2.1 Deformation theory generalities

Let $\Gamma$ be a profinite group satisfying the $\Phi_p$ -finiteness condition of Mazur [Reference MazurMaz89, § 1.1]. We recall fundamental facts about $\mathrm{Spf} {\mathbb{Z}}_p$ -formal schemes and stacks of two-dimensional representations, following [Reference Wang-EricksonWE18]in part. The reader is presumed to be familiar with the theory of pseudorepresentations and their deformation theory, which is developed in [Reference ChenevierChe14].Sometimes we take the liberty of discussing a pseudorepresentation as a ‘trace function’, using the theory of pseudocharacters, but these amount to the same thing by [Reference ChenevierChe14, Proposition 1.29].

• • Let $\widehat{\mathrm{Rep}}^{\square, \tilde\psi}$ denote the moduli functor of homomorphisms $\Gamma \to{\mathrm{GL}}_2(B)$ .

• • Let $\widehat{\mathrm{Rep}}^{\tilde \psi}$ denote the moduli groupoid of rank-2 projective B-modules V equipped with a homomorphism $\Gamma \to\mathrm{Aut}_B(V)$ and a trivialization of the determinant of V, $\wedge^2 V {\buildrel\sim\over\rightarrow} B$ .

• • Let $\mathrm{PsR}^{\tilde \psi}$ denote the moduli functor of two-dimensional pseudorepresentations $D : \Gamma \to B$ .

These moduli spaces admit natural morphisms $\widehat{\mathrm{Rep}}^{\square,\tilde\psi} \to \widehat{\mathrm{Rep}}^{\tilde\psi} \to\mathrm{PsR}^{\tilde\psi}$ , where the first arrow is compatible with a presentation of the stack $\widehat{\mathrm{Rep}}^{\tilde\psi}$ as $[\widehat{\mathrm{Rep}}^{\square,\tilde\psi}/{\mathrm{SL}}_2]$ . Here, the action of ${\mathrm{SL}}_2$ arises from its adjoint action on ${\mathrm{GL}}_2$ . The second arrow arises from associating a pseudorepresentation $D(\rho)$ to the action of $\Gamma$ on the B-module V by $\rho$ , using the characteristic polynomial coefficients of this action.

A two-dimensional pseudorepresentation $D : \Gamma \to B$ is called reducible when it has the form $D(\rho)$ for some $\rho$ of the form $\rho \simeq \nu_1 \oplus \nu_2$ for characters $\nu_i : \Gamma \to B^\times$ . Reducibility is a Zariski closed condition on each of these moduli spaces.From now on, we drop ‘two-dimensional’ from our terminology for pseudorepresentations.

The moduli functor $\mathrm{PsR}^{\tilde \psi}$ is known to be the disjoint union of formal spectra representing deformation functors of finite field-valued pseudorepresentations $D : \Gamma \to{\mathbb{F}}$ over their minimal field of definition ${\mathbb{F}}$ , a finite extension of ${\mathbb{F}}_p$ (see [Reference ChenevierChe14, Theorem F]). That is, if we write $\mathrm{Def}_{D}^{\tilde \psi} = \mathrm{Spf} R_{D}^{\tilde \psi}$ as the formal spectrum of the complete Noetherian local ring $R_{D}^{\tilde \psi}$ representing the deformation functor for D, the decomposition is expressible as

\[\mathrm{PsR}^{\tilde \psi} \cong \coprod_{D} \mathrm{Def}_{D}^{\tilde \psi}. \]

We write $\widehat{\mathrm{Rep}}^{\square,\tilde \psi}_{D}$ and $\widehat{\mathrm{Rep}}^{\tilde \psi}_{D}$ for the substack/subspace of $\widehat{\mathrm{Rep}}^{\square,\tilde \psi}$ and $\widehat{\mathrm{Rep}}^{\tilde\psi}$ over $\mathrm{Def}_{D}^{\tilde \psi}$ .

Any residual pseudorepresentation is induced by a unique (up to isomorphism) semisimple representation $\rho_D :\Gamma \to {\mathrm{GL}}_2({\mathbb{F}})$ over the same field of definition ${\mathbb{F}}$ as D. After a possible at most quadratic extension, we may assume that the irreducible summands of this semisimple representation are absolutely irreducible. In what follows, we replace ${\mathbb{F}}$ with a minimal such extension.

A residual pseudorepresentation $D : \Gamma \to {\mathbb{F}}$ is called multiplicity free when the irreducible summands of $\rho_D$ are pairwise distinct. This includes the case that $\rho_D$ is irreducible, in which case we also call D irreducible.

Next we introduce Cayley–Hamilton algebras; see [Reference ChenevierChe14, § 1] for a reference. We refer to a Cayley–Hamilton algebra over A (or with scalar ring A) as an A-algebra E equipped with a pseudorepresentation $D_E : E \to A$ satisfying the Cayley–Hamilton property;concisely, this property means that every element of E satisfies the characteristic polynomial determined by $D_E$ .

Definition 2.3. Let $E_D^{\tilde \psi}$ denote the universal Cayley–Hamilton algebra over D, which is given by

\[E_D^{\tilde \psi} := \frac{R_D^{\tilde \psi}[\![ \Gamma]\!]}{\mathrm{CH}(D^{u,\tilde \psi})}, \]

where $\mathrm{CH}(D^{u,\tilde \psi})$ denotes the minimal two-sided ideal that factors the universal deformation $D^{u,\tilde \psi} : R_D^{\tilde \psi}[\![\Gamma]\!] \to R_D^{\tilde\psi}$ and makes it satisfy the Cayley–Hamilton property. We also write $D_{E_D^{\tilde \psi}}: E_D^{\tilde\psi} \to R_D^{\tilde \psi}$ for the pseudorepresentation that $E_D^{\tilde\psi}$ is equipped with. The Cayley–Hamilton representation

\[\rho^{u, \tilde\psi} : \Gamma \to E_D^{\tilde\psi} \]

is universal in the sense that for any Cayley–Hamilton representation $\rho : \Gamma \to E$ with scalar ring A, if the induced pseudorepresentation $D_E \circ \rho: \Gamma \to A$ has constant residual pseudorepresentation D, then there exists a morphism of Cayley–Hamilton algebras $(f : E_D^{\tilde \psi} \to E$ , $R_D^{\tilde \psi} \to A)$ such that $\rho = f \circ \rho^{u,\tilde\psi}$ and the map $R_D^{\tilde \psi} \to A$ equals the map comingfrom the moduli interpretation of $R_D^{\tilde \psi}$ applied to $D_E \circ \rho$ .

In this paper, we almost always want to restrict the determinant of representations and pseudorepresentations. Writing:

• • $\psi : \Gamma \to {{{\mathcal{O}}}}^\times$ fora character deforming $\det D : \Gamma \to {\mathbb{F}}^\times$ ;

• • $\widehat{\mathrm{Rep}}^\square_D, \widehat{\mathrm{Rep}}_D, \mathrm{PsR}_D$ for moduli functors;

• • along with the following objects representing moduli problems with fixed determinant $\psi$ :

• • $E_D$ for the universal Cayley–Hamilton algebra;

• • with scalar ring $R_D$ ; and

• • universal representation $\rho^u : \Gamma \to E_D^\times$ over D.

