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# Big images of two-dimensional pseudorepresentations

Andrea Conti, Jaclyn Lang, Anna Medvedovsky

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Andrea Conti, Jaclyn Lang, Anna Medvedovsky. Big images of two-dimensional pseudorepresentations. Mathematische Annalen, 2022, 71 p. 10.1007/s00208-021-02345-w. hal-03965664

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BIG IMAGES OF TWO-DIMENSIONAL PSEUDOREPRESENTATIONS

ANDREA CONTI, JACLYN LANG, AND ANNA MEDVEDOVSKY

ABSTRACT. Bellaïche has recently applied Pink-Lie theory to prove that, under mild conditions, the image of a continuous 2-dimensional pseudorepresentation $\rho$ of a profinite group on a local pro-$p$ domain $A$ contains a nontrivial congruence subgroup of $SL_2(B)$ for a certain subring $B$ of $A$. We enlarge Bellaïche's ring and give this new $B$ a conceptual interpretation both in terms of *conjugate self-twists* of $\rho$, symmetries that constrain its image, and in terms of the *adjoint trace ring* of $\rho$, which we show is both more natural and the optimal ring for these questions in general. Finally, we use our purely algebraic result to recover and extend a variety of arithmetic big-image results for $GL_2$ Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and $p$-adic Hida or Coleman families of elliptic and Hilbert modular forms.

Let $\rho : \Pi \to \text{GL}_2(A)$ be a continuous representation of a profinite group $\Pi$ on a local pro-$p$ domain $A$. How big or small can the image of $\rho$ be? Arithmetic versions of this question, with $\rho$ the $p$-adic Galois representation attached to a cuspidal modular eigenform — or more often a compatible family of such $\rho$ considered adelically — have been considered since the 1960s, when Serre proved an adelic open-image result for the Galois representation on the Tate module of a non-CM elliptic curve [Ser68]. Serre's result was adapted by Ribet [Rib85](i) and then by Momose [Mom81] to the more delicate setting of modular forms, where certain symmetries naturally bound the size of the image. Recently Nekovář [Nek12] generalized their work to Hilbert modular forms.

In the 1990s Pink began a purely algebraic study of this kind of question by characterizing closed pro-$p$ subgroups of $\text{SL}_2(A)$ in terms of associated “Pink-Lie” algebras. Pink’s investigations fueled further exploration of arithmetic big-image questions, this time for Galois representations attached to $p$-adic Hida families of modular forms, by Hida [Hid15] and then, accounting for Ribet–Momose-type symmetries, by Lang (second author here) [Lan16]. Simultaneously, Bellaïche began adapting Pink-Lie theory to the (pseudo)representation setting, obtaining abstract big-image results, but in a form that was difficult to compare to the symmetry formulations of Ribet, Momose, and Lang.

In the present work we finally unite the two approaches, refining Bellaïche-Pink-Lie theory to relate $p$-adic big-image results to natural symmetry bounds in an abstract algebraic setting. We thereby recover the $p$-adic big-image results of Ribet, Momose, Nekovář, and Lang, improving the latter, under mild conditions on $\bar{p}$. We also obtain the first big-image results for Galois representations attached to $p$-adic Coleman families of modular forms (rather than for associated rigid-analytic Lie algebras, as in Conti(first author here)–Iovita–Tilouine [CIT16]), $p$-adic families of Hilbert modular forms, and Bianchi modular forms. Along the way we propose shifting our perspective towards formulating big-image results in terms of rings of definition of adjoint representations rather than in terms of rings fixed by symmetries. We emphasize that our results require absolutely no arithmetic input and are provably optimal. We explain how our results apply to a wide variety of modular Galois representations, and we anticipate that this framework can yield even more arithmetic fruit, from understanding *even* Galois representations to relating reducibility/dihedrality ideals and automorphic congruence modules. Finally we hope that the algebraic nature of our results might portend similar phenomena in higher dimension.

<footnote n="2010">2010 Mathematics Subject Classification. 11F11, 11F80, 11F85.</footnote>

<footnote>Key words and phrases. Representations of profinite groups, images of $p$-adic Galois representations, pseudorepresentations.</footnote>

<footnote>(i) Ribet's work appeared in a series of papers starting in 1975; [Rib85] is a convenient reference.</footnote>
**Acknowledgements.** The authors would like to thank Joël Bellaïche for helpful discussions about his work. We also thank Gebhard Böckle, Keith Conrad, Shaunak Deo, Mladen Dimitrov, Haruzo Hida, David Loeffler, Robert Pollack, and Jacques Tilouine for useful comments. Thanks to David Rohrlich for help with Clifford theory. Finally, we thank the anonymous referee, whose detailed comments improved both the content and the exposition of this article, not once but twice. All three authors express gratitude to the Max Planck Institute for Mathematics for creating space for us to get together; without the hospitality of the MPIM, this collaboration would not have begun.

The first author was supported by the DFG Forschergruppe 1920, the second and third authors by the National Science Foundation through awards DMS-1604148 (second author) and DMS-1703834 (third author). This article underwent extensive revisions under soft covid lockdown conditions. The third author thanks her husband and especially her nanny for months of childcare support.

CONTENTS

<table><tbody><tr><td>1. Introduction</td><td>2</td></tr><tr><td>2. Bellaïche-Pink-Lie theory</td><td>8</td></tr><tr><td>3. Fullness</td><td>17</td></tr><tr><td>4. Adjoint trace rings and conjugate self-twists</td><td>21</td></tr><tr><td>5. Optimality</td><td>28</td></tr><tr><td>6. Fullness for B<sub>ρ</sub>(E)</td><td>30</td></tr><tr><td>7. Lifting residual conjugate self-twists</td><td>33</td></tr><tr><td>8. Regularity and residual conjugate self-twists</td><td>38</td></tr><tr><td>9. Fullness peers: B<sub>ρ</sub>(E) and A<sup>Σ<sub>ρ</sub></sup></td><td>43</td></tr><tr><td>10. Main fullness results</td><td>52</td></tr><tr><td>11. Residually large representations</td><td>53</td></tr><tr><td>12. Applications to Galois representations</td><td>55</td></tr><tr><td>Appendix A. Algebraic sundries</td><td>62</td></tr><tr><td>References</td><td>70</td></tr></tbody></table>

1. INTRODUCTION

1.1. **The question.** Let $p$ be an odd prime, $A$ a local pro-$p$ domain with maximal ideal $\mathfrak{m}$ and (finite) residue field $\mathbb{F} := A/\mathfrak{m}$, and $\Pi$ a profinite group. Let $\rho: \Pi \to \text{GL}_2(A)$ be a continuous representation<sup>(ii)</sup> with the property that the residual representation $\bar{\rho} := \rho \mod \mathfrak{m}$ is semisimple and *multiplicity free*: either absolutely irreducible, or a sum of two distinct characters to $\mathbb{F}^\times$. Roughly, the objective is to show that the image of $\rho$ is as big as possible.

Note that if $\rho$, or its restriction to an index-2 subgroup of $\Pi$, is reducible, then the image of $\rho$ is both well understood and not big. Similarly, one cannot expect a big-image result when the image of $\rho$ is up to twist isomorphic to that of $\bar{\rho}$, as happens when $\rho$ arises from a modular form of weight one. Let us call these three kinds of representations *a priori small*. The a priori small representations are exactly those that are not strongly absolutely irreducible.

Suppose now that $\rho$ is not a priori small. We cannot expect $\rho$ to be surjective: even its determinant need not be surjective. Nor can we expect the image of $\rho$ to contain all of $\text{SL}_2(A)$, unless the image of $\bar{\rho}$ contains all of $\text{SL}_2(\mathbb{F})$. Following ideas of Hida, we settle on the notion of *fullness*. If $B$ is any ring and $\mathfrak{b} \subseteq B$ is any nonzero ideal, the subgroup of $\text{SL}_2(B)$ given by the kernel of reduction modulo $\mathfrak{b}$ is a *congruence subgroup* of $\text{SL}_2(B)$ (of level $\mathfrak{b}$):

$$ \Gamma_B(\mathfrak{b}) := \ker (\text{SL}_2(B) \to \text{SL}_2(B/\mathfrak{b})). $$

<footnote><sup>(ii)</sup>In fact we consider 2-dimensional pseudorepresentations, but we stick with representations for the introduction.</footnote>
If the image of $\rho$, up to conjugation, contains such a congruence subgroup, we say that $\rho$ is B-full. A key part of the big-image game is the search for an optimal fullness ring — or rather for an optimal equivalence class of fullness peers, rings that each contain an ideal of the other.

Historically, the constraints on the fullness rings of a representation $\rho$ have been described in terms of certain symmetries of $\rho$. If $\sigma$ is an automorphism of $A$ and $\eta$ is a character of $\Pi$, the pair $(\sigma, \eta)$ is a conjugate self-twist of $\rho$ if applying the automorphism gives the same representation as twisting by the character: $\sigma\rho \cong \eta \otimes \rho$. If $\rho$ has a nontrivial conjugate self-twist $(\sigma, \eta)$, then $\rho$ cannot be A-full: indeed, the equation

$$ (1) \qquad \sigma(\operatorname{tr} \rho(g)) = \operatorname{tr} \sigma\rho(g) = \eta(g) \operatorname{tr} \rho(g) $$

means that the trace of $\rho(g)$ is an eigenvector for $\sigma$ viewed as a linear map over the $\sigma$-invariant scalars. But the trace of a congruence subgroup of $A$ is not so constrained. Accordingly, the known arithmetic big-image results — of Ribet and Momose, of Nekovář, of Lang, described in Section 1.2 below — have all established fullness with respect to $A^{\Sigma_{\rho}}$, the subring of $A$ fixed by the conjugate self-twists of $\rho$.

Stepping outside the constraints of the arithmetic setting, however, reveals shortcomings of the $A^{\Sigma_{\rho}}$ perspective: $A^{\Sigma_{\rho}}$ may simply be too big for $\rho$ to be $A^{\Sigma_{\rho}}$-full. For one thing, $A$ itself may not see all the conjugate self-twists of $\rho$, as in Example 4.3, a failure of normality. Enlarging $A$ may still not suffice if $A$ has inseparable elements, as in Example 4.10, or worse yet, transcendental ones. The limitations are exactly those of Galois theory: there may not be enough automorphisms to carve down deep enough to a fullness ring with conjugate self-twists alone.

Rather than carving down from above, we propose building a fullness ring from below. Let $A_0$ be the *adjoint trace ring* of $\rho$, the closed subring of $A$ topologically generated by the elements $(\operatorname{tr} \rho(g))^2 / \det \rho(g)$ for $g \in \Pi$, the traces of the adjoint representation.

It is easy to see that this generating set is both twist-invariant and fixed by all conjugate self-twists. Thus $A_0$ acts as a base ring for the conjugate self-twist automorphisms, and the question of whether $A$ or its extensions have enough conjugate self-twists turns into the usual one of Galois theory: are there enough automorphisms to isolate the base? On the other hand, $A_0$ is a potential fullness ring free from the limitations of $A^{\Sigma_{\rho}}$ outlined above. Moreover, in all arithmetic settings where we recover existing fullness results, we show that $A_0$ and $A^{\Sigma_{\rho}}$ are always fullness peers, so that $A_0$- and $A^{\Sigma_{\rho}}$-fullness are equivalent. Our first theorem shows that $A_0$ is the optimal fullness ring in all cases.

**Theorem A** (Optimality theorem. See Theorem 5.3).

If $\rho$ is B-full for some ring B, then a fullness peer of B is contained in $A_0$.

Accompanying the $A_0$-optimality theorem, we present the main result of this paper: $A_0$-fullness. We assume that the pro-$p$ part of $\Pi$ is topologically finitely generated: for more on this $p$-finiteness condition of Mazur, always satisfied by characteristic-zero local Galois groups and global Galois groups with ramification restricted to a finite set of places, see Definition 2.6.

**Theorem B'** (Main fullness theorem, preliminary version. See Theorem B and Theorem 10.1). Suppose that $\bar{\rho}$ satisfies a mild condition. If $\rho$ is not a priori small, then $\rho$ is $A_0$-full.

For a more precise formulation and a detailed discussion of the mild condition, see Theorem B in Section 1.3 below. For now we merely note that the most limiting condition for applications is a *regularity assumption*: we require that the image of $\bar{\rho}$ contain a matrix whose eigenvalue ratio differs from $\pm 1$ but is contained in residue field of $A_0$. For the most general version of our theorem, which is formulated for images of *pseudorepresentations* in the sense of Chenevier [Che14] as studied by Bellaïche [Bel19], see Theorem 10.1.

The last stand-alone result of this paper is a refinement of Theorem B' in the case where the residual image of $\rho$ is large. Let $\mathbb{E}$ be the residue field of $A_0$; here we assume that $\#\mathbb{E} \ge 7$.

**Theorem C** (See Corollary 11.2). If $\operatorname{Im} \bar{\rho} \supseteq \operatorname{SL}_2(\mathbb{E})$ then $\operatorname{Im} \rho$ contains $\operatorname{SL}_2(A_0)$ up to conjugation.
1.2. **History.** We now survey the history of big-image results, both arithmetic and algebraic, using the terminology introduced above, to situate **Theorem B'** in context. In all of the theorems stated in Section 1.2, $A_0$-fullness is equivalent to $A^{\Sigma_\rho}$-fullness.

1.2.1. *Classical modular forms.* The big-image line of inquiry began in the late 1960s, when Serre showed that if $\rho$ comes from the $p$-adic Tate module (including for $p=2$) of a non-CM elliptic curve over a number field $F$, so that $\Pi = \text{Gal}(\bar{F}/F)$ and $A = A_0 = \mathbb{Z}_p$, then $\rho$ is $\mathbb{Z}_p$-full [Ser68, Theorem IV.2.2].$^{(iii)}$

In the 80s, Ribet and Momose generalized Serre's theorem to elliptic modular forms. Let $f$ be a cuspidal non-CM eigenform of weight at least 2. Given a prime $p$ and an embedding $\iota_p: \overline{\mathbb{Q}} \to \overline{\mathbb{Q}}_p$, one can associate to $f$ a 2-dimensional Galois representation $\rho = \rho_{\iota_p}$ of $\Pi = \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ over the ring of integers $A$ of a finite extension $\mathbb{Q}_p$.

**Theorem 1.1** ([Rib85, Theorem 3.1], [Mom81, Theorem 4.1]; see also **Theorem 12.2**).

*For all but finitely many primes p, the representation ρ is A^{Σρ}-full.*$^{(iv)}$

More recently, Nekovář generalized **Theorem 1.1** to representations coming from Hilbert modular forms, in which case $\Pi$ is the absolute Galois group of a totally real number field and $A$ is still a finite extension of $\mathbb{Z}_p$ [Nek12, Appendices B.3–B.6].

Our main theorem (**Theorem B'**) recovers the at-$p$ statements of both the Ribet–Momose and the Nekovář results, under the assumption that residual representation satisfies our regularity condition. See Section 12.1 and Section 12.2 for details.

1.2.2. *Families of p-adic modular forms.* Although we have stated the work of Serre, Ribet, Momose, and Nekovář for a fixed prime $p$ to better fit our $p$-adic framework, all of these theorems are actually adelic open-image results proved using geometric methods. Much work has been done to generalize such theorems to groups other than $\text{GL}_2$, but that is not the direction that interests us. Rather, we are interested in fixing $p$ and deforming representations $p$-adically, which necessitates a completely different approach. There has been some progress in this direction in special cases. Recall that we are assuming throughout that $p \neq 2$.

First we suppose that $\rho$ arises from a non-CM cuspidal Hida family. In this case $\Pi = \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ and $A$ is a finite domain over $\Lambda := \mathbb{Z}_p[\mathcal{X}]$. When $A$ is a constant extension of $\Lambda$ and the image of $\bar{\rho}$ contains $\text{SL}_2(\bar{F}_p)$, Boston [MW86, Proposition 3] and Fischman [Fis02, Theorem 4.8] show that the image of $\rho$ contains $\text{SL}_2(A^{\Sigma_{\rho}})$, hence $\rho$ is $A^{\Sigma_{\rho}}$-full.$^{(v)}$

More recently, Hida proved that if $\bar{\rho}$ is locally-at-$p$ multiplicity free then $\rho$ is $\Lambda$-full [Hid15, Theorem I], but his work did not relate $\Lambda$ to $A_0$ or conjugate self-twists of $\rho$. Lang then improved Hida's result from $\Lambda$-fullness to $A^{\Sigma_\rho}$-fullness under the assumption that $\bar{\rho}$ is absolutely irreducible, proving the following result.

**Theorem 1.2** ([Lan16, Theorem 2.4]; see also **Theorem 12.11**). *If $\rho$ arises from a non-CM cuspidal Hida family, and $\bar{\rho}$ is absolutely irreducible and satisfies additional multiplicity-freeness conditions locally at $p$ then $\rho$ is $A^{\Sigma_\rho}$-full.*

The case when $\rho$ arises from a Coleman family was studied by Conti–Iovita–Tilouine [CIT16]. In this case we again have $\Pi = \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ and $A$ is a domain finite over $\Lambda$. In [CIT16, Theorem 6.2] it is proved that, under hypotheses similar to those in **Theorem 1.2**, a certain rigid analytic Lie algebra attached to $\text{Im} \rho$ contains that of a congruence subgroup of $A^{\Sigma_\rho}$. This strongly suggests that $\rho$ should be $A^{\Sigma_\rho}$-full, though this statement does not follow from [CIT16].

<footnote>$^{(iii)}$Serre's result is better known as an open-image theorem; and in fact he shows much more: the image of all the $p$-adic Tate modules for all $p$ at once is open adelically.</footnote>

<footnote>$^{(iv)}$Like Serre, Ribet and Momose prove stronger adelic big-image results. See Section 12.1 for more details.</footnote>

<footnote>$^{(v)}$All the works in Section 1.2.2 consider only conjugate self-twists that fix $\Lambda$; see Section 12.4 for details.</footnote>
1.2.3. Abstract $p$-adic representations. Both Hida [Hid15] and Lang [Lan16] rely in a key way on results of Pink [Pin93] classifying, for odd $p$, pro-$p$ subgroups of $\text{SL}_2(A)$ in terms of a correspondence with purely algebraically defined “Pink-Lie algebras”. The analogous role in [CIT16] is played by rigid-analytic Lie theory, whence the different form of the conclusion in that case. Although the conclusions of the big-image theorems in all of [Hid15, Lan16, CIT16] are stated in terms of pure algebra — a feature that is most clear in the fullness results of [Hid15] and [Lan16] — nonetheless all of their proofs are arithmetic in nature: they rely on special information about the restriction of $\rho$ to the local Galois group at $p$, and they use the results of Ribet and Momose as input.

In contrast, Bellaïche in [Bel19] studies the image of $\rho: \Pi \to \text{GL}_2(A)$ in a purely algebraic way. More precisely, he applies Pink's theory from [Pin93] to images of 2-dimensional (pseudo)representations with constant determinant, that is, with $\det \rho$ equal to the Teichmüller lift of $\det \bar{\rho}$. Bellaïche's main application is to density results for mod-$p$ modular forms, but along the way he also proves the following theorem, under the same $p$-finiteness assumption on $\Pi$ (Definition 2.6).

**Theorem 1.3** ([Bel19, Theorem 7.2.3]; see also Theorem 2.23). Suppose that the image of $\bar{\rho}$ contains an element with eigenvalues in $\mathbb{F}_{p}^{\times}$ whose ratio is not $\pm 1$. If $\rho$ has constant determinant and is not a priori small, then there is a subring $\mathcal{B}_{\rho}(\mathbb{F}_{p})$ of $A$ such that $\rho$ is $\mathcal{B}_{\rho}(\mathbb{F}_{p})$-full.

Bellaïche's ring $\mathcal{B}_{\rho}(\mathbb{F}_{p})$ is defined as the subring of $A$ topologically generated by a $\mathbb{Z}_{p}$-module $I_{1}(\rho)$ created out of the Pink-Lie algebra of $\text{Im} \rho$. See Theorem 2.23 and the discussion following it for the definition of the generating set $I_{1}(\rho)$ and the ring $\mathcal{B}_{\rho}(\mathbb{F}_{p})$.

Unfortunately, as Bellaïche himself notes, it is not straightforward to relate the ring $\mathcal{B}_{\rho}(\mathbb{F}_{p})$ to the rings $A_{0}$ or $A^{\Sigma_{\rho}}$ from previous results. Indeed, he gives no conceptual interpretation of $\mathcal{B}_{\rho}(\mathbb{F}_{p})$ at all. The goal of the present work is to refine the definition of $\mathcal{B}_{\rho}(\mathbb{F}_{p})$ by enlarging scalars and then give it a conceptual interpretation. Under mild assumptions we thus recover, and in the case of $p$-adic families improve, the results mentioned above in a uniform and purely algebraic way. We point out that prior to Bellaïche's work, Hida's work was the only fullness result when $\bar{\rho}$ is reducible and $\rho$ comes from a $p$-adic family of modular forms. In the case of Coleman families, a true fullness result was not previously known. Additionally, we obtain first results in other $\text{GL}_2$-contexts, including Galois representations attached to Bianchi modular forms and to $p$-adic families of Hilbert modular forms. See Section 12 for all the details.

1.3. Main theorem. We now state our main fullness theorem in more detail. Recall that $A$ is a local pro-$p$ domain with maximal ideal $\mathfrak{m}$ and residue field $\mathbb{F}$ and $\Pi$ is a $p$-finite profinite group. Let $\rho: \Pi \to \text{GL}_2(A)$ be a representation with mod-$\mathfrak{m}$ reduction $\bar{\rho}$. Let $A_0$ be the adjoint trace ring of $\rho$ and $\mathbb{E}$ its residue field, so that $\mathbb{E} \subseteq \mathbb{F} \subseteq \bar{\mathbb{F}}_{\rho}$. We say that $\bar{\rho}$ is *regular* if there is some $g_0 \in \Pi$ such that $\bar{\rho}(g_0)$ has eigenvalues $\lambda_0, \mu_0 \in \bar{\mathbb{F}}^{\times}$ with $\lambda_0 \mu_0^{-1} \in \mathbb{E}^{\times} \setminus \{\pm 1\}$. See Remark 2.20 for an analysis of this condition. For the notion of *goodness*, see Definition 8.10.

**Theorem B** (Main fullness theorem; see also Theorem 10.1).

Assume that $\bar{\rho}$ is regular. If the projective image of $\bar{\rho}$ is isomorphic to $S_4$, assume that $\bar{\rho}$ is good. If $\rho$ is not a priori small, then $\rho$ is $A_0$-full.

In fact we prove something slightly more general in that we can replace $\rho$ by a pseudodeformation $(t, d): \Pi \to A$ of $\bar{\rho}$: see Theorem 10.1 for a precise statement. Recall that Theorem B is provably optimal, in the sense that, if $\rho$ is $B$-full for some subring $B$ of $A$, then a fullness peer of $B$ is contained in $A_0$ (see Theorem A or Theorem 5.3).

Let us point out some features of the statement of Theorem B. First, the group $\Pi$ can be quite general. For that reason, representations coming from Hilbert modular forms and their $p$-adic families are no more difficult than representations coming from elliptic modular forms or their $p$-adic families. Similarly, since $\Pi$ need not be the absolute Galois group of a number field, the notion of oddness does not play a role in the paper. In particular, Theorem B applies to deformations of $\bar{\rho}$ when $\bar{\rho}$ is an even Galois representation.
The proof of Theorem B proceeds via Bellaïche-Pink-Lie theory with various refinements and improvements. As in Bellaïche [Bel19], we linearize the pro-$p$ normal core of the image of a constant-determinant $\rho$ by considering its Pink-Lie algebra $L$. But rather than building up an $A_0$-structure on $L$ directly and interpreting this for $\mathrm{Im}\,\rho$, we proceed by showing fullness in turn for a sequence of fullness peer rings, first $\mathcal{B}_{\rho}(\mathbb{E})$, and then $A^{\Sigma_{\rho}}$, and finally $A_0$, which then proves $A_0$-fullness for all the twists of $\rho$. More precisely, the argument proceeds in steps as follows.

(1) We show that fullness rings and adjoint trace rings are unchanged under twisting by a character (Corollary 3.12 and Proposition 4.8), so that we may assume that $\rho$ has constant determinant.

(2) For constant-determinant $\rho$, we show that $A_0$ and $A^{\Sigma_{\rho}}$ are fullness peers (Corollary 4.21). Therefore it suffices to prove $A^{\Sigma_{\rho}}$-fullness in the constant-determinant case.

(3) We refine Theorem 1.3 to show that a constant-determinant $\rho$ is $\mathcal{B}_{\rho}(\mathbb{E})$-full (Corollary 6.6). Here $\mathcal{B}_{\rho}(\mathbb{E})$ is the $W(\mathbb{E})$-algebra generated by the same $\mathbb{Z}_p$-module $I_1(\rho)$ as Bellaïche's $\mathcal{B}_{\rho}(\mathbb{F}_p)$ — see discussion following Theorem 1.3 above. Although a small improvement, this is crucial for the next step, and it allows our regularity hypothesis to be weaker than that of Bellaïche.

(4) We show that for $\rho$ with constant determinant, $\mathcal{B}_{\rho}(\mathbb{E})$ and $A^{\Sigma_{\rho}}$ are fullness peers (Corollary 9.15), so that $\rho$ is $A^{\Sigma_{\rho}}$-full.

More precisely, we show that $\mathcal{B}_{\rho}(\mathbb{E})$ is always contained in $A^{\Sigma_{\rho}}$, and the discrepancy between them is explained on the residue field by a failure of conjugate self-twists of $\bar{\rho}$ to lift to $\rho$. The residue field of $\mathcal{B}_{\rho}(\mathbb{E})$ is $\mathbb{E} = \mathbb{F}^{\Sigma_{\bar{\rho}}}$, the fixed field of residual twists (Corollary 4.13), whereas the residue field of $A^{\Sigma_{\rho}}$ is $\mathbb{F}^{\Sigma_{\rho}}$, the field fixed by those residual twists that lift to twists of $\rho$. We show both that $A^{\Sigma_{\rho}} = \mathcal{B}_{\rho}(\mathbb{F}^{\Sigma_{\rho}})$ and that the elements of $\mathbb{F}^{\Sigma_{\rho}}$ “missing” in $\mathcal{B}_{\rho}(\mathbb{E})$ show up in its fraction field. This is the most delicate step of the argument.

We do not show that non-constant-determinant $\rho$ are $A^{\Sigma_{\rho}}$-full: Examples 4.3 and 4.10 show that such a result is impossible in general, even under favorable regularity conditions. The key point is that $A^{\Sigma_{\rho}}$ depends on $\rho$ itself and may chance after twisting, whereas $A_0$ depends only on the projective representation of $\rho$. Nor do we show that $A_0 = A^{\Sigma_{\rho}}$ for constant-determinant $\rho$ — they are merely fullness peers: see Example 4.22.

The circuitous nature of our argument may well be merely a matter of our historical bias: the known arithmetic big-image results are formulated in terms of conjugate self-twists, so our original motivation was to relate Bellaïche's $\mathcal{B}_{\rho}(\mathbb{F}_p)$ to $A^{\Sigma_{\rho}}$. The advantages of $A_0$ revealed themselves only later. A more direct argument for proving Theorem B is the subject of our current investigation.

## 1.4. Structure of the paper.
This article is informally organized into parts as follows.

**Background:** Section 2. We review known material: pseudorepresentations, generalized matrix algebras, Pink-Lie theory, and Bellaïche's recent results from [Bel19]. We also introduce our notion of regularity.

**Our philosophy:** In Sections 3 to 5 we present and justify our approach: our goal is to show that $\rho$ is full with respect to the adjoint trace ring $A_0$, which is both the optimal fullness result and equivalent to the historically familiar $A^{\Sigma_{\rho}}$-fullness of known applications.

- Section 3 discusses the basic properties of fullness and fullness peer rings. We prove that fullness is twisting-invariant (Corollary 3.12).

- Section 4 studies the relationship between $A_0$ and conjugate self-twists, paying particular attention to constant-determinant, and nearly so, settings where conjugate self-twists carve out the fraction field of $A_0$ (Corollary 4.20), so that $A_0$ and $A^{\Sigma_{\rho}}$ are fullness peers.

- Section 5 proves two related optimality results: every fullness ring has a fullness peer contained in $A_0$ (Corollary 5.2) and is also fixed by all conjugate self-twists (Theorem 5.4).
**Technical results:** Sections 6 to 9 are the technical heart of the paper. We prove that constant-determinant $\rho$ are $\mathcal{B}_{\rho}(\mathbb{E})$-full, which implies $A^{\Sigma_{\rho}}$-fullness, which in turn guarantees $A_0$-fullness.

- In Section 6 we show that the Bellaïche-Pink-Lie algebra $L$ attached to $\rho$, a priori only a $\mathbb{Z}_p$-module, is in fact a $W(\mathbb{E})$-module. Under certain regularity assumptions, $\rho$ is therefore $\mathcal{B}_{\rho}(\mathbb{E})$-full. This completes Step (3) of the proof.

- In Section 7 we study residual conjugate self-twists and their lifting properties via the universal deformation ring — how does $\Sigma_{\bar{\rho}}$ compare to $\Sigma_{\rho}$?

- Section 8 explores how the regularity of $\bar{\rho}$ imposes structure on residual conjugate self-twists in preparation for Section 9. We also introduce the *goodness* constraint.

- Section 9 is the technical heart of the technical heart of the paper. Its main goal is to prove that $\mathcal{B}_{\rho}(\mathbb{E})$ has the same field of fractions as $A^{\Sigma_{\rho}}$ to show that they are fullness peers (Corollary 9.15), completing Step (4) of the proof. This turns out to be intimately related to the lifting properties of conjugate self-twists of $\bar{\rho}$ to $\rho$ explored in Section 7.

