma 4.4. If $(p-1) \nmid i$ , then $(\mathbb {T}_i)_{\mathfrak {n}}/(p)$ is the largest quotient of $(\mathbb {T}_i)_{\mathfrak {n}}$ acting faithfully on $M_i(\ell ,\mathbb {F}_p)_{\mathfrak {n}}$ .

Proof. As p is odd and $w_{\ell }$ is an involution, it follows that the image of $w_{\ell }$ in $(\mathbb {T}_i)_{\mathfrak {n}}$ is either $1$ or $-1$ . Since $i> 0$ , we get, from [22, Corollary 2.1.4], a perfect pairing

$$\begin{align*}G: (\mathbb{T}_i)_{\mathfrak{n}} \times M_i(\ell,\mathbb{Z}_p)_{\mathfrak{n}} \to \mathbb{Z}_p\end{align*}$$

which sends $(T,f)$ to $a_1(Tf)$ , where $a_1(Tf)$ is the coefficient of q in the q-expansion of $Tf$ .

Let $\overline {\mathbb {T}_i}$ be the largest quotient of $(\mathbb {T}_i)_{\mathfrak {n}}$ acting faithfully on $M_i(\ell ,\mathbb {F}_p)_{\mathfrak {n}}$ . As $(p-1) \nmid i$ , we know that no nonzero modular form in $M_i(\ell ,\mathbb {F}_p)$ has constant Fourier expansion i.e., (see the discussion on Page $459$ of [14] for more details). Hence, by applying [22, Corollary 2.1.4] again, we get that the map

$$\begin{align*}\bar{G}: \overline{\mathbb{T}_i} \times M_i(\ell,\mathbb{F}_p)_{\mathfrak{n}} \to \mathbb{F}_p\end{align*}$$

which sends $(T,\bar {f})$ to $a_1(T\bar {f})$ is a perfect pairing.

As G is a perfect pairing, we get that the $\mathbb {Z}_p$ -ranks of $(\mathbb {T}_i)_{\mathfrak {n}}$ and $M_i(\ell ,\mathbb {Z}_p)_{\mathfrak {n}}$ are equal. Recall that $M_i(\ell ,\mathbb {F}_p)_{\mathfrak {n}}$ is the reduction of $M_i(\ell ,\mathbb {Z}_p)_{\mathfrak {n}}$ modulo p. So, we conclude that the $\mathbb {F}_p$ -dimension of $M_i(\ell ,\mathbb {F}_p)_{\mathfrak {n}}$ is same as the $\mathbb {Z}_p$ -rank of $M_i(\ell ,\mathbb {Z}_p)_{\mathfrak {n}}$ . Since $\bar {G}$ is a perfect pairing, the $\mathbb {F}_p$ -dimensions of $\overline {\mathbb {T}_i}$ and $M_i(\ell ,\mathbb {F}_p)_{\mathfrak {n}}$ are the same. Therefore, the $\mathbb {F}_p$ -dimensions of $\overline {\mathbb {T}_i}$ and $(\mathbb {T}_i)_{\mathfrak {n}}/(p)$ are equal. As $(\mathbb {T}_i)_{\mathfrak {n}}/(p)$ surjects onto $\overline {\mathbb {T}_i}$ , we infer that $(\mathbb {T}_i)_{\mathfrak {n}}/(p) \simeq \overline {\mathbb {T}_i}$ .

We will now relate the principality of $I^{\text {eis},0}$ with that of $I^{\text {eis}}$ . This result will be used in the proof of Theorem 5.5. Recall that $F : \mathbb {T}_{\mathfrak {m}} \to \mathbb {T}^0_{\mathfrak {m}}$ is the map induced from the natural surjective map $\mathbb {T} \to \mathbb {T}^0$ .

Lemma 4.5. $I^{\text {eis}}$ is principal if and only if $I^{\text {eis},0}$ is principal.

Proof. Since $I^{\text {eis},0}$ is the ideal generated by $F(I^{\text {eis}})$ , it follows that $I^{\text {eis},0}$ is principal if $I^{\text {eis}}$ is principal.

Now suppose $I^{\text {eis},0}$ is principal. From Lemma 3.4 and Lemma 4.1, we know that there exists a surjective map . Moreover, we can choose this map so that $f(x), f(y) \in I^{\text {eis}}$ , and hence $I^{\text {eis}} = (f(x),f(y))$ . As $I^{\text {eis},0}$ is principal and F is surjective, we get, using Nakayama’s lemma, that $I^{\text {eis}}$ is either $(F(f(x)))$ or $(F(f(y)))$ . Hence, $\ker (F)$ contains either $f(y)-rf(x)$ for some $r \in \mathbb {T}_{\mathfrak {m}}$ or $f(x)-r'f(y)$ for some $r' \in \mathbb {T}_{\mathfrak {m}}$ .

Suppose $f(y)-rf(x) \in \ker (F)$ for some $r \in \mathbb {T}_{\mathfrak {m}}$ . Recall that $f(y)-rf(x) \in I^{\text {eis}}$ . Suppose the $\mathbb {Z}_p$ -rank of $M_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ is d. Note that the Eisenstein subspace $E_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ of $M_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ has $\mathbb {Z}_p$ -rank $1$ and the cuspidal subspace $S_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ of $M_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ has $\mathbb {Z}_p$ -rank $d-1$ (see [26, Section 2.2] for more details).

Now $E_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}} \cap S_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}} = \{0\}$ . Hence, if $g \in M_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ , then there exist $g' \in E_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ , $h \in S_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ and an integer $n \geq 0$ such that $g =\dfrac {g'-h}{p^n}$ . Thus, if $\sigma \in I^{\text {eis}}$ , then $\sigma (g')=0$ , $\sigma (h) \in S_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ and $\sigma (g) \in M_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ . Therefore, we conclude that $\sigma (g) \in S_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ , and hence $\sigma .M_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}} \subset S_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ .

Since $f(y)-rf(x) \in I^{\text {eis}} \cap \ker (F)$ , $(f(y)-rf(x))^2.M_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}=0$ i.e. $(f(y)-rf(x))^2=0$ . Recall that $\mathbb {T}_{\mathfrak {m}}$ is reduced. Therefore, we get that $f(y)-rf(x)=0$ . As $I^{\text {eis}}=(f(x),f(y))$ , we conclude that $I^{\text {eis}}$ is principal.

Using the same argument as above, we get that if $f(x)-r'f(y) \in \ker (F)$ for some $r' \in \mathbb {T}_{\mathfrak {m}}$ , then $I^{\text {eis}}$ is a principal ideal. This finishes the proof of the lemma.

We will now consider $I^{\text {eis}}$ as a $\mathbb {T}_{\mathfrak {m}}$ -module and determine its annihilator. This result will be crucially used in the proofs of Theorem 5.5 and Part (1) of Theorem B.

Lemma 4.6. The annihilator of the $\mathbb {T}_{\mathfrak {m}}$ -module $I^{\text {eis}}$ is $\ker (F)$ .