In what follows, we continue with this convention as we introduce new moduli functors and rings.

2.2 Algebraization of moduli functors and groupoids

A main result of [Reference Wang-EricksonWE18, § 3] is that all of the formal moduli spaces or groupoids of representations of $\Gamma$ with residual pseudorepresentation D have a natural $R_D$ -algebraic model of finite type. The source of this algebraization is the following finiteness result.

Using the universality of $E_D$ , one can use the moduli $\mathrm{Rep}^\square(E_D), \mathrm{Rep}(E_D)$ of (non-topological) compatible representations of $E_D$ as an $R_D$ -algebraic model for $\widehat{\mathrm{Rep}}_D^\square,\widehat{\mathrm{Rep}}_D$ . That is, $\widehat{\mathrm{Rep}^\square(E_D)} \cong \widehat{\mathrm{Rep}}_D^\square$ and $\widehat{\mathrm{Rep}(E_D)} \cong \widehat{\mathrm{Rep}}_D$ , completing with respect to the maximal ideal of $R_D$ .

• • Let $\mathrm{Rep}^\square(E)$ be the $\mathrm{Spec}B$ -functor of compatible representations of E.

• • Let $\mathrm{Rep}(E)$ be the $\mathrm{Spec} B$ -groupoid which associates to a B-algebra C a projective rank-2 C-module V, an isomorphism $\wedge^2 V {\buildrel\sim\over\rightarrow}C$ , and a compatible representation of E on V, as follows.

As in Definition 2.1, $\mathrm{Rep}(E) \cong [\mathrm{Rep}^\square(E)/{\mathrm{SL}}_2]$ under the adjoint action of ${\mathrm{SL}}_2$ .

We also record the self-duality of the universal vector bundle on $\mathrm{Rep}(E)$ .

2.3 GMAs and adapted representations in the multiplicity-free reducible case

When D is multiplicity free and reducible, arising from the representation $\chi_1 \oplus \chi_2 : \Gamma \to{\mathrm{GL}}_2({\mathbb{F}})$ , any lift of the two canonical orthogonal ordered idempotents of ${\mathbb{F}} \times {\mathbb{F}}$ over $E_D \to {\mathbb{F}} \times{\mathbb{F}}$ amounts to a $2\times2$ generalized matrix $R_D$ -algebra ( $R_D$ -GMA) structure on $E_{D}$ (see [Reference ChenevierChe14, Theorem 2.22]). We simply use the term ‘GMA’ to refer to a $2\times 2$ GMA.

See [Reference Bellache and ChenevierBC09, § 1.3] for generalities on GMAs. In particular, using coordinates coming from these ordered idempotents, we get an isomorphism

(2.1) \begin{equation}E_D = \begin{pmatrix}(E_D)_{1,1} & (E_D)_{1,2} \\(E_D)_{2,1} & (E_D)_{2,2}\end{pmatrix}\cong\begin{pmatrix}R_D & B_D \\C_D &R_D\end{pmatrix}\!,\end{equation}

where there is an implicit $R_D$ -bilinear cross-diagonal multiplication with corresponding map

\[ B_D \otimes_{R_D} C_D \to R_D \]

giving rise to an $R_D$ -algebra structure on $E_D$ . The pseudorepresentation $E_D \to R_D$ naturally arising from the GMA structure is equal to $D_{E_D}: E_D \to R_D$ (see [Reference Wang-EricksonWE18, Proposition 2.23]) and is Cayley–Hamilton, making any GMA a Cayley–Hamiltonalgebra. And the reducibility ideal of $R_D$ , which cuts out the locus of reducible pseudorepresentations in $\mathrm{Spec} R_D$ , equals the image of the cross-diagonal multiplication map.

We recall the following general notions from [Reference Bellache and ChenevierBC09, § 1.3], where B is a commutative ring and E is a B-GMA.

• • Let $\mathrm{Rep}^{\square,\mathrm{Ad}}(E)$ denote the $\mathrm{Spec}B$ -functor of adapted representations of E.

• • Let $\mathrm{Rep}^{\mathrm{Ad}}(E)$ denote the $\mathrm{Spec} B$ -groupoid whose value on C consists of an ordered pair of rank-1 projective C-modules $(V_1, V_2)$ equipped with an isomorphism $V_1 \otimes V_2 {\buildrel\sim\over\rightarrow} C$ and a homomorphism of B-GMAs (so, in particular, they preserve the ordered idempotents) $E \to \mathrm{End}_C(V_1 \oplus V_2)$ .

One may check that $\mathrm{Rep}^{\mathrm{Ad}}(E) \cong[\mathrm{Rep}^{\square,\mathrm{Ad}}(E)/T]$ , where T is the standard diagonal torus in ${\mathrm{SL}}_2$ , acting via the adjoint representation on $M_2$ .We fix the isomorphism $T \cong {\mathbb{G}}_m$ that makes ${\mathbb{G}}_m$ act on the B (upper-right) coordinate by $2\in X^*({\mathbb{G}}_m) \cong {\mathbb{Z}}$ and the C coordinate by $-2$ .

In the following theorem, we let A denote the B-algebra representing $\mathrm{Rep}^\square(E)$ and likewise let S represent $\mathrm{Rep}^{\square,\mathrm{Ad}}(E)$ .

1. (1) Adapted representations of E are compatible representations.

2. (2) The resulting map $\mathrm{Rep}^{\square,\mathrm{Ad}}(E) \hookrightarrow\mathrm{Rep}^\square(E)$ is a closed immersion of affine B-schemes; we have the corresponding surjection $A\twoheadrightarrow S$ .

3. (3) The morphism of part (2) descends to an isomorphism of algebraic stacks

   \[\mathrm{Rep}^{\mathrm{Ad}}(E) = [\mathrm{Rep}^{\square, \mathrm{Ad}}(E)/T] \cong [\mathrm{Rep}^\square(E)/{\mathrm{SL}}_2] = \mathrm{Rep}(E).\]

4. (4) The GIT quotient scheme $\mathrm{Rep}^{\square,\mathrm{Ad}}(E) /\!/ {\mathbb{G}}_m$ isnaturally isomorphic to $\mathrm{Spec} B$ . Equivalently, in ring-theoretic terms, there are natural isomorphisms

   \[B \cong A^{{\mathrm{SL}}_2} \cong S^{{\mathbb{G}}_m}. \]

5. (5) Moreover, the natural action of ${\mathrm{SL}}_2$ on $M_2(A)$ (respectively, ${\mathbb{G}}_m$ on $M_2(S)$ ) and the natural maps $E \to M_2(A) \twoheadrightarrow M_2(S)$ produce isomorphisms

   \[E \cong M_2(A)^{{\mathrm{SL}}_2} \cong M_2(S)^{{\mathbb{G}}_m}. \]

6. (6) If E is finitely generated as a B-algebra, then all of these schemes and stacks are of finite type over $\mathrm{Spec} B$ .

We also use the following result of Bellaïche and Chenevier.