**Interpretation and applications:** Sections 10 to 12 interpret our results for general $\rho$ and explain how to apply them to various modular form contexts in detail.

- In Section 10 we derive our main fullness results: Theorem 10.1 or Theorem 10.3.

- Section 11 is independent of the main thrust of the paper; it gives an improvement on previous very-big-image results, showing that the image of $\rho$ contains $\text{SL}_2(A_0)$ if it does so residually.

- Section 12 explains in detail how and to what extent Theorem B recovers and refines known big-image results about Galois representations arising from modular forms and their $p$-adic families. We also apply Theorem B to obtain new results for Galois representations attached to Bianchi modular forms (Section 12.3), Coleman families of elliptic modular forms (Section 12.5), and $p$-adic families of Hilbert modular forms (Section 12.6).

**Appendix:** Appendix A houses a variety of lemmas on representation theory and commutative algebra for which we failed to find convenient references in the literature, in particular the statement that semisimple representations with isomorphic adjoints are isomorphic up to twist (Theorem A.10). No claims to originality here.

**1.5. Leitfaden suggestions.** We propose different levels of interaction with this paper for different readers. Those who are merely interested in results and applications should read the present Section 1 and skip to Section 12. Those also curious about our methods should additionally skim Section 2, and then read Sections 3 to 5 and Section 10. Readers who have the stomach for the technical weeds should begin the same way, but then also brave Sections 6 to 9. Finally, any of the previous types of readers who are interested in very-big-image results (Im $\rho$ contains an $\text{SL}_2$ if it does so residually) should peek at Section 11.

**1.6. Notation.** We establish some notation and conventions. All rings are unital. Given any ring $R$ (not necessarily commutative), we will let $R^\times$ denote the multiplicative group of invertible elements in $R$. For brevity, we call a commutative ring $B$ containing a subring $C$ an *extension of C*, *finite* if $B$ is module-finite as a $C$-algebra. If $B$ is a domain, then $Q(B)$ denotes its field of fractions. If $F$ is a field, then $\bar{F}$ (respectively, $F^{\text{sep}}$) denotes a fixed algebraic (respectively, separable) closure.

If $C \subseteq B$ is an extension of rings, write $\text{Aut}(B)$ for the group of ring automorphisms of $B$ and $\text{Aut}(B/C)$ for the subgroup fixing $C$ pointwise. For $\Sigma \subseteq \text{Aut}(B)$, write $B^\Sigma$ for the subring pointwise fixed by every $\sigma \in \Sigma$. If $\sigma \in \text{Aut}(B)$ and $f: X \to B$ is any set map, write $\sigma f: X \to B$ for the map $x \mapsto \sigma(f(x))$.

For any integer $n$, let $\zeta_n$ denote a primitive $n^{\text{th}}$ root of unity. Given a finite field $\mathbb{F}$ of size $q$ a power of a prime $p$, its ring of Witt vectors, isomorphic to $\mathbb{Z}_p[\zeta_{q-1}]$, will be denoted by $W(\mathbb{F})$. Let $s: \mathbb{F}^\times \to W(\mathbb{F})^\times$ be the Teichmüller lift. We extend it to $s: \mathbb{F} \to W(\mathbb{F})$ by defining $s(0) := 0$. If $A$
is a W(ℝ)-algebra, we use W(ℝ) to denote the image of W(ℝ) in A under the structure map. In particular, s may be viewed as being A-valued by composing with the structure map.

Throughout the paper we fix a prime $p \neq 2$. The ring $A$ will always denote a local pro-$p$ commutative ring with maximal ideal $\mathfrak{m}$ and residue field $\mathbb{F}$, which is then automatically a finite extension of $\mathbb{F}_p$. A closed subring of such an $A$ is also automatically local and pro-$p$. (vi) Since 2 is invertible in $A$, we can always take square roots of elements $x \in 1 + \mathfrak{m}$ via the formula

$$ \sqrt{x} := \sum_{n=0}^{\infty} \binom{1/2}{n} (x-1)^n. $$

In particular, when we write $\sqrt{x}$, we always choose the root congruent to 1 modulo $\mathfrak{m}$. In general, the profinite topology on $A$ is coarser than the $\mathfrak{m}$-adic topology, but if such an $A$ is noetherian, then the profinite topology coincides with the $\mathfrak{m}$-adic topology, so that $A$ is a complete local noetherian ring [dSL97, Proposition 2.4]. In this case, every finite $A$-module is equipped with its natural $\mathfrak{m}$-adic $A$-module topology, which is compatible on submodules by the Artin-Rees lemma — see [AM69, Theorem 10.11]. In particular, every ideal in $A$ is closed. By the Cohen structure theorem [Eis95, Theorem 7.7], if $A$ is noetherian then it is a quotient of $W(ℝ)\llbracket x_1, \dots, x_n \rrbracket$ for some $n$. If $A$ is additionally a domain then $A$ enjoys the so-called $N2$ (or sometimes “japanese”) property: the integral closure of $A$ in a finite extension of its field of fractions is finite over $A$ [Mat70, Chapter 12, proof of Corollary 2], and hence a pro-$p$ local noetherian domain in its own right. Note that in this case there need not be a topology on $Q(A)$ under which $Q(A)$ is a topological field containing $A$ as a closed subring: since $A$ is compact, the existence of any such topology would mean that $Q(A)$ is a locally compact field, so that $Q(A)$ is a finite extension of $\mathbb{F}_p$, $Q_p$, or $\mathbb{F}_p((X))$ [RV91, Theorem 4.12]. But our $A$ are more general.

If $M$ is a subset of a $W(ℝ)$-module $N$, then we will write $W(ℝ)M$ for the $W(ℝ)$-linear span of $M$ in $N$. When $R$ is a topological ring and $S$ a subring of $R$, we say that $S$ is topologically generated by a set $X$ if $S$ is the smallest closed subring of $R$ containing $X$. Similarly, we can talk about an additive subgroup or a $W(ℝ)$-algebra topologically generated by a set.

Finally, $\Pi$ always denotes a $p$-finite profinite group (Definition 2.6). If $\bar{\rho}: \Pi \to GL_2(ℝ)$ is a representation over a finite field $F$, we can compose $\bar{\rho}$ with the natural projection $\mathbb{P}: GL_2(ℝ) \to PGL_2(ℝ)$. We shall refer to the image of $\Pi$ under the composition $\mathbb{P} \circ \bar{\rho}$ as the projective image of $\bar{\rho}$. It is well known that the projective image of $\bar{\rho}$ is cyclic, dihedral, or isomorphic to $A_4$, $S_4$, $A_5$ or one of $PSL_2(\mathbb{F}')$ or $PGL_2(\mathbb{F}')$ for some subfield $\mathbb{F}'$ of $F$ ([Dic58, Chapter XII] or see [Bel19, Section 3.1]). If $\mathbb{P}\bar{\rho}(\Pi) \cong A_4$ (respectively, $S_4$, $A_5$), we say that $\bar{\rho}$ is tetrahedral (respectively, octahedral, icosahedral). If $\bar{\rho}$ is tetrahedral, octahedral, or icosahedral, then we say that $\bar{\rho}$ is exceptional. If $\mathbb{P}\bar{\rho}(\Pi)$ contains $PSL_2(\mathbb{F}_p)$ and $\bar{\rho}$ is not exceptional, then we say that $\bar{\rho}$ is large. Be warned that there are exceptional isomorphisms $PSL_2(\mathbb{F}_3) \cong A_4$, $PGL_2(\mathbb{F}_3) \cong S_4$, $PSL_2(\mathbb{F}_5) \cong A_5$.

## 2. BELLAÏCHE-PINK-LIE THEORY

In this section we introduce the basic objects of study — 2-dimensional pseudorepresentations and their associated realizations over generalized matrix algebras — along with the primary tools we use to study them: Pink-Lie algebras and Bellaïche’s structure theorem (Theorem 2.23). The main reference for everything in this section is Bellaïche’s paper [Bel19], which we refer to for most proofs. The only exception is that our definition of regularity (Definition 2.19) is weaker than that of Bellaïche.

<footnote>(vi) Let $B$ be closed subring of $A$. Its ideal $\mathfrak{m}_B := B \cap \mathfrak{m}$ is maximal since $\mathbb{F}_B := B/\mathfrak{m}_B$ is a subring of $F$. Given any $\alpha \in A$ one can check that the sequence $\{\alpha^{p^n}\}_n$ converges and that its limit is $s(\bar{\alpha})$, the Teichmüller lift of the image of $\alpha$ in $F$. Since $B$ is closed, it thus contains both $s(\mathbb{F}_B^\times)$ and inverses of elements of $1 + \mathfrak{m}_B$. Therefore every element of $B - \mathfrak{m}_B$ is invertible, and $B$ is local. And as a closed subgroup of a pro-$p$ group, $B$ is automatically pro-$p$.</footnote>
**2.1. Pseudorepresentations.** In this section we summarize the definitions and notation related to two-dimensional pseudorepresentations, algebraic gadgets introduced by Chenevier in [Che14] (where they are called “determinants”) to mimic the behavior of trace and determinant functions of true 2-dimensional representation of groups. We follow [Che14, Example 1.8] and [Bel19, 2.1.1] for our definitions.

### 2.1.1. Abstract pseudorepresentations.

**Definition 2.1.** A (2-dimensional) pseudorepresentation of a group $G$ over a commutative ring $B$ is a pair of functions $t: G \to B$ and $d: \Pi \to B^\times$ such that

(1) $t(1) = 2;$

(2) $d(gh) = d(g)d(h)$ for all $g, h \in G;$

(3) $t(gh) = t(hg)$ for all $g, h \in G;$

(4) $t(gh) + d(h)t(gh^{-1}) = t(g)t(h)$ for all $g, h \in G.$

If $G$ is a topological group and $B$ is a topological ring, we say that a pseudorepresentation $(t, d): G \to B$ is continuous if $t$ and $d$ are continuous maps.

One can verify that if $\rho: G \to GL_2(B)$ is a (continuous) representation, then $(\text{tr} \rho, \text{det} \rho)$ is a (continuous) 2-dimensional pseudorepresentation. Conversely, if $B$ is an algebraically closed field, then every pseudorepresentation $(t, d): G \to B$ is carried by a unique semisimple representation $G \to GL_2(B)$. If $2$ is invertible in $B$, then a 2-dimensional pseudorepresentation $(t, d)$ is determined by $t$ alone: setting $h = g$ in (3) above yields $d(g) = \frac{t(g)^2-t(g^2)}{2}$. In this way, pseudorepresentations generalize earlier work of Wiles [Wil90], Taylor [Tay91], and Rouquier [Rou96] on pseudocharacters, which mimic representations by keeping track only of a trace function.

If $(t, d): \Pi \to B$ is a (continuous) pseudorepresentation and $\chi: \Pi \to B^\times$ is a (continuous) character, then $(\chi t, \chi^2 d)$ is also a (continuous) pseudorepresentation, called the *twist of* $(t, d)$ by $\chi$. We say $(t, d)$ is *reducible* if $t = \chi_1 + \chi_2$ with $\chi_i: \Pi \to B^\times$ characters. Otherwise $(t, d)$ is *irreducible*. We say $(t, d)$ is *dihedral* if it is irreducible and there is a nontrivial character $\eta: \Pi \to B^\times$ such that $(\eta t, \eta^2 d) = (t, d)$.

The kernel of a pseudorepresentation $(t, d): G \to B$ is

$$ \ker(t, d) := \{g \in G : d(g) = 1 \text{ and } t(gx) = t(x) \text{ for all } x \in G\} \subseteq G. $$

This is a normal subgroup of $G$, closed if $(t, d)$ is continuous. Moreover, $(t, d)$ factors through the quotient $\Pi/\ker(t, d)$.

**2.1.2. Pseudorepresentations of profinite groups over pro-$p$ rings.** We henceforth assume that all pseudorepresentations are continuous if both the group and the ring have topologies. Recall that $A$ is a local pro-$p$ commutative ring with maximal ideal $\mathfrak{m}$ and residue field $\mathbb{F}$ and $\Pi$ is a profinite group.

If $(t, d): \Pi \to A$ is a pseudorepresentation, its *residual* pseudorepresentation $(\bar{t}, \bar{d}) : \Pi \to \mathbb{F}$ is obtained by composing $(t, d)$ with reduction modulo $\mathfrak{m}$. Because the Brauer group of $\mathbb{F}$ is trivial, the semisimple representation $\bar{\rho}$ carrying $(\bar{t}, \bar{d})$ is always realizable over $\mathbb{F}$ (see also Lemma A.12).

**Definition 2.2.** A pseudorepresentation $(t, d): \Pi \to A$ has *constant determinant* if $d$ is the Teichmüller lift of its reduction modulo $\mathfrak{m}$: that is, $d = s(\bar{d})$. Since $A^\times \cong s(\mathbb{F}^\times) \times 1 + \mathfrak{m}$, we see that $d$ is always the product of $s(\bar{d})$ and a pro-$p$ character $d_1: \Pi \to 1 + \mathfrak{m}$. The twist of $(t, d)$ by $d_1^{-1/2}$ is the unique constant-determinant pseudorepresentation with the same $\bar{\rho}$: this is the *constant determinant twist* of $(t, d)$.

If $(t, d): \Pi \to A$ is a pseudorepresentation, we call the subring of $A$ topologically generated by $t(\Pi)$ the *trace ring* of $(t, d)$. Note that the residue field of the trace ring is the trace ring of $(\bar{t}, \bar{d})$ or $\bar{\rho}$. Sometimes we need the residue field $\mathbb{F}$ of $A$ to be a quadratic extension of the trace ring of
($\bar{t}, \bar{d}$). Thus if $(t, d): \Pi \to A$ is a pseudorepresentation and $\mathbb{F}$ is the residue field of $A$, we define the *trace algebra* of $(t, d)$ to be the $W(\mathbb{F})$-subalgebra of $A$ topologically generated by $t(\Pi)$. Thus the residue field does not change when restricting the codomain of a pseudorepresentation to its trace algebra. Both the trace ring and the trace algebra are pro-$p$ local rings (Footnote (vi)).

**Definition 2.3.** We call a pseudorepresentation $(t, d): \Pi \to A$ is a priori small if it is reducible or dihedral, or if the kernel of its constant-determinant twist is equal to the kernel of $\bar{\rho}$.

If $A$ is a domain it turns out that the a priori small notion coincides with a certain weak kind of reducibility. Recall that a representation $\rho: G \to GL_2(F)$ over a field $F$ is strongly (absolutely) irreducible if $\rho|_H$ is (absolutely) irreducible for any finite-index subgroup $H$ of $G$.

**Proposition 2.4.** Let $(t, d): \Pi \to A$ be a pseudorepresentation to a local pro-$p$ domain $A$ with field of fractions $K$, and let $\rho: \Pi \to GL_2(\bar{K})$ be the semisimple representation carrying $(t, d)$. The following are equivalent:

(1) $(t, d)$ is a priori small;

(2) $\rho$ is reducible, dihedral, or its constant-determinant twist has finite image;

(3) $\rho$ is not strongly irreducible.

Note that if any twist of $\rho: \Pi \to GL_2(\bar{K})$ has finite image, then the constant-determinant twist does; equivalently, the image of the projective representation $\mathbb{P}\rho: \Pi \to PGL_2(\bar{K})$ is finite.

*Proof*. Since all the notions in question are twist-invariant, we may replace $(t, d)$ and $\rho$ by constant-determinant twists. Clearly (1) $\implies$ (2) $\implies$ (3). To see that (3) $\implies$ (2), we follow [Rib75, Theorem 2.3]. Suppose $\rho|_H$ reducible for some finite-index subgroup $H$ of $\Pi$. Up to replacing $H$ with its normal core (i.e., the intersection of all conjugates, which is still of finite index in $\Pi$), we may assume that $H$ is normal in $\Pi$, so that by Clifford's theorem [Cra19, Theorem 7.1.1], $\rho|_H$ is semisimple, and hence has abelian image. If $\rho(H)$ is not contained in the center of $GL_2(\bar{K})$ then $H$ contains a semisimple element $h$ with distinct $\rho$-eigenvalues, and $\rho(H)$ is contained in the maximal torus centralizing $\rho(h)$. Moreover, since $H$ is normal in $\Pi$, all of $\rho(\Pi)$ is contained in the normalizer of $\rho(h)$, so that $\rho(H)$ has index 1 or 2 in $\rho(\Pi)$, and $\rho$ is either reducible or dihedral. On the other hand, if $\rho|_H$ is scalar, then since its trace is $A$-valued, $\rho|_H = \alpha \oplus \alpha$ for some character $\alpha: H \to A^\times$; since $\rho$ has constant determinant, $\alpha^2 = s(\bar{\alpha}^2)$, so that $\rho|_H$ takes values in the finite set of prime-to-$p$ roots of unity in $A$, whence $\rho$ has finite image.

For (2) $\implies$ (1) it suffices to consider $(t, d)$ residually exceptional or large, in which case by Rouquier-Nyssen $\rho$ is the base change of a representation $\rho_A: \Pi \to GL_2(A)$ with $\ker \rho_A = \ker(t, d)$ (see also Proposition 2.11 and Remark 2.12). We show that if $\rho$ has finite image, then reduction modulo the pro-$p$ subgroup $1 + M_2(\mathfrak{m})$ induces an isomorphism $\rho(\Pi) \cong \bar{\rho}(\Pi)$. If $A = \mathbb{F}$ there is nothing to show, so we can assume that $A$ is infinite. (Indeed, if $A$ has characteristic zero, then $A$ contains $\mathbb{Z}_p$; otherwise, $A$ is a local $\mathbb{F}$-algebra with residue field $\mathbb{F}$, and any such ring that is also finite over $\mathbb{F}$ is equal to $\mathbb{F}$.) We claim that the projective image of $\rho$ is isomorphic to that of $\bar{\rho}$: If $A$ has characteristic zero, then the finite subgroups of $GL_2(K)$ are the same as those of $GL_2(\mathbb{C})$ [Ser72, Proposition 16], hence the projective image of $\rho$ is isomorphic to $A_4, S_4$ or $A_5$. And if $A$ has characteristic $p$, then the finite subgroups of $GL_2(A)$ are all defined over $\mathbb{F}$, because the eigenvalues of any finite-order element are roots of unity. In any case, the kernel of the reduction map $\rho_A(\Pi) \to \bar{\rho}(\Pi)$ is pro-$p$, so that it can be seen on the map $\mathbb{P}\rho_A(\Pi) \to \bar{\rho}(\Pi)$. But none of $A_4, S_4, A_5, PSL_2(\mathbb{F}')$, or $PGL_2(\mathbb{F}')$ have normal subgroups of $p$-power order for $p > 2$. $\square$

2.1.3. *Pseudodeformations*. Fix a continuous semisimple representation $\bar{\rho}: \Pi \to GL_2(\mathbb{F})$.

**Definition 2.5.** We say that a pseudorepresentation $(t, d): \Pi \to A$ is a pseudodeformation of $\bar{\rho}$ if $(t, d) = (\operatorname{tr} \bar{\rho}, \det \bar{\rho}) \bmod \mathfrak{m}$.
Let $\mathcal{C}$ be the category of local pro-$p$ commutative rings with residue field $\mathbb{F}$, which have a natural $W(\mathbb{F})$-algebra structure, and with morphisms being local continuous $W(\mathbb{F})$-algebra homomorphisms. We are interested in the deformation functors

$$F: \mathcal{C} \to \underline{\text{SET}}$$
$$A \mapsto \{(t, d) : \Pi \to A \text{ pseudodeformation of } \bar{\rho}\}.$$

and

$$G: \mathcal{C} \to \underline{\text{SET}}$$
$$A \mapsto \{(t, d) \in F(A) : d = s(\det \bar{\rho})\}.$$

These functors are always representable. In order for the representing ring to be noetherian, we need to impose a finiteness condition on $\Pi$ due to Mazur, which we now recall.

**Definition 2.6.** [Maz89, §1.1] A profinite group $\Pi$ satisfies the $p$-finiteness condition or is $p$-finite if, for every open subgroup $\Pi_0$ of $\Pi$, the set $\text{Hom}(\Pi_0, \mathbb{F}_p)$ is finite.

It is well known that $F$ is represented by a pro-$p$ local noetherian $W(\mathbb{F})$-algebra $\tilde{\mathcal{A}}$ whenever $\Pi$ is a $p$-finite profinite group. See, for example, [Che14, Proposition 3.3] or [Böc13, Proposition 2.3.1]. In particular, the trace algebra of any pseudorepresentation of a $p$-finite profinite group on a local pro-$p$ ring is a quotient of $\tilde{\mathcal{A}}$ and hence noetherian. Let $(t^{\text{univ}}, d^{\text{univ}}): \Pi \to \tilde{\mathcal{A}}$ be the universal pseudodeformation of $\bar{\rho}$. It is easy to see that the constant-determinant condition is a deformation condition. Indeed, let $\mathfrak{a}$ be the ideal of $\tilde{\mathcal{A}}$ topologically generated by $\{d^{\text{univ}}(g) - s(\det \bar{\rho}(g)): g \in \Pi\}$. Then $\mathcal{A} := \tilde{\mathcal{A}}/\mathfrak{a}$ represents $G$. In particular, $\mathcal{A}$ is also a pro-$p$ local noetherian $W(\mathbb{F})$-algebra with residue field $\mathbb{F}$. We use $(T, d): \Pi \to \mathcal{A}$ to denote the universal constant-determinant pseudodeformation.

**Definition 2.7.** If $\mathbb{F}'$ is a subfield of $\mathbb{F}$, then we say that a 2-dimensional semisimple representation $\bar{\rho}$ is multiplicity free over $\mathbb{F}'$ if either $\bar{\rho}$ is absolutely irreducible or $\bar{\rho} \cong \chi_1 \oplus \chi_2$ such that $\chi_1, \chi_2: \Pi \to \mathbb{F}'^\times$ are distinct characters.

The following notion of admissibility, introduced by Bellaïche, plays a central role in [Bel19].

**Definition 2.8.** [Bel19, Section 5.2] A tuple $(\Pi, \bar{\rho}, t, d)$ is an admissible pseudodeformation over $A$ if the following conditions are satisfied:

(1) $\Pi$ is a $p$-finite profinite group;

(2) $\bar{\rho}: \Pi \to \text{GL}_2(\mathbb{F})$ is a continuous representation that is multiplicity free over $\mathbb{F}$;

(3) $(t, d): \Pi \to A$ is a continuous pseudodeformation of $\bar{\rho}$;

(4) $d(g) \in s(\mathbb{F}^\times)$ for all $g \in \Pi$, that is, $(t, d)$ has constant determinant;

(5) $A$ is the trace algebra of $(t, d)$.

A local pro-$p$ $A$ accepting an admissible pseudodeformation is a complete noetherian local ring.

2.2. **GMAs and $(t, d)$-representations.** It is natural to ask when a given pseudodeformation $(t, d): \Pi \to A$ arises as the trace and determinant of an actual representation $\rho: \Pi \to \text{GL}_2(A)$. This has been studied in great generality; see the introduction of Chenevier's paper [Che14] for a thorough history. Bellaïche and Chenevier [BC09, Section 1.4] have shown that, under the residual multiplicity-free assumption, $(t, d)$ always comes from a representation if one allows something more general than matrix algebras for the target. In Section 2.2 we summarize Bellaïche's [Bel19, Section 2], where he specializes his work with Chenevier to the 2-dimensional setting. All proofs that can be found in Bellaïche's work are omitted.
**Definition 2.9.** A generalized matrix algebra (GMA) over a commutative ring $A$ is given by a tuple of data $(A, B, C, m)$, where $B$ and $C$ are $A$-modules, $m: B \otimes_A C \to A$ is a morphism of $A$-modules satisfying

$$m(b \otimes c)b' = m(b' \otimes c)b \text{ and } m(b \otimes c')c = m(b \otimes c)c' \text{ for all } b, b' \in B, c, c' \in C.$$

Given such data, define the $A$-module $R := A \oplus B \oplus C \oplus A = (A_B)$ and give $R$ a multiplicative structure via

$$(\begin{pmatrix} a & b \\ c & d \end{pmatrix}) (\begin{pmatrix} a' & b' \\ c' & d' \end{pmatrix}) := \begin{pmatrix} aa' + m(b \otimes c') & ab' + bd' \\ a'c + dc' & dd' + m(b' \otimes c) \end{pmatrix} \quad \text{for } a, a', d, d' \in A, b, b' \in B, c, c' \in C,$$

so that $R$ has the structure of an $A$-algebra via the ring homomorphism $a \mapsto (a_0^0) \in R$. We refer to the GMA given by $(A, B, C, m)$ simply by $R$.

A morphism of GMAs $(A, B, C, m) \to (A', B', C', m')$ (with associated $A$-algebras $R$ and $R'$) is a triple $(f_A, f_B, f_C)$ consisting of a ring morphism $f_A: A \to A'$ and two $A'$-module morphisms $f_B: B \otimes_{A, f_A} A' \to B'$, $f_C: C \otimes_{A, f_A} A' \to C'$ such that $f_A \circ m = m' \circ (f_B \otimes f_C)$. The data $(f_A, f_B, f_C)$ defines in a natural way an $A$-algebra morphism $\psi: R \to R'$; we say that $\psi$ is associated with $(f_A, f_B, f_C)$.

If $A$ is a topological ring and $B, C$ are topological $A$-modules, then $R$ inherits a natural topology, and we call $R$ a topological GMA if $m$ is continuous. We say that $R$ is faithful if $m$ is nondegenerate as a pairing of $A$-modules. As with matrix algebras, we have the notion of a trace and determinant on a GMA $R$ given by $\text{tr}(\begin{pmatrix} a & b \\ c & d \end{pmatrix}) = a + d$ and $\text{det}(\begin{pmatrix} a & b \\ c & d \end{pmatrix}) = ad - m(b \otimes c)$.

The following lemma shows that when $A$ is a domain, faithful GMAs can be embedded into a matrix algebra over the field of fractions of $A$.

**Lemma 2.10.** [Bel19, Lemmas 2.2.2, 2.2.3] Assume that $A$ is a domain with field of fractions $K$ and that $R = (A_C^B)$ is a faithful GMA over $A$. Then there exist embeddings of $A$-modules $B, C \hookrightarrow K$ such that (identifying $B, C$ with their images in $K$), $m: B \otimes_A C \to A$ is induced by multiplication in $K$. In particular, if $BC \neq 0$, then $R \otimes_A K$ is isomorphic over $K$ as a GMA to $M_2(K)$.

We recall the following result of Bellaïche, which explains that any residually multiplicity-free pseudorepresentation can be realized as the trace of a GMA-valued representation.

**Proposition 2.11.** [Bel19, Proposition 2.4.2] Let $\bar{\rho}: \Pi \to \text{GL}_2(\mathbb{F})$ be multiplicity free over $\mathbb{F}$. Let $(t, d): \Pi \to A$ be a pseudodeformation of $\bar{\rho}$.

(1) There exists a faithful GMA $R$ over $A$ and a morphism of groups $\rho: \Pi \to R^\times$ such that $\text{tr} \rho = t$, $\text{det} \rho = d$, and $A\rho(\Pi) = R$.

(2) If $(\rho, R)$ and $(\rho', R')$ are as in (1), then there is a unique isomorphism of $A$-algebras $\Psi: R \to R'$ such that $\Psi \circ \rho = \rho'$.

(3) If $g_0 \in \Pi$ such that $\bar{\rho}(g_0)$ has distinct eigenvalues $\lambda_0, \mu_0 \in \mathbb{F}^\times$, then there exists $(\rho, R)$ as in (1) such that $\rho(g_0)$ is diagonal and $\rho(g_0) \equiv \begin{pmatrix} \lambda_0 & 0 \\ 0 & \mu_0 \end{pmatrix} \mod \mathfrak{m}$.

(4) If $g_0 \in \Pi$ and $(\rho, R), (\rho', R')$ are as in (3), then the unique isomorphism of $A$-algebras $\Psi: R \to R'$ such that $\Psi \circ \rho = \rho'$ is associated with an isomorphism of GMAs.

(5) If $\bar{\rho}$ is irreducible and $(\rho, R)$ is as in (1), then $R = (A, B, C, m, R)$ is isomorphic to $M_2(A)$ as a GMA over $A$. If $\bar{\rho}$ is reducible, then $BC \subseteq \mathfrak{m}$.

(6) If $(\rho, R)$ are as in (1), then $\text{ker} \rho = \text{ker}(t, d)$.

(7) Assume that $A$ is noetherian and $\Pi$ is $p$-finite. If $(t, d)$ is continuous, then for $(\rho, R)$ as in (1), $R$ is of finite type as an $A$-module. If $R$ is given its unique topology as an $A$-algebra, then $\rho$ is continuous.
Remark 2.12. When $\bar{\rho}$ is absolutely irreducible, Proposition 2.11 allows us to identify the GMA $R$ with the matrix algebra $M_2(A)$. We follow Bellaïche in always implicitly making such an identification. In particular, in the dihedral case, elements of $B$ and $C$ are viewed as elements of $A$. $\triangle$

Following Bellaïche, we make the following definitions.

**Definition 2.13.** [Bel19, Definition 2.4.3] A representation $\rho: \Pi \to R^\times$ satisfying condition (1) in Proposition 2.11 is called a $(t, d)$-representation. If in addition $\rho$ satisfies condition (3), then we say that $\rho$ is adapted to $(g_0, \lambda_0, \mu_0)$.

In fact, it is often useful to have the following strengthening of Proposition 2.11(3).