Proof. Suppose $\alpha \in \ker (F)$ and $\beta \in I^{\text {eis}}$ . Recall, from the proof of Lemma 4.5, that $\beta M_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}} \subset S_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}}$ . So $\alpha \beta M_k(\ell ,\mathbb {Z}_p)_{\mathfrak {m}} =0$ . Therefore, it follows that $\alpha \beta =0$ for all $\beta \in I^{\text {eis}}$ and $\alpha \in \ker (F)$ .

On the other hand, suppose $\alpha \in \mathbb {T}_{\mathfrak {m}}$ and $\alpha I^{\text {eis}}=0$ . As $\mathbb {T}_{\mathfrak {m}}$ is reduced, $\alpha \not \in I^{\text {eis}}$ . Since $\mathbb {T}_{\mathfrak {m}}$ has Krull dimension $1$ and $\mathbb {T}_{\mathfrak {m}}/I^{\text {eis}} \simeq \mathbb {Z}_p$ , $I^{\text {eis}}$ is a minimal prime ideal of $\mathbb {T}_{\mathfrak {m}}$ . Let $\mathcal {S}$ be the set of all minimal prime ideals of $\mathbb {T}_{\mathfrak {m}}$ which are different from $I^{\text {eis}}$ . As $\mathbb {T}^0_{\mathfrak {m}}$ is nonzero, $\mathcal {S}$ is nonempty. So, if $P \in \mathcal {S}$ , then $I^{\text {eis}} \not \subset P$ , and hence $\alpha \in P$ . Thus, $\alpha \in \cap _{P \in \mathcal {S}} P$ .

Now we have $I^{\text {eis}}\ker (F)=0$ . Therefore, if $P \in \mathcal {S}$ , then $\ker (F) \subset P$ . As $\mathbb {T}_{\mathfrak {m}}$ is a local ring of Krull dimension $1$ , $\mathcal {S}$ is the set of all primes of $\mathbb {T}_{\mathfrak {m}}$ which are minimal over $\ker (F)$ . Since $\mathbb {T}^0_{\mathfrak {m}} \simeq \mathbb {T}_{\mathfrak {m}}/\ker (F)$ is reduced, we conclude that $\cap _{P \in \mathcal {S}} P =\ker (F)$ . Therefore, we get that $\alpha \in \ker (F)$ .

5 Main results

We are now ready to prove our main results. We know, from [19] and [26], that $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p))) \geq 1$ .

Theorem 5.1. Suppose $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p))) = 1$ . Then $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}}) = 1$ if and only if $c_0 \cup a_0 \neq 0$ .

Proof. As $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p))) = 1$ , Lemma 4.3 implies that $I^{\text {eis}}$ is principal. Let $x_0 \in \mathbb {T}_{\mathfrak {m}}$ be a generator of $I^{\text {eis}}$ . Then $\mathfrak {m} = (p,x_0)$ and $(p,\mathfrak {m}^2)=(p,x_0^2)$ . So $\mathbb {T}_{\mathfrak {m}}/(p,\mathfrak {m}^2) = \mathbb {T}_{\mathfrak {m}}/(p,x_0^2) \simeq \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ . Let $f : \mathbb {T}_{\mathfrak {m}} \to \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ be the map obtained by composing the isomorphism obtained above with the natural surjective map $\mathbb {T}_{\mathfrak {m}} \to \mathbb {T}_{\mathfrak {m}}/(p,\mathfrak {m}^2)$ . Now Lemma 4.3 also implies that the pseudo-representation $(t,d) : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ obtained by composing $(\tau _{\ell },\delta _{\ell })$ with the surjective map $f : \mathbb {T}_{\mathfrak {m}} \to \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ is reducible.

If $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}})> 1$ , then $\dim (\text {tan}(\mathbb {T}^0_{\mathfrak {m}}/(p))) \geq 1$ . As $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p))) = 1$ , we get that $\dim (\text {tan}(\mathbb {T}^0_{\mathfrak {m}}/(p))) = 1$ . Therefore, the map f factors through $\mathbb {T}^0_{\mathfrak {m}}$ . Thus Lemma 4.2 implies that $(t,d)$ arises from a nontrivial first order p-ordinary deformation of $\bar \rho _{c_0}$ with determinant $\omega _p^{k-1}$ . Hence, Lemma 3.11 implies that $c_0 \cup a_0=0$ .

Now suppose $c_0 \cup a_0 =0$ . Recall that $\phi : R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ is the surjective morphism induced by $(\operatorname {tr}(\rho ^{\text {univ},\ell }), \det (\rho ^{\text {univ},\ell }))$ . As $(t,d)$ is reducible, Lemma 3.12 implies that there exists a map $f' : R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell ) \to \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ such that the following diagram commutes:

Hence, $\phi _{\mathbb {T}}(\ker (\phi )) \subset (p,x_0^2)$ .

On the other hand, Lemma 4.2 implies that the following diagram commutes:

So $\phi _{\mathbb {T}}(\ker (\phi )) \subset \ker (F)$ .

Now suppose $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}}) = 1$ . As $I^{\text {eis}}=(x_0)$ , [26, Theorem $\mathrm{5.1.2}$ ] and [19, Proposition II. $9.6$ ] imply that $F(x_0)=p^{\nu +v_p(k)}.u$ for some $u \in \mathbb {Z}_p^{\times }$ (see [26, Remark $\mathrm{5.1.3}$ ] for more details). Since the image of $x_0$ in $\mathbb {T}_{\mathfrak {m}}/(p)$ generates its cotangent space, it follows that $\ker (F) = (x_0-p^{\nu +v_p(k)}.u)$ .

Note that $J_0$ , which is the Eisenstein ideal of $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ , is the inverse image of $I^{\text {eis}}$ under the surjective map $\phi _{\mathbb {T}}$ . Then $I_0$ is generated by $\phi (J_0)$ , and Lemma 3.10 implies that there is a $y \in \ker (\phi )$ such that $y = p^e +y_0$ with $y_0 \in J_0$ and $e \leq \nu +v_p(k)$ . So $\phi _{\mathbb {T}}(y) = p^e + \phi _{\mathbb {T}}(y_0) \in \ker (F)$ . As $y_0 \in J_0$ , $\phi _{\mathbb {T}}(y_0) \in (x_0)$ , and hence $\phi _{\mathbb {T}}(y_0)=x_0\alpha $ for some $\alpha \in \mathbb {T}_{\mathfrak {m}}$ . So $p^e + x_0\alpha \in (x_0-p^{\nu +v_p(k)}.u)$ . As $\mathbb {T}_{\mathfrak {m}}/(x_0) \simeq \mathbb {Z}_p$ , it follows that $p^e+x_0\alpha = (x_0-p^{\nu +v_p(k)}.u)(x_0\beta + p^{e'}u')$ for some $e' \geq 0$ , $u' \in \mathbb {Z}_p^{\times }$ and $\beta \in \mathbb {T}_{\mathfrak {m}}$ , and hence $p^e = -uu'p^{\nu +v_p(k)+e'}$ .