Proposition 2.10. [Reference Bellache and ChenevierBC09, Proposition 1.3.13, Remark 1.3.15]. Writing $E_{i,j}$ for the R-GMA coordinates of E as in (2.1), $1 \leq i,j \leq 2$ , there are canonical isomorphisms of graded R-modules

\[E_{i,j} \simeq S_{2(j-i)} \]

such that the coordinate-wise multiplication maps $E_{i,j}\otimes_R E_{j,k} \to E_{i,k}$ are compatible with the multiplication law of S. In particular, $S_0 = R = E_{1,1} = E_{2,2}$ , and S is generated as an R-algebra by $E_{1,2}$ and $E_{2,1}$ .

Next, we apply these equivalences to the case of the universal Cayley–Hamilton algebra $E_D$ with scalar ring being the universal pseudorepresentation ring $R_D$ . First, we set up notation.

Notation 2.11. Let $A_D$ denote the finitely generated $R_D$ -algebra representing $\mathrm{Rep}^\square(E_D)$ , with ${\mathfrak{m}}_D$ -adic completion $\hat A_D$ . When D is reducible and multiplicity free, let us write $S_D$ as the ring representing the $R_D$ -algebraic moduli functor $\mathrm{Rep}^{\square,\mathrm{Ad}}(E_D)$ , and let $\hat S_D$ denote its ${\mathfrak{m}}_D$ -adic completion. (We remark that $\hat A_D$ and $\hat S_D$ are not local, in general.) We have the following diagram of rings and moduli functors (and the top row of vertical arrows in the left diagram are pseudorepresentations).

We use the $T \cong {\mathbb{G}}_m$ -action on $\mathrm{Spec} S_D$ to consider $S_D$ tobe a ${\mathbb{Z}}$ -graded algebra $S_D = \bigoplus_{i \in {\mathbb{Z}}} S_{D,i}$ . In fact, it is a ${\mathbb{Z}}$ -graded $R_D$ -algebra because characteristic polynomial functions are adjoint invariants, moreover $R_D \cong S_{D,0}$ , by Theorem 2.9.

From now on, our notational convention is that the subscript ‘D’ is dropped.

2.3.1 Irreducible case

When D and $\rho = \rho_D$ are irreducible, it is well known that deformation theory of $\rho$ is identical to that of D (see e.g. [Reference ChenevierChe14, Theorem 2.22]). That is, the natural homomorphism $R \to R_\rho$ , where $R_\rho$ is the universal deformation ring of $R_\rho$ , is an isomorphism; also, $E \simeq M_2(R)$ .

In this case, the stacks above can be expressed in terms of the universal deformation ring $R_\rho$ and the universal lifting ring $R_\rho^\square$ as follows:

\[\widehat{\mathrm{Rep}} \cong [\mathrm{Spf} R_\rho/\mu_2], \quad \widehat{\mathrm{Rep}}^\square \cong \mathrm{Spf} R_\rho^\square,\quad\mathrm{Rep}(E) \cong [\mathrm{Spec} R_\rho/\mu_2], \]

where the implicit adjoint action of $\mu_2 \subset {\mathbb{G}}_m \cong T \subset{\mathrm{SL}}_2$ is trivial. In general, $R^\square_\rho$ is a further completion of $\hat A$ at the maximal ideal of $\hat A$ associated to $\rho$ ; in the irreducible case, $R^\square_\rho$ and $\hat A$ are isomorphic.

Remark 2.12. The remaining trivial action of $\mu_2$ reflects the kernel of the adjoint action of ${\mathrm{SL}}_2$ , making $\mathrm{Rep}(E)$ a $\mu_2$ -gerbe. It reflects that the collection of rank-2 vector bundles V (over scalar ring C) with a fixed isomorphism $\wedge^2 V {\buildrel\sim\over\rightarrow} C$ , such that $\mathrm{End}_C(V) \simeq M_2(C)$ , admits a twisting action by the group of isomorphism classes of line bundles whose squares are trivial.

2.4 Coherent sheaves on stacks, and duality

In § 3 we describe the stacks $\mathrm{Rep}(E)$ for the four different types of finite field-valued pseudorepresentations of $\Gamma_{{{{\mathbb{Q}_{p}}}}}$ that are of interest to us, as well as the coherent sheaves on $\mathrm{Rep}(E)$ that we use. For this reason, we record a few general recollections about coherent sheaves on stacks, specialized to the situation we encounter.

2.4.3 Categories of sheaves on X

We now apply this to coherent and quasicoherent sheaves on X. Recall that $\mathrm{QCoh}(X)$ is a Grothendieck abelian category; see e.g. [Sta18, Tag 0781] (though one can give a much more direct proof in this special case). We then define $\mathcal{D}^+_{ qcoh}(X)$ and $\mathcal{D}_{ qcoh}(X)$ as $\mathcal{D}^+(\mathrm{QCoh}(X))$ and $\mathcal{D}(\mathrm{QCoh}(X))$ , respectively. We also define $\mathcal{D}^b_{ coh}(X)$ as the full subcategory of $\mathcal{D}^+(\mathrm{QCoh}(X))$ of complexes whose cohomology is bounded, and coherent in each degree.

Remark 2.13. A different, perhaps more standard, definition of the unbounded derived category of quasicoherent sheaves on X is as the unbounded derived category of complexes of lisse-étale $\mathcal{O}_X$ -modules with quasicoherent cohomology. Unlike the situation of abelian categories, these different definitions can produce genuinely different categories, as we now recall. Let us use $\mathcal{D}^\prime_{ qcoh}(X)$ to denote the category constructed using the lisse-étale site; $\mathcal{D}^\prime_{ qcoh}(X)$ is not used anywhere else in this paper. The relationship between $\mathcal{D}_{ qcoh}(X)$ and $\mathcal{D}^\prime_{qcoh}(X)$ is as follows: there is a natural functor $\mathcal{D}_{ qcoh}(X) \to\mathcal{D}^\prime_{ qcoh}(X)$ which identifies $\mathcal{D}^\prime_{qcoh}(X)$ as the left completion of $\mathcal{D}_{qcoh}(X)$ ; see [Reference Hall, Neeman and RydhHNR19, Remark C.4]. This functor can fail to be an equivalence. Indeed, in the situation of § 3.3, where $G={\mathrm{SL}}_2$ , the functor is not full by [Reference Hall, Neeman and RydhHNR19, Theorem 1.3]. Moreover, in this situation, the category $\mathcal{D}_{ qcoh}(X)$ is not compactly generated (this also follows from the results of [Reference Hall, Neeman and RydhHNR19]). These issues do not arise for the bounded-below derived category, cf. e.g. [Reference Arinkin and BezrukavnikovAB10, Claim 2.7].