**Lemma 2.14.** Let $\bar{\rho}: \Pi \to GL_2(\mathbb{F})$ be multiplicity free over $\mathbb{F}$ and $\lambda_0 \neq \mu_0 \in \mathbb{F}^\times$ be the eigenvalues of an element in $\mathrm{Im} \bar{\rho}$. Let $(t, d): \Pi \to A$ be a pseudodeformation of $\bar{\rho}$. Then there exists $g_0 \in \Pi$ and a $(t, d)$-representation $\rho$ adapted to $(g_0, \lambda_0, \mu_0)$ such that $\rho(g_0) = \begin{pmatrix} s(\lambda_0) & 0 \\ 0 & s(\mu_0) \end{pmatrix}$.

*Proof.* Let $g'_0 \in \Pi$ be any element such that $\bar{\rho}(g'_0)$ has eigenvalues $\lambda_0, \mu_0$. Then Proposition 2.11(3) guarantees the existence of a $(t, d)$-representation $\rho: \Pi \to R^\times$ adapted to $(g'_0, \lambda_0, \mu_0)$. By [Bel19, Theorem 6.2.1], it follows that $\begin{pmatrix} s(\lambda_0) & 0 \\ 0 & s(\mu_0) \end{pmatrix} \in \mathrm{Im} \rho$. Let $g_0$ be any element in $\rho^{-1}(\begin{pmatrix} s(\lambda_0) & 0 \\ 0 & s(\mu_0) \end{pmatrix})$. Then $\rho$ is a $(t, d)$-representation adapted to $(g_0, \lambda_0, \mu_0)$ and $\rho(g_0) = \begin{pmatrix} s(\lambda_0) & 0 \\ 0 & s(\mu_0) \end{pmatrix}$. $\square$

**2.3. Pink-Lie algebras.** In Section 2.3 we recall Pink’s theory relating pro-$p$ subgroups of $SL_2(A)$ to closed Lie subalgebras of $sl_2(A)$ [Pin93]. In fact, we use Bellaïche’s generalization to GMAs [Bel19, Section 4].

Recall that $A$ is a local pro-$p$ ring with $p \neq 2$. The assumption that $p \neq 2$ is critical for Pink’s theory. We denote by $\mathfrak{m}$ the maximal ideal of $A$. Fix a compact topological GMA $R = \begin{pmatrix} A & B \\ C & A \end{pmatrix}$ over $A$. (The compactness condition is satisfied, for instance, when $R$ is finite as an $A$-module.) Write

$$SR^\times := \{r \in R^\times : \det r = 1\}.$$

Let $\mathrm{rad} R$ be the Jacobson radical of $R$, and $R^1 := 1 + \mathrm{rad} R$. We let $SR^1 := SR^\times \cap R^1$, which is a closed normal pro-$p$ subgroup of $R^\times$. See [Bel19, Remark 4.2.1] for an explicit description of these objects. We mention here that in the case when $BC = A$ there is by [Bel19, Lemma 2.2.1] an isomorphism of GMAs $R \cong M_2(A)$ that we can use to identify $\mathrm{rad} R$ with $\mathfrak{m}M_2(A)$ and $R/\mathrm{rad} R$ with $M_2(\mathbb{F})$, while if $BC \subset \mathfrak{m}$ then $\mathrm{rad} R = \begin{pmatrix} \mathfrak{m} & B \\ C & \mathfrak{m} \end{pmatrix}$ and $R/\mathrm{rad} R = \begin{pmatrix} \mathbb{F} & 0 \\ 0 & \mathbb{F} \end{pmatrix}$.

Given any subset $S$ of $R$, we write

$$S^0 := \{s \in S : \mathrm{tr} s = 0\}.$$

Then $(\mathrm{rad} R)^0$ has a Lie algebra structure with bracket given by $[r_1, r_2] := r_1 r_2 - r_2 r_1$.

For any topological group $G$ and closed subgroup $H$ of $G$, write $(G, H)$ for the smallest closed subgroup of $G$ containing $\{g^{-1}h^{-1}gh : g \in G, h \in H\}$. Fix a closed subgroup $\Gamma \subseteq SR^1$. Recall that the lower central series of $\Gamma$ is defined by $\Gamma_1 := \Gamma$ and define $\Gamma_{n+1} := (\Gamma, \Gamma_n)$. We describe how Pink associates a filtration of Lie algebras to $\Gamma$ [Pin93, Section 2].

Define a function

$$\Theta: R^\times \to R^0$$

$$r \mapsto r - \frac{\mathrm{tr} r}{2},$$

where $(\mathrm{tr} r)/2$ is regarded as a scalar via the structure morphism $A \to R$. Let $L(\Gamma) = L_1(\Gamma)$ be the (additive) subgroup of $(\mathrm{rad} R)^0$ topologically generated by $\Theta(\Gamma)$. For $n \geq 2$, define $L_n(\Gamma)$ recursively as the subgroup of $(\mathrm{rad} R)^0$ topologically generated by the set

$$\{xy - yx : x \in L_1(\Gamma), y \in L_{n-1}(\Gamma)\}.$$
Although the $L_n(\Gamma)$ are a priori only subgroups of $(\text{rad } R)^0$, Pink shows that they are closed under Lie brackets and form a descending filtration, as summarized in the following proposition, which is due to Pink when $R = M_2(A)$ [Pin93, Proposition 3.1, Proposition 2.3] and to Bellaïche in the GMA case [Bel19, Proposition 4.7.1].

**Proposition 2.15.** For all $n \ge 1$, we have $L_{n+1}(\Gamma) \subseteq L_n(\Gamma)$. In particular, each $L_n(\Gamma)$ is a Lie subalgebra of $(\text{rad } R)^0$.

We emphasize that, a priori, each $L_n(\Gamma)$ is just a $\mathbb{Z}_p$-module, even if the ring $A$ is very large. The point of Section 6 is to prove that, under mild conditions, $L_n(\Gamma)$ is in fact an algebra over an (in general) much larger ring.

Conversely, given a closed Lie subalgebra $L$ of $(\text{rad } R)^0$, define $H(L) := \Theta^{-1}(L) \cap SR^1$. Let $H_n := H(L_n(\Gamma))$. A priori, $H(L)$ is only a subset of $SR^1$. However, we have the following theorem of Pink [Pin93, Proposition 2.4, Theorem 2.7], which was generalized to GMAs by Bellaïche [Bel19, Theorem 4.7.3].

**Theorem 2.16.** We have that $H_n$ is a pro-$p$ subgroup of $SR^1$. Furthermore, $\Gamma$ is a normal subgroup of $H_1$, and $H_1/\Gamma$ is abelian. For $n \ge 2$, we have $H_n = \Gamma_n$.

*Remark 2.17.* Pink’s construction satisfies the following two important properties.

(1) It is functorial with respect to surjective ring homomorphisms. Namely, let $\mathfrak{a}$ be a closed ideal of $A$ and $\varphi: R \to R/\mathfrak{a}R$ the natural projection. Then for all $n \ge 1$ we have

$$ \varphi(L_n(\Gamma)) = L_n(\varphi(\Gamma)). $$

(2) Pink’s Lie algebra $L_n(\Gamma)$ is closed under conjugation by the normalizer of $\Gamma$ in $R^\times$. This follows easily from the definitions since $\Theta$ is invariant under conjugation. $\triangle$

See Lemma 3.9 for an example calculating $L_n(\Gamma)$ when $\Gamma$ is a congruence subgroup.

## 2.4. Decomposability and regularity.
In order to prove fullness theorems, it is useful to be able to decompose Pink’s Lie algebra according to its entries. In Section 2.4 we define this precisely and then define regularity, which will turn out to ensure that the Lie algebras of the representations we work with are decomposable.

**Definition 2.18.** [Bel19, Section 4.9] Let $R$ be a GMA over $A$ and $L$ a closed subspace of $(\text{rad } R)^0$. We say that $L$ is *decomposable* if

$$ (a \ b \ c \ d) \in L \text{ implies that } (a \ b \ 0 \ 0) \in L \text{ and } (0 \ b \ c \ d) \in L. $$

We say that $L$ is *strongly decomposable* if $L$ is decomposable and

$$ (a \ b \ c \ d) \in L \text{ implies that } (0 \ b \ 0 \ 0) \in L \text{ and } (0 \ 0 \ 0 \ 0) \in L. $$

If $L_n(\Gamma) \subseteq R = (A \ B \ C \ D)$ is decomposable, we write

$$ I_n(\Gamma) := \{a \in A : (a \ 0 \ 0) \in L_n(\Gamma)\}, $$

$$ \nabla_n(\Gamma) := \{(0 \ b \ c \ d) \in L_n(\Gamma)\}, $$

$$ B_n(\Gamma) := \{b \in B : \exists c \in C \text{ such that } (0 \ b \ c \ d) \in L_n(\Gamma)\}, $$

$$ C_n(\Gamma) := \{c \in C : \exists b \in B \text{ such that } (0 \ b \ c \ d) \in L_n(\Gamma)\}. $$

Eventually, $L$ will be a Pink-Lie algebra associated to some admissible pseudodeformation of $\bar{\rho}: \Pi \to \text{GL}_2(\mathbb{F})$. Regularity is a condition on $\bar{\rho}$ that will allow us to decompose $L$, as we will see in Section 6.
Let $\mathbb{E}$ be the subfield of $\mathbb{F}$ generated by $\{(\text{tr } \bar{\rho}(g))^2/ \det \bar{\rho}(g): g \in \Pi\}$; equivalently, $\mathbb{E}$ is generated by the traces of $\text{ad } \bar{\rho}$. If $\lambda_g, \mu_g$ are the eigenvalues of $\bar{\rho}(g)$, then we see that $(\text{tr } \bar{\rho}(g))^2/ \det \bar{\rho}(g) = \lambda_g \mu_g^{-1} + \lambda_g^{-1} \mu_g + 2$. Hence $\mathbb{E}$ is generated over $\mathbb{F}_p$ by the set

$$ (2) \qquad \{\lambda\mu^{-1} + \lambda^{-1}\mu: \lambda, \mu \text{ are the eigenvalues of } \bar{\rho}(g) \text{ for some } g \in \Pi\}. $$

In particular, $g$ will not contribute to $\mathbb{E}$ if the multiplicative order of $\lambda_g \mu_g^{-1}$ is strictly less than 5. Using this reasoning, it is straightforward to calculate $\mathbb{E}$ when $\bar{\rho}$ exceptional. Namely, if $\bar{\rho}$ is tetrahedral or octahedral, then $\mathbb{E} = \mathbb{F}_p$. If $\bar{\rho}$ is icosahedral, then $\mathbb{E} = \mathbb{F}_p$ if $p = 5$ and $\mathbb{E} = \mathbb{F}_p(\zeta_5 + \zeta_5^{-1}) = \mathbb{F}_p(\sqrt{5})$ otherwise.

**Definition 2.19.** Let $\bar{\rho}: \Pi \to \text{GL}_2(\mathbb{F})$ be a semisimple representation. We say that $\bar{\rho}$ is *regular* if there exists $g_0 \in \Pi$ such that $\bar{\rho}(g_0)$ has eigenvalues $\lambda_0, \mu_0 \in \bar{\mathbb{F}}_p^\times$ satisfying $\lambda_0 \mu_0^{-1} \in \mathbb{E}^\times \setminus \{\pm 1\}$. We call $g_0$ a *regular element* for $\bar{\rho}$. If in addition $\lambda_0, \mu_0 \in \mathbb{E}^\times$, then we say that $\bar{\rho}$ is *strongly regular*.

**Definition 2.19** is weaker than Bellaïche's definition of regularity [Bel19, Definition 7.2.1], where the eigenvalues $\lambda_0$ and $\mu_0$ are required in addition to belong to $\mathbb{F}_p$. Examining (2), we see that the only way $\bar{\rho}$ can fail to be regular is if, for every matrix in $\text{Im } \bar{\rho}$ with eigenvalues $\lambda, \mu$, either $\lambda\mu^{-1} = \pm 1$ or the unique quadratic extension of $\mathbb{E}$ is $\mathbb{E}(\lambda\mu^{-1})$.

**Remark 2.20.** Let us analyze regularity depending on the projective image of $\bar{\rho}$. With notation as in **Definition 2.19**, note that the order of $\lambda_0 \mu_0^{-1}$ in $\mathbb{E}^\times$ corresponds to the order of $\bar{\rho}(g_0)$ in the projective image of $\bar{\rho}$.

(1) If $\bar{\rho}$ is large, then $\bar{\rho}$ is regular. Indeed, $\mathbb{P}\bar{\rho}(\Pi)$ contains $\text{PSL}_2(\mathbb{E})$ up to conjugation. Since $\bar{\rho}$ is not exceptional, $\mathbb{E}^\times$ contains an element $x$ such that $x^2 \neq \pm 1$. Then the image of $\bar{\rho}$ contains, up to conjugation, a scalar multiple of $(\begin{smallmatrix} x & 0 \\ 0 & x^{-1} \end{smallmatrix})$, which satisfies the regularity property.

(2) If $\bar{\rho}$ is tetrahedral and $p > 3$, then a regular element must map to a 3-cycle in the projective image of $\bar{\rho}$, since the other elements of $A_4$ have order at most 2. Thus in this case regularity is equivalent to $\zeta_3 \in \mathbb{E} = \mathbb{F}_p$, which is equivalent to $p \neq 2 \bmod 3$. By a similar argument we see that if $\bar{\rho}$ is octahedral and $p > 3$, then regularity is equivalent to one of $\zeta_3$ or $\zeta_4$ being in $\mathbb{E} = \mathbb{F}_p$, which is equivalent to $p \neq 11 \bmod 12$. If $\bar{\rho}$ is icosahedral and $p \neq 5$, then regularity is equivalent to one of $\zeta_3$ or $\zeta_5$ being in $\mathbb{E} = \mathbb{F}_p(\sqrt{5})$, which is equivalent to $p \neq 14 \bmod 15$.

(3) If $\bar{\rho} \cong \varepsilon \oplus \delta$, then $\bar{\rho}$ is regular if and only if $\varepsilon\delta^{-1}$ takes values in $\mathbb{E}^\times$ (Lemma 8.1).

(4) If $\bar{\rho} = \text{Ind}_{\Pi_0}^\Pi \chi$ is dihedral, then elements in $\Pi \setminus \Pi_0$ have projective order 2, and so any regular element must lie in $\Pi_0$. Furthermore, elements in $\Pi \setminus \Pi_0$ have trace 0, and so the field $\mathbb{E}$ associated to $\bar{\rho}$ is the same as the field $\mathbb{E}$ associated to $\bar{\rho}|_{\Pi_0}$. Hence we are reduced to the previous case when $\bar{\rho}$ is reducible.

(5) If the projective image of $\bar{\rho}$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ or $(\mathbb{Z}/2\mathbb{Z})^2$, or if $\mathbb{E} = \mathbb{F}_3$, then $\bar{\rho}$ is never regular. In particular, if $p = 3$ and $\bar{\rho}$ is tetrahedral or octahedral, then $\bar{\rho}$ is not regular. If $p = 5$ and $\bar{\rho}$ is icosahedral, then $\mathbb{P}\bar{\rho}(\Pi)$ is conjugate to $\text{PSL}_2(\mathbb{F}_5)$. Thus $\mathbb{E} = \mathbb{F}_5$ and any potential regular element has eigenvalues in $(\mathbb{F}_5^\times)^2 = \{\pm 1\}$, so $\bar{\rho}$ is not regular in this case. $\triangle$

**2.5. Bellaïche's results.** The purpose of this section is to state Bellaïche's main results that form the basis for our work in this paper. We state them in slightly less generality than [Bel19, Section 6]. As before, $A$ denotes a local pro-$p$ ring with maximal ideal $\mathfrak{m}$ and residue field $\mathbb{F}$. In particular, $A$ is naturally a topological $W(\mathbb{F})$-algebra.

Let $R$ be a faithful GMA over $A$. Recall the description of $R/\text{rad } R$ that we gave in the beginning of **Section 2.3**. We define $s: R/\text{rad } R \to R$ by

$$ s \begin{pmatrix} a & b \\ c & d \end{pmatrix} := \begin{cases} \begin{pmatrix} s(a) & s(b) \\ s(c) & s(d) \end{pmatrix} & \text{if } R = M_2(A) \\ \begin{pmatrix} s(a) & 0 \\ 0 & s(d) \end{pmatrix} & \text{else.} \end{cases} $$
Note that in the latter case, we have a priori that $b = c = 0$.

Let us fix an admissible pseudodeformation $(\Pi, \bar{\rho}, t, d)$ over $A$. If $p = 3$, let us assume that $\bar{\rho}$ is not tetrahedral. By Proposition 2.11, there exists a $(t, d)$-representation $\rho: \Pi \to R^\times$. Given such a $(t, d)$-representation, write $G = G_\rho := \rho(\Pi)$ and $\Gamma = \Gamma_\rho := G \cap SR^1$. Furthermore, let $\bar{G}$ denote the image of $G$ modulo rad $R$. (Note that the image of $\bar{G}$ under an embedding $R/\text{rad} R \to GL_2(\mathbb{F})$ is a conjugate of $\bar{\rho}(\Pi)$.) We will write $L_n(\rho) := L_n(\Gamma_\rho)$ and analogously for $I_n(\rho), \nabla_n(\rho), B_n(\rho), C_n(\rho)$.

Bellaïche chooses his $(t, d)$-representations very carefully in order to give a nice description of their Pink-Lie algebras. How this is done depends upon the projective image of $\bar{\rho}$. Since $\bar{\rho}$ is multiplicity free over $\mathbb{F}$, we can let $\lambda_0 \neq \mu_0 \in \bar{\mathbb{F}}_p^\times$ be the eigenvalues of a matrix $x_0 \in \text{Im} \bar{\rho}$ chosen such that the following conditions are satisfied:

* if $\bar{\rho}$ is large, then $(\lambda_0\mu_0^{-1})^2 \neq 1$ and $\lambda_0, \mu_0 \in \bar{\mathbb{F}}_p^\times$;

* if $p = 3$ and $\bar{\rho}$ is octahedral, then $\lambda_0\mu_0^{-1}$ is a primitive fourth root of unity;

* if $p = 5$ and $\bar{\rho}$ is icosahedral, then $\lambda_0\mu_0^{-1}$ is a primitive third root of unity;

* if $\bar{\rho}$ is exceptional and does not belong to one of the previous to scenarios, then $\lambda_0\mu_0^{-1}$ is a primitive third, fourth, or fifth root of unity;

* otherwise, the multiplicative order of $\lambda_0\mu_0^{-1}$ is equal to the maximal order of an element in the projective image of $\bar{\rho}$.

**Definition 2.21.** Suppose $(\Pi, \bar{\rho}, t, d)$ is an admissible pseudodeformation. We say that a $(t, d)$-representation $\rho$ is *well adapted* if

(1) $\rho$ is adapted to an element $g_0$ such that $\rho(g_0) = \begin{pmatrix} s(\lambda_0) & 0 \\ 0 & s(\mu_0) \end{pmatrix}$, where $\lambda_0, \mu_0$ satisfy the relevant property listed above;

(2) if the projective image of $\bar{\rho}$ is dihedral and nonabelian, then $\bar{G}$ contains a matrix of the form $\begin{pmatrix} 0 & b \\ c & 0 \end{pmatrix}$ with $bc^{-1} \in \bar{\mathbb{F}}_p^\times$ and $s(\bar{G}) \subseteq \text{Im} \rho$.

Bellaïche shows that well-adapted $(t, d)$-representations always exist, provided that one is willing to replace $\mathbb{F}$ by a quadratic extension in the dihedral case [Bel19, Proposition 6.3.2, Lemma 6.8.2].

Define $\mathbb{F}_q$ as in the table below. We will see in Lemma 8.1 that if $\bar{\rho}$ is regular and reducible or dihedral, then $\mathbb{F}_q$ can be taken to be $\mathbb{E}$. If $\bar{\rho}$ is not projectively cyclic or dihedral, then $\mathbb{F}_q \subseteq \mathbb{E}$ by definition. (In the $A_5$ case, this follows from the calculation that $\mathbb{E} = \bar{\mathbb{F}}_p(\sqrt{5})$ prior to Definition 2.19.)

**Remark 2.22.** Our definition of $\mathbb{F}_q$ differs from that of Bellaïche when $\bar{\rho}$ is exceptional. If $\bar{\rho}$ is tetrahedral, then Bellaïche defines $\mathbb{F}_q = \mathbb{F}_p(\zeta_3)$. If $\bar{\rho}$ is octahedral, he defines $\mathbb{F}_q$ to be $\mathbb{F}_p(\zeta_4)$ if the ratio $\lambda_0\mu_0^{-1}$ chosen prior to Definition 2.21 has order 3 and $\mathbb{F}_p(\zeta_4)$ if that ratio has order 4. If $\bar{\rho}$ is icosahedral, then he defines $\mathbb{F}_q = \mathbb{F}_p(\zeta_5)$. The key property that Bellaïche needs is that $\bar{\rho}$ can be conjugated so that its image lies in $Z \cdot GL_2(\mathbb{F}_q)$ and $\begin{pmatrix} \lambda_0 & 0 \\ 0 & \mu_0 \end{pmatrix} \in \text{Im} \bar{\rho}$, where $Z$ is the group of scalar matrices in $\mathbb{F}$ (cf. [Bel19, Lemma 6.8.5]). This change of definition will be justified in Lemma 8.12. △

<table><thead><tr><th>the projective image of <span style="text-decoration: overline;">ρ</span> is</th><th><span style="text-decoration: overline;">F</span><sub>q</sub></th></tr></thead><tbody><tr><td>cyclic of order <i>m</i> or dihedral of order <i>2<i>m</i></sub></td><td>any subfield of <span style="text-decoration: overline;">F</span> such that (<i>m</i>, <i>q</i> &minus; 1) &gt; 2</td></tr><tr><td>exceptional</td><td><span style="text-decoration: overline;">E</span>(<i>λ</i><sub>0</sub><i>μ</i><sub>0</sub><sup>&minus;1</sup>)</td></tr><tr><td>otherwise</td><td><span style="text-decoration: overline;">F</span><sub><i>p</i></sub></td></tr></tbody></table>

The following theorem summarizes Bellaïche's results describing the structure of $W(\mathbb{F}_q)L_1(\rho)$ from [Bel19, Section 6]. We recall from Remark 2.12 that in the dihedral case, elements in $B$ and $C$ can be viewed as elements of $A$.
**Theorem 2.23** (Bellaïche). Let $(\Pi, \bar{\rho}, t, d)$ be an admissible pseudodeformation such that the projective image of $\bar{\rho}$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$ nor $(\mathbb{Z}/2\mathbb{Z})^2$. Then every well-adapted $(t, d)$-representation $\rho: \Pi \to \mathbb{R}^\times$ with $R = \begin{pmatrix} A & B \\ C & A \end{pmatrix}$ has the following properties:

(1) $L_1(\rho)$ is decomposable;

(2) the ring $A$ is equal to

$$ \begin{cases} W(\mathbb{F}) + W(\mathbb{F})I_1(\rho) + W(\mathbb{F})I_1(\rho)^2 + W(\mathbb{F})B_1(\rho) & \text{if } \bar{\rho} \text{ is projectively dihedral} \\ W(\mathbb{F}) + W(\mathbb{F})I_1(\rho) + W(\mathbb{F})I_1(\rho)^2 & \text{otherwise;} \end{cases} $$

(3) $W(\mathbb{F})C_1(\rho) = C$ and $W(\mathbb{F})B_1(\rho) = B$;

(4) up to possibly replacing $\rho$ with its conjugate by a certain matrix $(\begin{smallmatrix} 1 & 0 \\ 0 & a \end{smallmatrix})$ with $a \in A^\times$ when $\bar{\rho}$ is exceptional or large, $W(\mathbb{F}_q)L_1(\rho)$ is equal to

$$ \begin{pmatrix} W(\mathbb{F}_q)I_1(\rho) & W(\mathbb{F}_q)B_1(\rho) \\ W(\mathbb{F}_q)C_1(\rho) & W(\mathbb{F}_q)I_1(\rho) \end{pmatrix}^0. $$

Furthermore

(i) $(W(\mathbb{F}_q)I_1(\rho))^3 \subseteq W(\mathbb{F}_q)I_1(\rho)$;

(ii) if $\bar{\rho}$ is not reducible, then $W(\mathbb{F}_q)C_1(\rho) = W(\mathbb{F}_q)B_1(\rho)$;

(iii) if $\bar{\rho}$ is exceptional or large, then $W(\mathbb{F}_q)B_1(\rho) = W(\mathbb{F}_q)I_1(\rho)$ and $(W(\mathbb{F}_q)I_1(\rho))^2 \subset W(\mathbb{F}_q)I_1(\rho)$.

For a subfield $\mathbb{F}'$ of $\mathbb{F}$, we shall often refer to the $W(\mathbb{F}')$-subalgebra of $A$ generated by $I_1(\rho)$. We denote it by $\mathcal{B}_\rho(\mathbb{F}')$, which is simply equal to $W(\mathbb{F}') + W(\mathbb{F}')I_1(\rho) + W(\mathbb{F}')I_1(\rho)^2$ whenever $\mathbb{F}_q \subseteq \mathbb{F}'$. When $\bar{\rho}$ is not reducible or dihedral, we have $\mathcal{B}_\rho(\mathbb{F}') = W(\mathbb{F}') + W(\mathbb{F}')I_1(\rho)$ by Theorem 2.23(ii, iii). In particular, Theorem 2.23 says that $A = \mathcal{B}_\rho(\mathbb{F})$ unless $\bar{\rho}$ is projectively dihedral, in which case $A = \mathcal{B}_\rho(\mathbb{F}) + W(\mathbb{F})B_1(\rho)$.

Bellaïche uses Theorem 2.23 to deduce that, under certain hypotheses, the representation $\rho$ is $\mathcal{B}_\rho(\mathbb{F}_p)$-full. See Theorem 1.3 or [Bel19, Theorem 7.2.3] for a precise statement of his result. The first step in our main theorem is to improve this to $\mathcal{B}_\rho(\mathbb{E})$-fullness in Section 6. But first we discuss fullness, conjugate self-twists, and the connections between them in detail.

### 3. FULLNESS

In this section we explore the notion of fullness, our measure of the size of the image of a continuous (pseudo)representation on a noetherian local pro-$p$ ring $A$. Here we additionally assume that $A$ is a domain, with field of fractions $K$.

**3.1. Fullness for (pseudo)representations.** Let $B$ be any ring. For any nonzero $B$-ideal $\mathfrak{b}$, let

$$ \Gamma_B(\mathfrak{b}) := \ker(\mathrm{SL}_2(B) \to \mathrm{SL}_2(B/\mathfrak{b})) = \{ \begin{pmatrix} 1+a & b \\ c & 1+d \end{pmatrix} \in \mathrm{SL}_2(B) : a, b, c, d \in \mathfrak{b} \} $$

be the congruence subgroup of $\mathrm{SL}_2(B)$ of level $\mathfrak{b}$.

**Definition 3.1.** Let $G$ be a subgroup of $\mathrm{GL}_2(K)$. For a subring $B$ of $K$ we say that $G$ is $B$-full if there exists a nonzero $B$-ideal $\mathfrak{b}$ and $x \in \mathrm{GL}_2(K)$ such that

$$ x^{-1}Gx \supseteq \Gamma_B(\mathfrak{b}). $$

A $\mathrm{GL}_2(K)$-valued representation is $B$-full if its image is $B$-full. If $(t, d): \Pi \to A$ is a pseudorepresentation, we say $(t, d)$ is $B$-full if there exists a $(t, d)$-representation $\rho: \Pi \to \mathbb{R}^\times$ such that $\iota \circ \rho$ is $B$-full, where $\iota$ is an embedding of $R$ into $M_2(K)$. Such an $\iota$ exists by Lemma 2.10; note that by replacing $\iota$ by a conjugate embedding we may insist that $\Gamma_B(\mathfrak{b}) \subset \mathrm{GL}_2(K)$ is contained in $\iota(\mathbb{R}^\times)$ on the nose. We will say $B$ is a $(t, d)$-fullness ring if $(t, d)$ is $B$-full.
The notion of fullness, which has appeared in earlier incarnations in [Hid15, last introductory paragraph] and [Lan16, Definition 2.2], is analogous to Bellaïche's notion of "congruence large-image" [Bel19, Definition 7.2.1]. We now show that fullness is well defined for pseudorepresentations and gives compatible notions for representations and pseudorepresentations.

**Lemma 3.2.** Let $(t, d): \Pi \to A$ be a pseudorepresentation and $\rho: \Pi \to \text{GL}_2(K)$ a representation whose trace takes values in $A$. Let $B$ be any subring of $K$.

(1) If there exists a $(t, d)$-representation that is $B$-full, then every $(t, d)$-representation is $B$-full.

(2) The representation $\rho$ is $B$-full if and only if its pseudorepresentation $(\text{tr} \rho, \text{det} \rho)$ is $B$-full.

*Proof.* To prove (1), let $\rho: \Pi \to R^\times$ and $\rho': \Pi \to R'^\times$ be two $(t, d)$-representations. We just have to verify that the $A$-algebra isomorphism $\Psi: R \to R'$ such that $\rho' = \Psi \circ \rho$ from Proposition 2.11 is given by conjugation by an element of $\text{GL}_2(K)$. Consider $\Psi \otimes 1: R \otimes_A K \to R' \otimes_A K$ and recall that $R \otimes_A K \cong M_2(K) \cong R' \otimes_A K$ by Lemma 2.10. By the Skolem-Noether theorem, it follows that $\Psi \otimes 1$ (and hence $\Psi$) is conjugation by an element of $\text{GL}_2(K)$.