Since $e \leq \nu + v_p(k)$ , we get that $e'=0$ , and hence $p^{e'}u'+x_0\beta \in \mathbb {T}_{\mathfrak {m}}^{\times }$ . Therefore, we conclude that $\mathfrak {m} = (p, p^e+x_0\alpha )$ . But $p^e+x_0\alpha \in \phi _{\mathbb {T}}(\ker (\phi )) \subset (p,x_0^2)$ . Hence, we see that p generates the maximal ideal of $\mathbb {T}_{\mathfrak {m}}$ which is a contradiction as $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p)))=1$ . Therefore, we conclude that if $c_0 \cup a_0=0$ , then $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}})> 1$ . This finishes the proof of the theorem.

Corollary 5.2. If $k=2$ , then $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}}) = 1$ if and only if $c_0 \cup a_0 \neq 0$ .

Proof. If $k=2$ , then [19, Proposition II. $16.6$ ] implies that $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p))) = 1$ . So the corollary follows from Theorem 5.1.

Corollary 5.3. If $k>2$ and $c_0 \cup b_0 \neq 0$ , then $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}}) = 1$ if and only if $c_0 \cup a_0 \neq 0$ .

Proof. If $k>2$ and $c_0 \cup b_0 \neq 0$ , then Lemma 3.4, Lemma 3.6 and Lemma 4.1 together imply that $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p))) = 1$ . Hence, the corollary follows from Theorem 5.1.

Let $\xi ^{\prime}_{\text {MT}} \in \mathbb {F}_p$ be the derivative of the Mazur–Tate $\zeta $ function defined by Wake in [26, Section $\mathrm{1.2.2}$ ].

Corollary 5.4. If $c_0 \cup b_0 \neq 0$ , then $\xi ^{\prime}_{\text {MT}} \neq 0$ if and only if $c_0 \cup a_0 \neq 0$ .

Proof. If $c_o \cup b_0 \neq 0$ , then we know that $I^{\text {eis}}$ is principal. Hence, the corollary follows from Corollary 5.3 and [26, Theorem $\mathrm{1.2.4}$ ].

Note that the statement of Corollary 5.4 is purely elementary but its proof is not elementary. We are not aware of any direct proof of this corollary.

We now move on to the case of $c_0 \cup b_0 =0$ .

Theorem 5.5. If $c_0 \cup b_0 = 0$ and $p \mid k$ , then $I^{\text {eis},0}$ is not principal and $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}})> 1$ .

Proof. If $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}}) = 1$ , then every ideal of $\mathbb {T}^0_{\mathfrak {m}}$ is principal, and hence $I^{\text {eis},0}$ is principal. So it suffices to prove that $I^{\text {eis},0}$ is not principal to prove the theorem. Suppose $c_0 \cup b_0=0$ , $p \mid k$ and $I^{\text {eis},0}$ is principal. By Lemma 4.5, we get that $I^{\text {eis}}$ is principal. Since $\mathbb {T}_{\mathfrak {m}}/I^{\text {eis}} \simeq \mathbb {Z}_p$ , it follows that $\mathbb {T}_{\mathfrak {m}}$ is a quotient of , and hence $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p)))=1$ .

Let $A=\begin {pmatrix} \mathbb {T}_{\mathfrak {m}} & B \\ C & \mathbb {T}_{\mathfrak {m}}\end {pmatrix}$ be the faithful GMA over $\mathbb {T}_{\mathfrak {m}}$ and $\rho : G_{\mathbb {Q},p\ell } \to A^{\times }$ be the representation attached to $(\tau _{\ell },\delta _{\ell })$ by Lemma 3.2. By Lemma 4.3, we know that the first order deformation of $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ arising from $\mathbb {T}_{\mathfrak {m}}$ is reducible. So Lemma 3.5 implies that $\mathfrak {m}$ is generated by p and x, where $\rho (i_{\ell }) = \begin {pmatrix} 1+x & b_{\ell } \\ c_{\ell } & 1-x\end {pmatrix}$ . As $\det (\rho (i_{\ell }))=1$ , we get that $b_{\ell }c_{\ell } = -x^2$ .

Let $g_{\ell }$ be a lift of $\text {Frob}_{\ell }$ in $G_{\mathbb {Q}_{\ell }}$ and suppose $\rho (g_{\ell }) = \begin {pmatrix} a & b \\ c & d\end {pmatrix}$ . Now $\rho (g_{\ell }i_{\ell }g_{\ell }^{-1})=\rho (i_{\ell })^{\ell }$ . As $b_{\ell }c_{\ell } = -x^2$ , it follows that $\rho (i_{\ell })^{\ell } = \begin {pmatrix} 1+\ell x & \ell b_{\ell } \\ \ell c_{\ell } & 1-\ell x\end {pmatrix} $ . So we have

$$\begin{align*}\begin{pmatrix} a & b \\ c & d\end{pmatrix}\begin{pmatrix} 1+x & b_{\ell} \\ c_{\ell} & 1-x\end{pmatrix} = \begin{pmatrix} 1+\ell x & \ell b_{\ell} \\ \ell c_{\ell} & 1-\ell x\end{pmatrix} \begin{pmatrix} a & b \\ c & d\end{pmatrix}.\end{align*}$$

Thus $b_{\ell }c+d(1-x) =\ell bc_{\ell }+ d(1-\ell x)$ . Now part (4) of Lemma 3.2 implies that $B=\mathbb {T}_{\mathfrak {m}}b_{\ell }$ and so $b=rb_{\ell }$ for some $r \in \mathbb {T}_{\mathfrak {m}}$ . Since $b_{\ell }c_{\ell }=-x^2$ , there exists an $r' \in \mathbb {T}_{\mathfrak {m}}$ such that

(4) $$ \begin{align} b_{\ell}c = dx(1-\ell)+x^2r'.\end{align} $$

Let $I=BC$ . So, by Lemma 3.3, we know that $\operatorname {tr}(\rho )\pmod {I}$ is reducible. Moreover, if $\rho (g) = \begin {pmatrix} a_g & b_g \\ c_g & d_g\end {pmatrix}$ for $g \in G_{\mathbb {Q},p\ell }$ , then the map $\chi _1 : G_{\mathbb {Q},p\ell } \to (\mathbb {T}_{\mathfrak {m}}/I)^{\times }$ sending g to $a_g \pmod {I}$ is a character of $G_{\mathbb {Q},p\ell }$ lifting the trivial character. Recall, from Part (4) of Lemma 3.2, that the representation $\rho $ is tamely ramified at $\ell $ . So, the character $\chi _1$ is also tamely ramified at $\ell $ . Therefore, by the Kronecker–Weber theorem, we get that the order of $\chi _1(I_{\ell })$ divides $\ell -1$ . Hence, $\chi _1(i_{\ell })^{\ell -1}=1$ which means $(1+x)^{\ell -1} \pmod {I}=1$ (i.e., $(1+x)^{\ell -1}-1 \in I$ ).

We know that $x^2 \in I$ . Since $(1+x)^{\ell -1}-1 \in I$ , we get that $p^{\nu }x \in I$ . On the other hand, Lemma 3.3 and [26, Theorem $\mathrm{5.1.1}$ ] imply that . Since we have already seen that $\mathfrak {m} = (p,x)$ , there is a surjective morphism sending X to x. Hence, by combining all this, we get that $I=(x^2,p^{\nu }x)$ .