Remark 2.14. The following remarks about coherent duality are useful. Assume that A is Gorenstein. This is the case, for example, when A is regular or equal to $B/(f)$ where B is regular and f is a non-zero divisor [Sta18, Tags 0AWX and 0BJJ]; this covers all cases we encounter. Assume further that A has a dualizing complex (in all cases we consider, this easily follows from the explicit descriptions of the rings that we give, together with [Sta18, Tag0BFR]). Since we assumed that A is Gorenstein, A itself (in degree 0) is a dualizing complex for A (see [Sta18, Tag 0DW9]). If M and N are G-equivariant A-modules, let $\mathrm{Hom}_A(M,N)$ denote the (not necessarily G-equivariant) A-modulehomomorphisms from M to N, with its induced G-action. This is the internal Hom in $\mathrm{QCoh}(X)$ and we use the notation

\[\mathrm{\underline{Hom}}(M,N) := \mathrm{Hom}_A(M,N). \]

Indeed, $\mathrm{\underline{Hom}}(-,-)$ denotes the internal Hom in anycategory where it exists. Moreover, we let $\mathrm{R\underline{Hom}}(M,N) :=\mathrm{RHom}_A(M,N)$ denote thederived functors of Hom. Then

\[\mathrm{R\underline{Hom}}(-,{{{\mathcal{O}}}}_{X}) = \mathrm{RHom}_A(-,A) : \mathcal{D}^b_{ coh}(X) \to \mathcal{D}^b_{ coh}(X) \]

is an exactFootnote ⁷ involution,Footnote ⁸ i.e. an antiequivalence whose square is naturally isomorphic to the identity. Here A is viewed as a G-equivariant A-module. In particular, we obtain an exactFootnote ⁹ involution

\[\mathrm{\underline{Hom}}(-,\mathcal{O}_X) = \mathrm{Hom}_A(-,A) : \mathrm{MCM}(X) \to \mathrm{MCM}(X), \]

where $\mathrm{MCM}(X) \subseteq \mathrm{Coh}(X)$ is the exact full subcategory of maximal Cohen–Macaulay modules (a G-equivariant finitely generated A-module is (maximal)Cohen–Macaulay if the underlying A-module is (maximal) Cohen–Macaulay). To simplify the notation, we write

\[M^\ast := \mathrm{\underline{Hom}}(M,\mathcal{O}_X) \]

for the coherent dual of $M \in \mathrm{MCM}(X)$ .

3.1 Supersingular case

In the supersingular case, we know from the discussion of § 2.3.1 that:

• • $R \cong R_{\rho} \hookrightarrow S \hookrightarrow \hat S$ ;

• • $A \hookrightarrow \hat A \hookrightarrow R^\square_\rho$ ;

• • $E \simeq M_2(R)$ ;

• • $\mathfrak{X} \cong [\mathrm{Spec} R/\mu_2]$ , where $\mu_2$ acts trivially.

Theorem 3.2. There are isomorphisms

\[R \cong R_{\rho} \simeq {{{\mathcal{O}}}} [\![ X_1, X_2, X_3]\!], \]

and a choice of isomorphism $E \cong M_2(R)$ gives rise to an isomorphism between the maps $R\to A \twoheadrightarrow S$ and

\[{{{\mathcal{O}}}} [\![ X_1, X_2, X_3]\!] \to {{{\mathcal{O}}}} [\![ X_1, X_2, X_3]\!][{\mathrm{PGL}}_2] \twoheadrightarrow {{{\mathcal{O}}}}[\![ X_1, X_2, X_3]\!][{\mathbb{G}}_m], \]

where the implicit closed immersion ${\mathbb{G}}_m \hookrightarrow {\mathrm{PGL}}_2$ is the standard torus.

3.2 Generic principal series

In this case, $\rho \simeq \chi_1 \oplus \chi_2$ where $\chi_1\chi_2^{-1} \neq 1, \omega^{\pm1}$ . Because $\chi_1\neq\chi_2$ , we may and do choose the additional structures discussed in § 2.3, such as a GMA structure on E and the resulting adapted moduli functor represented by S. In particular, we use a ${\mathbb{Z}}$ -grading of S to represent the $T= {\mathbb{G}}_m$ -action on the moduli scheme $\mathrm{Spec} S =\mathrm{Rep}^{\mathrm{Ad},\square}(E)$ of adapted representations.

Theorem 3.4. There is an isomorphism of ${\mathbb{Z}}$ -graded rings

\[S \cong \mathcal{O} [\![ a_0, a_1, bc]\!] [b, c], \]

where b has graded degree 2, c has graded degree $-2$ ,and the remaining generators have degree 0. The pseudodeformation ring R is the degree-0 subring $R = S_0 \subset S$ ,

\[R \cong {{{\mathcal{O}}}}[\![ a_0, a_1, bc]\!], \]

and its reducibility ideal is generated by bc. The universal Cayley–Hamilton algebra admits R-GMA form

\[E \cong\begin{pmatrix}R & Rb \\Rc & R\end{pmatrix}\!, \]

where the cross-diagonal multiplication is given by

\[Rb \times Rc \to Rbc \subset R, \quad (xb,yc) \mapsto xybc. \]

Proof. We know from Theorem 2.9 that $R =S_0$ . Straightforward calculations in local Galois cohomology yield that

\[H^2({\mathbb{Q}}_p, \mathrm{ad} \rho) =\begin{pmatrix}H^2({\mathbb{Q}}_p, {\mathbb{F}}) & H^2({\mathbb{Q}}_p, \chi_1\chi_2^{-1})\\H^2({\mathbb{Q}}_p, \chi_2\chi_1^{-1}) & H^2({\mathbb{Q}}_p, {\mathbb{F}})\end{pmatrix} = 0, \]

by the genericity assumption $\chi_1\chi_2^{-1} \neq 1, \omega^{\pm 1}$ . Therefore, the deformation theory of $\rho$ is unobstructed. Likewise, the tangent space (mod $\varpi$ ) of $\mathfrak{X}$ at $\rho$ decomposes as

\[H^1({\mathbb{Q}}_p, \mathrm{ad}^0 \rho) \cong H^1({\mathbb{Q}}_p, \chi_1\chi_2^{-1}) \oplus H^1({\mathbb{Q}}_p, {\mathbb{F}}) \oplus H^1({\mathbb{Q}}_p, \chi_2\chi_1^{-1}), \]

whose summands have ${\mathbb{F}}$ -dimensions $1\oplus 2 \oplus 1$ . Then, [Reference Wang-EricksonWE20, Theorem 11.3.3] (see Proposition 3.25) establishes an explicit presentation of $\hat S/\varpi \hat S$ as

\[\hat S/\varpi \hat S \cong {\mathbb{F}}[b,c] [\![ \bar a_0, \bar a_1, \bar b \bar c]\!], \]

where $\{\bar b\}$ , $\{\bar a_0, \bar a_1\}$ , and $\{\bar c\}$ are dual bases to the summands of $H^1({\mathbb{Q}}_p, \mathrm{ad}^0 \rho)$ , respectively, and where these bases have graded degrees 2, 0, and $-2$ , respectively. In this way, $\hat S/\varpi \hat S$ is the coordinate ring of a formal algebraic stack in which each element of its defining colimit is a quotient stack of $\hat S/(\varpi,(a_0, a_1, bc)^n)$ by the action of ${\mathbb{G}}_m$ .

By Proposition 2.10 and Nakayama’s lemma, E has off-diagonal summands generated by lifts b of $\bar b$ and c of $\bar c$ , respectively, of identical graded degrees. Likewise, denote by $a_i \in R$ lifts of $\bar a_i \in \bar R$ for $i=0,1$ . Thus (again using Proposition 2.10), we arrive at a presentation of graded rings $\mathcal{O}[\![ a_0, a_1, bc]\!] [b,c] \twoheadrightarrow S$ that we wish to show is an isomorphism.