For (2), let $(t, d) = (\text{tr} \rho, \text{det} \rho)$, which is a pseudorepresentation over $A$ by assumption. Let $r: \Pi \to R^\times$ be a $(t, d)$-representation, and embed $R$ into $\text{GL}_2(K)$ by Lemma 2.10, thus viewing $r$ as valued in $\text{GL}_2(K)$. Note that fullness of $\rho$ (respectively, $r$) implies that $\rho$ (respectively, $r$) is absolutely irreducible since this is true for the inclusion representation of a congruence subgroup in $\text{GL}_2(K)$. As $(\text{tr} \rho, \text{det} \rho) = (t, d) = (\text{tr} r, \text{det} r)$ and either $\rho$ or $r$ is absolutely irreducible, by the Brauer-Nesbitt Theorem it follows that $\rho$ and $r$ are conjugate by a matrix in $\text{GL}_2(K)$. Since fullness is defined up to conjugation in $\text{GL}_2(K)$, it follows that $\rho$ is $B$-full if and only if $r$, and hence $(t, d)$, is $B$-full. $\square$

The next two propositions suggest that we can restrict our attention to fullness rings that are closed subrings of the trace algebra. These ideas are made precise Proposition 3.4 and Corollary 3.7 below. Fix a continuous pseudorepresentation $(t, d): \Pi \to A$.

**Proposition 3.3.** If $(t, d)$ is $B$-full for some subring $B$ of $K$, then the $\mathbb{Z}$-linear span of $t(\Pi)$ contains a nonzero $B$-ideal. In particular, the trace algebra $A_t$ of $t$ contains a nonzero $B$-ideal.

*Proof.* Let $0 \neq \mathfrak{b}$ be an $B$-ideal such that $\Gamma_B(\mathfrak{b})$ is contained in the image of some $(t, d)$-representation. We claim that pairwise products of elements of $\mathfrak{b}$ are all in the set $\{t(g) - 2 : g \in \Pi\}$, so that $\mathfrak{b}^2$ is contained in the trace algebra $A_t$. Indeed, an element of $\Gamma_B(\mathfrak{b})$ is of the form $\binom{1+a}{c} \mathfrak{b}^{1+d}$ with $a, b, c, d \in \mathfrak{b}$ such that $a+d+ad-bc=0$. In particular, for any $b, c \in \mathfrak{b}$, taking $d = bc$ and $a = 0$ shows that

$$ \text{tr}(\binom{1}{c} \mathfrak{b}^b) = 2 + bc \in t(\Pi). $$

Since $2 = t(1) \in t(\Pi)$ and elements of the form $bc$ generate $\mathfrak{b}^2$, we see that the $\mathbb{Z}$-span of $t(\Pi)$ contains $\mathfrak{b}^2$, which is nonzero since it is the square of a nonzero ideal of a domain. $\square$

**Proposition 3.4.** Let $(t, d): \Pi \to A$ be a continuous pseudorepresentation. If $(t, d)$ is $B$-full for a subring $B$ of $A$, then $(t, d)$ is also full for the closure $\bar{B}$ of $B$ in $A$. Conversely, suppose that $(t, d)$ is $\bar{B}$-full. If the image of $a$ $(t, d)$-representation contains a congruence subgroup of $\text{SL}_2(\bar{B})$ whose level is the closure $\bar{\mathfrak{b}}$ of an ideal $\mathfrak{b}$ of $B$, then $(t, d)$ is also $B$-full.

*Proof.* We show that for a closed subgroup $G$ of the unit group $R^\times$ of a faithful GMA $R$ over $A$ equipped with an embedding $\iota: R \hookrightarrow M_2(K)$, if $\iota(G)$ contains $\Gamma_B(\mathfrak{b})$ for some nonzero ideal $\mathfrak{b}$ of $B$, then $\iota(G)$ also contains $\Gamma_{\bar{B}}(\bar{\mathfrak{b}})$ for the closure $\bar{\mathfrak{b}}$ of $\mathfrak{b}$. Since $A$ is noetherian, and both $R$ and $M_2(A)$ are finite $A$-algebras, the preimage $S := \iota^{-1}(M_2(A))$ is a finite, hence closed, $A$-subalgebra of $R$. Moreover, the induced map $\iota|_S: S \to M_2(A)$ is a homeomorphism onto its image by the compatibility of topologies on finite $A$-modules (see p. 8). In particular, any closed subset of $R$ containing $\iota^{-1}(\Gamma_B(\mathfrak{b}))$ will also contain its closure $\iota^{-1}(\Gamma_{\bar{B}}(\bar{\mathfrak{b}}))$. The converse claim is clear since $\bar{\mathfrak{b}}$ is nonzero only if $\mathfrak{b}$ is. $\square$
**3.2. Fullness peers.** A pseudorepresentation may be $B$-full for more than one choice of ring $B$, even if $B$ is a closed subring of $A$. For example, any pseudorepresentation that happens to be full for $\mathbb{Z}_{p^2} = W(\mathbb{F}_{p^2})$ is also full for the order $\mathbb{Z}_p + p\mathbb{Z}_{p^2} \subset \mathbb{Z}_{p^2}$. We say two subrings $B_1, B_2$ of $K$ are fullness peers if every nonzero ideal of $B_1$ contains a nonzero ideal of $B_2$ and vice versa, in which case $B_1$-fullness is equivalent to $B_2$-fullness. One easily checks that fullness peerage is an equivalence relation on subrings of $K$. The next lemma gives a criterion for establishing when nested domains are fullness peers.

**Lemma 3.5.** Let $B_1 \subseteq B_2$ be domains. The following conditions are equivalent:

(1) $B_1$ contains a nonzero ideal of $B_2$;

(2) there exists $y \in B_1 \setminus \{0\}$ such that $yB_2 \subseteq B_1$;

(3) $B_1$ and $B_2$ are fullness peers.

These equivalent conditions imply that $B_2$ and $B_1$ have the same field of fractions. If moreover $B_1$ is noetherian, then conditions (1,2,3) are equivalent to:

(4) $Q(B_2) = Q(B_1)$ and $B_2$ is a finite $B_1$-algebra.

*Proof.* For (1) implies (2), take $y$ to be any nonzero element of the nonzero ideal of $B_2$ contained in $A_1$. If (2) holds, then an arbitrary nonzero ideal $\mathfrak{b}$ of $A_1$ contains $(yB_2)\mathfrak{b}$, which is a nonzero ideal of $B_2$, implying (3). Clearly (3) implies (1).

To see that $Q(B_2) = Q(B_1)$ under any of (1,2,3), note that any $x \in B_2$ can be written as $(yx)/y \in Q(B_1)$ with $y$ as in (2).

For the rest of the proof, assume that $B_1$ is noetherian. Suppose first that any of (1,2,3) holds. If $J$ is a non-zero ideal of $B_2$ contained in $B_1$, then $J$ is a finitely generated $B_1$-module. By replacing $J$ with a smaller $B_2$-ideal, we can assume that $J$ is principal in $B_2$, that is, $J = bB_2$ for some $b \in B_1$. Now choose a finite set of generators $\{bx_1, \dots, bx_n\}$ of $bB_2$ as an $B_1$-module, with $x_1, \dots, x_n$ in $B_2$. Then, for every $y$ in $B_1$, $by$ is a linear combination $\sum_i a_i x_i$ for some $a_i \in B_1$, which means that $y = \sum_i a_i x_i$, so the set $\{x_1, \dots, x_n\}$ generates $B_2$ as an $B_1$-module.

Conversely, suppose that (4) is satisfied. Let $x_1, \dots, x_n$ be generators for $B_2$ as an $B_1$-module. Write $x_i = b_{i1}/b_{i2}$ with $b_{ij} \in B_1 \setminus \{0\}$. Set $b = \prod_{i=1}^n b_{i2} \in B_1 \setminus \{0\}$. Then $bx_i \in B_1$ for all $i$, and it follows that $bB_2 \subseteq A_1$, proving (2). $\square$

**Question 3.6.** Note that $\mathbb{Z}_p[[X]]$ and its non-noetherian subring $\mathbb{Z}_p + p\mathbb{Z}_p[[X]]$ and are fullness peers even though the extension is not finite. Could profiniteness substitute for finiteness in (4) above? That is, if $B_2$ is a local noetherian pro-$p$ domain, and $B_1 \subset B_2$ is a closed subring with the same field of fractions, are $B_1$ and $B_2$ necessarily fullness peers? $\triangle$

**Corollary 3.7.** Let $(t, d) : \Pi \to A$ be a pseudorepresentation. If $(t, d)$ is $B$-full for some subring $B$ of $K$, then $B \cap A$ is a fullness peer of $B$.

*Proof.* By Proposition 3.3, the trace algebra of $(t, d)$, and hence $A$, contains a nonzero ideal of $B$. Therefore so does $B \cap A$, and by Lemma 3.5, $B \cap A \subseteq B$ is an extension of fullness peers. $\square$

**Corollary 3.7** together with Proposition 3.4 allows us to restrict our attention to fullness for closed subrings of $A$, though we continue to point out features of the general case for completeness.

**Corollary 3.8.** Let $B$ be a complete local noetherian domain. Then all the extensions of $B$ contained in the normalization $B^{\text{norm}}$ are fullness peers of $B$.

*Proof.* Since $B^{\text{norm}}$ is the integral closure of $B$ in its field of fractions, $B^{\text{norm}}$ is finite over $B$ by the $N2$ property (see p. 8), hence a noetherian $B$-module. Therefore any ring $C$ with $B \subseteq C \subseteq B^{\text{norm}}$ is finite over $B$. Fullness peerage then follows from Lemma 3.5. $\square$
**3.3. Fullness and twisting.** A key property of fullness, shown in Corollary 3.12, is that it does not change when we twist a pseud representation by a character. This will allow us to do all of our technical work in the setting of constant determinant pseudorepresentations where we have Bellaiche's Theorem 2.23 available. The proof of the twist invariance of fullness relies on a calculation of the Pink-Lie algebras of a congruence subgroup. Let $B$ be a local pro-$p$ domain. For a closed nonzero $B$-ideal $\mathfrak{b}$, define

$$ \mathfrak{sl}_2(\mathfrak{b}) := \{ \begin{pmatrix} a & b \\ c & -a \end{pmatrix} : a, b, c \in \mathfrak{b} \} \subset M_2(\mathfrak{b}). $$

**Lemma 3.9** (cf. [Bel19, Proposition 4.8.2]). Let $\mathfrak{b}$ be a closed ideal of $B$. Then $\Gamma_B(\mathfrak{b})$ is a closed pro-$p$ subgroup of $GL_2(B)$ and $L_n(\Gamma_B(\mathfrak{b})) = \mathfrak{sl}_2(\mathfrak{b}^n)$.

*Proof.* For $x = \begin{pmatrix} 1+a & b \\ c & 1+d \end{pmatrix} \in \Gamma_B(\mathfrak{b})$ one has $\Theta(x) = \begin{pmatrix} \frac{a-d}{2} & b \\ c & \frac{d-a}{2} \end{pmatrix} \in \mathfrak{sl}_2(\mathfrak{b})$, so $L_1(\Gamma_B(\mathfrak{b})) \subseteq \mathfrak{sl}_2(\mathfrak{b})$.

In particular, for any $b, c \in \mathfrak{b}$, we have $\Theta(\begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix}) = (\begin{pmatrix} 0 & b \\ 0 & 0 \end{pmatrix})$ and $\Theta(\begin{pmatrix} 1 & 0 \\ c & 1 \end{pmatrix}) = (\begin{pmatrix} 0 & 0 \\ c & 0 \end{pmatrix})$. For $a \in \mathfrak{b}$ we have $(\begin{pmatrix} 1+2a & -2a \\ 2a & 1-2a \end{pmatrix}) \in \Gamma_B(\mathfrak{b})$, and so

$$ \Theta \begin{pmatrix} 1+2a & -2a \\ 2a & 1-2a \end{pmatrix} = \begin{pmatrix} a & -2a \\ 2a & -a \end{pmatrix} = \begin{pmatrix} a & 0 \\ 0 & -a \end{pmatrix} + \Theta \begin{pmatrix} 1 & -2a \\ 0 & 1 \end{pmatrix} + \Theta \begin{pmatrix} 1 & 0 \\ 2a & 1 \end{pmatrix}. $$

It follows that $\mathfrak{sl}_2(\mathfrak{b})$ is contained in the additive subgroup generated by $\Theta(\Gamma_B(\mathfrak{b}))$. Since $\mathfrak{sl}_2(\mathfrak{b})$ is closed in $\mathfrak{sl}_2(B)$, it follows that $\mathfrak{sl}_2(\mathfrak{b}) = L_1(\Gamma_B(\mathfrak{b}))$.

It is straightforward to calculate by induction on $n$ that the subgroup topologically generated by

$$ \{xy - yx : x \in \mathfrak{sl}_2(\mathfrak{b}), y \in \mathfrak{sl}_2(\mathfrak{b}^n)\} $$

is $\mathfrak{sl}_2(\mathfrak{b}^{n+1})$. That is, $L_n(\Gamma_B(\mathfrak{b})) = \mathfrak{sl}_2(\mathfrak{b}^n)$ for all $n \ge 1$.

□

**Corollary 3.10.** Let $\mathfrak{b}$ be a closed $B$-ideal different from $B$. Then $(\Gamma_B(\mathfrak{b}), \Gamma_B(\mathfrak{b})) = \Gamma_B(\mathfrak{b}^2)$. If $\#\mathbb{B} > 3$, this also holds for $\mathfrak{b} = B$.

*Proof.* First assume that $\mathfrak{b} \neq B$. By Theorem 2.16,

$$ (\Gamma_B(\mathfrak{b}), \Gamma_B(\mathfrak{b})) = \Theta^{-1}(L_2(\Gamma_B(\mathfrak{b}))) \cap \Gamma_B(\mathfrak{m}). $$

By Lemma 3.9, $L_2(\Gamma_B(\mathfrak{b})) = \mathfrak{sl}_2(\mathfrak{b}^2)$.

Clearly $\Gamma_B(\mathfrak{b}^2) \subseteq \Theta^{-1}(\mathfrak{sl}_2(\mathfrak{b}^2)) \cap \Gamma_B(\mathfrak{m})$. We compute $\Theta^{-1}(\begin{pmatrix} a & b \\ c & -a \end{pmatrix}) \cap \Gamma_B(\mathfrak{m})$ for $\begin{pmatrix} a & b \\ c & -a \end{pmatrix} \in \mathfrak{sl}_2(\mathfrak{b}^2)$. If $\begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \in \Theta^{-1}(\begin{pmatrix} a & b \\ c & -a \end{pmatrix}) \cap \Gamma_B(\mathfrak{m})$ then we must have $\beta = b, \gamma = c, \alpha - \delta = 2a$, and $1 = \alpha\delta - \beta\gamma$. From this one calculates that $\alpha = a \pm \sqrt{1+a^2+bc}$ and $\delta = -a \pm \sqrt{1+a^2+bc}$. But only one of these possibilities has $\alpha \equiv 1 \equiv \delta \mod \mathfrak{m}$ and thus is in $\Gamma_B(\mathfrak{m})$. That is, there is a unique element in $\Theta^{-1}(\begin{pmatrix} a & b \\ c & -a \end{pmatrix}) \cap \Gamma_B(\mathfrak{m})$. It follows that $\Theta^{-1}(\mathfrak{sl}_2(\mathfrak{b}^2)) \cap \Gamma_B(\mathfrak{m}) = \Gamma_B(\mathfrak{b}^2)$, as desired.

We now prove that $(\mathrm{SL}_2(B), \mathrm{SL}_2(A)) = \mathrm{SL}_2(A)$ when $\#\mathbb{B} > 3$. By the first statement in the corollary, we know that $\Gamma_B(\mathfrak{m}^2) \subseteq (\mathrm{SL}_2(B), \mathrm{SL}_2(B))$, so we may assume that $\mathfrak{m}^2 = 0$. Furthermore, the residual image of $(\mathrm{SL}_2(B), \mathrm{SL}_2(B))$ is $(\mathrm{SL}_2(\mathbb{B}), \mathrm{SL}_2(\mathbb{B}))$, which is equal to $\mathrm{SL}_2(\mathbb{B})$. Therefore, it suffices to show that $(\begin{pmatrix} 1+a & b \\ c & 1-a \end{pmatrix}) \in (\mathrm{SL}_2(B), \mathrm{SL}_2(B))$ for any $a, b, c \in \mathfrak{m}$. Since $\mathfrak{m}^2 = 0$, we can decompose

$$ (\begin{pmatrix} 1+a & b \\ c & 1-a \end{pmatrix}) = (\begin{pmatrix} 1+a & 0 \\ 0 & 1-a \end{pmatrix}) (\begin{pmatrix} 1 & b \\ c & 1 \end{pmatrix}). $$

Let $x \in B^\times$ such that $x^2 \neq 1 \mod \mathfrak{m}$, which exists since $\#\mathbb{B} > 3$. Note that for any $\beta, \gamma \in \mathfrak{m}$ we have

$$ \begin{pmatrix} 1 & b \\ c & 1 \end{pmatrix} = \begin{pmatrix} 1 & b(1-x^2)^{-1} \\ c(1-x^2)^{-1} & 1 \end{pmatrix} \begin{pmatrix} x & 0 \\ 0 & x^{-1} \end{pmatrix} \begin{pmatrix} 1 & -b(1-x^2)^{-1} \\ -c(1-x^2)^{-1} & 1 \end{pmatrix} \begin{pmatrix} x^{-1} & 0 \\ 0 & x \end{pmatrix} \\ \in (\mathrm{SL}_2(B), \mathrm{SL}_2(B)) $$

and

$$ (\begin{pmatrix} 1+a & 0 \\ 0 & 1-a \end{pmatrix}) = (\begin{pmatrix} 1+a & 0 \\ 0 & 1-a \end{pmatrix}) (\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}) (\begin{pmatrix} 1-a & 0 \\ 0 & 1+a \end{pmatrix}) (\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}) \in (\mathrm{SL}_2(B), \mathrm{SL}_2(B)). $$
It follows that $\Gamma_B(\mathfrak{m}) \subseteq (\mathrm{SL}_2(B), \mathrm{SL}_2(B))$ and hence that $\mathrm{SL}_2(B)$ is its own topological derived subgroup. $\square$

Having calculated the derived subgroup of a congruence subgroup, we can now prove that fullness for closed subrings of $A$ is inherited by restrictions to finite-index and coabelian subgroups.

**Proposition 3.11.** Suppose that the pseudorepresentation $(t, d) : \Pi \to A$ is $B$-full for a subring $B$ of $K$. Let $\Pi_0$ be a closed normal subgroup of $\Pi$ so that $\Pi/\Pi_0$ is abelian. Then $(t|_{\Pi_0}, d|_{\Pi_0})$ is also $B$-full. The same is true if $\Pi_0$ is a closed finite-index subgroup of $\Pi$ so long as $B$ is not finite.

*Proof.* First, assume that $B$ is a closed subring of $A$.

Let $\rho: \Pi \to R^\times$ be a $(t, d)$-representation such that $\rho(\Pi)$ contains $\Gamma_B(\mathfrak{b})$ for some nonzero $B$-ideal $\mathfrak{b}$. Write $G := \rho(\Pi)$ and let $G_0 := \rho(\Pi_0)$. If $\Pi_0$ is coabelian, then $G/G_0$ is abelian, so that $G_0$ contains the derived subgroup $[G, G]$. In particular, $G_0$ contains $[\Gamma_B(\mathfrak{b}), \Gamma_B(\mathfrak{b})]$, which is $\Gamma_B(\mathfrak{b}^2)$ by **Corollary 3.10**. Since $A$ is a domain, $\mathfrak{b}^2$ is nonzero if $\mathfrak{b}$ is.

Suppose alternatively that $\Pi_0$ is finite index in $\Pi$ and $B$ is not finite. Replacing $\mathfrak{b}$ by $\mathfrak{m}_A \cap \mathfrak{b}$ (that is, if $\mathfrak{b} = B$, we replace $\mathfrak{b}$ by the maximal ideal $\mathfrak{m}_A \cap B$ of $B$, nonzero by the assumption on $B$), we note that $\Gamma_B(\mathfrak{b})$ is contained in $\rho(\Gamma)$ for $\Gamma := \ker \bar{\rho} \subseteq \Pi$. Let $\Gamma_0$ be the normal core of $\Gamma \cap \Pi_0$ inside $\Gamma$, so that $\Gamma_0$ is a finite-index normal subgroup of $\Gamma$ contained in $\Pi_0$. Since $\Gamma$ is pro-$p$, and hence pro-solvable, and $\Gamma/\Gamma_0$ is finite, $\Gamma_0$ must contain the $n^{\text{th}}$ derived subgroup of $\Gamma$ for some $n \ge 1$. Therefore $\rho(\Gamma_0)$, and hence $\rho(\Pi_0)$, contains the $n^{\text{th}}$ closed derived subgroup of $\Gamma_B(\mathfrak{b})$, namely $\Gamma_B(\mathfrak{b}^{2n})$ (**Corollary 3.10**). This last is again a nontrivial congruence subgroup of $\mathrm{SL}_2(B)$.

For arbitrary $B$, **Corollary 3.7** allows us to assume that $B$ is a subring of $A$. If $(t, d)$ is $B$-full of level $\mathfrak{b}$ for some ideal $\mathfrak{b}$ of $B$, then the argument above and the first part of **Proposition 3.4** tell us that $(t|_{\Pi_0}, d|_{\Pi_0})$ is $\bar{B}$-full of level $\bar{\mathfrak{b}}^{2n}$ for some $n \ge 1$. This ideal is the closure of the ideal $\mathfrak{b}^{2n}$ of $B$, so we are done by the second part of **Proposition 3.4**. $\square$

We conclude that fullness is unchanged under twisting.

**Corollary 3.12.** Let $(t, d) : \Pi \to A$ be a pseudorepresentation and $\chi : \Pi \to A^\times$ a continuous character. If $(t, d)$ is $B$-full for some subring $B$ of $K$, then $(\chi t, \chi^2 d)$ is also $B$-full.

*Proof.* Follows from **Proposition 3.11** by setting $\Pi_0 := \ker \chi$. $\square$

## 4. ADJOINT TRACE RINGS AND CONJUGATE SELF-TWISTS

Having established the terms of the investigation — finding congruence subgroups for fullness rings (**Definition 3.1**) contained in images of GMA-valued representations (**Proposition 2.11**) — in this section we search for optimal (fullness peerage equivalence classes of) fullness rings. Our first stop is the ring fixed by the conjugate self-twists of $(t, d)$ (**Definition 4.1** below), symmetries that naturally limit its image. Although this fixed-by-twist-automorphisms subring is a fullness ring for the historical big-image results that serve as our inspiration, one cannot expect fullness with respect to the ring fixed by conjugate self-twists in the general setting. Indeed, there may not be enough automorphisms to carve down to a fullness ring as illustrated in **Examples 4.3** and 4.10, reflecting the limits of Galois theory.

Instead of trying to cut out a fullness ring from above, we build one from below by considering the trace ring $A_0$ of the adjoint pseudorepresentation. This *adjoint trace ring* (**Definition 4.6** below) does not change when twisting $(t, d)$ by a character, and moreover is morally expected to be pointwise fixed by all conjugate self-twists. Because of topological considerations, we do not actually show that $A_0$ is fixed by all conjugate self-twists (**Corollary 4.24**) until after we prove our main fullness result, so this idea is merely a guiding principle — except for $(t, d)$ whose determinant is $A_0$-constant (**Definition 4.14**), a condition expected to be satisfied by all intended applications.
The main result of this section is Corollary 4.20: if $(t, d)$ has $A_0$-constant-determinant, then $A_0$ and the ring fixed by conjugate self-twists are fullness peers. We crucially use this fullness peerage result when deriving our $A_0$-fullness results for certain constant-determinant pseudorepresentations satisfying our mild conditions (Corollary 9.16), which we then propagate to all such pseudorepresentations using the twist-invariance of $A_0$ (Theorem 10.1).

**4.1. Conjugate self-twists.** Recall that $A$ is a local pro-$p$ noetherian ring and $\Pi$ is a $p$-finite profinite group. Fix a continuous pseud representation $(t, d) : \Pi \to A$ with trace algebra $A_t$.

**Definition 4.1.** If $A$ is a domain, let $B$ be a domain extending it; otherwise let $B = A$. A $(B$-valued) conjugate self-twist of $(t, d)$ is a pair $(\sigma, \eta)$, where $\sigma$ is an automorphism of $B$ as a $\mathbb{Z}_p$-algebra and $\eta : \Pi \to B^\times$ is a character. We also consider $\tilde{\Sigma}_t(B/C)$, the conjugate self-twists whose automorphisms $\Sigma_t(B/C)$ fix a subring $C$ of $B$.

The set of all $B$-valued conjugate self-twists of a pseud representation forms a group $\tilde{\Sigma}_t(B)$, with composition law $(\sigma_1, \eta_1) \circ (\sigma_2, \eta_2) = (\sigma_1\sigma_2, \eta_1\eta_2)$ and inverse $(\sigma, \eta)^{-1} = (\sigma^{-1}, \sigma^{-1}\eta^{-1})$. Forgetting the character is therefore a group homomorphism $\tilde{\Sigma}_t(B) \to \mathrm{Aut}_{\mathbb{Z}_p}(B)$ whose image we denote $\Sigma_t(B)$. If $A$ is a domain and $\rho : \Pi \to \mathrm{GL}_2(A)$ is a semisimple representation, we use the notation $\Sigma_\rho(B) := \Sigma_{\mathrm{tr}\rho}(B)$ and $\tilde{\Sigma}_\rho(B) := \tilde{\Sigma}_{\mathrm{tr}\rho}(B)$.

The kernel of the forget-the-character map are the *dihedral* conjugate self-twists $\Sigma_t^{\mathrm{di}}$:

$$1 \to \Sigma_t^{\mathrm{di}} \to \tilde{\Sigma}_t(B) \to \Sigma_t(B) \to 1.$$

If $(1, \eta)$ is a nontrivial dihedral conjugate self-twist, then one can check that $H := \ker \eta$ is an index-2 subgroup of $G$, that $(t|_H, d|_H)$ is a sum of two characters, and that $(t, d)$ is carried by the induction of either of them. In particular $\eta$ is quadratic so that $\Sigma_t^{\mathrm{di}}$ does not depend on $B$. Moreover, when $B$ is a field, $H$, and hence $\eta$, is uniquely defined by $(t, d)$ unless the projective image of the semisimple $\rho$ carrying $(t, d)$ is the Klein-4 group, in which case there are three possibilities for $H$. In other words, when $B$ is a field, $\Sigma_t^{\mathrm{di}}$ is abelian dihedral if $\rho$ is projectively dihedral, cyclic of order 2 if $\rho$ is reducible of projective order 2, and trivial in all other cases. See Lemmas A.6 and A.7, and the proof of Lemma 7.1 for details.

**Proposition 4.2.** If $\chi : \Pi \to A^\times$ is a continuous character and $B$ is an extension of $A$, then

$$\tilde{\Sigma}_t(B) \cong \tilde{\Sigma}_{\chi_t}(B).$$

*Proof.* The map $(\sigma, \eta) \mapsto (\sigma, \sigma^\chi \chi^{-1}\eta)$ realizes the isomorphism. $\square$

<figure><img src="image_3.png" /></figure>

If a conjugate self-twist $(\sigma, \eta)$ happens to be $A$-valued, then the automorphism $\sigma$ is automatically continuous: since $A$ is local, algebraic automorphisms automatically send the maximal ideal to itself, and since $A$ is noetherian, the maximal ideal defines the topology. It turns out that an $A$-valued conjugate self-twist character $\eta$ must also be continuous — see Proposition 4.4(3) below. But in general we cannot expect that all conjugate self-twists can be restricted to ones defined over $A$, as seen in Example 4.3 below illustrating a failure of normality in the sense of Galois theory.

**Example 4.3.** For any odd prime $p$, let $\Pi$ be the subgroup of $\mathrm{GL}_2(\mathbb{Z}_p[\sqrt[p]{p}])$ generated by $\mathrm{GL}_2(\mathbb{Z}_p)$ and the scalar matrix $1 + \sqrt[p]{p}$. Let $\rho: \Pi \to \mathrm{GL}_2(\mathbb{Z}_p[\sqrt[p]{p}])$ be the inclusion, a representation with trace algebra $A = \mathbb{Z}_p[\sqrt[p]{p}]$. Note that $\Sigma_\rho(A)$ is trivial, since $A$ has no automorphisms — but that's not the whole story. Let $B = \mathbb{Z}_p[\sqrt[p]{p}, \zeta_p]$ and consider the automorphism $\sigma$ in $\mathrm{Aut}(B)$ sending $\sqrt[p]{p}$ to $\zeta_p \sqrt[p]{p}$, together with the character $\eta : \Pi \to B^\times$ with kernel $\mathrm{GL}_2(\mathbb{Z}_p)$ sending $1 + \sqrt[p]{p}$ to $(1 + \zeta_p \sqrt[p]{p})(1 + \sqrt[p]{p})^{-1}$. Then $(\sigma, \eta)$ is a $B$-valued conjugate self-twist of $\rho$. One can show that $\rho$ is not $A$-full (tr $\rho$ does not span an ideal of $A$; see Proposition 3.3). On the other hand, $\rho$ is visibly $\mathbb{Z}_p$-full, corresponding to the fact that $\mathbb{Z}_p$ is the ring fixed by $\Sigma_\rho(B)$. $\triangle$
Because it is not immediately clear that we can demand that relevant extensions of $A$ be endowed with a sensible topology (if $A$ is a domain we already cannot expect a topology on $Q(A)$: see p. 8), there is no way to require conjugate self-twists to be continuous. On the other hand, since trace algebras are topologically generated rings, they only behave well under conjugate self-twists satisfying continuity conditions. As a result, much of this section is devoted to finding settings where conjugate self-twists of constant-determinant pseudorepresentations satisfying the mild conditions of Theorem B are continuous: see Corollary 4.24. Write $A_t^{\text{alg}}$ for the $W(\mathbb{F})$-subalgebra of $A$ algebraically generated by $t(\Pi)$, so that the trace algebra $A_t$ is the closure of $A_t^{\text{alg}}$ in $A$.