Now as we are assuming $c_0 \cup b_0 = 0$ , the proof of Lemma 3.7 implies that $b_0|_{G_{\mathbb {Q}_{\ell }}} = 0$ .

Let $C'=C/\mathbb {T}_{\mathfrak {m}}c_{\ell }$ . Then, following the proof of part (4) of Lemma 3.2, we get an injective map $\psi : \text {Hom}(C'/\mathfrak {m} C', \mathbb {F}_p) \to H^1(G_{\mathbb {Q},p},\omega _p^{k-1})$ . So if its image is nonzero, then it is generated by $b_0$ . From the construction of $\psi $ along with the fact $b_0|_{G_{\mathbb {Q}_{\ell }}} =0$ , we see that the image of c in $C'/\mathfrak {m} C'$ is $0$ . So $c \in \mathfrak {m} C + \mathbb {T}_{\mathfrak {m}} c_{\ell }$ . Therefore, $b_{\ell }c \in (x^2,p^{\nu +1}x)$ as $\mathfrak {m} =(p,x)$ and $BC=(x^2,p^{\nu }x)$ .

Now $d \in \mathbb {T}_{\mathfrak {m}}^{\times }$ . Hence, from (4), we get $x(p^{\nu }+p^{\nu +1}z + xr")=0$ for some $z \in \mathbb {Z}_p$ and $r" \in \mathbb {T}_{\mathfrak {m}}$ . From Lemma 4.6, we know that the annihilator of $I^{\text {eis}}$ is $\ker (F)$ . As $(x)=I^{\text {eis}}$ , it follows that $p^{\nu }+p^{\nu +1}z + xr" \in \ker (F)$ .

So, $|\mathbb {T}^0_{\mathfrak {m}}/(F(x))| \leq p^{\nu }$ . Since $(x)=I^{\text {eis}}$ , it follows that $(F(x)) = I^{\text {eis},0}$ . We know, from [26, Theorem $\mathrm{5.1.2}$ ], that $\mathbb {T}^0_{\mathfrak {m}}/F(x) = \mathbb {T}^0_{\mathfrak {m}}/I^{\text {eis},0} \simeq \mathbb {Z}/p^{\nu +v_p(k)}\mathbb {Z}$ . Since $p \mid k$ , $v_p(k)>0$ , and hence this gives us a contradiction. Therefore, we get $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}})> 1$ , which proves the theorem.

We will now prove Theorem 5.5 without the assumption that $p \mid k$ . We will crucially use the theory of modular forms modulo p (recalled in §4) along with Theorem 5.5 in its proof. Note that if $i \geq 2$ , then the action of $T_p$ on $M_i(\ell ,\mathbb {F}_p)$ coincides with the action of the operator U considered in [15, Section $1$ ] (note that the prime $\ell $ of [15] corresponds to the prime p in our context).

Theorem 5.6. If $c_0 \cup b_0 = 0$ and $k> 2$ , then $I^{\text {eis},0}$ is not principal and $\operatorname {\mathrm {rank}}_{\mathbb {Z}_p}(\mathbb {T}^0_{\mathfrak {m}})> 1$ .

Proof. We have already proved the theorem for $p \mid k$ . So assume $p \nmid k$ and $c_0 \cup b_0=0$ . Recall that it suffices to prove that $I^{\text {eis},0}$ is not principal.

Now let $k'$ be an integer such that $k'> k$ , $p-1 \mid (k'-k)$ and $p \mid k'$ . Let $\mathbb {T}'$ be the $\mathbb {Z}_p$ -subalgebra of $\text {End}_{\mathbb {Z}_p}(M_{k'}(\ell ,\mathbb {Z}_p))$ generated by the Hecke operators $T_q$ for primes $q \neq \ell $ and the Atkin–Lehner operator $w_{\ell }$ at $\ell $ . Let $\mathbb {T}^{\prime}_{\mathfrak {m}}$ be the completion of $\mathbb {T}'$ at its maximal ideal generated by the set $\{p, w_{\ell }+1, T_q - (1+q^{k'-1}) \mid q \neq \ell \text { is a prime}\}$ .

Let $\mathcal {E}_{p-1}$ be the Eisenstein series of level $1$ and weight $p-1$ such that the constant term of the q-expansion $\mathcal {E}_{p-1}(q)$ of $\mathcal {E}_{p-1}$ is $1$ . Note that . Let $\overline {\mathcal {E}_{p-1}(q)} \in M_{p-1}(\ell ,\mathbb {F}_p)$ be the reduction of $\mathcal {E}_{p-1}(q)$ modulo p. Then we know that $\overline {\mathcal {E}_{p-1}(q)} = 1$ . Therefore, by using the multiplication by $(\overline {\mathcal {E}_{p-1}(q)})^{\frac {k'-k}{p-1}}$ map, we can identify $M_k(\ell ,\mathbb {F}_p)$ as a subspace of $M_{k'}(\ell ,\mathbb {F}_p)$ , and we will denote this subspace by $M_k(\ell ,\mathbb {F}_p)$ as well.

As $k> 2$ , $k'> p+1$ , by [15, Lemma $1.9$ ] we know that there exists an integer $n>0$ such that $T_p^n(M_{k'}(\ell ,\mathbb {F}_p)) \subset M_k(\ell ,\mathbb {F}_p)$ . Hence, after localizing at $\mathfrak {m}$ , we get that $T_p^n(M_{k'}(\ell ,\mathbb {F}_p)_{\mathfrak {m}}) \subset M_k(\ell ,\mathbb {F}_p)_{\mathfrak {m}}$ . As $T_p -1 -p^{k'-1} \in \mathfrak {m}$ , it follows that $T_p \in (\mathbb {T}^{\prime}_{\mathfrak {m}})^{\times }$ . Therefore, $T_p$ is an invertible operator on $M_{k'}(\ell ,\mathbb {F}_p)_{\mathfrak {m}}$ . So, it follows that $M_k(\ell ,\mathbb {F}_p)_{\mathfrak {m}} = M_{k'}(\ell ,\mathbb {F}_p)_{\mathfrak {m}}$ .

As $p-1 \mid (k'-k)$ and $p-1 \nmid k$ , it follows that $p-1 \nmid k'$ . Hence, by Lemma 4.4, the largest quotient of $\mathbb {T}'$ (resp. of $\mathbb {T}$ ) acting faithfully on $M_{k'}(\ell ,\mathbb {F}_p)_{\mathfrak {m}}$ (resp. on $M_{k}(\ell ,\mathbb {F}_p)_{\mathfrak {m}}$ ) is $\mathbb {T}^{\prime}_{\mathfrak {m}}/(p)$ (resp. $\mathbb {T}_{\mathfrak {m}}/(p)$ ). Since $M_k(\ell ,\mathbb {F}_p)_{\mathfrak {m}} = M_{k'}(\ell ,\mathbb {F}_p)_{\mathfrak {m}}$ , $\mathbb {T}^{\prime}_{\mathfrak {m}}/(p) \simeq \mathbb {T}_{\mathfrak {m}}/(p)$ .