The vanishing of $H^2$ implies that the completion $A^\wedge_\rho$ of A at its maximal ideal corresponding to $\rho$ is formally smooth over $\mathrm{Spf}{\mathbb{Z}}_p$ , since $A^\wedge_\rho$ is the framed deformation ring at $\rho$ .Therefore, $\mathfrak{X}^\wedge_\rho$ and $\mathrm{Spec} S^\wedge_\rho$ (the respective completions at the point corresponding to $\rho$ ) are formally smooth at $\rho$ as well, since these spaces are connected by smooth presentations $\mathrm{Spec} A \to \mathfrak{X} \leftarrow\mathrm{Spec} S$ as quotient stacks by ${\mathrm{SL}}_2$ and ${\mathbb{G}}_m$ , respectively. Because $S^\wedge_\rho$ is formally smooth, then so is S, because $\mathfrak{X}$ is coherently complete at $\rho$ (by [Reference Alper, Hall and RydhAHR25, Theorem 1.6]). Indeed, the equivalence of categories of coherent sheaves of [Reference Alper, Hall and RydhAHR25], i.e. a ‘formal GAGA’ result, implies an equivalence of ideal sheaves under completion and therefore an equivalence of closed subschemes; this implication is just like the classical case for formal schemes, in which an equivalence of closed subschemes [Reference GrothendieckGro61, Corollary 5.1.8] is deduced from a formal GAGA result [Reference GrothendieckGro61, Corollary 5.1.3].Therefore, the presentation map is an isomorphism, as desired.

We wish to discuss some line bundles on $\mathfrak{X} = \mathrm{Rep}(E)$ , which we present, by Theorem 2.9, as $\mathfrak{X} \cong [\mathrm{Spec} S/{\mathbb{G}}_m]$ . In particular, coherent sheaves (respectively, vector bundles) on $\mathfrak{X}$ are equivalent to finitely generated ${\mathbb{Z}}$ -graded S-modules (respectively, finitely generated ${\mathbb{Z}}$ -graded S-modules, which are projective as S-modules), where we regard S as a graded ring as in Theorem 3.4. We refer back to § 1.8 for our notational conventions regarding graded rings and modules. For $m\in {\mathbb{Z}}$ , we define the graded S-module $L_m$ as S(m), i.e.

\[(L_m)_k = S_{m+k}. \]

This is a line bundle on $\mathfrak{X}$ . If $\mathcal{V}$ is the vector bundle on $\mathfrak{X}$ underlying the universal representation, then we observe that its corresponding graded S-module is $L_1 \oplus L_{-1}$ . From this, we get the following theorem.

The claim about $\mathrm{End}(\mathcal{V})$ amounts to the conclusion of Theorem 2.9. To make this clear, we compute

\[\mathrm{Hom}_S(L_m, L_n)_0 = \mathrm{Hom}_S(S,L_{n-m})_0 = S_{n-m} \]

for all $m,n\in {\mathbb{Z}}$ to see that

\[\mathrm{End}_S(L_1 \oplus L_{-1})_0 = \begin{pmatrix} L_0 & L_2 \\ L_{-2} & L_0 \end{pmatrix}_0 = \begin{pmatrix} R & Rb \\ Rc & R \end{pmatrix}\]

and one easily checks that the multiplication matches.

3.3 Non-generic case I

In this case, the underlying pseudorepresentations are deformations of the trivial pseudorepresentation, and the determinant is trivial, after twisting. The pseudodeformation ring R and the Cayley–Hamilton algebra E were studied by Paškūnas[Reference PaškūnasPaš13, Appendix A and 9], where it was shown that they are equal to the corresponding object for the maximal pro-p quotient $\mathcal{G}$ of $\Gamma$ (see [Reference PaškūnasPaš13, Corollaries A.3 and A.4]). It is well known that $\mathcal{G}$ is a free pro-p group on two generators, which greatly helps in the study of R and E. Continue denoting by $\mathcal{V}$ the vector bundle of the universal representation on $\mathfrak{X}$ . We let $\mathrm{Spec} A$ be the affine scheme representing $\mathrm{Rep}^\square(E)$ . In this section, we prove the following result.

Theorem 3.8. The natural map $R \to A^{{\mathrm{SL}}_2}$ is an isomorphism, and $\mathrm{Ext}^i(\mathcal{V},\mathcal{V})=0$ for all $i\geq1$ . Moreover, the natural map

\[E \to M_2(A)^{{\mathrm{SL}}_2} = \mathrm{End}(\mathcal{V}) \]

is an isomorphism.

We prove the statements in Theorem 3.8 in the order they are mentioned. For the first and second part, we make use of the notion of a good filtration on algebraic representations of reductive groups over $\overline{\mathbb{F}}_p$ , which is summarized briefly in [Reference Fargues and ScholzeFS24, § VIII.5.1]. For details on standard constructions in the representation theory of algebraic groups we refer to [Reference JantzenJan03].

Let $H/\overline{\mathbb{F}}_p$ be a connected reductive group and let $T \subseteq B \subseteq H$ be a maximal torus and a Borel subgroup of H, respectively. For a dominant weight $\lambda$ , let $\mathcal{O}(\lambda)$ denote the corresponding standard line bundle on $H/B$ and set

\[\nabla_\lambda := H^0(H/B,\mathcal{O}(\lambda)). \]

A descending filtration $(V_i)$ ( $i \in{\mathbb{Z}}$ ) of H-subrepresentations of an H-representation V is said to be good if the successive quotients $V_i/V_{i-1}$ are isomorphic to direct sums of $\nabla_\lambda$ . Given a total ordering $0=\lambda_0,\lambda_1,\dots$ of the dominant weights, compatible with the dominance ordering, we can choose $V_i$ to be the maximal subrepresentation of V with weights $\lambda_j$ for $j\leq i$ , and V has a good filtration if and only if $V_i/V_{i-1}$ is isomorphic to a direct sum of copies of $\nabla_{\lambda_i}$ . An H-representation V has a good filtration if andonly if $H^i(H,V\otimes \nabla_\lambda) = 0$ for all $i\geq 1$ and all $\lambda$ (see [Reference DonkinDon81]). In particular, $H^i(H,V) = 0$ for all $i\geq1$ if V has a good filtration.

To prove that $R = A^{{\mathrm{SL}}_2}$ , we begin by recalling the following result of Donkin [Reference DonkinDon92, § 3.1]. For simplicity, we specialize to ${\mathrm{GL}}_2$ , which is the case we need. For any $r \in {\mathbb{Z}}_{\geq1}$ and any function $\sigma : \{1,\dots,r \} \to\{1,2\}$ , define a function

\[t_{r,\sigma}(g_1,g_2) := \mathrm{trace}(g_{\sigma(1)} \dots g_{\sigma(r)}) \]

on ${\mathrm{GL}}_2^2$ . Moreover, set $d_i(g_1,g_2)= \det(g_i)$ for $i=1,2$ .

From this, we deduce the following corollary.