**Proposition 4.4.** Let $(\sigma, \eta)$ be a $B$-valued conjugate self-twist.

(1) If $(t, d)$ has constant determinant, then $\eta$ is $W(\mathbb{F})$-valued and finite order.

(2) If $\eta$ is $A_t^{\text{alg}}$-valued (for example, if $\eta$ is $W(\mathbb{F})$-valued), then $\sigma$ restricts to an automorphism of $A_t^{\text{alg}}$, and there is an $A_t$-valued conjugate self-twist $(\sigma', \eta)$ satisfying $\sigma'|_{A_t^{\text{alg}}} = \sigma|_{A_t^{\text{alg}}}$.

(3) If $\eta$ is $A$-valued, then $\eta$ is continuous.

*Proof.* (1) We follow [Mom81, 1.5]. Recall that $d = s(\bar{d})$ has finite order by the assumption. Since $\sigma d = \eta^2 d$ we have $\eta^2 = \sigma dd^{-1}$. Since $d$ is finite order, $\sigma d$ must be a power of $d$. We now claim that $\sigma dd^{-1}$ has a power of $d$ as a square root, so that $\eta$ is different from a power of $d$ by at most a quadratic character and hence takes values in $W(\mathbb{F})$. Indeed, if $d$ has odd order, then $d$ itself has a power-of-$d$ square root, so that any power of $d$ has the same property. And if $d$ has even order, then since $\sigma$ preserves orders, $\sigma d$ must be an odd power of $d$; which means that $\sigma dd^{-1}$ is an even power of $d$ and hence has a power-of-$d$ square root.

(2) Given that $\eta$ and $t$ are both $A_t^{\text{alg}}$-valued, $\sigma t = \eta t$ is $A_t^{\text{alg}}$-valued as well. Since $B$ is either a domain or a local ring and $\sigma$ is a $\mathbb{Z}_p$-algebra homomorphism, it follows that $\sigma$ permutes $W(\mathbb{F})$, and hence it permutes $A_t^{\text{alg}}$ as well. Moreover, the action of $\sigma$ is continuous on $A_t^{\text{alg}}$ in the topology from $A_t$: with $\mathfrak{m}^{\text{alg}} := \mathfrak{m} \cap A_t^{\text{alg}}$, we have $\sigma(\mathfrak{m}^{\text{alg}}) \subseteq \eta(\Pi)\mathfrak{m}^{\text{alg}} \subseteq \mathfrak{m}^{\text{alg}}$. Therefore $\sigma|_{A_t^{\text{alg}}}$ extends uniquely to a continuous automorphism $\sigma'$ of the closure $A_t$ of $A_t^{\text{alg}}$. Finally, since $\sigma$ and $\sigma'$ agree on $t(\Pi)$, the pair $(\sigma', \eta)$ is still a conjugate self-twist, this time $A_t$-valued.

(3) First, we claim that $\ker(t, d) \subseteq \ker \eta$, so that $\eta$ factors through $G := \Pi/\ker(t, d)$. Indeed, for $g \in \ker(t, d)$ we have in particular $t(g) = 2$ so that $\eta(g) = \sigma t(g)/t(g) = 1$. Since $\ker(t, d)$ is closed, $G = \Pi/\ker(t, d)$ is still $p$-finite profinite, and we check continuity of $\eta$ as a character on $G$. Moreover $\Gamma := \ker \bar{\rho}/\ker(t, d) \subseteq G$ is a finite-index pro-$p$ subgroup of $G$ [Che14, Lemma 3.8]. By $p$-finiteness, $\Gamma$ is topologically finitely generated, so that any finite-index subgroup of $\Gamma$ is open in $\Gamma$ [Ser97, §4.2 exercise 6(d)], and hence in $G$. Now use the fact that $A^\times \cong \mathbb{F}^\times \times (1+\mathfrak{m})$ to write $\eta = \eta^{(p)}\eta_p$, where $\eta_p : G \to 1+\mathfrak{m}$ is a pro-$p$ character and $\eta^{(p)} : G \to \mathbb{F}^\times$ has prime-to-$p$ order. Then $\eta^{(p)}$ is continuous because its kernel contains the open pro-$p$ subgroup $\Gamma$. And $\eta_p$ is continuous because the preimage $U \subseteq \Gamma$ of any open subgroup of $1+\mathfrak{m}$ along $\eta_p|_{\Gamma}$ is finite index in $\Gamma$, hence open in $G$. Therefore $\eta$ is continuous. $\square$

**Corollary 4.5.** If $(t, d)$ has constant determinant, then any conjugate self-twist $(\sigma, \eta)$ of $(t, d)$ has $\eta$ continuous, finite-order, and $W(\mathbb{F})$-valued. Moreover, there exists an automorphism $\sigma'$ of $A_t$ agreeing with $\sigma$ on $A_t^{\text{alg}}$ so that $(\sigma', \eta)$ is an $A_t$-valued conjugate self-twist of $(t, d)$.

A posteriori after our main fullness results, we will deduce that, at least for most constant-determinant $(t, d)$ that are not a priori small, $\sigma$ restricts to an automorphism of $A_t$, necessarily continuous, and consequently $\sigma' = \sigma|_{A_t}$: see Theorems 5.4 and 10.1.

4.2. **The adjoint trace ring.** Quite generally, if $V$ is a 2-dimensional vector space over a field $F$, then the adjoint action of matrices $m \in \text{GL}(V)$ on $\text{End}_F(V)^0$ — that is, the conjugation action
on the vector-space of trace-zero endomorphisms — has trace $(\mathrm{tr} \, m)^2 / \det m$ and factors through PGL($V$) since scalars act trivially. By analogy, we define the adjoint-trace elements and the adjoint-trace ring attached to $(t, d)$.

**Definition 4.6.** The adjoint trace ring of $(t, d)$, denoted $A_{0,t}$ or simply $A_0$, is the closed subring of $A$ topologically generated by the adjoint-trace elements $\mathrm{tr} \, t(g) := t(g)^2/d(g)$ for $g \in \Pi$.

**Proposition 4.7.** *The adjoint trace ring $A_0$ of any pseudorepresentation of a $p$-finite profinite group is a local noetherian pro-$p$ ring. In particular, it is $N_2$ (see p. 8).*

*Proof.* By definition, $A_0$ is a closed subring of the local pro-$p$ ring $A$, so that $A_0$ is also local and pro-$p$. Moreover, by construction $A_0$ is the trace ring of a pseudodeformation of $\mathrm{ad}^0 \bar{\rho}$, where $\bar{\rho}$ is the semisimple residual representation carrying $(t, d)$. Since $\Pi$ is $p$-finite, the universal deformation ring of the pseudorepresentation associated to $\mathrm{ad}^0 \bar{\rho}$ is noetherian, hence so is its quotient $A_0$. $\square$

Adjoint-trace elements don't change when $(t(g), d(g))$ is replaced by $(\alpha t(g), \alpha^2 d(g))$ for nonzero scalars $\alpha$. This has two consequences. First, the adjoint-trace ring is unchanged under twisting.

**Proposition 4.8.** *If $\chi : \Pi \to A$ is a continuous character, then $A_{0,t} = A_{0,\chi t}$.*

*Proof.* The adjoint-trace elements of $(t, d)$ and of $(\chi t, \chi^2 t)$ are the same: $\frac{t(g)^2}{d(g)} = \frac{(\chi(g)t(g))^2}{\chi^2(g)d(g)}$. $\square$

To state the second, we define $A_{0,t}^{\mathrm{alg}} \subseteq A_{0,t}$ as the subring generated algebraically rather than topologically by the values of $t^2/d$, so that by definition $A_{0,t}$ is the closure of $A_{0,t}^{\mathrm{alg}}$ in $A$.

**Proposition 4.9.** *If $(\sigma, \eta)$ is an arbitrary conjugate self-twist of $(t, d)$, then $\sigma$ fixes $A_{0,t}^{\mathrm{alg}}$ pointwise.*

*Proof.* The adjoint-trace elements are fixed by any $\sigma$. Indeed, for $g \in \Pi$,

$$ \sigma(t(g)^2/d(g)) = \sigma(t(g))^2/\sigma(d(g)) = (\eta(g)t(g))^2/\eta(g)^2d(g) = t(g)^2/d(g). \quad \square $$

In spite of Proposition 4.9, there may not be enough automorphisms to cut out $A_0$ or $A_0^{\mathrm{alg}}$, as the following example illustrates.

**Example 4.10.** Let $G = \mathrm{GL}_2(\mathbb{F}_p[x])$ and let $\rho : G \to \mathrm{GL}_2(\mathbb{F}_p[x])$ be the inclusion representation. Define $\chi : G \to \mathbb{F}_p[x^{1/p}]^\times$ as follows: for $g \in G$ let $d_g := \det g \in \mathbb{F}_p[x]$, and set $\chi(g) := d_g(x^{1/p})$. Then $\chi$ is a character to $A := \mathbb{F}_p[x^{1/p}]$, and we consider the representation $\rho \otimes \chi : G \to \mathrm{GL}_2(A)$. Since $\rho$ has no nontrivial conjugate self-twists — or, said more precisely, for any extension $B$ of $A_0 := \mathbb{F}_p[x]$, we have $\tilde{\Sigma}_\rho(B/A_0) \cong \mathrm{Aut}(B/A_0)$, each appearing with trivial character — the conjugate self-twists of $\rho \otimes \chi$ defined over an extension $B$ of $A$ are all of the form $(\sigma, \sigma\chi/\chi)$ for $\sigma \in \mathrm{Aut}(B/A_0)$ (Proposition 4.2). Since any automorphism that fixes $A_0$ also fixes $A$, the ring fixed by all conjugate self-twists of $\rho \otimes \chi$ is $A$ itself. But in this case it's clear by inspection that $A$ is not a fullness ring (see, for example Proposition 3.3). On the other hand $A_0$ is a fullness ring — the image of $\rho \otimes \chi$ contains $\mathrm{SL}_2(A_0)$. $\triangle$

In the next section we show that, working over fields rather than rings, one can always cut out the analogue of $A_0$ with conjugate self-twists provided the usual conditions from Galois theory hold.

**4.3. Interlude: the theory over abstract fields.** In this subsection we switch gears to a topology-free setting, where we show that under Galois-theoretically favorable conditions, the *ad-joint trace field* (defined below) is exactly the fixed field of all conjugate self-twists. This result allows us to give an interpretation of the residue field of $A_0$ as the fixed field of residual conjugate self-twists, and is both inspiration and key in deducing an analogous result for (a generalization of) constant-determinant pseudorepresentations in our pro-$p$ setting in the next subsection — a lodestar and a tool.
Let $F$ be an arbitrary field whose characteristic is not 2, $G$ an abstract group, and $(t, d) : G \to F$ a pseudorepresentation. For an extension $L$ of $F$, define an $L$-valued conjugate self-twist $(\sigma, \eta)$ of $(t, d)$ as in **Definition 4.1**, but where $\sigma$ is simply a field automorphism rather than a morphism of $\mathbb{Z}_p$-algebras. Write $\tilde{\Sigma}_t(L)$ for the set of $L$-valued conjugate self-twists of $(t, d)$, and $\Sigma_t(L)$ for the group of automorphisms appearing in $\tilde{\Sigma}_t(L)$. Define the *adjoint-trace field* of $(t, d)$ to be the subfield $F_0 := F_{0,t}$ generated by the adjoint-trace elements $t(g)^2/d(g)$ over all $g \in G$. Then as in **Proposition 4.9**, the adjoint trace field $F_{0,t}$ is fixed by all conjugate self-twist automorphisms, so that $F_{0,t} \subseteq L^{\Sigma_t(L)}$. The following theorem gives conditions that guarantee reverse containment.

**Theorem 4.11.** Let $L$ be a separably closed extension of $F$, and $E \subseteq L$ an extension of $F_{0,t}$. Then

$$ \Sigma_t(L/E) = \mathrm{Aut}(L/E). $$

In particular, $\Sigma_t(F^{\mathrm{sep}}) = \mathrm{Aut}(F^{\mathrm{sep}}/F_{0,t})$.

*Proof.* To show that $\mathrm{Aut}(L/E) \subseteq \Sigma_t(L/E)$, we start with $\sigma \in \mathrm{Aut}(L/E)$ and produce a twist character. Let $\rho$ be a semisimple representation over a finite extension $F'$ of $F$ carrying $(t, d)$. Since $\sigma$ fixes the adjoint trace algebra $F_0 \subseteq E$, we have $\mathrm{tr} \, \mathrm{ad} \, \sigma\rho = \mathrm{tr} \, \mathrm{ad} \, \rho$. Moreover, $\mathrm{ad} \, \rho$ is semisimple if $\rho$ is. By Brauer-Nesbitt, therefore, the multiplicities of irreducible representations of $G$ inside $\mathrm{ad} \, \rho$ and inside $\mathrm{ad} \, (\sigma\rho)$ are equal in (the prime field of) $F$. If $F$ has characteristic 0 or characteristic $p \ge 5$, then we can already conclude that $\mathrm{ad} \, \rho \cong \mathrm{ad} \, (\sigma\rho)$. If $F$ has characteristic 3, then we split off a trivial character acting on the center using $\mathrm{ad} \, \rho = 1 \oplus \mathrm{ad}^0 \rho$ and need only eliminate the case that $\mathrm{ad}^0 \rho \cong \phi^{\oplus 3}$ and $\mathrm{ad}^0 \sigma\rho \cong \sigma\phi^{\oplus 3}$ for some character $\phi$ with $\phi \neq \sigma\phi$. But since at least one of the eigenvalues of $\mathrm{ad}^0 \rho(g)$ is 1 for any $g \in G$ (see **Lemma A.4**), this is impossible. Finally, **Theorem A.10** says that, since $\mathrm{ad} \, \rho \cong \mathrm{ad} \, \sigma\rho$ as representations over $L$, there is a character $\eta : G \to L^\times$ such that $\sigma\rho \cong \eta \otimes \rho$. Therefore $(\sigma, \eta)$ is a conjugate self-twist!

The second statement is a special case of the first since every conjugate self-twist fixes $F_{0,t}$. $\square$

**Corollary 4.12.** (1) If $F/F_{0,t}$ is a separable extension, then $(F^{\mathrm{sep}})^{\Sigma_t(F^{\mathrm{sep}})} = F_{0,t}$.

(2) Suppose that $L$ is a separably closed extension of $F$, and $E \subseteq L$ is an extension of $F_{0,t}$.

$$ \textit{If } L/E \textit{ is separable, then } L^{\Sigma_t(L/E)} = E. $$

Both statements follow from **Theorem 4.11**. The conditions of **Corollary 4.12(2)** are nontrivially satisfied, for example, if $F_{0,t} = \mathbb{Q}$, $F = \mathbb{Q}(\sqrt[3]{2})$, $E = \mathbb{Q}_p$ and $L = \overline{\mathbb{Q}}_p$. Or see **Theorem 10.3(4)**.

In particular **Corollary 4.12(1)** for $F = \mathbb{F}$ finite applies in our pro-$p$ setting. In this case, every pseudorepresentation $(t, d) : \Pi \to F$ is carried by a unique semisimple $\bar{\rho} : \Pi \to \mathrm{GL}_2(\mathbb{F})$. Write $\mathbb{E}$ for the adjoint trace field $\mathbb{F}_{0,\bar{\rho}}$.

**Corollary 4.13.** $\mathbb{E} = \overline{\mathbb{F}}^{\Sigma_{\bar{\rho}}}(\overline{\mathbb{F}}) = \mathbb{F}^{\Sigma_{\bar{\rho}}}(\mathbb{F})$

*Proof.* The first equality is **Corollary 4.12(1)**. The second follows from **Corollary 4.5**, which implies that every conjugate self-twist of $\bar{\rho}$ restricts to one defined over $\mathbb{F}$. $\square$

Our next goal is to limit a setting where the pro-$p$ analogue of the extension $F/F_{0,t}$ is Galois, so that we can obtain analogues of **Theorem 4.11** and **Corollary 4.12(1)**.

4.4. $A_0$-constant determinant and simple conjugate self-twists. We now return to our pro-$p$ setting and consider a restriction on the determinant of $(t, d)$ that shares many properties with the constant-determinant case, but is mild enough to be expected to be satisfied by all our arithmetic applications. This constraint allows us to restrict our attention to conjugate self-twists valued in the trace algebra.

**Definition 4.14.** We say that $(t, d)$ has $A_0$-constant determinant if $d$ is the product of an $A_0$-valued character and a character of finite prime-to-$p$ order. Said another way, $d$ is $A_0$-constant if its pro-$p$ part $d_1$ takes values in $A_0$. If $(t, d)$ has $A_0$-constant determinant, we call the conjugate
self-twists in $\tilde{\Sigma}_t(A_t)$ simple. For brevity we write $\tilde{\Sigma}_t$ and $\Sigma_t$ for these simple conjugate self-twists in Sections 7 through 9.

By automatic continuity, all automorphisms in $\Sigma_t(A_t)$ fix $A_0$, so that $\Sigma_t(A_t) = \Sigma_t(A_t/A_0)$. The following lemma shows that twist characters are also continuous.

**Lemma 4.15.** Let $(t, d) : \Pi \to A$ be an $A_0$-constant-determinant pseudorepresentation, and let $(t', d')$ be its constant-determinant twist obtained by twisting off the pro-p part $d_1$ of $d$. Then:

(1) $A_t = A_{t'}$

(2) $\tilde{\Sigma}_t(A_t) = \tilde{\Sigma}_{t'}(A_{t'})$

(3) If $(\sigma, \eta)$ in $\tilde{\Sigma}_t(A_t)$ is a simple conjugate self-twist, then $\eta$ is $W(\mathbb{F})^\times$-valued and continuous.

*Proof.* First note that $(t, d)$ and $(t', d')$ have the same adjoint trace algebra $A_0$, since the latter is twist invariant (Proposition 4.8). Write $\chi$ for $d_1^{-1/2}$, a continuous $A_0$-valued character. Note that $A_0 \subset A_t \cap A_{t'}$. Since $\chi$ and $\chi^{-1}$ are both valued in $A_0 \subset A_t \cap A_{t'}$, we can move back and forth using $t' = \chi t$ and $t = \chi^{-1}t$. That is $A_{t'} = A_{\chi t} \subseteq A_t$ and $A_t = A_{\chi^{-1}t'} \subseteq A_{t'}$. Part (1) follows. For (2), because $\chi$ is fixed by $\sigma$, the map $\tilde{\Sigma}_t(A_t) \to \tilde{\Sigma}_{\chi t'}(A_t)$ from Proposition 4.2 sending $(\sigma, \eta)$ to $(\sigma, \sigma\chi\chi^{-1}\eta)$ is the identity. For (3), use (2) and Corollary 4.5, or redo Proposition 4.4(1) directly and use Proposition 4.4(3). $\square$

**4.5. $K/K_0$ as a Galois extension.** Finally we assume that $A$ is a (local pro-$p$) domain and fix an $A_0$-constant-determinant pseudorepresentation $(t, d) : \Pi \to A$. By replacing $A$ by $A_t$, we may assume that $A$ is the trace algebra of $(t, d)$; let $K$ be the fraction field of $A$. Recall that $\bar{\rho}$ is the semisimple representation $\Pi \to \text{GL}_2(\mathbb{F})$ carrying $(\bar{t}, \bar{d})$. Let $A_0$ be the adjoint trace ring of $(t, d)$ with fraction field $K_0$; write $\mathbb{E}$ for the adjoint trace ring of $\bar{\rho}$. Here $\mathbb{F}$ is the residue field of $A$ and $\mathbb{E}$ is the residue field of $A_0$. Let $\Gamma := \ker \bar{\rho} \subset \Pi$. This is a finite-index normal subgroup of $\Pi$, and we can restrict $(t, d)$ to $\Gamma$ to obtain $(t|_{\Gamma}, d_1)$.

**Lemma 4.16.** We have $t(\Gamma) \subset A_0$.

*Proof.* For $\gamma \in \Gamma$, we have $t(\gamma) = 2+m$ for some $m \in \mathfrak{m}$. The corresponding adjoint-trace element is $t(\gamma)^2/d_1(\gamma) = d_1(\gamma)^{-1}(4+4m+m^2) \in A_0$. Since $d_1(\gamma)^{-1} \in A_0$ by assumption, so is $4+4m+m^2$. But any closed subring containing $4+4m+m^2$ contains $2+m = t(\gamma)$ as well since 2 is invertible. $\square$

**Proposition 4.17.** *A is a finite $A_0$-algebra.*

*Proof.* The subgroup $\Gamma := \ker \bar{\rho}$ is a finite-index normal subgroup of $\Pi$. Moreover, $A_{t|_{\Gamma}} \subseteq A_0$ (Lemma 4.16). Thus $K_0$ contains the trace field of $t|_{\Gamma}$, so that by Lemma A.12 there is an at-most quadratic extension $L_0$ over $K_0$ over which we can define a representation carrying $(t|_{\Gamma}, d_1)$. By the proof of Proposition A.13, there is a finite extension $L$ over $L_0$ containing $t(\Pi)$. In fact, letting $A^{\text{alg}}$ be the $W(\mathbb{F})A_0$-submodule of $A$ (algebraically, not topologically) generated by $t(\Pi)$, we have $A^{\text{alg}} \subseteq L$. Note that $A$ is the closure of $A^{\text{alg}}$.

We claim that $A^{\text{alg}}$ is integral over $A_0$. For every $g \in \Pi$, the element $t(g)$ is a square root of $t(g)^2$, which is in $A_0[d] \subseteq W(\mathbb{F})A_0$. Since $A^{\text{alg}}$ is generated by the $t(g)$ over $W(\mathbb{F})A_0$, the former is integral over the latter. The claim follows.

Finally, let $B$ be the integral closure of $A_0$ in $L$. Since $A_0$ is noetherian and $N_2$ (Proposition 4.7), $B$ is a finite, hence noetherian, $A_0$-algebra. Because $B$ contains $A^{\text{alg}}$ by integrality, the latter is also finite over $A_0$. But this means that $A^{\text{alg}}$ is a sum of finitely many compact submodules of $A$, so compact itself, hence closed. In other words, $A^{\text{alg}} = A$. Therefore $A$ is finite over $A_0$. $\square$

**Corollary 4.18.** *A is a multiquadratic extension of $W(\mathbb{F})A_0$. More precisely, there is an integer $n$ prime to $p$, elements $a_1, \dots, a_r$ of $A_0$, and integers $k_1, \dots, k_r$ modulo $n$ so that for $\zeta = \zeta_n$ a generator of $s(\mathbb{F}^\times)$ we have*

$$ (3) \qquad A = A_0[\zeta, \sqrt{\zeta^{k_1} a_1}, \dots, \sqrt{\zeta^{k_r} a_r}]. $$
**Proof.** Let $n$ be the order of $\mathbb{F}^\times$, so that $\zeta$ generates $W(\mathbb{F})$ and $A_0[\zeta] = W(\mathbb{F})A_0$. Since $A_0$ contains

$$a(g) := d_1(g)t^2(g)/d(g) = t^2(g)/s(\bar{d}(g)) = t^2(g)/\zeta^{k(g)}$$

for every $g \in \Pi$ and $k(g)$ depending on $g$, we see that $A$ is topologically generated over $W(\mathbb{F})A_0$ by $t(g) = \sqrt{\zeta^{k(g)}a(g)}$. By Proposition 4.17 and compactness only finitely many of these are needed. $\square$

**Theorem 4.19.** The extension $K$ over $K_0$ is finite abelian, with

$$\mathrm{Gal}(K/K_0) = \mathrm{Aut}(A/A_0) = \Sigma_t(A).$$

We give two different arguments both fundamentally rooted in Theorem 4.11.

**First proof of Theorem 4.19.** Since $K = K_0(\zeta, \sqrt{\zeta^{k_1}a_1}, \dots, \sqrt{\zeta^{k_r}a_r})$ by Corollary 4.18 (and using the same notation), it is clear that $\mathrm{Aut}(K/K_0) = \mathrm{Aut}(A/A_0)$. To see that $K/K_0$ is Galois, note that it is a subextension of $K_0(\zeta_{2n}, \sqrt{a_1}, \dots, \sqrt{a_r})/K_0$, which is a compositum of abelian extensions and hence abelian as well. Here we use the fact that the characteristic of (the prime field of) $K$ is either zero or an odd prime $p$, so that $K_0(\zeta_{2n})/K_0$ for $n$ prime to $p$ and the $K_0(\sqrt{a_i})/K_0$ are all separable and hence abelian. Therefore $K/K_0$ is abelian with $\mathrm{Gal}(K/K_0)$ a subgroup of $(\mathbb{Z}/2\mathbb{Z})^r \times (\mathbb{Z}/n\mathbb{Z})^\times$.

To see that $\mathrm{Aut}(A/A_0) = \Sigma_t(A)$, start with $\sigma \in \mathrm{Aut}(A/A_0)$ and proceed as in the proof of Theorem 4.11 to produce a character $\eta : \Pi \to K^\times$ with $(\sigma, \eta)$ a conjugate self-twist of $(t, d)$. By Lemma 4.15, $\eta$ is $A$-valued, so that $(\sigma, \eta)$ is in $\Sigma_t(A)$. $\square$

**Second proof of Theorem 4.19.** As in the first proof, it is clear that $K$ is separable over $K_0$. First suppose that $(t, d)$ has constant determinant. To show that $K/K_0$ is normal, take an embedding $\sigma$ of $K$ into $\bar{K}$ over $K_0$. Proceed as in the proof of Theorem 4.11 to get a character $\eta : \Pi \to K^\times$ so that $(\sigma, \eta)$ is a $\bar{K}$-valued conjugate self-twist. Use Corollary 4.5 to find $\sigma'$ so that $(\sigma', \eta)$ is an $A$-valued conjugate self-twist. Note that $\sigma'|_{W(\mathbb{F})A_0} = \sigma|_{W(\mathbb{F})A_0}$ since both $\sigma$ and $\sigma'$ fix $A_0$ and act the same way on $W(\mathbb{F})$ by construction. Therefore $\sigma'\sigma^{-1}$ is an embedding of $K$ that fixes $Q(W(\mathbb{F})A_0) = K_0(\zeta)$. Since $K/K_0(\zeta)$ is a multiquadratic extension, it is Galois, so $\sigma'\sigma^{-1}$ sends $K$ to itself. But $\sigma'$ also sends $K$ to itself by construction, so $\sigma$ must be an automorphism of $K$. In fact, using that $\sigma'\sigma^{-1}$ fixes $K_0$ and $t(\Pi)$, we see that $\sigma|_A = \sigma'$ and thus $\mathrm{Gal}(K/K_0) = \Sigma_t(A)$. The fact that $\Sigma_t(A)$ is abelian in the constant-determinant case follows from Corollary 7.2 and the diagram following Corollary 7.4. In general when $(t, d)$ has $A_0$-constant determinant, the result follows from the constant determinant-case and Lemma 4.15. $\square$

**Corollary 4.20.** We have $K_0 = K^{\Sigma_t(A)}$. Moreover, $A^{\Sigma_t(A)}$ is a finite extension of $A_0$ with the same field of fractions and the same normalization.

**Proof.** The first statement follows from Theorem 4.19. Since $A_0 \subseteq A^{\Sigma_t(A)} \subseteq K^{\Sigma_t(A)} = K_0$, the field of fractions of $A^{\Sigma_t(A)}$ is $K_0$. Finally, since $A$ is integral over $A_0$, so is $A^{\Sigma_t(A)}$. $\square$

Corollaries 3.8 and 4.20 imply the following fullness comparison result for two key subrings of $A$.

**Corollary 4.21.** The rings $A_0$ and $A^{\Sigma_t(A)}$, as well as their normalizations, are all fullness peers.

The example below shows that proper containment $A_0 \subsetneq A^{\Sigma_t(A)}$ is possible: $A_0$ and $A^{\Sigma_t(A)}$ need not have the same residue field.

**Example 4.22.** Define

$$\bar{G} := \left\{ \begin{pmatrix} a & b \\ b^p & a^p \end{pmatrix} \right\} \cup \left\{ \begin{pmatrix} a & b \\ -b^p & -a^p \end{pmatrix} \right\} \subset \mathrm{GL}_2(\mathbb{F}_{p^2}),$$

so that in each set above $a$ and $b$ are in $\mathbb{F}_{p^2}$ satisfying $N(a) \neq N(b)$. Let $\tilde{G} \subset \mathrm{GL}_2(\mathbb{F}_{p^2}[X])$ be the set of matrices whose determinants are in $\mathbb{F}_{p^2}$, and let $G \subset \tilde{G}$ be the set of invertible matrices that
are residually in $\overline{G}$. Finally, let $\Pi := G$ and set $\rho : \Pi \to \text{GL}_2(\mathbb{F}_{p^2}[\mathbb{X}])$ to be the inclusion map, a constant-determinant representation. Then the trace algebra $A$ of $\rho$ is $A = \mathbb{F}_{p^2}[\mathbb{X}]$ — indeed, fix a generator $\beta$ of $\mathbb{F}_{p^2}^{\times}$, so that $N(\beta) = \beta^{p+1} \neq 1$. Then for every $a \in \mathbb{F}_{p^2}^{\times}$, $G$ contains both

$$ g_a := \begin{pmatrix} 1 & \beta \\ \beta^p + \beta^{-1}aX & 1 + aX \end{pmatrix} \quad \text{and} \quad h_a := \begin{pmatrix} a & 0 \\ 0 & -a^p \end{pmatrix}, $$

so that $A$ contains both $\text{tr} g_a = 2 + aX$ and $\text{tr} h_a = a - a^p$. A similar computation shows that the adjoint trace ring is $A_0 = \mathbb{F}_p + X\mathbb{F}_{p^2}[\mathbb{X}]$: on one hand, $A_0$ contains

$$ (\text{tr} g_a)^2 / \det g_a = (4 + 4aX + a^2X^2)(1 - N(\beta))^{-1} $$

for every $a \in \mathbb{F}_{p^2}^{\times}$, and on the other hand every element of $A_0$ is residually in $\mathbb{F}_p$, reflecting the fact that $\bar{\rho}$ admits a conjugate self-twist and illustrating Corollary 4.13. Since $\rho$ itself has no conjugate self-twist — any automorphism of $A$ appearing in a conjugate self-twist must fix $A_0$ and hence extends to the identity on $K_0$, which contains $A$ — we have $A^{\Sigma_{\rho}(A)} = A$. $\triangle$

We close with a technical observation necessary for our a posteriori justification for focusing on $A$-valued conjugate self-twists for $A_0$-constant-determinant pseudorepresentations.