Now suppose $I^{\text {eis}}$ is principal. Then $\mathbb {T}_{\mathfrak {m}}$ is a quotient of , which means $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p))) = 1$ . By combining Theorem 5.5 and Lemma 4.5, we get that the Eisenstein ideal of $\mathbb {T}^{\prime}_{\mathfrak {m}}$ is not principal. Therefore, Lemma 4.3 implies that $\dim (\text {tan}(\mathbb {T}^{\prime}_{\mathfrak {m}}/(p)))> 1$ , and hence $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p)))> 1$ . This gives us a contradiction. Thus, we conclude that $I^{\text {eis}}$ is not principal. So by Lemma 4.5, we get that $I^{\text {eis},0}$ is not principal. This finishes the proof of the theorem.

We will now prove Corollaries A, B and C. We begin with the proof of Corollary A.

Proof of Corollary A

By combining Theorem 5.5 and Theorem 5.6, we get that if $I^{\text {eis},0}$ is principal, then $c_0 \cup b_0 \neq 0$ . Now suppose $c_0 \cup b_0 \neq 0$ . Then, by combining Lemma 3.4, Lemma 3.6 and Lemma 4.1, we get that $\dim (\text {tan}(\mathbb {T}_{\mathfrak {m}}/(p)))=1$ . So, Lemma 4.3 implies that $I^{\text {eis}}$ is principal. Hence, by Lemma 4.5, we get that $I^{\text {eis},0}$ is principal. If Vandiver’s conjecture holds for p, then we know, from Lemma 3.7, that $c_0 \cup b_0 \neq 0$ if and only if $\prod _{i=1}^{p-1}(1-\zeta _p^i)^{i^{k-2}} \in (\mathbb {Z}/\ell \mathbb {Z})^{\times }$ is not a p-th power. This proves the corollary.

Before proving Corollary B and Corollary C, we first prove a result that relates vanishing of cup product with class groups.

Recall that we denoted $ \mathbb {Q}(\zeta _{\ell }^{(p)},\zeta _p)$ by K and denoted its class group of by $\text {Cl}(K)$ . Let L be the unramified abelian extension of K such that $\text {Gal}(L/K) = \text {Cl}(K)/\text {Cl}(K)^p$ . Note that L is also Galois over $\mathbb {Q}$ and $\text {Gal}(L/K)$ is a normal subgroup of $\text {Gal}(L/\mathbb {Q})$ . As $\text {Gal}(L/K) = \text {Cl}(K)/\text {Cl}(K)^p$ is abelian, we get an action of $\text {Gal}(K/\mathbb {Q})$ on it. Now $\text {Gal}(K/\mathbb {Q}) = \text {Gal}(\mathbb {Q}(\zeta _{\ell }^{(p)})/\mathbb {Q}) \times \text {Gal}(\mathbb {Q}(\zeta _p)/\mathbb {Q})$ . So we have an action of $\text {Gal}(\mathbb {Q}(\zeta _p)/\mathbb {Q})$ on $\text {Cl}(K)/\text {Cl}(K)^p$ . Denote by $(\text {Cl}(K)/\text {Cl}(K)^p)[\omega _p^{1-k}]$ the subspace of $\text {Cl}(K)/\text {Cl}(K)^p$ on which $\text {Gal}(\mathbb {Q}(\zeta _p)/\mathbb {Q})$ acts by $\omega _p^{1-k}$ .

Proposition 5.7. Suppose k is an even integer, $p-1 \nmid k$ , and the $\omega _p^{1-k}$ -component of the p-part of the class group of $\mathbb {Q}(\zeta _p)$ is trivial. The following are equivalent:

1. (1) $c_0 \cup a_0 =0$ ,

2. (2) $\dim ((\text {Cl}(K)/\text {Cl}(K)^p)[\omega _p^{1-k}]) \geq 2$ ,

3. (3) The image of $\prod _{i=1}^{\ell -1}i^{(\sum _{j=1}^{i-1}j^{k-1})}$ in $(\mathbb {Z}/\ell \mathbb {Z})^{\times }$ is a p-th power.

Proof. As we are assuming that the $\omega _p^{1-k}$ -component of the p-part of the class group of $\mathbb {Q}(\zeta _p)$ is trivial, the equivalence between parts (2) and (3) follows from [17, Theorem $1.9$ ].

To prove that part (1) implies part (2), we follow the proof of [28, Proposition $\mathrm{11.1.1}$ ]. Suppose $c_0 \cup a_0=0$ . Therefore, there exists a representation $\rho : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_3(\mathbb {F}_p)$ such that

$$\begin{align*}\rho(g) = \begin{pmatrix} \omega_p^{1-k}(g) & c_0(g) & F(g) \\ 0 & 1 & a_0(g) \\ 0 & 0 & 1\end{pmatrix} \text{ for all } g \in G_{\mathbb{Q},p\ell}.\end{align*}$$

Here $F : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p$ is a cochain such that the coboundary of $-F$ is $c_0 \cup a_0$ . From the proof of Lemma 3.11, it follows that we can change F by a suitable element of $H^1(G_{\mathbb {Q},p},\omega _p^{1-k})$ to assume $F(G_{\mathbb {Q}_p})=0$ .

Let M be the extension of $\mathbb {Q}$ fixed by $\ker (\rho )$ . So $\text {Gal}(M/K) \simeq \mathbb {Z}/p\mathbb {Z} \times \mathbb {Z}/p\mathbb {Z}$ and its image under $\rho $ is $ \{\begin {pmatrix} 1 & a & b \\ 0 & 1 & 0 \\ 0 & 0 & 1\end {pmatrix} \mid a, b \in \mathbb {F}_p\}$ . As $c_0$ is unramified at p and $F(I_p)=0$ , it follows that M is unramified over all primes of K lying above p.

As $\omega _p^{1-k}(I_{\ell }) =1$ , it follows that $\rho (I_{\ell })$ is a p-group, and hence $\rho $ is tamely ramified at $\ell $ . This means that $|\rho (I_{\ell })|=p$ . So the image of $I_{\ell }$ in $\text {Gal}(M/\mathbb {Q})$ has cardinality p and the image of $I_{\ell }$ in $\text {Gal}(K/\mathbb {Q})$ also has cardinality p. Hence, M is unramified over all primes of K lying above $\ell $ . Therefore, we conclude that M is an unramified extension of K.

From the description of $\rho $ and the description of $\rho (\text {Gal}(M/K))$ , it follows that $\text {Gal}(\mathbb {Q}(\zeta _p)/\mathbb {Q})$ acts via $\omega _p^{1-k}$ on $\text {Gal}(M/K)$ . As $\text {Gal}(M/K) \simeq \mathbb {Z}/p\mathbb {Z} \times \mathbb {Z}/p\mathbb {Z}$ , it follows that

$$\begin{align*}\dim((\text{Cl}(K)/\text{Cl}(K)^p)[\omega_p^{1-k}]) \geq 2\end{align*}$$

which shows that part (1) implies part (2).