Proof. We may regard $\mathcal{O}[{\mathrm{SL}}_2^2]$ as a ${\mathrm{GL}}_2$ -representation, acting by diagonal conjugation; clearly $\mathcal{O}[{\mathrm{SL}}_2^2]^{{\mathrm{GL}}_2}=\mathcal{O}[{\mathrm{SL}}_2^2]^{{\mathrm{SL}}_2}$ . Therestriction map $\mathcal{O}[{\mathrm{GL}}_2^2]\to \mathcal{O}[{\mathrm{SL}}_2^2]$ is surjective, and is the first part of a Koszul resolution

\[0 \to \mathcal{O}[{\mathrm{GL}}_2^2] \to \mathcal{O}[{\mathrm{GL}}_2^2]^2 \to \mathcal{O}[{\mathrm{GL}}_2^2] \to \mathcal{O}[{\mathrm{SL}}_2^2]\to 0, \]

since ${\mathrm{SL}}_2^2$ is a complete intersection in ${\mathrm{GL}}_2^2$ cut out by the equations $d_1=d_2=1$ . If $\mathcal{O}[{\mathrm{GL}}_2^2]$ has vanishing higher cohomology, the Koszul resolution together with elementary considerations of long exact sequences in cohomology shows that $\mathcal{O}[{\mathrm{GL}}_2^2]^{{\mathrm{GL}}_2} \to \mathcal{O}[{\mathrm{SL}}_2^2]^{{\mathrm{GL}}_2}$ issurjective, and the result then follows from Theorem 3.10. Therefore, it remains to show that $\mathcal{O}[{\mathrm{GL}}_2^2]$ has vanishing higher cohomology. By [Reference van derKallenvdK15, Theorem 10.5], each $H^i({\mathrm{GL}}_2,\mathcal{O}[{\mathrm{GL}}_2^2])$ is a finitely generated module over the finitely generated $\mathcal{O}$ -algebra $\mathcal{O}[GL_2^2]^{{\mathrm{GL}}_2}$ , so it suffices to show that $H^i({\mathrm{GL}}_2,\mathcal{O}[{\mathrm{GL}}_2^2])\otimes_{\mathcal{O}}\overline{L} = 0$ and $H^i({\mathrm{GL}}_2,\mathcal{O}[{\mathrm{GL}}_2^2])\otimes_{\mathcal{O}}\overline{{\mathbb{F}}} = 0$ for $i\geq 1$ . We have $H^i({\mathrm{GL}}_2,\mathcal{O}[{\mathrm{GL}}_2^2])\otimes_{\mathcal{O}}\overline{L} =H^i({\mathrm{GL}}_2,\overline{L}[{\mathrm{GL}}_2^2]) = 0$ for $i\geq 1$ , where the first equality comes from [Reference JantzenJan03, Proposition I.4.18] and the second comes from ${\mathrm{GL}}_2$ being reductive and $\overline{L}$ having characteristic 0. Finally, we also have $H^i({\mathrm{GL}}_2,\mathcal{O}[{\mathrm{GL}}_2^2])\otimes_{\mathcal{O}}\overline{{\mathbb{F}}} \hookrightarrow H^i({\mathrm{GL}}_2,\overline{{\mathbb{F}}}[{\mathrm{GL}}_2^2]) = 0$ for $i\geq 1$ , where the injection comes from [Reference JantzenJan03, Proposition I.4.18] and the equality $H^i({\mathrm{GL}}_2,\overline{{\mathbb{F}}}[{\mathrm{GL}}_2^2]) = 0$ holds because $\overline{{\mathbb{F}}}[{\mathrm{GL}}_2^2]$ has a good filtration, by [Reference Fargues and ScholzeFS24, Corollary VIII.5.7].

Let F be the free group on two generators; its pro-p completion is $\mathcal{G}$ . Attached to F, we have its ${\mathrm{SL}}_2$ -representation variety, which is isomorphic to ${\mathrm{SL}}_2^2 = \mathrm{Spec} A_F$ , its character variety $\mathrm{Spec} A_F^{{\mathrm{SL}}_2}$ (i.e. the GIT quotient ${\mathrm{SL}}_2^2 /\!/ {\mathrm{SL}}_2$ ), and its moduli variety of pseudorepresentations $\mathrm{Spec} R_F$ , all taken over the base $\mathcal{O}$ . There is a canonical map $R_F \to A_F^{{\mathrm{SL}}_2}$ , which is an adequate homeomorphism by [Reference EmersonEme18, Theorem 6.0.5(iv)] (cf. the ${\mathrm{GL}}_d$ case in [Reference Wang-EricksonWE18, Theorem 2.20]). By Theorem 3.10, it is also surjective. To show that $R_F \to A_F^{{\mathrm{SL}}_2}$ is an isomorphism, it therefore suffices to prove that $R_F$ is reduced. In fact, we may compute $R_F$ .

Proof. The second part follows from the first part and the discussion above, so it remains to prove the first part. We obtain a map $\phi : \mathcal{O}[s_1,s_2,s_3] \to R_F$ by sending, at the level of B-points for B an arbitrary $\mathcal{O}$ -algebra, a pseudorepresentation $T: F \to B$ to the tuple $(T(\gamma),T(\delta),T(\gamma\delta))$ , where $\gamma$ and $\delta$ are generators of F. Since a two-dimensional pseudorepresentation T over $\mathcal{O}$ with trivial determinant satisfies the identity

(3.1) \begin{equation}T(g^{-1}h) - T(g)T(h) + T(gh) = 0\end{equation}

(see [Reference ChenevierChe14, Lemma 1.9])Footnote ¹⁰ for any $g,h\in F$ , [Reference PaškūnasPaš13, Lemma 9.10] implies that $\phi$ induces an injection $f \mapsto f \circ\phi$ at the level of functors of points (cf. [Reference PaškūnasPaš13, Corollary 9.11]). Proving that $f\mapsto f \circ \phi$ is surjective on B-points for any $\mathcal{O}$ -algebra B (and, hence, that $\phi$ is an isomorphism) is then equivalent to showing that there is a map $\psi : R_F \to\mathcal{O}[s_1,s_2,s_3]$ such that $\psi \circ\phi$ is the identity. In terms of the moduli problem, this means that we need to construct a pseudorepresentation $T^{\mathrm{univ}} : F \to \mathcal{O}[s_1,s_2,s_3]$ with $T(\gamma) = s_1$ , $T(\delta) = s_2$ , and $T(\gamma\delta) = s_3$ .

For the sake of brevity,Footnote ¹¹ we construct $T^{\mathrm{univ}}$ from the representation $\rho : \mathcal{G} \to {\mathrm{SL}}_2(C)$ constructed in [Reference PaškūnasPaš13, Proposition 9.8]. Here,C is a ring that is finite over $\mathcal{O} [\![ t_1, t_2, t_3 ]\!]$ , and thetrace $T_\rho = \mathrm{tr}(\rho)$ satisfies $T_\rho(\gamma) = 2 + 2t_1$ , $T_\rho (\delta) =2+2t_2$ , and $T_\rho(\gamma\delta) = 2+2t_3$ . Let $T^\prime$ denote the restriction of $T_\rho$ from $\mathcal{G}$ to F. By (3.1) and [Reference PaškūnasPaš13, Lemma 9.10], $T^\prime$ takes values in $\mathcal{O}[t_1,t_2,t_3]$ . A simple change of variables then gives the desired pseudorepresentation $T^{\mathrm{univ}}$ .