**Proposition 4.23.** Let $(\sigma, \eta)$ be any conjugate self-twist of an $A_0$-constant-determinant pseudorepresentation $(t, d)$. Its trace algebra $A$ is $\sigma$-stable if and only if $\sigma$ fixes $A_0$ pointwise. Both of these conditions are satisfied if $(t, d)$ is $A_0$-full.

*Proof.* If $\sigma$ restricts to an automorphism of $A$, then $\sigma|_A$ is automatically continuous, so that $\sigma$ fixing $A_0^{\text{alg}}$ pointwise per Proposition 4.9 is enough to conclude that $\sigma$ fixes all of $A_0$. Conversely, if $A_0$ is fixed by $\sigma$, then since $A$ is finite over $A_0$ in this setting (Proposition 4.17), $A_0^{\text{alg}}$ is dense in $A$, and $A$ is noetherian, we can express $A = a_1A_0 + \dots + a_nA_0$ for $a_i \in A_0^{\text{alg}}$, so that $\sigma(A) \subseteq A$.

If $(t, d)$ is $A_0$-full, then $A_0^{\text{alg}}$ contains some nonzero $A_0$-ideal $b_0$ (Proposition 3.3). The expression $A = a_1A_0 + \dots + a_nA_0$ for $a_i \in A_0^{\text{alg}}$ implies that $b_0A \subseteq A_0^{\text{alg}}$, whence $A_0^{\text{alg}}$ and $A$ are fullness peers, and in particular have the same field of fractions (Lemma 3.5). Since $A \subseteq Q(A_0^{\text{alg}})$, the values of $\sigma$ on $A$ are entirely determined by the restriction of $\sigma$ to $A_0^{\text{alg}}$. Therefore the action of $\sigma$ on $A$ coincides with the action on $A$ by the continuous extension of $\sigma|_{A_0^{\text{alg}}}$ guaranteed by Corollary 4.5. Thus $\sigma$ restricts to an automorphism of $A$, as claimed. $\square$

**Corollary 4.24.** Let $(t, d) : \Pi \to A$ be any pseudorepresentation. If $(t, d)$ is full for its adjoint trace ring $A_0$, then every arbitrary conjugate self-twist of $(t, d)$ fixes $A_0$ pointwise.

*Proof.* Let $(\sigma, \eta)$ be a conjugate self-twist of $(t, d)$, and let $(t', d')$ be its constant-determinant twist. Then $(\sigma, \eta')$ is a conjugate self-twist of $(t', d')$ for some character $\eta'$ (Proposition 4.2). Since neither fullness nor $A_0$ is changed under twisting (Corollary 3.12 and Proposition 4.8), the constant-determinant twist $(t', d')$ is still $A_0$-full, and hence by Proposition 4.23 $\sigma$ fixes $A_0$ pointwise. $\square$

Combined with our main $A_0$-fullness result (Theorem 10.1), Corollary 4.24 allows us to conclude a posteriori that under mild conditions on non-a-priori-small $(t, d)$, all conjugate self-twists of $(t, d)$ fix all of $A_0$, resolving the topological concerns we have danced around in Sections 4.1 and 4.2. Of course one continues to hope for a less circuitous argument that works for all $(t, d)$.

## 5. OPTIMALITY

Armed with the notions of conjugate self-twists and the adjoint trace ring from Section 4, we prove two related optimality results. We show that the adjoint trace ring of a pseudorepresentation $(t, d)$ contains a fullness peer of any fullness ring for $(t, d)$ (Theorem 5.3). In this way we establish the adjoint trace ring as the optimal fullness ring up to fullness peerage. We also show that every fullness ring is fixed by all conjugate self-twists (Theorem 5.4), generalizing Corollary 4.24.
As usual, $A$ is a local noetherian pro-$p$ domain carrying a continuous pseudorepresentation $(t, d) : \Pi \to A$ of a $p$-finite profinite group $\Pi$. We assume that $A$ is a domain with field of fractions $K$. Also let $A_0$ and $K_0$ be the adjoint trace ring of $(t, d)$ (Definition 4.1) and its fraction field, respectively. Recall that $(t, d)$ is $B$-full for any subring $B$ of $K$ if there exists a $(t, d)$-representation $\rho : \Pi \to R^\times$ and an embedding $R \hookrightarrow M_2(K)$ such that via this embedding $\text{Im} \rho \supseteq \Gamma_B(\mathfrak{b})$ for some nonzero ideal $\mathfrak{b}$ of $B$ (Definition 3.1).

We first show that fullness rings are fixed by conjugate self-twist automorphisms whose characters are continuous, or at least nearly so.

**Proposition 5.1.** Suppose $(t, d)$ is $B$-full for some subring $B$ of $A$. Let $(\sigma, \eta)$ be a conjugate self-twist of $(t, d)$ valued in any domain extending $A$. If $\ker \eta$ contains a subgroup $H$, closed and normal in $\Pi$, such that $\Pi/H$ is abelian, then $\sigma$ fixes $B$ pointwise.

In particular, if $(\sigma, \eta)$ is $A$-valued, then $\sigma$ always fixes $B$ pointwise.

*Proof.* By Proposition 3.4 we may replace $B$ by its closure in $A$. On one hand, since $H \subseteq \ker \eta$, we have that $t(H)$ is fixed by $\sigma$. On the other hand, by Proposition 3.11, $(t|_H, d|_H)$ is still $B$-full, so that by Proposition 3.3 applied to $(t|_H, d|_H): H \to A$, the $\mathbb{Z}$-span of $t(H)$ contains some nonzero $B$-ideal $\mathfrak{b}$. Therefore every element of $\mathfrak{b}$ is fixed by $\sigma$. Since any element of $B$ is a ratio of elements of $\mathfrak{b}$ (if $b \neq 0$ is in $\mathfrak{b}$, then $x = \frac{xb}{b}$ for any $x \in B$), every element of $B$ is fixed by $\sigma$.

If $(\sigma, \eta)$ is $A$-valued, then $\eta$ is continuous (Proposition 4.4(3)) so that we can take $H = \ker \eta$. $\square$

**Corollary 5.2.** If $(t, d)$ is $B$-full for some subring $B$ of $K$, then $B \subseteq K_0$.

*Proof.* Replace $(t, d)$ with its constant-determinant twist (Corollary 3.12 and Proposition 4.8) and let $A_t$ be its trace algebra. Replace $B$ by its fullness peer $B \cap A_t$ (Corollary 3.7 and Lemma 3.5). The result now follows from Proposition 5.1 using Theorem 4.19. $\square$

Our main optimality theorem is an improvement on Corollary 5.2: any fullness ring is not only contained in $K_0$, it in fact has a fullness peer subring contained in $A_0$.

**Theorem 5.3.** Suppose $(t, d)$ is $B$-full for some subring $B$ of $K$. Then $B \cap A_0$ is a fullness peer of $B$ contained in $A_0$.

*Proof.* First suppose that $B$ is finite. Then $B$ is a finite field; its only nonzero ideal is itself, and the only congruence subgroup of $SL_2(B)$ is $SL_2(B)$ itself. In this case, $B$ has no proper subring fullness peers and we show directly that $B$ is contained in $A_0$. Indeed, $K$, and hence $A$, is now an $\mathbb{F}_p$-algebra, and it follows that $\bar{p}$ is also $B$-full. Now Proposition 5.1 applied to $\bar{p}$ tells us that $B$ is fixed by all conjugate self-twist automorphisms of $\bar{p}$: in other words, $B \subseteq \mathbb{E}$ (Corollary 4.13), which is the residue field of, and hence here contained in, $A_0$.

Now assume that $B$ is not finite. Use Corollary 3.12 to replace $(t, d)$ by its constant-determinant twist and let $A_t \subseteq A$ be its trace algebra. Replace $B$ by its fullness peer $B \cap A_t$ (Corollary 3.12 and Proposition 4.8). Let $\Gamma = \ker \bar{\rho}$, a finite-index closed subgroup of $\Pi$. By Proposition 3.11, the restriction $(t|_{\Gamma}, d|_{\Gamma})$ is $B$-full. By Proposition 3.3, $A_0$, which contains the trace algebra of $\Gamma$ (Lemma 4.16), contains a nonzero ideal of $B$. Therefore so does $B \cap A_0$, making $B \cap A_0 \subseteq B$ an extension of fullness peers by Lemma 3.5. $\square$

We do not know whether we can sharpen Theorem 5.3 to conclude that any closed fullness ring $B$ contained in $A_0$ with $Q(B) = K_0$ must in fact be a fullness peer of $A_0$. This would follow from an affirmative answer to Question 3.6.

We close with an observation that fullness rings are always fixed by conjugate self-twist automorphisms. Corollary 4.24 has already established this for $A_0$; Theorem 5.4 below is a generalization.

**Theorem 5.4.** Let $(t, d) : \Pi \to A$ be a pseudorepresentation. If $(t, d)$ is $B$-full for some subring $B$ of $K$, then $B$ is fixed by any conjugate self-twists valued in any domain extending $A$ containing $B$.
Note that any automorphism valued in a domain E containing A extends uniquely to an automorphism of Q(E), which contains K and hence B.

*Proof.* Let $(t', d')$ be the constant determinant twist associated to $(t, d)$, obtained by twisting off the pro-$p$ part $d_1$ of $d$. As in Proposition 4.2, if $(\sigma, \eta)$ is an E-valued conjugate self-twists $(t, d)$ for some domain E extending A, then $(\sigma, \sigma(d_1^{-1/2})d_1^{1/2}\eta)$ is a E-valued conjugate self-twist of $(t', d')$. By Corollary 4.5, $\chi = \sigma(d_1^{-1/2})d_1^{1/2}\eta$ is continuous and $W(\mathbb{F})$-valued. Note that $\ker d_1 = \ker d_1^{1/2} = \ker \sigma(d_1^{-1/2})$ is closed since $d_1$ is also continuous. Thus $\ker d_1 \cap \ker \chi$ is a closed subgroup of $\ker \eta$. Moreover, since the order of $\chi$ is prime to $p$ and $d_1$ is pro-$p$ we get that

$$ \Pi/(\ker d_1 \cap \ker \chi) \cong \Pi/\ker d_1 \times \Pi/\ker \chi $$

is abelian. Thus we can take $H = \ker d_1 \cap \ker \chi$ in Proposition 5.1 to deduce that $B \cap A$ is fixed by $\sigma$. As $B \cap A$ and $B$ are fullness peers, $\sigma$ thus fixes an ideal of $B$, hence all of $B$. $\square$

## 6. FULLNESS FOR $\mathcal{B}_\rho(\mathbb{E})$

Throughout this section we fix a local pro-$p$ ring A, not necessarily a domain, with residue field $\mathbb{F}$. Fix an admissible pseudodeformation $(\Pi, \bar{\rho}, t, d)$ over A. Let $A_0$ be the adjoint trace ring of $(t, d)$ and $\mathbb{E}$ the residue field of $A_0$.

**6.1. $L_2(\rho)$ is a $W(\mathbb{E})$-module.** Recall that, a priori, Pink's construction only gives Lie algebras that are $\mathbb{Z}_p$-modules. The goal of Section 6.1 is to show that if $\rho$ is a $(t, d)$-representation, then in fact its associated Lie algebras are modules over $W(\mathbb{E})$ (Proposition 6.2). Although this is a minor improvement on $\mathbb{Z}_p$ (indeed, it is no improvement at all if $W(\mathbb{E}) = \mathbb{Z}_p$), it is an essential input for proving the results of Section 9.

We assume throughout Section 6.1 that the eigenvalues of $\bar{\rho}(g)$ are in $\mathbb{F}^\times$ for all $g \in \Pi$. This requires at most replacing $\mathbb{F}$ by its unique quadratic extension.

Let $\lambda \neq \mu \in \mathbb{F}^\times$ be the eigenvalues of a matrix in $\mathrm{Im}\,\bar{\rho}$. By Lemma 2.14, there is a $(t, d)$-representation $\rho_{\lambda,\mu}: \Pi \to R_{\lambda,\mu}^\times$ and $g_{\lambda,\mu} \in \Pi$ such that

$$ \rho_{\lambda,\mu}(g_{\lambda,\mu}) = \begin{pmatrix} s(\lambda) & 0 \\ 0 & s(\mu) \end{pmatrix}. $$

Recall that $G_{\rho_{\lambda,\mu}} := \mathrm{Im}\,\rho_{\lambda,\mu}$, $\Gamma_{\rho_{\lambda,\mu}} := G_{\rho_{\lambda,\mu}} \cap \mathrm{SR}_{\lambda,\mu}^1$, and $L_n(\rho_{\lambda,\mu}) := L_n(\Gamma_{\rho_{\lambda,\mu}})$. Since $\lambda \neq \mu$, the Lie algebra $L_1(\rho_{\lambda,\mu})$ is decomposable [Bel19, Corollary 6.2.2]. Note that although the Teichmüller map $s$ is not additive, it is easy to check that $W(\mathbb{F}_p(\lambda\mu^{-1} + \lambda^{-1}\mu)) = \mathbb{Z}_p[s(\lambda)s(\mu)^{-1} + s(\lambda)^{-1}s(\mu)]$, a fact that we make frequent use of in Lemma 6.1 below.

**Lemma 6.1.** With the notation introduced at the beginning of Section 2.5, we have

(1) $\nabla_1(\rho_{\lambda,\mu}), B_1(\rho_{\lambda,\mu}), C_1(\rho_{\lambda,\mu})$, and $L_2(\rho_{\lambda,\mu})$ are $W(\mathbb{F}_p(\lambda\mu^{-1} + \lambda^{-1}\mu))$-modules;

(2) if the projective image of $\bar{\rho}$ contains $\mathrm{PSL}_2(\mathbb{F}_p)$ and $p \ge 7$, then $I_1(\rho_{\lambda,\mu})$ is a $W(\mathbb{F}_p(\lambda\mu^{-1} + \lambda^{-1}\mu))$-module; after possibly replacing $\rho_{\lambda,\mu}$ with its conjugate by a certain $\begin{pmatrix} 1 & 0 \\ 0 & a \end{pmatrix}$ with $a \in A^\times$, one has that $L_1(\rho_{\lambda,\mu})$ is a $W(\mathbb{F}_p(\lambda\mu^{-1} + \lambda^{-1}\mu))$-module.

*Proof.* Note that

$$ L_2(\rho_{\lambda,\mu}) = [I_1(\rho_{\lambda,\mu}) \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \nabla_1(\rho_{\lambda,\mu})] + [\nabla_1(\rho_{\lambda,\mu}), \nabla_1(\rho_{\lambda,\mu})]. $$

Furthermore, from their definitions on page 14, it's clear that $B_1(\rho_{\lambda,\mu})$ and $C_1(\rho_{\lambda,\mu})$ inherit whatever structure $\nabla_1(\rho_{\lambda,\mu})$ has. Therefore it suffices to show that $\nabla_1(\rho_{\lambda,\mu})$ is a $W(\mathbb{F}_p(\lambda\mu^{-1} + \lambda^{-1}\mu))$-module.

To prove that

$$ (s(\lambda)s(\mu)^{-1} + s(\lambda)^{-1}s(\mu))\nabla_1(\rho_{\lambda,\mu}) \subseteq \nabla_1(\rho_{\lambda,\mu}), $$
recall that $L_1(\rho_{\lambda,\mu})$ is closed under conjugation by $G_{\rho_{\lambda,\mu}}$ (in fact, by any element in the normalizer of $\Gamma_{\rho_{\lambda,\mu}}$). In particular, it is closed under conjugation by $\begin{pmatrix} s(\lambda) & 0 \\ 0 & s(\mu) \end{pmatrix}$ and $\begin{pmatrix} s(\lambda)^{-1} & 0 \\ 0 & s(\mu)^{-1} \end{pmatrix}$. Using this, a short matrix calculation shows that if $\begin{pmatrix} 0 & b \\ c & 0 \end{pmatrix} \in \nabla_1(\rho_{\lambda,\mu})$ then

$$ \begin{pmatrix} 0 & s(\lambda)s(\mu)^{-1}b \\ s(\lambda)^{-1}s(\mu)c & 0 \end{pmatrix}, \begin{pmatrix} 0 & s(\lambda)^{-1}s(\mu)b \\ s(\lambda)s(\mu)^{-1}c & 0 \end{pmatrix} \in \nabla_1(\rho_{\lambda,\mu}). $$

Therefore $(s(\lambda)s(\mu)^{-1} + s(\lambda)^{-1}s(\mu))\nabla_1(\rho_{\lambda,\mu}) \subseteq \nabla_1(\rho_{\lambda,\mu})$.

Finally, if the projective image of $\bar{\rho}$ contains $\text{PSL}_2(\mathbb{F}_p)$ and $p \ge 7$, then by Theorem 2.23, up to replacing $\rho_{\lambda,\mu}$ with its conjugate by a certain $\begin{pmatrix} 1 & 0 \\ 0 & a \end{pmatrix}$ with $a \in A^\times$, we have

$$ L_1(\rho_{\lambda,\mu}) = \begin{pmatrix} I_1(\rho_{\lambda,\mu}) & I_1(\rho_{\lambda,\mu}) \\ I_1(\rho_{\lambda,\mu}) & I_1(\rho_{\lambda,\mu}) \end{pmatrix}^0, $$

and thus $B_1(\rho_{\lambda,\mu}) = I_1(\rho_{\lambda,\mu}) = C_1(\rho_{\lambda,\mu})$. (Note that conjugation by $\begin{pmatrix} 1 & 0 \\ 0 & a \end{pmatrix}$ with $a \in A^\times$ does not change $I_1(\rho_{\lambda,\mu})$.) In particular, $L_1(\rho_{\lambda,\mu})$ is strongly decomposable. By the second statement of the lemma, we see that $I_1(\rho_{\lambda,\mu})$ is a $W(\mathbb{F}_p(\lambda\mu^{-1} + \lambda^{-1}\mu))$-module. The above description of $L_1(\rho_{\lambda,\mu})$ shows that it is also a $W(\mathbb{F}_p(\lambda\mu^{-1} + \lambda^{-1}\mu))$-module. $\square$

**Proposition 6.2.** Let $\rho: \Pi \to R^\times$ be a $(t, d)$-representation. Then $L_n(\rho)$ is a $W(\mathbb{E})$-module for all $n \ge 2$. If the projective image of $\bar{\rho}$ contains $\text{PSL}_2(\mathbb{F}_p)$ and $p \ge 7$, then $L_1(\rho)$ is a $W(\mathbb{E})$-module.

*Proof.* By the generating set for $\mathbb{E}$ given in (2) in Section 2.4, it suffices to show that $L_n(\rho)$ is closed under multiplication by $s(\lambda)s(\mu)^{-1} + s(\lambda)^{-1}s(\mu)$ for all $\lambda, \mu \in \mathbb{F}_p^\times$ that are distinct eigenvalues of an element in $\text{Im } \bar{\rho}$. Fix such $\lambda, \mu$. Let $\rho_{\lambda,\mu}: \Pi \to R_{\lambda,\mu}^\times$ be the $(t, d)$-representation over $A$ described prior to Lemma 6.1. Let us assume furthermore that, in the case when $\bar{\rho}$ is not projectively cyclic or dihedral, that we have already replaced $\rho_{\lambda,\mu}$ by its relevant diagonal conjugate so that the description of $W(\mathbb{F}_q)L_1(\rho_{\lambda,\mu})$ from Theorem 2.23 applies to $\rho_{\lambda,\mu}$.

Since $\rho: \Pi \to R^\times$ and $\rho_{\lambda,\mu}: \Pi \to R_{\lambda,\mu}^\times$ are both $(t, d)$-representations over $A$, it follows from Proposition 2.11 that there is a unique $A$-algebra isomorphism $\Psi: R_{\lambda,\mu} \to R$ such that $\rho = \Psi \circ \rho_{\lambda,\mu}$. We claim that this implies that $L_n(\rho) = \Psi(L_n(\lambda, \mu))$ for all $n \ge 1$. If this is true, then $L_2(\rho)$ is closed under multiplication by $s(\lambda)s(\mu)^{-1} + s(\lambda)^{-1}s(\mu) \in A$ since $L_2(\rho_{\lambda,\mu})$ is by Lemma 6.1 and $\Psi$ is an $A$-algebra homomorphism. Since $L_2(\rho)$ is a $W(\mathbb{E})$-module, it follows immediately from the definition that $L_n(\rho)$ is a $W(\mathbb{E})$-module for all $n \ge 2$. Furthermore, the argument in this paragraph applies to $L_1(\rho)$ under the assumption that the projective image of $\bar{\rho}$ contains $\text{PSL}_2(\mathbb{F}_p)$ for $p \ge 7$.

To see that $L_n(\rho) = \Psi(L_n(\rho_{\lambda,\mu}))$, note that $G_\rho = G_{\Psi \circ \rho_{\lambda,\mu}} = \Psi(G_{\rho_{\lambda,\mu}})$. Since $\Psi$ is an algebra morphism, it follows that $\Psi(\text{rad } R_{\lambda,\mu}) = \text{rad } R$. Furthermore, since $\rho$ and $\rho_{\lambda,\mu}$ are both $(t, d)$-representations, it follows that $\Psi$ preserves determinants. Therefore $\Psi(SR_{\lambda,\mu}^1) \supset SR^1$. Since $\Psi$ is a continuous algebra homomorphism, it follows directly from the definition of $\Theta$ that $\Psi(L_1(\rho_{\lambda,\mu})) = L_1(\rho)$ and hence $\Psi(L_n(\rho_{\lambda,\mu})) = L_n(\rho)$ for all $n \ge 1$. $\square$

6.2. $L_2(\rho)$ is a $\mathcal{B}_\rho(\mathbb{E})$-module. In Section 6.2 we use Bellaïche's work to show that, for any well-adapted $(t, d)$-representation $\rho$, $L_n(\rho)$ is a module over a ring comparable to $A$. This is the key input into Corollary 6.6, which is our improvement on Bellaïche's fullness theorem.

**Proposition 6.3.** Let $\rho$ be a $(t, d)$-representation adapted to $(g_0, \lambda_0, \mu_0)$. Then

(1) $L_2(\rho)$ is a module over $W(\mathbb{E})[I_1(\rho)^2] := W(\mathbb{E}) + W(\mathbb{E})I_1(\rho)^2$;

(2) if $n \ge 1$ and $L_n(\rho)$ is strongly decomposable, then $L_{n+1}(\rho)$ is a module over $\mathcal{B}_\rho(\mathbb{E})$;

(3) if the projective image of $\bar{\rho}$ contains $\text{PSL}_2(\mathbb{F}_p)$ for $p \ge 7$, then up to replacing $\rho$ with its conjugate by some $\begin{pmatrix} 1 & 0 \\ 0 & a \end{pmatrix}$ with $a \in A^\times$, $L_1(\rho)$ is a module over $\mathcal{B}_\rho(\mathbb{E})$.

*Proof.* Since $\rho$ is adapted to $(g_0, \lambda_0, \mu_0)$, it follows that $L_1(\rho)$ is decomposable [Bel19, Corollary 6.2.2]. Note that $[I_1(\rho)(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}), \nabla_1(\rho)] \subset \nabla_1(\rho)$ since $L_1(\rho)$ is a Lie algebra. That is, for all $a \in I_1(\rho)$ and $\begin{pmatrix} 0 & b \\ c & 0 \end{pmatrix} \in \nabla_1(\rho)$, we have $2a(\begin{pmatrix} 0 & b \\ -c & 0 \end{pmatrix}) \in \nabla_1(\rho)$. To prove the first statement, we can apply this fact
a second time to $\alpha \in I_1(\rho)$ and $2a(\begin{smallmatrix} 0 & b \\ -c & 0 \end{smallmatrix})$ to see that $4a\alpha(\begin{smallmatrix} 0 & b \\ c & 0 \end{smallmatrix}) \in \nabla_1(\rho)$. Therefore $\nabla_1(\rho)$ is closed under multiplication by $I_1(\rho)^2$. Since

$$L_2(\rho) = [I_1(\rho)(\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}), \nabla_1(\rho)] + [\nabla_1(\rho), \nabla_1(\rho)],$$

we see that $L_2(\rho)$ is closed under multiplication by $I_1(\rho)^2$.

For the second statement, if $L_n(\rho)$ is strongly decomposable, then we can write

$$L_n(\rho) = I_n(\rho)(\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}) \oplus B_n(\rho)(\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix}) \oplus C_n(\rho)(\begin{smallmatrix} 0 & 0 \\ 1 & 0 \end{smallmatrix}).$$

By calculating $[(\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}), (\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix})]$ and $[(\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}), (\begin{smallmatrix} 0 & 0 \\ 1 & 0 \end{smallmatrix})]$, we find that $I_1(\rho)B_n(\rho) = B_{n+1}(\rho) \subset B_n(\rho)$ and $I_1(\rho)C_n(\rho) = C_{n+1}(\rho) \subset C_n(\rho)$. Therefore $B_n(\rho), C_n(\rho)$ are closed under multiplication by $I_1(\rho)$. Since

$$L_{n+1}(\rho) = [B_n(\rho)(\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix}), C_n(\rho)(\begin{smallmatrix} 0 & 0 \\ 1 & 0 \end{smallmatrix})] + [I_1(\rho)(\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}), B_n(\rho)(\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix})] + [I_1(\rho)(\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}), C_n(\rho)(\begin{smallmatrix} 0 & 0 \\ 1 & 0 \end{smallmatrix})],$$

it follows that $L_{n+1}(\rho)$ is closed under multiplication by $I_1(\rho)$.

The first two results now follow from Proposition 6.2. The last statement follows from Theorem 2.23 and Proposition 6.2. □

Remark 6.4. It would be nice to remove the assumption that $L_1(\rho)$ is strongly decomposable and still conclude that $L_2(\rho)$ is a $\mathcal{B}_\rho(\mathbb{E})$-module, but we do not see a way to do this. △

### 6.3. Regularity implies $\mathcal{B}_\rho(\mathbb{E})$-fullness.
The goal of Section 6.3 is to establish a slightly stronger version of [Bel19, Theorem 7.2.3], which is Bellaïche's Theorem 1.3 of the introduction. We do so in Corollary 6.6 below. Our result is different from that of Bellaïche mainly in that we can weaken his definition of regularity and enlarge his ring $\mathcal{B}_\rho(\mathbb{F}_p)$ to $\mathcal{B}_\rho(\mathbb{E})$.

Throughout Section 6.3 the ring $A$ will be a local pro-$p$ domain with residue field $\mathbb{F}$ and field of fractions $K$. We fix an admissible pseudodeformation $(\Pi, \bar{\rho}, t, d)$ over $A$ throughout this section. If $\rho$ is a $(t, d)$-representation that is adapted to some $(g_0, \lambda_0, \mu_0)$, then $L_1(\rho)$ is decomposable by [Bel19, Corollary 6.2.2]. Thus $I_1(\rho)$ is defined. We write $K_1$ for the field of fractions of $\mathcal{B}_\rho(\mathbb{E})$.

**Proposition 6.5.** Assume that $\bar{\rho}$ is regular. Let $\rho: \Pi \to R^\times$ be a $(t, d)$-representation adapted to $(g_0, \lambda_0, \mu_0)$ for a regular element $g_0$ such that $\rho(g_0) = \begin{pmatrix} s(\lambda_0) & 0 \\ 0 & s(\mu_0) \end{pmatrix}$. If $B_1(\rho), C_1(\rho) \neq 0$, then $\rho$ is $\mathcal{B}_\rho(\mathbb{E})$-full.

*Proof.* It is easy to see that the eigenvalues of $\rho(g_0) = \begin{pmatrix} s(\lambda_0) & 0 \\ 0 & s(\mu_0) \end{pmatrix}$ acting on $L_n(\rho)$ by conjugation are $1, s(\lambda_0)s(\mu_0^{-1}), s(\lambda_0^{-1})s(\mu_0)$, which are distinct elements of $W(\mathbb{E})^\times$ since $g_0$ is a regular element. Since $L_n(\rho)$ is a $W(\mathbb{E})$-module for $n \ge 2$ by Proposition 6.2, it follows that $L_n(\rho)$ is the direct sum of the eigenspaces for the conjugation action of $\rho(g_0)$. Thus, $L_n(\rho)$ is strongly decomposable for $n \ge 2$. By Proposition 6.3, it follows that $L_n(\rho)$ is an $\mathcal{B}_\rho(\mathbb{E})$-module for $n \ge 3$.

Since $A$ is a domain, we may view $R$ inside of $M_2(K)$ by Lemma 2.10. Note that if $B_1(\rho), C_1(\rho) \neq 0$, then since $I_n(\rho), B_n(\rho), C_n(\rho) \subset K$, it follows that $I_n(\rho), B_n(\rho)$, and $C_n(\rho)$ are nonzero for all $n \ge 1$. In particular, $I_3(\rho), B_3(\rho), C_3(\rho)$ are nonzero $\mathcal{B}_\rho(\mathbb{E})$-modules.