We will now prove that part (2) implies part (1). Suppose $\dim ((\text {Cl}(K)/\text {Cl}(K)^p)[\omega _p^{1-k}]) \geq 2$ . Let M be the subfield of L such that $\text {Gal}(M/K) = (\text {Cl}(K)/\text {Cl}(K)^p)[\omega _p^{1-k}]$ . Note that M is also Galois over $\mathbb {Q}$ .

So $V := \text {Gal}(M/K)$ is an $\mathbb {F}_p$ vector space on which the cyclic p-group $\text {Gal}(\mathbb {Q}(\zeta _{\ell }^{(p)})/\mathbb {Q})$ acts. Let $\alpha $ be a generator of $\text {Gal}(\mathbb {Q}(\zeta _{\ell }^{(p)})/\mathbb {Q})$ . Let $M'$ be the subfield of M such that $\text {Gal}(M'/K) \simeq V/(\alpha -1)V$ . As $(\alpha -1)V$ is a subspace of V stable under the action of $\text {Gal}(K/\mathbb {Q})$ , it follows that $(\alpha -1)V$ is a normal subgroup of $\text {Gal}(M/\mathbb {Q})$ , and hence $M'$ is also Galois over $\mathbb {Q}$ . As $\text {Gal}(\mathbb {Q}(\zeta _{\ell }^{(p)})/\mathbb {Q})$ acts trivially on $\text {Gal}(M'/K)$ , it follows that $\text {Gal}(M'/\mathbb {Q}(\zeta _p))$ is an abelian p-group. Note that $\text {Gal}(\mathbb {Q}(\zeta _p)/\mathbb {Q})$ acts on $\text {Gal}(M'/K)$ via $\omega _p^{1-k}$ and it acts trivially on $\text {Gal}(K/\mathbb {Q}(\zeta _p))$ . Hence, $\text {Gal}(M'/\mathbb {Q}(\zeta _p)) \simeq \mathbb {F}_p^{\oplus r}$ for some $r \geq 2$ and as a $\text {Gal}(\mathbb {Q}(\zeta _p)/\mathbb {Q})$ -representation, $\text {Gal}(M'/\mathbb {Q}(\zeta _p)) \simeq \mathbb {F}_p(\omega _p^{1-k})^{\oplus r-1} \oplus \mathbb {F}_p$ .

Therefore, from subfields of $M'$ , we get $r-1$ elements of $H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k})$ which are linearly independent over $\mathbb {F}_p$ . Now the prime of $\mathbb {Q}(\zeta _p)$ lying above p is unramified in K. As $M'$ is unramified over K, it follows that the prime of $\mathbb {Q}(\zeta _p)$ lying above p is also unramified in $M'$ . So the $r-1$ elements of $H^1(G_{\mathbb {Q},p\ell },\omega _p^{1-k})$ arising from subfields of $M'$ are all unramified at p. Hence, Lemma 2.4 implies that $r-1=1$ . Therefore, $M'$ is a $\mathbb {Z}/p\mathbb {Z}$ extension of K. Since $c_0$ generates the space of classes of $H^1(G_{\mathbb {Q},p\ell },\omega _p^{k-1})$ which are unramified at p, we get that $M'=K.K_{\bar \rho _{c_0}}$ , where $K_{\bar \rho _{c_0}}$ is the extension of $\mathbb {Q}$ fixed by $\ker (\bar \rho _{c_0})$ .

Now let $M"$ be the subfield of M such that $\text {Gal}(M"/K) \simeq V/(\alpha -1)^2V$ . By our assumption, we have $\dim (V) \geq 2$ and we have just proved that $\dim (V/(\alpha -1)V) = 1$ . So, we have $\dim (V/(\alpha -1)^2V) \geq 2$ and $\text {Gal}(M"/\mathbb {Q}(\zeta _p))$ is not abelian as $\alpha $ does not act trivially on $V/(\alpha -1)^2V$ . Note that $K.K_{\bar \rho _{c_0}}=M' \subset M"$ . Denote the image of $G_{\mathbb {Q}_{\ell }}$ in $\text {Gal}(M"/\mathbb {Q})$ by $D_{\ell }$ . As $\ell $ splits completely in $\mathbb {Q}(\zeta _p)$ , it follows that $D_{\ell }$ lies in $\text {Gal}(M"/\mathbb {Q}(\zeta _p))$ . Since $M"$ is unramified over K and K is abelian over $\mathbb {Q}$ , it follows that $D_{\ell }$ is abelian.

Suppose $c_0|_{G_{\mathbb {Q}_{\ell }}} \neq \beta a_0|_{G_{\mathbb {Q}_{\ell }}}$ for any $\beta \in \mathbb {F}_p$ . Note that $c_0|_{G_{\mathbb {Q}_{\ell }}} \neq 0$ and $a_0|_{G_{\mathbb {Q}_{\ell }}} \neq 0$ . So the image of $D_{\ell }$ in $\text {Gal}(M'/\mathbb {Q}(\zeta _p))$ has cardinality $p^2$ , and hence this image is all of $\text {Gal}(M'/\mathbb {Q}(\zeta _p))$ . Now $\text {Gal}(M"/K)$ is an abelian group and it is normal in $\text {Gal}(M"/\mathbb {Q}(\zeta _p))$ with their quotient given by $\text {Gal}(K/\mathbb {Q}(\zeta _p)) \simeq \mathbb {Z}/p\mathbb {Z}$ . As

$$\begin{align*}\text{Gal}(M"/M') = (\alpha-1)V/(\alpha-1)^2V \subset \text{Gal}(M"/K),\end{align*}$$

$\text {Gal}(K/\mathbb {Q}(\zeta _p))$ acts trivially on it. Hence, $\text {Gal}(M"/M')$ is in the center of $\text {Gal}(M"/\mathbb {Q}(\zeta _p))$ .

We have already seen that $D_{\ell }$ gets mapped surjectively on $\text {Gal}(M'/\mathbb {Q}(\zeta _p))$ under the surjective map $\text {Gal}(M"/\mathbb {Q}(\zeta _p)) \to \text {Gal}(M'/\mathbb {Q}(\zeta _p))$ . Therefore, $\text {Gal}(M"/\mathbb {Q}(\zeta _p)) = D_{\ell }.\text {Gal}(M"/M')$ . As $D_{\ell }$ is abelian and $\text {Gal}(M"/M')$ is in the center of $\text {Gal}(M"/\mathbb {Q}(\zeta _p))$ , it follows that $\text {Gal}(M"/\mathbb {Q}(\zeta _p))$ is abelian. But this gives a contradiction as we have already seen that $\text {Gal}(M"/\mathbb {Q}(\zeta _p))$ is not abelian.

So we have $c_0|_{G_{\mathbb {Q}_{\ell }}} = \beta a_0|_{G_{\mathbb {Q}_{\ell }}}$ for some $\beta \in \mathbb {F}_p$ . This means that $c_0|_{G_{\mathbb {Q}_{\ell }}} \cup a_0|_{G_{\mathbb {Q}_{\ell }}} = 0$ . From [25, Proposition $\mathrm{2.4.1}$ ], we get that $c_0 \cup a_0 =0$ which completes the proof of the proposition.