For the second part, we first note that $\rho_B$ factors through the Cayley–Hamilton quotient $E_R$ of R[F] with respect to the specialization of the universal pseudorepresentation to R. Since R[F] is finitely generated over R, $E_R$ is a finite R-module [Reference Wang-EricksonWE18, Proposition 2.13], and is in particular ${\mathfrak{m}}_R$ -adically complete. For each $n \ge1$ , the quotient $E_R \otimes_R R/{\mathfrak{m}}_R^n$ is a finite-length R-module, so the map $R[F] \to E_R \otimes_R R/{\mathfrak{m}}_R^n$ factors through $R[F/H]$ for a finite quotient $F/H$ of F (consider the induced map $F \to (E_R \otimes_R R/{\mathfrak{m}}_R^n)^\times$ ). The proof of [Reference ChenevierChe14, Lemma 3.8] now shows that we can take $F/H$ to be a p-group. Indeed, we have a Cayley–Hamilton pseudorepresentation $D_n: E_R \otimes_R R/{\mathfrak{m}}_R^n \to R/{\mathfrak{m}}_R^n$ and [Reference ChenevierChe14, Lemma 2.10 and Theorem 2.16]implies that the radical $\mathcal{R}$ of $E_R \otimes_RR/{\mathfrak{m}}_R^n$ satisfies $(E_R \otimes_RR/{\mathfrak{m}}_R^n)/\mathcal{R}\cong M_2(k)$ , with the induced representation of F equalto the trivial representation. In particular, the image of $F/H$ in $E_R \otimes_R R/{\mathfrak{m}}_R^n$ lies in $1+\mathcal{R}$ , which is a p-group. Taking the limit over n shows that themap $R[F] \to E_R$ factors through $R[\![ \mathcal{G}]\!]$ , and we are done.

Proof. From the definitions, $A_F$ is the representation ring for the Cayley–Hamilton quotient $E_F$ of $R_F[F]$ with respect to the universal pseudorepresentation $R_F[F] \to R_F$ , and A is the representation ring for the Cayley–Hamilton quotient E of $R [\![ \mathcal{G} ]\!] \to R$ . By compatibility of Cayley–Hamilton quotients with base change [Reference ChenevierChe14, § 1.17], $A_F\otimes_{R_F} R$ is the representation ring for the Cayley–Hamilton quotient $E_F \otimes_{R_F}R$ of R[F] with respect to the pseudorepresentation $R[F] \to R$ . In particular, to show that $A = A_F \otimes_{R_F}R$ it suffices to show that the natural R-linear map $E_F \otimes_{R_F} R \to E$ induces bijections

\[\iota_B : \mathrm{Rep}^\Box(E)(B) \to \mathrm{Rep}^\Box(E_F \otimes_{R_F}R)(B)\]

for all R-algebras B. Since F is dense in $\mathcal{G}$ , the map $E_F \otimes_{R_F} R \to E$ has dense image, which implies that it is surjective since both sides are finite R-modules. This gives injectivity of $\iota_B$ , and surjectivity then follows from Lemma 3.13(2).

Next we record some properties of cohomology that we need.

Proof. Since R is flat over $R_F$ , we may write $R= \varinjlim_j R_F^{m_j}$ as a direct limit of finitely generated free modules by Lazard’s theorem.Since cohomology commutes with direct limits [Reference JantzenJan03, Lemma I.4.17], we see that

\begin{align*}H^i({\mathrm{SL}}_2,V \otimes_{R_F}R) &= H^i({\mathrm{SL}}_2, \varinjlim_j V \otimes_{R_F} R_F^{m_j}) = \varinjlim H^i({\mathrm{SL}}_2, V \otimes_{R_F} R_F^{m_j}) \\&= \varinjlim H^i({\mathrm{SL}}_2, V) \otimes_{R_F} R_F^{m_j} = H^i({\mathrm{SL}}_2,V)\otimes_{R_F}R,\end{align*}

as desired. That $R = A^{{\mathrm{SL}}_2}$ then follows by setting $V=A_F$ , $i=0$ , and using Proposition 3.12 and Corollary 3.14. Finally, by [Reference van derKallenvdK15, Theorem 10.5], $H^i({\mathrm{SL}}_2,V)$ is a finitely generated $R_F$ -module, and hence $H^i({\mathrm{SL}}_2,V\otimes_{R_F}R) =H^i({\mathrm{SL}}_2,V) \otimes_{R_F}R$ is a finitely generated R-module.

This proves that $R = A^{SL_2}$ , as desired. We can now prove that $\mathrm{Ext}^i(\mathcal{V},\mathcal{V}) = 0$ for $i \geq1$ .

Proof. Let ad denote the adjoint representation of ${\mathrm{GL}}_2$ , restricted to ${\mathrm{SL}}_2$ , which is a direct sum of induced representations. Then we have

\[\mathrm{Ext}^i(\mathcal{V},\mathcal{V}) = H^i({\mathrm{SL}}_2, A \otimes \mathrm{ad}) = H^i({\mathrm{SL}}_2, A_F \otimes \mathrm{ad})\otimes_{R_F} R, \]

where the last isomorphism follows from Lemma 3.15, and it is a finitely generated R-module. Since R is local, it suffices to prove that $H^i({\mathrm{SL}}_2,A_F \otimes \mathrm{ad}) \otimes_{\mathcal{O}}\overline{{\mathbb{F}}} =0$ for $i \geq 1$ . This cohomology group injects into $H^i({\mathrm{SL}}_2,(A_F \otimes \mathrm{ad})\otimes_{\mathcal{O}}\overline{{\mathbb{F}}})$ by [Reference JantzenJan03,Proposition I.4.18], and $H^i({\mathrm{SL}}_2,(A_F \otimes \mathrm{ad})\otimes_{\mathcal{O}}\overline{{\mathbb{F}}}) =0$ since $A_F\otimes_{\mathcal{O}}\overline{{\mathbb{F}}}$ has a good filtration by [Reference Fargues and ScholzeFS24, CorollaryVIII.5.7].

It remains to prove that $E \to \mathrm{End}(\mathcal{V})$ is an isomorphism. As above,

\[\mathrm{End}(\mathcal{V}) = (A \otimes \mathrm{ad})^{{\mathrm{SL}}_2} = M_2(A)^{{\mathrm{SL}}_2}. \]

We start by looking at the problem after inverting $\varpi$ . Then $A[1/\varpi]$ is the representation ring for the Cayley–Hamilton algebra $E[1/\varpi]$ and hence, by [Reference ProcesiPro87, Theorem 2.6], the natural map

\[E[1/\varpi] \to M_2(A[1/\varpi])^{{\mathrm{SL}}_2} \]

is an isomorphism. Since $M_2(A[1/\varpi])^{{\mathrm{SL}}_2} =M_2(A)^{{\mathrm{SL}}_2}[1/\varpi]$ , we see that $E \to M_2(A)^{{\mathrm{SL}}_2}$ is an isomorphism after inverting $\varpi$ . To prove that it is an isomorphism on the nose, we need to study the map more explicitly.