Define

$$R_1 := \begin{pmatrix} \mathcal{B}_\rho(\mathbb{E}) & B_3(\rho) \\ C_3(\rho) & \mathcal{B}_\rho(\mathbb{E}) \end{pmatrix}.$$

Then $R_1$ is a faithful GMA over $\mathcal{B}_\rho(\mathbb{E})$. By the proof of [Bel19, Lemma 2.2.2], if $0 \neq b_0 \in B_3(\rho)$ and $x = \begin{pmatrix} 1 & 0 \\ 0 & b_0 \end{pmatrix}$, it follows that $xR_1x^{-1} \subseteq \text{GL}_2(K_1)$. Thus, by replacing $\rho$ with $x\rho x^{-1}$, which is still a $(t, d)$-representation adapted to $(g_0, \lambda_0, \mu_0)$ that sends $g_0$ to $\begin{pmatrix} s(\lambda_0) & 0 \\ 0 & s(\mu_0) \end{pmatrix}$, we may assume that $B_3(\rho), C_3(\rho) \subseteq K_1$. (Note that $I_1(\rho) = I_1(x\rho x^{-1})$.)

Note that any nonzero $\mathcal{B}_\rho(\mathbb{E})$-submodule of $K_1$ contains a nonzero element of $\mathcal{B}_\rho(\mathbb{E})$ and thus contains a non-zero $\mathcal{B}_\rho(\mathbb{E})$-ideal. Therefore there exists a nonzero $\mathcal{B}_\rho(\mathbb{E})$-ideal $\mathfrak{b}$ contained in $I_3(\rho) \cap$
$B_3(\rho) \cap C_3(\rho)$. Hence $\mathfrak{sl}_2(\mathfrak{b}) \subseteq L_3(\rho)$. Using Theorem 2.16 we deduce that $\Gamma_{\mathcal{B}_\rho(\mathbb{E})}(\mathfrak{b}) \subset \mathrm{Im} \rho$ and $\rho$ is $\mathcal{B}_\rho(\mathbb{E})$-full. $\square$

**Corollary 6.6.** Assume that $\bar{\rho}$ is regular. Let $\rho$ be a well-adapted $(t, d)$-representation adapted to $(g_0, \lambda_0, \mu_0)$ for a regular element $g_0$ such that $\rho(g_0) = \begin{pmatrix} s(\lambda_0) & 0 \\ 0 & s(\mu_0) \end{pmatrix}$. If $(t, d)$ is not a priori small, then $(t, d)$ is $\mathcal{B}_\rho(\mathbb{E})$-full.

*Proof.* By Proposition 6.5 it suffices to show that $B_1(\rho), C_1(\rho) \neq 0$. We do this by analyzing the different possibilities for $\bar{\rho}$. By Lemma A.5, we see that either $\bar{\rho}$ is reducible, dihedral, or $\mathrm{ad}^0 \bar{\rho}$ is irreducible. Assume first that we are in the last case. The group $\Gamma$ is equipped with a decreasing normal filtration

$$ \Gamma_n := \Gamma \cap \Gamma_A(\mathfrak{m}^n) $$

whose quotients $\Gamma_n/\Gamma_{n+1}$ have the structure of $\mathbb{F}_p$-vector spaces with an action of $\bar{G}$ by conjugation. Moreover, the map $\gamma \mapsto \gamma - 1$ gives a $\bar{G}$-equivariant embedding $\Gamma_n/\Gamma_{n+1} \hookrightarrow \mathfrak{sl}_2(\mathfrak{m}^n/\mathfrak{m}^{n+1})$; write $V_n$ for its image. The assumption that $(t, d)$ is not a priori small implies that $\Gamma$ is nontrivial, so that $\Gamma_n/\Gamma_{n+1}$, and hence $V_n$, is nontrivial for some $n \ge 1$. To see that $B_1(\rho)$ (respectively, $C_1(\rho)$) is nonzero, it suffices to show that this $V_n$ contains an element whose upper right (respectively, lower left) entry is nonzero. This can be checked on the $\mathbb{F}$-span of $V_n$. Choosing an $\mathbb{F}$-basis $x_1, \dots, x_d$ of $\mathfrak{m}^n/\mathfrak{m}^{n+1}$ gives a $\bar{G}$-equivariant splitting $\mathfrak{sl}_2(\mathfrak{m}^n/\mathfrak{m}^{n+1}) = \bigoplus_{i=1}^d \mathfrak{sl}_2(\mathbb{F})x_i$. Since $V_n$ is nontrivial, there is some $i$ such that the projection $W$ of $\mathbb{F}V_n$ to $\mathfrak{sl}_2(\mathbb{F})x_i$ is nonzero. Then $W$ is a stable subspace of $\mathfrak{sl}_2(\mathbb{F})x_i$, which is simple since $\mathrm{ad}^0 \bar{\rho}$ is irreducible. Thus $W = \mathfrak{sl}_2(\mathbb{F})x_i$ and $W$, and hence $\mathbb{F}V_n$, contains an element whose upper right (respectively, lower left) entry is nonzero. (vii)

Now suppose that $\bar{\rho}$ is reducible. Since $\rho$ is well adapted by assumption, it follows that $\rho$ is adapted to $(g_0, \lambda_0, \mu_0)$, where $\bar{\rho}(g_0)$ generates the projective image of $\bar{\rho}$. In particular, $\rho$ is automatically adapted to a regular element. Suppose for contradiction that $C_1(\rho) = 0$ (respectively, $B_1(\rho) = 0$). Then $\Gamma_\rho$ is contained in the upper (respectively, lower) triangular matrices. By [Bel19, Theorem 6.2.1], we know that $s(\bar{G}) \subset G_\rho$ since $\rho$ is well adapted. Thus $G_\rho = s(\bar{G})\Gamma_\rho$. But then $G_\rho$ is contained in the upper (respectively, lower) triangular matrices, and hence $\rho$ is reducible. Therefore $t$ is the sum of two continuous characters $\Pi \to A^\times$, which contradicts our assumption that it is not a priori small. Thus $B_1(\rho), C_1(\rho) \neq 0$ if $\bar{\rho}$ is reducible.

Finally suppose that $\bar{\rho}$ is dihedral. By Lemma A.7 there is a unique subgroup $\Pi_0$ of index 2 in $\Pi$ such that $\bar{\rho} \cong \mathrm{Ind}_{\Pi_0}^\Pi \chi$ for some character $\chi: \Pi_0 \to \mathbb{F}^\times$. Applying the reducible case to $\bar{\rho}|_{\Pi_0}$, we see that either $\rho|_{\Pi_0}$ is reducible or $B_1(\rho|_{\Pi_0}), C_1(\rho|_{\Pi_0}) \neq 0$. The first possibility is not allowed by hypothesis, so we must have $B_1(\rho|_{\Pi_0}), C_1(\rho|_{\Pi_0}) \neq 0$. But $B_1(\rho|_{\Pi_0}) \subseteq B_1(\rho)$ and $C_1(\rho|_{\Pi_0}) \subseteq C_1(\rho)$, which proves the desired result when $\bar{\rho}$ is dihedral. $\square$

## 7. LIFTING RESIDUAL CONJUGATE SELF-TWISTS

In Section 7 we study the $(A_t$-valued) conjugate self-twists of constant-determinant pseudorepresentations. In particular, we show in Section 7.2 that they are all controlled by those of $\bar{\rho}$, and all residual conjugate self-twists lift to the universal constant-determinant pseudodeformation ring. Having shown in Section 7.1 that the group of residual conjugate self-twists is finite and abelian, we deduce the same for any constant determinant pseudorepresentation. Finally, in Section 7.3 we study special phenomena that arise when $\bar{\rho}$ is dihedral and hence there may be conjugate self-twists that are residually dihedral.

Throughout Section 7 we only consider simple conjugate self-twists. Thus we write $\tilde{\Sigma}_t = \tilde{\Sigma}_t(A_t)$ and $\tilde{\Sigma}_{\bar{\rho}}$ for $\tilde{\Sigma}_{\bar{\rho}}(\mathbb{F})$; similarly for $\Sigma_t$ and $\Sigma_{\bar{\rho}}$.

<footnote>(vii)Together with the observation that $B_n(\rho), C_n(\rho) \neq 0$ for all $n$ if it is true for $n = 1$ from Proposition 6.5, the argument here gives an independent proof of the (2) $\implies$ (1) implication in Proposition 2.4.</footnote>
7.1. **Residual conjugate self-twists.** Fix a semisimple representation $\bar{\rho}: \Pi \to \text{GL}_2(\mathbb{F})$, and recall that $\mathbb{P}: \text{GL}_2(\mathbb{F}) \to \text{PGL}_2(\mathbb{F})$ denotes the natural projection. For Section 7.1 only, we do not require $\bar{\rho}$ to be residually multiplicity-free. We begin by studying $\Sigma_{\bar{\rho}}^{\text{di}}$ and use that to show that $\tilde{\Sigma}_{\bar{\rho}}$ is finite and abelian.

**Lemma 7.1.**

(1) If $\text{Im} \mathbb{P}\bar{\rho}$ is not dihedral or cyclic of order 2, then $\Sigma_{\bar{\rho}}^{\text{di}}$ is trivial.

(2) If $\text{Im} \mathbb{P}\bar{\rho}$ is either a nonabelian dihedral group or has order 2, then $\Sigma_{\bar{\rho}}^{\text{di}}$ has order 2.

(3) If $\text{Im} \mathbb{P}\bar{\rho}$ is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$, then $\Sigma_{\bar{\rho}}^{\text{di}}$ is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$.

*Proof.* We claim that if $\bar{\rho}$ is irreducible, then the following sets are in bijection:

(a) $\Sigma_{\bar{\rho}}^{\text{di}} \setminus \{(1, 1)\}$;

(b) subgroups $\Pi_0$ of $\Pi$ such that $[\Pi: \Pi_0] = 2$ and $\bar{\rho}(\Pi_0)$ is abelian;

(c) subgroups $H$ of $\mathbb{P}\bar{\rho}(\Pi)$ such that $[\mathbb{P}\bar{\rho}(\Pi): H] = 2$ and $H$ is abelian.

Indeed, the maps between them can be described as follows. Given $(1, \eta) \in \Sigma_{\bar{\rho}}^{\text{di}} \setminus \{(1, 1)\}$, let $\Pi_0 := \ker \eta$. The fact that $[\Pi: \Pi_0] = 2$ follows from Lemma A.7, and Lemma A.6 shows that $\bar{\rho}(\Pi_0)$ is abelian. Conversely, given $\Pi_0$ as in (b), let $\eta_{\Pi_0}: \Pi \to \Pi/\Pi_0 \cong \{\pm 1\}$ be the natural projection. Note that $\bar{\rho}|_{\Pi_0}$ is reducible since $\bar{\rho}(\Pi_0)$ is abelian. Let $\chi: \Pi_0 \to \mathbb{F}^\times$ be a constituent of $\bar{\rho}|_{\Pi_0}$. Then $\bar{\rho} \cong \text{Ind}_{\Pi_0}^{\Pi} \chi$ by Frobenius reciprocity since $\bar{\rho}$ is irreducible. Thus $(1, \eta_{\Pi_0}) \in \Sigma_{\bar{\rho}}^{\text{di}} \setminus \{(1, 1)\}$ by Lemma A.7.

Given $\Pi_0$ as in (b), let $H := \mathbb{P}\bar{\rho}(\Pi_0)$. Given $H$ as in (c), let $\Pi_0 := \mathbb{P}\bar{\rho}^{-1}(H)$. It is clear that $[\Pi: \Pi_0] = 2$. That $\bar{\rho}(\Pi_0)$ is abelian follows from the fact that $H$ is abelian and scalar matrices commute with everything.

When $\bar{\rho}$ is irreducible, the lemma now follows from counting subgroups as in (c) in each of the possible projective images of $\bar{\rho}$. (The fact that elements in $\Sigma_{\bar{\rho}}^{\text{di}}$ have order at most 2 follows from the fact that $\det \bar{\rho} = \eta^2 \det \bar{\rho}$ and so $\eta^2 = 1$.)

Finally, suppose that $\bar{\rho} = \varepsilon \oplus \delta$. If $(1, \eta) \in \Sigma_{\bar{\rho}}^{\text{di}}$ and $\eta$ is nontrivial, then we must have $\eta\varepsilon = \delta$ and $\eta\delta = \varepsilon$. Thus

$$ \varepsilon\delta^{-1} = \eta = \delta\varepsilon^{-1}, $$

which implies that $\varepsilon\delta^{-1}$ has order 2. But the projective image of $\bar{\rho}$ is isomorphic to the image of $\varepsilon\delta^{-1}$. Thus $\Sigma_{\bar{\rho}}^{\text{di}}$ is trivial unless the projective image of $\bar{\rho}$ has order 2, in which case there is one nontrivial element. $\square$

**Corollary 7.2.** *The group $\tilde{\Sigma}_{\bar{\rho}}$ is finite and abelian.*

*Proof.* Since $\mathbb{F}$ is a finite field, there are only finitely many automorphisms of $\mathbb{F}$. Any two elements in $\tilde{\Sigma}_{\bar{\rho}}$ with the same automorphism differ by an element of $\Sigma_{\bar{\rho}}^{\text{di}}$, which is finite by Lemma 7.1.

To see that $\tilde{\Sigma}_{\bar{\rho}}$ is abelian, fix a generator $\sigma$ of the cyclic group $\Sigma_{\bar{\rho}} = \text{Gal}(\mathbb{F}/\mathbb{E})$. Let $\eta$ be a character such that $(\sigma, \eta) \in \tilde{\Sigma}_{\bar{\rho}}$. Then $\tilde{\Sigma}_{\bar{\rho}}$ is generated by $\{(\sigma, \eta\eta'): (1, \eta') \in \Sigma_{\bar{\rho}}^{\text{di}}\}$. Since $\eta'$ is at most quadratic by Lemma 7.1, the action of $\sigma$ on $\eta'$ is trivial and hence one easily checks that any two of the generators commute. $\square$

7.2. **Lifting conjugate self-twists.** Let $\Pi$ be a profinite group satisfying the $p$-finiteness condition. Fix a multiplicity-free representation $\bar{\rho}: \Pi \to \text{GL}_2(\mathbb{F})$. Recall from Section 2.1 that there is a local pro-$p$ noetherian $W(\mathbb{F})$-algebra $\mathcal{A}$ with maximal ideal $\mathfrak{m}_{\mathcal{A}}$ and residue field $\mathbb{F}$ and a pseudo-deformation $(T, d): \Pi \to \mathcal{A}$ that is universal among all constant-determinant pseudodeformations of $\bar{\rho}$. The purpose of Section 7.2 is to show that every conjugate self-twist of $\bar{\rho}$, and in fact of every constant-determinant pseudodeformation of $\bar{\rho}$, can be lifted to a conjugate self-twist of $(T, d)$ (see Proposition 7.3 and Corollary 7.4 below).
Since we are working only with constant-determinant pseudodeformations, we shall identify any $\mathbb{F}$-valued character $\eta$ with the $W(\mathbb{F})$-valued character $s(\eta)$. Furthermore, we will consider $\eta$ as being valued in any $W(\mathbb{F})$-algebra via the structure map. If $\sigma$ is an automorphism of $\mathbb{F}$, we write $W(\sigma)$ the automorphism of $W(\mathbb{F})$ induced by $\sigma$.

We introduce some notation that will be used in the proof of Proposition 7.3. For any $W(\mathbb{F})$-algebra $A$, let $A^\sigma := A \otimes_{W(\mathbb{F}),W(\sigma)} W(\mathbb{F})$, where $W(\mathbb{F})$ is considered as a $W(\mathbb{F})$-algebra via $W(\sigma)$. We can equip $A^\sigma$ with two different $W(\mathbb{F})$-algebra structures by letting $W(\mathbb{F})$ act either on the first or second factor of the tensor product. In what follows, we refer to these actions respectively as the first or second $W(\mathbb{F})$-algebra structure on $A^\sigma$. Let $\iota(\sigma, A): A \to A^\sigma$ be the natural map given by $\iota(\sigma, A)(a) = a \otimes 1$. It is an isomorphism of rings with inverse given by $\iota(\sigma^{-1}, A^\sigma)$ since $(A^\sigma)^{-1}$ can be naturally identified with $A$ as a $W(\mathbb{F})$-algebra. Furthermore, $\iota(\sigma, A)$ is a morphism of $W(\mathbb{F})$-algebras with respect to the first structure on $A^\sigma$. Note that if we view $A^\sigma$ with respect to its second $W(\mathbb{F})$-algebra structure, its residue field is $\mathbb{F} \otimes_{\mathbb{F},\sigma} \mathbb{F}$, which is identified with $\mathbb{F}$ via $x \otimes y \mapsto \sigma(x)y$. The proof of the following proposition is a more streamlined treatment of the arguments in [Lan16, Section 2].

**Proposition 7.3.** Let $(\sigma, \eta) \in \tilde{\Sigma}_{\bar{\rho}}$. Then there is an automorphism $\tilde{\sigma}$ of $\mathcal{A}$ such that $(\tilde{\sigma}, \eta) \in \tilde{\Sigma}_T$ and $\tilde{\sigma}$ induces $\sigma$ modulo $\mathfrak{m}_{\mathcal{A}}$. Furthermore, for any $w$ in the image of $W(\mathbb{F})$ in $\mathcal{A}$, we have $\tilde{\sigma}(w) = W(\sigma)(w)$.

Note that any such $\tilde{\sigma}$ is necessarily unique, because it is determined by the character $\eta$.

*Proof.* Note that $\eta T: \Pi \to \mathcal{A}$ is the universal constant-determinant pseudodeformation of $\eta \otimes \bar{\rho} \cong \sigma \bar{\rho}$. We claim that, considering $\mathcal{A}^{-1}$ with its second $W(\mathbb{F})$-algebra structure, $\iota(\sigma^{-1}, \mathcal{A}) \circ (\eta T)$ is a constant-determinant pseudodeformation of $\bar{\rho}$. Indeed, reducing $\iota(\sigma^{-1}, \mathcal{A}) \circ (\eta T)$ modulo the maximal ideal of $\mathcal{A}^{-1}$ gives

$$ \iota(\sigma^{-1}, \mathbb{F}) \circ (\eta \operatorname{tr} \bar{\rho}) = \sigma \operatorname{tr} \bar{\rho} \otimes_{\mathbb{F}, \sigma^{-1}} 1 = 1 \otimes \operatorname{tr} \bar{\rho}, $$

which is identified with $\operatorname{tr} \bar{\rho}$ under the identification of $\mathbb{F} \otimes_{\mathbb{F},\sigma^{-1}} \mathbb{F}$ with $\mathbb{F}$ discussed prior to the proposition.

By universality, there is a unique $W(\mathbb{F})$-algebra homomorphism $\alpha: \mathcal{A} \to \mathcal{A}^{-1}$, where $\mathcal{A}^{-1}$ is given its second $W(\mathbb{F})$-algebra structure, such that

$$ \alpha \circ T = \iota(\sigma^{-1}, \mathcal{A}) \circ (\eta T). $$

Since $\iota(\sigma, \mathcal{A}^{-1})$ is the inverse of $\iota(\sigma^{-1}, \mathcal{A})$, we have that

$$ (4) \qquad \iota(\sigma, \mathcal{A}^{-1}) \circ \alpha \circ T = \eta T. $$

Define $\tilde{\sigma} := \iota(\sigma, \mathcal{A}^{-1}) \circ \alpha$, which is a ring endomorphism of $\mathcal{A}$. The relation (4) implies that $\tilde{\sigma}$ is an automorphism of $\mathcal{A}$ since the image of $T$ topologically generates $\mathcal{A}$ as a $W(\mathbb{F})$-module [Bel19, Proposition 5.3.3] and $\eta$ takes values in $W(\mathbb{F})$. The relation (4) also shows that $(\tilde{\sigma}, \eta) \in \tilde{\Sigma}_T$.

Finally, let $w \in W(\mathbb{F})$. Since $\alpha$ is a $W(\mathbb{F})$-algebra homomorphism with respect to the second $W(\mathbb{F})$-algebra structure on $\mathcal{A}^{-1}$, we have that

$$ \begin{align*} \tilde{\sigma}(w) &= \iota(\sigma, \mathcal{A}^{-1}) \circ \alpha(w) = \iota(\sigma, \mathcal{A}^{-1})(1 \otimes w) \\ &= \iota(\sigma^{-1}, \mathcal{A})^{-1}(W(\sigma)(w) \otimes 1) = W(\sigma)(w). \end{align*} \quad \square $$

For the rest of Section 7.2, fix a local pro-$p$ $W(\mathbb{F})$-algebra $A$ with residue field $\mathbb{F}$ and a constant-determinant pseudodeformation $(t, d): \Pi \to A$ of $\bar{\rho}$. Assume that $A$ is the $W(\mathbb{F})$-algebra generated by $t(\Pi)$. Let $\alpha_t: A \to A$ be the unique $W(\mathbb{F})$-algebra homomorphism such that $\alpha \circ T = t$ given by universality. The following corollary shows that conjugate self-twists of $(t, d)$ also lift to conjugate self twists of $(T, d)$.
**Corollary 7.4.** Given $(\sigma, \eta) \in \tilde{\Sigma}_t$, there is a unique $(\tilde{\sigma}, \eta) \in \tilde{\Sigma}_T$ such that $\alpha_t \circ \tilde{\sigma} = \sigma \circ \alpha_t$.

*Proof.* Let $\bar{\sigma}$ denote the automorphism of $\mathbb{F}$ induced by $\sigma$. Let $\tilde{\sigma}$ be the automorphism of $\mathcal{A}$ given by Proposition 7.3 lifting $\bar{\sigma}$ to $\mathcal{A}$. Then we just have to show that $\alpha_t \circ \tilde{\sigma} = \sigma \circ \alpha_t$. Note that $\sigma$ acts by $W(\bar{\sigma})$ on the image of $W(\mathbb{F})$ in $\mathcal{A}$, so $\sigma^{-1} \circ \alpha \circ \tilde{\sigma}$ is a $W(\mathbb{F})$-algebra homomorphism. Thus by universality, it suffices to show that $t = \sigma^{-1} \circ \alpha_t \circ \tilde{\sigma} \circ T$. Since $\eta$ takes values in $W(\mathbb{F})$ and $\alpha_t$ is a $W(\mathbb{F})$-algebra homomorphism, we have that

$$ \sigma^{-1} \circ \alpha_t \circ \tilde{\sigma} \circ T = \sigma^{-1} \circ \alpha_t(\eta T) = \sigma^{-1}(\eta t) = \sigma^{-1}(\sigma(t)) = t. \quad \square $$

We end Section 7.2 with some observations about the consequences of Proposition 7.3 and Corollary 7.4. They give the following commutative diagram with exact rows.

<figure><img src="image_4.png" /></figure>

We write

$$ \beta_t : \Sigma_t \to \Sigma_{\bar{\rho}} $$

for the composition of the vertical maps on the right in the above diagram. It is induced by the composition $\tilde{\beta}_t : \tilde{\Sigma}_t \to \tilde{\Sigma}_{\bar{\rho}}$ of the middle maps, which reflects the fact that every conjugate self-twist of $(t, d)$ induces a conjugate self-twist of $\bar{\rho}$. Combining Corollary 7.4 with Corollary 7.2, we see that $\tilde{\Sigma}_t$ is a finite abelian group for any constant-determinant pseudodeformation $(t, d)$ of $\bar{\rho}$.

In this paper, we will only be concerned with pseudodeformations $(t, d)$ of $\bar{\rho}$ that are not a priori small. Under this assumption, if $t \neq \text{tr} \bar{\rho}$ then $\Sigma_t^{\text{di}} = 1$ and $\Sigma_T^{\text{di}} = 1$ by Lemma 7.1(1). In particular, $\tilde{\Sigma}_t = \Sigma_t$ and $\tilde{\Sigma}_T = \Sigma_T$, so (except for $\bar{\rho}$) a conjugate self-twist $(\sigma, \eta)$ is determined uniquely by the automorphism $\sigma$.

**7.3. The dihedral case.** As usual, let $\Pi$ be a $p$-finite profinite group, $A$ a local pro-$p$ ring and $(t, d) : \Pi \to A$ a constant determinant pseudorepresentation with trace algebra $A$. Throughout this section we assume that its associated residual representation $\bar{\rho}$ is dihedral with nonabelian projective image. We also fix any well-adapted $(t, d)$-representation $\rho$, which can be taken to be valued in $\text{GL}_2(A)$ since $\bar{\rho}$ is absolutely irreducible Proposition 2.11(5). This case requires special care for two related reasons. First, it is the only case when $A$ is not generated simply by $I_1(\rho)$ as a $W(\mathbb{F})$-algebra; one also needs to include $B_1(\rho)$ in the generating set by Theorem 2.23. (As explained in Remark 2.12, it makes sense to view $B_1(\rho)$ as a subset of $A$ in this case.) Second, this is the only case when $\Sigma_{\bar{\rho}}^{\text{di}}$ is nontrivial and hence $\text{ker } \beta_t$ can be nontrivial. As we will see, a nontrivial element in $\text{ker } \beta_t$ necessarily behaves quite differently from all other conjugate self-twists, because its action cannot be seen on the residue field. In Proposition 7.5 we explain how a nontrivial element in $\text{ker } \beta_t$ interacts with $I_1(\rho)$ and $B_1(\rho)$. In contrast, when $\text{ker } \beta_t = 1$ we show that $\mathcal{B}_{\rho}(\mathbb{F})$ has the same fraction field as $A = \mathcal{B}_{\rho}(\mathbb{F}) + W(\mathbb{F})B_1(\rho)$.

We use $\tau$ to refer to the nontrivial element of $\text{ker } \beta_t$, should it exist. For $\varepsilon \in \{+, -\}$, let

$$ A^{\varepsilon} := \{a \in A : \tau a = \varepsilon a\}. $$

**Proposition 7.5.** Assume that $\bar{\rho}$ is projectively dihedral and nonabelian. Suppose there exists $1 \neq \tau \in \text{ker } \beta_t$. If $\rho : \Pi \to \text{GL}_2(A)$ is a well-adapted $(t, d)$-representation, then $A^+ = \mathcal{B}_{\rho}(\mathbb{F})$ and $A^- = W(\mathbb{F})B_1(\rho)$.
Proof. Since $A = A^+ \oplus A^-$ and $A = \mathcal{B}_\rho(\mathbb{F}) + W(\mathbb{F})B_1(\rho)$ by Bellaïche's Theorem 2.23, it suffices to show that $\tau$ acts trivially on $I_1(\rho)$ and by $-1$ on $B_1(\rho)$. Let $\eta: \Pi \to \{\pm 1\}$ be the unique quadratic character such that $\bar{\rho} \cong \bar{\rho} \otimes \eta$. (It is unique by Lemma 7.1 since the projective image of $\bar{\rho}$ is not isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$). Since $\tau \in \ker \beta_t$, it follows that $\eta$ must be the character such that $(\tau, \eta) \in \tilde{\Sigma}_t$.

We first prove that $I_1(\rho)$ is fixed by $\tau$. As usual, let $\Gamma := \mathrm{Im} \rho \cap \Gamma_A(\mathfrak{m})$. Recall that by definition $I_1(\rho)$ is the $\mathbb{Z}_p$-module topologically generated by $\{\alpha - \delta: \binom{1+\alpha}{c} \binom{b}{1+\delta} \in \Gamma\}$. Let $g \in \ker \eta$. Since $\rho$ is well adapted, we can write

$$ \rho(g) = \gamma \cdot \begin{pmatrix} s(\lambda) & 0 \\ 0 & s(\mu) \end{pmatrix} $$

for some $\gamma \in \Gamma$ and $\lambda, \mu \in \mathbb{F}^\times$. Write $\gamma = \binom{1+\alpha}{c} \binom{b}{1+\delta}$ with $\alpha - \delta = 2a$, $\binom{a}{c} \binom{b}{-a} \in L_1(\rho)$ and $0 = \alpha + \delta + \alpha\delta - bc$. Then we have

$$ \rho(g) = \begin{pmatrix} s(\lambda)(1+\alpha) & s(\mu)b \\ s(\lambda)c & s(\mu)(1+\delta) \end{pmatrix}. $$

Since $g \in \ker \eta$, it follows that

$$ s(\lambda)(1 + \alpha) + s(\mu)(1 + \delta) = \mathrm{tr} \rho(g) = \tau(\mathrm{tr} \rho(g)) = s(\lambda)(1 + \tau\alpha) + s(\mu)(1 + \tau\delta), $$

since $\tau$ acts trivially on $W(\mathbb{F})$. Thus, we obtain

$$ s(\lambda_0 \mu_0^{-1})(\alpha - \tau\alpha) = \tau\delta - \delta $$

for all $\lambda, \mu$ such that $\binom{\lambda}{0} \binom{0}{\mu} \in \mathrm{Im} \bar{\rho}$. As the projective image of $\bar{\rho}$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$ or $(\mathbb{Z}/2\mathbb{Z})^2$, it follows that when $g$ varies $\lambda\mu^{-1}$ takes at least two distinct values in $\mathbb{F}^\times$. Thus it follows that $\alpha - \tau\alpha = 0 = \tau\delta - \delta$. Since $I_1(\rho)$ is generated by $\alpha - \delta$ with $\alpha, \delta$ as above, it follows that $I_1(\rho)$ is fixed by $\tau$.