Proof of Corollary B and Corollary C

Corollary B follows directly by Corollary 5.2 and Proposition 5.7. Corollary C follows directly by Theorem 5.6, Proposition 5.7 and Lemma 3.7.

We will end this article by proving the $R=\mathbb {T}$ theorems mentioned in the introduction (Theorem B).

Proof of Theorem B

Part (1): Recall that we denoted by $(T,D) : G_{\mathbb {Q},p\ell } \to R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ the universal pseudo-representation deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ . Let $A=\begin {pmatrix} R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) & B \\ C & R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )\end {pmatrix}$ be the faithful GMA over $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ and $\rho : G_{\mathbb {Q},p\ell } \to A^{\times }$ be the representation attached to $(T,D)$ by Lemma 3.2. Suppose $\rho (i_{\ell }) =\begin {pmatrix} 1+x & b_{\ell } \\ c_{\ell } & 1-x \end {pmatrix}$ and $\rho (g_0) = \begin {pmatrix} a_0 & 0\\ 0 & d_0 \end {pmatrix}$ . Let $g_{\ell }$ be a lift of $\text {Frob}_{\ell }$ in $G_{\mathbb {Q}_{\ell }}$ and suppose $\rho (g_{\ell }) =\begin {pmatrix} a & b \\ c & d \end {pmatrix}$ .

Now Lemma 3.6 implies that $\dim (\text {tan}(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/(p)))=1$ . So the proof of Lemma 3.4 implies that any p-ordinary, $\ell $ -unipotent pseudo-representation $(t,d) : G_{\mathbb {Q},p\ell } \to \mathbb {F}_p[\epsilon ]/(\epsilon ^2)$ with determinant $\omega _p^{k-1}$ which deforms $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ is reducible. Thus, Lemma 3.5 implies that $m = (p,x)$ where m is the maximal ideal of $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ . As $\operatorname {tr}(\rho (g_0i_{\ell }))-\operatorname {tr}(\rho (g_0)) = (a_0-d_0)x \in J_0$ and $a_0-d_0 \in (R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ))^{\times }$ , it follows that $x \in J_0$ . Since $m=(p,x)$ and $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )/J_0 \simeq \mathbb {Z}_p$ , we get that $J_0=(x)$ .

Note that the relation $\rho (g_{\ell }i_{\ell }g_{\ell })^{-1} = \rho (i_{\ell })^{\ell }$ implies that $ab_{\ell } + b(1-x) = b(1+\ell x) + \ell b_{\ell } d$ (see the proof of Theorem 5.5 for more details). So we have $b_{\ell }(a-\ell d) = bx(1+\ell )$ which means $b_{\ell }c_{\ell }(a-\ell d) = bc_{\ell }x(1+\ell )$ . By part (4) of Lemma 3.2, we have $B=R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )b_{\ell }$ . Thus, $b = rb_{\ell }$ for some $r \in R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ . As $b_{\ell }c_{\ell }=-x^2$ , we get that $x^2(a - \ell d) = x^3r(1+\ell )$ . Since $J_0 = (x)$ is the Eisenstein ideal of $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ , Lemma 3.3 implies that $a \equiv 1 \pmod {(x)}$ and $d \equiv \chi _p^{k-1}(\text {Frob}_{\ell }) \pmod {(x)}$ . Therefore, we get $x^2(1-\ell ^k + xr')=x^3r(1+\ell )$ which means $x^2(1 - \ell ^k + xr")=0$ .

Denote the image of $r \in R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ in $(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ))^{\text {red}}$ by $\bar r$ . As $J_0$ is a prime ideal of $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ , it contains the nilradical of $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell )$ . Let $J^{\text {red}}_0$ be the image of $J_0$ in $(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ))^{\text {red}}$ . So $J^{\text {red}}_0=(\bar x)$ . From the previous paragraph, we get that $\bar {x}^2(1 - \ell ^k + \overline {xr"})=0$ . Therefore, $\bar {x}(1 - \ell ^k + \overline {xr"})=0$ . As $1-\ell ^k = p^{\nu +v_p(k)}.u$ for some $u \in \mathbb {Z}_p^{\times }$ . Hence, $|J^{\text {red}}_0/(J^{\text {red}}_0)^2| \leq p^{\nu +v_p(k)}$ .

As $\mathbb {T}_{\mathfrak {m}}$ is reduced, the map $\phi _{\mathbb {T}} : R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to \mathbb {T}_{\mathfrak {m}}$ factors through $(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ))^{\text {red}}$ to get a map $\phi ^{\prime}_{\mathbb {T}} : (R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ))^{\text {red}} \to \mathbb {T}_{\mathfrak {m}}$ . Note that $\phi _{\mathbb {T}}$ is a surjective morphism of augmented $\mathbb {Z}_p$ -algebras, and hence $\phi ^{\prime}_{\mathbb {T}}$ is also a surjective morphism of augmented $\mathbb {Z}_p$ -algebras. The kernels of the surjective morphisms $(R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ))^{\text {red}} \to \mathbb {Z}_p$ and $\mathbb {T}_{\mathfrak {m}} \to \mathbb {Z}_p$ are $J^{\text {red}}_0$ and $I^{\text {eis}}$ , respectively.

From Lemma 4.6, it follows that the annihilator of $I^{\text {eis}}$ is $\ker (F)$ . Therefore, $\mathbb {T}_{\mathfrak {m}}/(I^{\text {eis}} +\ker (F)) \simeq \mathbb {T}^0_{\mathfrak {m}}/I^{\text {eis},0}$ and [26, Theorem $\mathrm{5.1.2}$ ] and [19, Proposition II. $9.6$ ] imply that $\mathbb {T}^0_{\mathfrak {m}}/I^{\text {eis},0} \simeq \mathbb {Z}/p^{\nu +v_p(k)}\mathbb {Z}$ . So, $|J^{\text {red}}_0/(J^{\text {red}}_0)^2| \leq |\mathbb {T}^0_{\mathfrak {m}}/I^{\text {eis},0}|$ . Hence, Wiles–Lenstra numerical criterion ([11, Criterion I]) implies that $\phi ^{\prime}_{\mathbb {T}}$ is an isomorphism of local complete intersection rings. This proves the first part of the theorem.

Part (2): We now move on to the second part of the theorem. Let $(T',D') : G_{\mathbb {Q},p\ell } \to R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell )$ be the universal pseudo-representation deforming $(\operatorname {tr}(\bar \rho _0),\det (\bar \rho _0))$ . Let $A=\begin {pmatrix} R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell ) & B \\ C & R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell )\end {pmatrix}$ be the faithful GMA over $R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell )$ and $\rho : G_{\mathbb {Q},p\ell } \to A^{\times }$ be the representation attached to $(T,D)$ by Lemma 3.2. Suppose $\rho (i_{\ell }) =\begin {pmatrix} 1+x & b_{\ell } \\ c_{\ell } & 1-x \end {pmatrix}$ . Let $g_{\ell }$ be a lift of $\text {Frob}_{\ell }$ in $G_{\mathbb {Q}_{\ell }}$ and suppose $\rho (g_{\ell }) =\begin {pmatrix} a & b \\ c & d \end {pmatrix}$ .