The proof that $\tau$ acts by $-1$ on $B_1(\rho)$ is similar. Namely, recall that $B_1(\rho)$ is topologically generated by $\{b, c \in A: \binom{1+\alpha}{c} \binom{b}{1+\delta} \in \Gamma\}$. Let $g \in \Pi \setminus \ker \eta$. Again since $\rho$ is well adapted, we can write

$$ \rho(g) = \gamma \cdot \begin{pmatrix} 0 & s(\lambda) \\ s(\mu) & 0 \end{pmatrix} $$

for some $\gamma \in \Gamma$, $\lambda, \mu \in \mathbb{F}^\times$. As above, write $\gamma = \binom{1+\alpha}{c} \binom{b}{1+\delta}$. Then we have

$$ \rho(g) = \begin{pmatrix} s(\mu)b & s(\lambda)(1+\alpha) \\ s(\mu)(1+\delta) & s(\lambda)c \end{pmatrix}. $$

Since $g \notin \ker \eta$, it follows that

$$ s(\mu)b + s(\lambda)c = \mathrm{tr} \rho(g) = -\tau(\mathrm{tr} \rho(g)) = -s(\mu)^{\tau}b - s(\lambda)^{\tau}c. $$

Thus

$$ s(\mu\lambda^{-1})(b + \tau b) = -(c + \tau c) $$

for all $\binom{0}{\mu} \binom{\lambda}{0} \in \mathrm{Im} \bar{\rho}$. Once again, since the projective image of $\bar{\rho}$ is not isomorphic to $\mathbb{Z}/2\mathbb{Z}$ or $(\mathbb{Z}/2\mathbb{Z})^2$, it follows that $\lambda\mu^{-1}$ takes at least two distinct values in $\mathbb{F}^\times$. Therefore $b + \tau b = 0 = c + \tau c$. Since $B_1(\rho)$ is generated by such $b$ and $c$, it follows that $\tau$ acts on $B_1(\rho)$ by $-1$. $\square$

In particular, we can always apply Proposition 7.5 to the universal pseudorepresentation $(T, d): \Pi \to \mathcal{A}$ of $\bar{\rho}$ since $\Sigma_T \cong \tilde{\Sigma}_{\bar{\rho}}$ and thus $\ker \beta_T$ is always nontrivial in the dihedral case whenever a nondihedral deformation exists. Fix a well-adapted $(T, d)$-representation $\rho^{\mathrm{univ}}: \Pi \to \mathrm{GL}_2(\mathcal{A})$. By universality, we have a $W(\mathbb{F})$-algebra homomorphism $\alpha_t: \mathcal{A} \to \mathcal{A}$ such that $t = \alpha_t \circ T$. Let $\rho = \rho_t$ be the well-adapted $(t, d)$-representation obtained by composing $\rho^{\mathrm{univ}}$ with the map $\mathrm{GL}_2(\mathcal{A}) \to \mathrm{GL}_2(\mathcal{A})$ induced by $\alpha_t$. Since the Pink-Lie algebra is functorial with respect to surjective ring homomorphisms, we see that $I_1(\rho^{\mathrm{univ}})$ (respectively, $B_1(\rho^{\mathrm{univ}})$) surjects onto $I_1(\rho)$ (respectively, $B_1(\rho)$).
For any subfield $\mathbb{F}' \subseteq \mathbb{F}$, let $\mathcal{A}' = \mathcal{B}_{\rho_{\text{univ}}}(\mathbb{F}') + W(\mathbb{F}')B_1(\rho^{\text{univ}})$. We have $\mathcal{A}' = (\mathcal{A}')^+ \oplus (\mathcal{A}')^-$ with $(\mathcal{A}')^+ = \mathcal{B}_{\rho_{\text{univ}}}(\mathbb{F}')$ and $(\mathcal{A}')^- = W(\mathbb{F}')B_1(\rho^{\text{univ}})$ by Proposition 7.5.

**Proposition 7.6.** Suppose that $A$ is a local pro-$p$ domain and $(t, d): \Pi \to A$ is a constant-determinant pseudodeformation of a dihedral $\bar{p}$ such that $\ker \beta_t = 1$. Let $\rho$ be a well-adapted $(t, d)$-representation obtained from a universal one as described above. Then for any subfield $\mathbb{F}' \subseteq \mathbb{F}$:

(1) $\mathcal{A}'$ and $\mathcal{B}_{\rho}(\mathbb{F}')$ are noetherian rings;

(2) $W(\mathbb{F}')B_1(\rho)$ and hence $\mathcal{B}_{\rho}(\mathbb{F}') + W(\mathbb{F}')B_1(\rho)$ are noetherian $\mathcal{B}_{\rho}(\mathbb{F}')$-modules;

(3) $\mathcal{B}_{\rho}(\mathbb{F}') + W(\mathbb{F}')B_1(\rho)$ and $\mathcal{B}_{\rho}(\mathbb{F}')$ have the same field of fractions.

*Proof.* For (1) note that $\mathcal{B}_{\rho}(\mathbb{F}')$ is the image of $(\mathcal{A}')^+$ under $\alpha_t$, which is noetherian if $\mathcal{A}'$ is by Lemma A.16. To see that $\mathcal{A}'$ is noetherian when $\mathbb{F}' = \mathbb{F}$, we have $\mathcal{A}' = \mathcal{A} = \mathcal{B}_{\rho_{\text{univ}}}(\mathbb{F}) + W(\mathbb{F})B_1(\rho^{\text{univ}})$ by Theorem 2.23, and hence $\mathcal{A}$ is noetherian by the $p$-finiteness of $\Pi$. Note that $\mathcal{A}$ is finite and integral over $\mathcal{B}_{\rho_{\text{univ}}}(\mathbb{F}') + W(\mathbb{F}')B_1(\rho^{\text{univ}})$ since this is true of $W(\mathbb{F})$ over $W(\mathbb{F}')$. Then $\mathcal{A}' = \mathcal{B}_{\rho_{\text{univ}}}(\mathbb{F}') + W(\mathbb{F}')B_1(\rho^{\text{univ}})$ is noetherian by [Eak86, Theorem 2].

The statements in (2) follow from the corresponding statements for $\rho^{\text{univ}}$, which in turn follow from Proposition A.18 since $\mathcal{A}'$ is noetherian.

As $\mathcal{B}_{\rho}(\mathbb{F}')$ is the image of $(\mathcal{A}')^+$ under $\alpha_t$ while $W(\mathbb{F}')B_1(\rho)$ is the image of $(\mathcal{A}')^-$ and $\mathcal{A}'$ is noetherian, (3) follows from Proposition A.19. $\square$

## 8. REGULARITY AND RESIDUAL CONJUGATE SELF-TWISTS

In this section we prepare the groundwork for Section 9 by studying conjugate self-twists of $\bar{\rho}$, particularly how they interact with regularity (Definition 2.19). In particular, $\bar{\rho}: \Pi \to \text{GL}_2(\mathbb{F})$ is a semisimple regular representation throughout this section, and after Section 8.1 it is always absolutely irreducible. We only consider simple conjugate self-twists in this section and thus write $\tilde{\Sigma}_{\bar{\rho}}$ for $\tilde{\Sigma}_{\bar{\rho}}(\mathbb{F})$ and similarly for $\Sigma_{\bar{\rho}}$. In Section 8.1 we see that when $\bar{\rho}$ is regular we may assume that it has no conjugate self-twists. We consider the restriction of $\bar{\rho}$ to the kernels of twist characters in Section 8.2, followed by some technical lemmas about the extension $\mathbb{F}/\mathbb{E}$ in Section 8.3. In Section 8.4 we define the condition of goodness when $\bar{\rho}$ is octahedral, which weighs on our main theorem. Section 8.4 culminates in Proposition 8.13 where we choose the basis of $\bar{\rho}$ that will be used throughout Section 9.

Recall that $\bar{\rho}$ is regular if $\text{Im} \bar{\rho}$ contains an element with eigenvalues $\lambda_0, \mu_0 \in \mathbb{F}^\times$ such that $\lambda_0\mu_0^{-1} \in \mathbb{E}^\times \setminus \{\pm 1\}$.

**8.1. Reducible regular representations.** We show that if $\bar{\rho}$ is reducible and regular, then one can eliminate the conjugate self-twists of $\bar{\rho}$ by twisting $\bar{\rho}$ by a character, making the proof of our main theorem especially easy in that case (cf. Section 9.1).

**Lemma 8.1.** Suppose that $\bar{\rho} = \varepsilon \oplus \delta$ and $\bar{\rho}$ is regular. If $(\sigma, \eta) \in \tilde{\Sigma}_{\bar{\rho}}$, then $\sigma\varepsilon = \eta\varepsilon$ and $\sigma\delta = \eta\delta$. In particular, $\varepsilon\delta^{-1}$ takes values in $\mathbb{E}$.

*Proof.* It suffices to show that if $\bar{\rho}$ is regular then we cannot have $\sigma\varepsilon = \eta\delta$ and $\sigma\delta = \eta\varepsilon$. If this were true, then we would have $\sigma\varepsilon\delta^{-1} = \eta = \sigma\delta\varepsilon^{-1}$, which implies that

$$ (5) \qquad \sigma(\varepsilon\delta^{-1}) = \delta\varepsilon^{-1}. $$

Since $\bar{\rho}$ is regular, there is some $g \in \Pi$ such that $\varepsilon(g)\delta(g)^{-1} \in \mathbb{E} \setminus \{\pm 1\}$. As $\mathbb{E}$ is fixed by $\sigma$ by Proposition 4.9, it follows from (5) that $\varepsilon(g)\delta(g)^{-1} = \pm 1$, a contradiction.

The last sentence in the statement of the lemma follows from the fact that, for any $\sigma \in \Sigma_{\bar{\rho}} = \text{Gal}(\mathbb{F}/\mathbb{E})$, we have $\sigma\varepsilon\varepsilon^{-1} = \eta = \sigma\delta\delta^{-1}$ and hence $\varepsilon\delta^{-1}$ is fixed by $\text{Gal}(\mathbb{F}/\mathbb{E})$. $\square$

**Corollary 8.2.** Suppose $\bar{\rho} = \varepsilon \oplus \delta$ and $\bar{\rho}$ is regular. Then $\bar{\rho}' := \bar{\rho} \otimes \delta^{-1}$ has no conjugate self-twists.
*Proof.* Since $\bar{\rho}$ is regular, its projective image cannot have order 2. Therefore it suffices to show that $\Sigma_{\bar{\rho}'}$ is trivial by Lemma 7.1. Let $\mathbb{F}'$ be the extension of $\mathbb{F}_p$ generated by the trace of $\bar{\rho}'$. Then $\Sigma_{\bar{\rho}'} = \text{Gal}(\mathbb{F}'/\mathbb{E})$, so it suffices to show that $\mathbb{F}' \subseteq \mathbb{E}$. But $\mathbb{F}'$ is generated by the values of $\varepsilon\delta^{-1}$, which takes values in $\mathbb{E}$ by Lemma 8.1. $\square$

## 8.2. Kernels of twist characters and regularity.
In this section we introduce the subgroup of $\Pi$ given by intersecting the kernels of all twist characters. It is often useful to restrict to this subgroup because doing so kills the conjugate self-twists and but retains fullness. In this section we study how this restriction affect the residual representation and regularity, first in the exceptional/large image case, then when $\bar{\rho}$ is dihedral. Define

$$ (6) \qquad \Pi_0(\bar{\rho}) := \bigcap_{(\sigma, \eta) \in \tilde{\Sigma}_{\bar{\rho}}} \ker \eta. $$

*Remark 8.3.* The quotient $\Pi/\Pi_0(\bar{\rho})$ is abelian. Indeed, $\Pi_0(\bar{\rho})$ is the kernel of the natural map diagonal map of $\Pi$ to $\prod_{(\sigma, \eta) \in \tilde{\Sigma}_{\bar{\rho}}} \Pi/\ker \eta$, so $\Pi/\Pi_0(\bar{\rho})$ can be embedded into an abelian group. $\triangle$

As we will now see, it is easiest to control $\bar{\rho}|_{\Pi_0(\bar{\rho})}$ when the order of $\det \bar{\rho}$ is a power of 2, which is an assumption we are forced to make in Section 9. We may always twist $\bar{\rho}$ by a character to assume that the order of $\det \bar{\rho}$ is a power of 2:

**Lemma 8.4.** Let $d: \Pi \to \mathbb{F}^\times$ be a character. Then there is a character $\chi: \Pi \to \mathbb{F}^\times$ such that the order of $d\chi^2$ is a power of 2.

*Proof.* The odd-order part of $d$ has a square root $\psi$. Take $\chi = \psi^{-1}$. $\square$

**Lemma 8.5.** Assume that $\bar{\rho}$ is exceptional or large. If the order of $\det \bar{\rho}$ is a power of 2, then $\bar{\rho}|_{\Pi_0(\bar{\rho})}$ is absolutely irreducible.

*Proof.* If $(\sigma, \eta) \in \tilde{\Sigma}_{\bar{\rho}}$, then $\eta^2$ is equal to a power of $\det \bar{\rho}$ and hence the order of $\eta$ is a power of 2. Thus $[\Pi : \Pi_0(\bar{\rho})]$ is a power of 2, and $\Pi/\Pi_0(\bar{\rho})$ is abelian by Remark 8.3.

By hypothesis, the projective image of $\bar{\rho}$ is isomorphic to one of $A_4$, $S_4$, $A_5$, $PSL_2(\mathbb{E})$, $PGL_2(\mathbb{E})$. None of $A_4$, $A_5$, $PSL_2(\mathbb{E})$ has a subgroup of 2-power index with abelian quotient. Both $S_4$ and $PGL_2(\mathbb{E})$ have a unique proper 2-power index subgroup with abelian quotient, namely $A_4$ and $PSL_2(\mathbb{E})$, respectively. Therefore the possible projective images of $\bar{\rho}|_{\Pi_0(\bar{\rho})}$ are the same as for $\bar{\rho}$, so $\bar{\rho}|_{\Pi_0(\bar{\rho})}$ is absolutely irreducible. $\square$

**Proposition 8.6.** Assume that $\bar{\rho}$ is regular dihedral, say $\bar{\rho} = \text{Ind}_{\Pi_0}^{\Pi} \chi$. Then $\bar{\rho}|_{\Pi_0(\bar{\rho})}$ is multiplicity free over $\mathbb{E}$. Furthermore, given $g \in \Pi_0$, we have $g \in \Pi_0(\bar{\rho})$ if and only if $\chi(g) \in \mathbb{E}^\times$.

*Proof.* Since $\bar{\rho}$ is regular, it follows from Lemma A.7 that there is a unique subgroup $\Pi_0$ of $\Pi$ of index 2 such that $\bar{\rho} \cong \text{Ind}_{\Pi_0}^{\Pi} \chi$ for some character $\chi: \Pi_0 \to \mathbb{F}^\times$. For any $h \in \Pi$, define $\chi^h: \Pi_0 \to \mathbb{F}^\times$ by $\chi^h(g) := \chi(h^{-1}gh)$. The character $\chi^h$ only depends on the class of $h$ in $\Pi/\Pi_0$. Fix an element $c \in \Pi \setminus \Pi_0$. Fix a generator $\sigma \in \Sigma_{\bar{\rho}} = \text{Gal}(\mathbb{F}/\mathbb{E})$, and choose $\eta$ such that $(\sigma, \eta) \in \tilde{\Sigma}_{\bar{\rho}}$. (Note that there are two choices for $\eta$, and they differ by the character $\eta_0: \Pi \to \Pi/\Pi_0 \cong \{\pm 1\}$.) Then $\Pi_0(\bar{\rho}) = \ker \eta_0 \cap \ker \eta$ since $\sigma$ generates $\Sigma_{\bar{\rho}}$. Therefore $\Pi_0(\bar{\rho}) = \ker \eta|_{\Pi_0}$.

Note that any regular element for $\bar{\rho}$ must be in $\Pi_0$ since elements in $\Pi \setminus \Pi_0$ have projective order 2. By applying Lemma 8.1 to $\bar{\rho}|_{\Pi_0}$, we find that $\sigma\chi = \eta\chi$ and $\sigma\chi^c = \eta\chi^c$. Hence $\eta|_{\Pi_0} = \sigma\chi\chi^{-1}$, so $g \in \Pi_0(\bar{\rho})$ if and only if $\chi(g) \in \mathbb{E}^\times$. In particular, $\ker \chi \subseteq \Pi_0(\bar{\rho})$. Furthermore, using the fact that $\sigma\chi\chi^{-1} = \eta|_{\Pi_0} = \sigma\chi^c(\chi^c)^{-1}$, we find that the character $\chi/\chi^c$ takes values in $\mathbb{E}^\times$.

We know that $\bar{\rho}|_{\Pi_0(\bar{\rho})}$ is multiplicity free over $\mathbb{E}$ if and only if there is some $g \in \Pi_0(\bar{\rho})$ such that $\chi(g) \neq \chi^c(g)$. If $\ker \chi \neq \ker \chi^c$, then we can choose $g \in \ker \chi \setminus \ker \chi^c$. Then $\chi(g) = 1 \neq \chi^c(g)$ and $g \in \Pi_0(\bar{\rho})$ by the previous paragraph. Therefore we may assume that $\ker \chi = \ker \chi^c$.
Let $n$ denote the order of $\chi$. Since $\ker \chi = \ker \chi^c$, we have that $\chi^c = \chi^a$ for some $a \in (\mathbb{Z}/n\mathbb{Z})^\times$. Note that $\chi^{c^2} = \chi$ since $c^2 \in \Pi_0$. Therefore

$$ \chi = \chi^{c^2} = (\chi^c)^c = (\chi^a)^c = (\chi^c)^a = (\chi^a)^a = \chi^{a^2}. $$

Fix $g_0 \in \Pi_0$ such that $\bar{\rho}(g_0)$ generates the projective image of $\bar{\rho}(\Pi_0)$. We will show that $h := g_0^{a-1}$ is in $\Pi_0(\bar{\rho})$ and $\chi(h) \in \mathbb{E}^\times$ with $\chi(h) \neq \chi^c(h)$. First we calculate, using the fact that $\chi^{a^2} = \chi$,

$$ \chi^c(h) = \chi^a(g_0^{a-1}) = \chi^{a^2}(g_0)\chi^{-1}(g_0) = 1. $$

Hence $h \in \ker \chi^c = \ker \chi \subseteq \Pi_0(\bar{\rho})$. On the other hand,

$$ \chi(h) = \chi(g_0^{a-1}) = \chi^c(g_0)\chi^{-1}(g_0). $$

We saw in the second paragraph that $\chi/\chi^c$ is an $\mathbb{E}$-valued character. Furthermore, $\chi^c(g_0)/\chi(g_0) \neq 1$ since $g_0$ was chosen has a generator of the projective image of $\bar{\rho}(\Pi_0)$, which is isomorphic to the image of $\chi/\chi^c$. $\square$

**Corollary 8.7.** Assume that $\bar{\rho}$ is regular and dihedral and that the order of $\det \bar{\rho}$ is a power of 2. Let $\sigma$ be a generator of $\Sigma_{\bar{\rho}}$ and $\eta: \Pi \to \mathbb{F}^\times$ a character such that $(\sigma, \eta) \in \tilde{\Sigma}_{\bar{\rho}}$. Then either $\Sigma_{\bar{\rho}}$ is trivial or $\bar{\rho}|_{\ker \eta}$ is absolutely irreducible.

*Proof.* Write $\bar{\rho} = \text{Ind}_{\Pi_0}^{\Pi_0} \chi$ and fix $c \in \Pi \setminus \Pi_0$. We shall make frequent use of Lemma A.7 in this proof without referencing it every time. We saw in the proof of Proposition 8.6 that $\Pi_0(\bar{\rho}) = \Pi_0 \cap \ker \eta$. Thus Proposition 8.6 implies that $\chi|_{\Pi_0 \cap \ker \eta} \neq \chi^c|_{\Pi_0 \cap \ker \eta}$.

If $\ker \eta \neq \Pi_0 \cap \ker \eta$, then $[\ker \eta: \Pi_0 \cap \ker \eta] = 2$ since $[\Pi: \Pi_0] = 2$. Thus $\bar{\rho}|_{\ker \eta} \cong \text{Ind}_{\Pi_0 \cap \ker \eta}^{\ker \eta} \chi|_{\Pi_0 \cap \ker \eta}$. Since $\chi|_{\Pi_0 \cap \ker \eta} \neq \chi^c|_{\Pi_0 \cap \ker \eta}$ it follows that $\bar{\rho}|_{\ker \eta}$ is irreducible.

If $\ker \eta = \Pi_0 \cap \ker \eta$ then $\Pi_0 \supseteq \ker \eta$ and $\Pi/\ker \eta$ is a cyclic group whose order is a power of 2 since $\eta^2$ is a power of $\det \bar{\rho}$. If $\Pi_0 \neq \ker \eta$, then there is a subgroup $\ker \eta \subseteq \Pi' \subset \Pi_0$ such that $[\Pi_0: \Pi'] = 2$. Note that $\chi|_{\Pi'} \neq \chi^c|_{\Pi'}$ since $\chi|_{\ker \eta} \neq \chi^c|_{\ker \eta}$. Then $\bar{\rho}|_{\Pi_0} \cong \text{Ind}_{\Pi'}^{\Pi_0} \chi|_{\Pi'}$ is irreducible, a contradiction since $\bar{\rho} \cong \text{Ind}_{\Pi_0}^{\Pi} \chi$. Thus we must have $\Pi_0 = \ker \eta$. Therefore $\bar{\rho} \cong \bar{\rho} \otimes \eta$ and so $\sigma$, and hence $\Sigma_{\bar{\rho}}$, is trivial. $\square$

8.3. $\mathbb{F}/\mathbb{E}$ when $\det \bar{\rho}$ is a power of 2. In Section 9 we will assume that the order of $\det \bar{\rho}$ is a power of 2. A large part of the reason for that assumption is that it guarantees that $[\mathbb{F}: \mathbb{E}]$ can be taken to be a power of 2 as well, as the next lemma shows. We need this in an induction argument in Section 9. Given any $\mathbb{F}$-valued function $f$ and any subfield $\mathbb{F}'$ of $\mathbb{F}$, let us write $\mathbb{F}'(f)$ for the subfield of $\mathbb{F}$ generated over $\mathbb{F}'$ by the values of $f$.

**Lemma 8.8.** Assume that the order of $\det \bar{\rho}$ is a power of 2. Then the degree of $\mathbb{F}_p(\text{tr} \bar{\rho})$ over $\mathbb{E}$ is a power of 2.

*Proof.* Let $d := \det \bar{\rho}$. Since the order of $d$ is a power of 2, the degree of $\mathbb{E}(d)$ over $\mathbb{E}$ is a power of 2. But, for an arbitrary $g' \in \Pi$, the extension $\mathbb{E}(\text{tr} \bar{\rho}(g'))$ is at most quadratic over $\mathbb{E}(d)$ because $\text{tr} \bar{\rho}(g')$ satisfies

$$ d(g')x^2 - (\text{tr} \bar{\rho}(g'))^2/d(g') \in \mathbb{E}(d)[x]. $$

The field $\mathbb{F}_p(\text{tr} \bar{\rho})$ is obtained from $\mathbb{E}(d)$ by adding finitely many values of $\text{tr} \bar{\rho}$. $\square$

In Section 9 we will be interested in gradings coming from conjugate self-twists. To be able to apply Lemma A.22 in those situations, we now verify one of the hypotheses.

**Lemma 8.9.** Assume that both the order of $\det \bar{\rho}$ and $[\mathbb{F}: \mathbb{E}]$ are powers of 2. If $n = \#\Sigma_{\bar{\rho}}$, then $\mathbb{F}$ contains a primitive $n^{\text{th}}$ root of unity. In particular, condition (*) from Appendix A.4 is satisfied.
**Proof.** Let $d := \det \bar{\rho}$, and write $2^s$ for the order of $d$. We have that $\mathbb{E}(d)$ contains a primitive $(2^s)^{\text{th}}$ root of unity. If $[\mathbb{F}: \mathbb{E}(d)] = 2^r$, then $\mathbb{F}$ contains a primitive $(2^{r+s})^{\text{th}}$ root of unity. On the other hand,

$$n = \#\Sigma_{\bar{\rho}} = [\mathbb{F}: \mathbb{E}] = 2^r [\mathbb{E}(d): \mathbb{E}].$$

Since $d$ has order $2^s$, it follows that $[\mathbb{E}(d): \mathbb{E}]$ divides $2^{s-1}$. Thus $n$ divides $2^{r+s-1}$, and so $\mathbb{F}$ contains a primitive $n^{\text{th}}$ root of unity. $\square$

**8.4. A good basis for $\bar{\rho}$.** We need to carefully choose a basis for $\bar{\rho}$ that has many good properties and will allow us to choose a good $(t, d)$-representation in Section 9.2. In this section we explain how to find this basis when $\bar{\rho}$ is exceptional or large. Let us first define an extra condition on octahedral representations.

**Definition 8.10.** We say a regular octahedral representation $\bar{\rho}$ is good if at least one of the following properties is satisfied:

(1) $p \equiv 1 \mod 3$;

(2) $\bar{\rho}$ is strongly regular;

(3) there is a regular element $g_0 \in \Pi$ such that $g_0^2 \in \Pi_0(\bar{\rho})$.

We shall need to know that if $\bar{\rho}$ is good, then twisting away the odd part of the determinant of $\bar{\rho}$ gives a representation that is also good.

**Lemma 8.11.** Let $\bar{\rho}: \Pi \to \text{GL}_2(\mathbb{F})$ be a good representation. Let $\chi: \Pi \to \mathbb{F}^\times$ be the unique odd-order character such that the order of $\chi^2 \det \bar{\rho}$ is a power of 2. Then $\bar{\rho} \otimes \chi$ is good.

**Proof.** First note that twisting by any character does not change the projective image, so $\bar{\rho} \otimes \chi$ is octahedral. Regularity is also invariant under twisting. The claim is clear if $p \equiv 1 \mod 3$, so we assume that $p \equiv 2 \mod 3$. The regularity assumption then implies that $\zeta_4 \in \mathbb{F}_p$ by Remark 2.20. As in the proof of Lemma 8.4, decompose $\det \bar{\rho} = d_1 d_2$, where $d_i: \Pi \to \mathbb{F}^\times$ are characters such that the order of $d_1$ is odd and the order of $d_2$ is a power of 2.

First suppose that $\bar{\rho}$ is strongly regular. Then there is a matrix $g_0 \in \Pi$ such that $\bar{\rho}(g_0)$ has eigenvalues $\lambda_0, \mu_0 \in \mathbb{E}^\times$ such that $\lambda_0 \mu_0^{-1} = \zeta_4$. We have $\lambda_0 \mu_0 = \det \bar{\rho}(g_0) = d_1(g_0)d_2(g_0)$. Note that any $\sigma \in \text{Gal}(\mathbb{F}/\mathbb{E})$ fixes $\lambda_0 \mu_0$ since $\lambda_0, \mu_0 \in \mathbb{E}$. Therefore $\sigma(d_1(g_0)d_2(g_0)) = d_1(g_0)d_2(g_0)$. But since $d_1(g_0)$ is an odd order root of unity and $d_2(g_0)$ is a 2-power order root of unity, it follows that $\sigma$ must fix both $d_1(g_0)$ and $d_2(g_0)$. Write $a$ for the order of $d_1$. Then $\chi = d_1^{-(a+1)/2}$ by the proof of Lemma 8.4. In particular, $\chi(g_0) \in \mathbb{E}^\times$. Thus the eigenvalues $\chi(g_0)\lambda_0$ and $\chi(g_0)\mu_0$ of $(\bar{\rho} \otimes \chi)(g_0)$ are in $\mathbb{E}$. Thus $g_0$ is a strongly regular element for $\bar{\rho} \otimes \chi$, as desired.

Finally, suppose that there is a regular element $g_0 \in \Pi$ such that $g_0^2 \in \Pi_0(\bar{\rho})$. Let $\sigma$ be a generator for $\text{Gal}(\mathbb{F}/\mathbb{E})$ and let $\eta: \Pi \to \mathbb{F}^\times$ such that $(\sigma, \eta) \in \tilde{\Sigma}_{\bar{\rho}}$. Then $\Pi_0(\bar{\rho}) = \ker \eta$ and $\Pi_0(\bar{\rho} \otimes \chi) = \ker \sigma \chi \chi^{-1} \eta$. Since $g_0^2 \in \Pi_0(\bar{\rho})$ and $\sigma \det \bar{\rho} = \eta^2 \det \bar{\rho}$, it follows that $\det \bar{\rho}(g_0) \in \mathbb{E}$. But $\det \bar{\rho}(g_0) = d_1(g_0)d_2(g_0)$, and since $d_1(g_0)$ is an odd order root of unity and $d_2(g_0)$ has 2-power order, it follows that both $d_1(g_0)$ and $d_2(g_0)$ are in $\mathbb{E}$. Therefore $\chi(g_0) = d_1^{-(a+1)/2}(g_0) \in \mathbb{E}$. Thus $g_0^2 \in \ker \sigma \chi \chi^{-1} \eta = \Pi_0(\bar{\rho} \otimes \chi)$. $\square$

Finally we describe the basis of $\bar{\rho}$ that we shall work with in Section 9. Let $Z$ denote the group of scalar matrices in $\text{GL}_2(\mathbb{F})$. The following lemma justifies our definition of $\mathbb{F}_q$ in Section 2.5 for exceptional representations.

**Lemma 8.12.** Up to conjugation, the image of $\bar{\rho}$ is contained in $Z \text{GL}_2(\mathbb{E})$. If $\mathbb{F}_q$ is an extension of $\mathbb{E}$ and $\lambda_0, \mu_0 \in \mathbb{F}_p^\times$ are eigenvalues of a matrix in the image of $\bar{\rho}$ such that $\lambda_0 \mu_0^{-1} \in \mathbb{F}_q$, then we may further conjugate $\bar{\rho}$ to assume that $(\lambda_0 \mu_0^0) \in \text{Im} \bar{\rho}$ and $\text{Im} \bar{\rho} \subseteq Z \text{GL}_2(\mathbb{F}_q)$.