Note that the pseudo-representation $(1+\chi _p^{k-1},\chi _p^{k-1}) : G_{\mathbb {Q},p\ell } \to \mathbb {Z}_p$ is unramified at $\ell $ . Hence, $J_0$ contains the kernel of the surjective map $R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell )$ . Let $J^{\prime}_0$ be the image of $J_0$ in $R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell )$ . Using the arguments of the first case of the theorem, we get that $J^{\prime}_0 = (x)$ .

Now the Steinberg-or-unramified at $\ell $ condition implies $\operatorname {tr}(\rho (g)(\rho (g_{\ell })-\ell ^{k/2})(\rho (i_{\ell })-1))=0$ for all $g \in G_{\mathbb {Q},p\ell }$ . As $R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell )[\rho (G_{\mathbb {Q},p\ell })] = A$ , it follows that $\operatorname {tr}(g'(\rho (g_{\ell })-\ell ^{k/2})(\rho (i_{\ell })-1))=0$ for all $g' \in A$ . Putting $g' = \begin {pmatrix} 1 & 0 \\ 0 & 0 \end {pmatrix}$ , we get $(a-\ell ^{k/2})x+bc_{\ell }=0$ .

As B is generated by $b_{\ell }$ (by part (4) of Lemma 3.2) and $b_{\ell }c_{\ell }=-x^2$ , it follows that $bc_{\ell }=x^2r$ for some $r \in R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell )$ . As $J^{\prime}_0=(x)$ , Lemma 3.3 implies that $a \equiv 1 \pmod {(x)}$ . Therefore, $(a-\ell ^{k/2})x+bc_{\ell }= x(1-\ell ^{k/2} - xr' + xr) = 0$ . As $v_p(1-\ell ^{k/2}) = \nu +v_p(k/2) = \nu + v_p(k)$ , it follows that $|J^{\prime}_0/(J^{\prime}_0)^2| \leq p^{\nu +v_p(k)}$ .

Note that $\psi _{\mathbb {T}}$ is a surjective morphism of augmented $\mathbb {Z}_p$ -algebras. The kernels of the surjective morphisms $R^{\text {pd},\text {st}}_{\bar \rho _0,k}(\ell ) \to \mathbb {Z}_p$ and $\mathbb {T}_{\mathfrak {m}} \to \mathbb {Z}_p$ are $J^{\prime}_0$ and $I^{\text {eis}}$ , respectively. We have already seen in the proof of part $1$ of the theorem that $\mathbb {T}_{\mathfrak {m}}/(I^{\text {eis}} +\ker (F)) \simeq \mathbb {T}^0_{\mathfrak {m}}/I^{\text {eis},0} \simeq \mathbb {Z}/p^{\nu +v_p(k)}\mathbb {Z}$ . Hence, we have $|J^{\prime}_0/(J^{\prime}_0)^2| \leq |\mathbb {T}^0_{\mathfrak {m}}/I^{\text {eis},0}|$ . So Wiles–Lenstra numerical criterion ([11, Criterion I]) implies that $\psi _{\mathbb {T}}$ is an isomorphism of local complete intersection rings. This proves the second part of the theorem.

Part (3): We now come to the last part of the theorem. By [10, Lemma $3.1$ ], one can choose the universal deformation $\rho ^{\text {univ},\ell } : G_{\mathbb {Q},p\ell } \to \mathrm {GL}_2(R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell ))$ of $\bar \rho _{c_0}$ such that $\rho ^{\text {univ},\ell }(g_0) = \begin {pmatrix} a_0 & 0 \\ 0 & d_0 \end {pmatrix}$ .

Suppose $\rho ^{\text {univ},\ell }(i_{\ell }) = \begin {pmatrix} 1+x & b_{\ell } \\ c_{\ell } & 1-x\end {pmatrix}$ . Let $\phi : R^{\text {pd},\text {ord}}_{\bar \rho _0,k}(\ell ) \to R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ be the surjective map induced by $(\operatorname {tr}(\rho ^{\text {univ},\ell }),\det (\rho ^{\text {univ},\ell }))$ . Now we know from the proof of the first part of the theorem that $J_0 = (T(g_0i_{\ell })-T(g_0))$ . The ideal generated by $\phi (J_0)$ is $I_0$ . So $I_0=(\operatorname {tr}(\rho ^{\text {univ},\ell }(g_0i_{\ell })) - \operatorname {tr}(\rho ^{\text {univ},\ell }(g_0))) = (x)$ . The ideal generated by $\phi _{\mathbb {T}^0}(I_0)$ is the Eisenstein ideal $I^{\text {eis},0}$ of $\mathbb {T}^0_{\mathfrak {m}}$ , so $I^{\text {eis},0} = (\phi _{\mathbb {T}^0}(x))$ .

As $I_0=(x)$ , it follows that the maximal ideal of $R^{\text {def},\text {ord}}_{\bar \rho _{c_0},k}(\ell )$ is generated by p and x. So we have a surjective map which sends X to x. Composing $F_1$ with $\phi _{\mathbb {T}^0}$ , we get a surjective map such that $F_2(X) = \phi _{\mathbb {T}^0}(x)$ . As is a UFD and $\mathbb {T}^0_{\mathfrak {m}}$ is finite and flat over $\mathbb {Z}_p$ , it follows that $\ker (F_2)$ is a principal ideal. Now [26, Theorem $\mathrm{5.1.2}$ ] and [19, Proposition II. $9.6$ ] imply that $\mathbb {T}^0_{\mathfrak {m}}/(\phi _{\mathbb {T}^0}(x)) = \mathbb {T}^0_{\mathfrak {m}}/I^{\text {eis},0} \simeq \mathbb {Z}/p^{\nu +v_p(k)}\mathbb {Z}$ . Hence, we can find a generator $\alpha $ of $\ker (F_2)$ such that $\alpha = p^{\nu +v_p(k)}+Xf(X)$ for some .

Since $I_0=(x) = (F_1(X))$ , Lemma 3.10 implies that there exists a $\beta \in \ker (F_1)$ such that $\beta =p^e + Xg(X)$ for some and $e \leq \nu +v_p(k)$ . As $\ker (F_1) \subset \ker (F_2)$ , it follows that there exists some such that $p^e + Xg(X) = h(X)(p^{\nu +v_p(k)}+Xf(X))$ . So the constant term of $h(x)$ is $p^{e'}$ for some $e' \geq 0$ . Now $e \leq \nu +v_p(k)$ , which means $e'=0$ . Hence, $h(X)$ is a unit which means $\ker (F_2) \subset \ker (F_1)$ . Therefore, we conclude that $\ker (F_2) = \ker (F_1)$ . As $F_1$ is surjective, it follows that $\phi _{\mathbb {T}^0}$ is injective. As $\phi _{\mathbb {T}^0}$ is also surjective, it follows that $\phi _{\mathbb {T}^0}$ is an isomorphsim which proves the final part of the theorem.