We begin this by recalling the structure of R and E from [Reference PaškūnasPaš13, 9.2]. Let $\gamma$ and $\delta$ be two generators of F.By [Reference PaškūnasPaš13, Proposition 9.12, Corollary 9.13] we have $R = \mathcal{O} [\![ t_1, t_2, t_3 ]\!]$ , where $2+2t_1$ is the trace of $\gamma$ , $2+2t_2$ is the trace of $\delta$ , and $2+2t_3$ is the trace of $\gamma \delta$ . The ring E is a free R-module of rank 4 by [Reference PaškūnasPaš13,Corollary 9.25], with a basis given by elements 1, u, v, and $uv-vu$ ,where

\[u = \gamma -1 - t_1 , \quad v = \delta - 1 - t_2; \]

recall that E is a quotient of $R [\![ \mathcal{G} ]\!]$ . The ring A may be described as a quotient of $R[a_i,b_i,c_i,d_i \mid i=1,2 \,]$ , where the universal representation $E \to M_2(A)$ sends

\[\gamma \mapsto \begin{pmatrix} 1+a_1 & c_1 \\ c_2 & 1+a_2 \end{pmatrix} \!, \quad \delta \mapsto \begin{pmatrix} 1+b_1 & d_1 \\ d_2 & 1+b_2\end{pmatrix}\!. \]

We have five relations. The first four come from the trace and determinant of the image of $\gamma$ and $\delta$ , and amount to

\[a_1 + a_2 = 2t_1, \quad b_1 + b_2 = 2t_2, \quad a_1 + a_2 + a_1 a_2 - c_1 c_2 = 0, \quad b_1 + b_2 + b_1 b_2 - d_1 d_2 = 0. \]

The fifth comes from the trace of $\gamma \delta$ , and is

\[a_1 + a_2 + b_1 + b_2 + a_1 b_1 + a_2 b_2 + c_1 d_2 + c_2 d_1 = 2 t_3. \]

Let us now write the map $E \to M_2(A)$ explicitly as an R-module map, using the basis 1, $\gamma-1$ , $\delta-1$ , and $uv-vu=\gamma \delta -\delta \gamma$ . Clearly 1 gets sent to the identity matrix, and from the descriptions above we see that

\[\gamma -1 \mapsto \begin{pmatrix} a_1 & c_1 \\ c_2 & a_2 \end{pmatrix} \!, \quad \delta -1 \mapsto \begin{pmatrix} b_1 & d_1 \\ d_2 & b_2\end{pmatrix}\!, \]

and hence

\[uv - vu \mapsto \begin{pmatrix} c_1 d_2 - c_2 d_1 & a_1 d_1 + b_2 c_1 - a_2 d_1 - b_1 c_1 \\ a_2 d_2 + b_1 c_2 - a_1 d_2 - b_2 c_2 & c_2 d_1 -c_1 d_2 \end{pmatrix}\!. \]

Now consider a general element $X = \lambda_1 + \lambda_2 (\gamma -1) + \lambda_3 (\delta -1) + \lambda_4 (\gamma \delta -\delta \gamma) \in E[1/\varpi]$ . It gets sent to

\[\begin{pmatrix} \lambda_1 + \lambda_2a_1 + \lambda_3 b_1 + \lambda_4 (c_1 d_2 - c_2 d_1) & \lambda_2 c_1 + \lambda_3 d_1 + \lambda_4 (a_1 d_1 +b_2 c_1 - a_2 d_1 - b_1 c_1) \\ \lambda_2 c_2 + \lambda_3 d_2 + \lambda_4 ( a_2 d_2 + b_1 c_2 - a_1 d_2 - b_2 c_2) & \lambda_1 + \lambda_2 a_2 + \lambda_3b_2 + \lambda_4 (c_2 d_1 - c_1 d_2) \end{pmatrix}\!. \]

These expressions are somewhat unwieldy to analyze. We instead consider the quotient C of A, introduced in [Reference PaškūnasPaš13, Definition 9.7], which is given by setting

\[c_1 = 1, \quad c_2 = 0, \quad d_1 = 0, \quad d_2 = 2t_3 - 2t_1 - 2t_2 - a_1 b_1 - a_2 b_2. \]

With this, one gets the presentation

\[C = \frac{R[a_1,a_2,b_1,b_2]}{(a_1 + a_2 -2t_1, a_1 a_2 + 2t_1, b_1 + b_2 -2t_2, b_1 b_2 + 2t_2) } \]

(we have changed some signs compared with [Reference PaškūnasPaš13], correcting apparent typos), which may be further simplified to

\[C = \frac{R[a_1,b_1]}{(a_1^2 - 2t_1 a_1 - 2t_1, b_1^2 - 2t_2 b_2 - 2t_2)}. \]

In particular, we see that C is a biquadratic extension of R and we get the following.

The composition of $E[1/\varpi] \to M_2(A[1/\varpi])$ with $M_2(A[1/\varpi]) \to M_2(C[1/\varpi])$ is then given by sending the general element $X= \lambda_1 + \lambda_2 (\gamma -1) + \lambda_3 (\delta -1) + \lambda_4 (\gamma \delta - \delta \gamma) \in E[1/\varpi]$ to

\[\begin{pmatrix} \lambda_1 + \lambda_2a_1 + \lambda_3 b_1 + \lambda_4 d_2 & \lambda_2 + \lambda_4 (b_2 - b_1) \\ \lambda_3 d_2 + \lambda_4 (a_2 d_2 - a_1 d_2) & \lambda_1 + \lambda_2 a_2 + \lambda_3 b_2 - \lambda_4 d_2 \end{pmatrix}\!. \]

With these preparations, we now prove the main theorem of this subsection.

Proof. We know that $E[1/\varpi ] \to M_2(A)^{{\mathrm{SL}}_2}[1/\varpi ]$ is an isomorphism and E is $\varpi$ -torsion free, since it is free over R. In particular, j is injective, so it remains to prove surjectivity. Note that A is $\varpi$ -torsion free as well, so by surjectivity of j after inverting $\varpi$ , it suffices to show that if an element $X = \lambda_1 + \lambda_2 u + \lambda_3 v +\lambda_4 (uv-vu) \in E[1/\varpi]$ as above has image $j(X) \in M_2(A)$ , then we must have $\lambda_i \in R$ for $i=1,\dots,4$ . If $j(X) \in M_2(A)$ , then its image in $M_2(C[1/\varpi])$ lies in $M_2(C)$ , i.e.

\[\begin{pmatrix} \lambda_1 + \lambda_2a_1 + \lambda_3 b_1 + \lambda_4 d_2 & \lambda_2 + \lambda_4 (b_2 - b_1) \\ \lambda_3 d_2 + \lambda_4 (a_2 d_2 - a_1 d_2) & \lambda_1 + \lambda_2 a_2 + \lambda_3 b_2 - \lambda_4 d_2 \end{pmatrix} \in M_2(C). \]

Looking at the top-right corner, we see that

\[\lambda_2 + \lambda_4 (b_2 - b_1) = (\lambda_2 + 2t_2 \lambda_4) - 2 \lambda_4 b_1 \in C. \]

By Lemma 3.17, we deduce first that $\lambda_4 \in R$ and then that $\lambda_2 \in R$ . Applying this to the top-left corner, we see that $\lambda_1 + \lambda_3 b_1 \in C$ and hence, by Lemma 3.17 again, we see that $\lambda_1,\lambda_3 \in R$ . This finishes the proof.

We finish this section by describing a free resolution of the left E-module ${{{\mathcal{O}}}}_{\mathbf{1}}$ given by the quotient (of ${{{\mathcal{O}}}}$ -algebras) $E \xrightarrow{f}{{{\mathcal{O}}}}$ with $f(g - 1) =0$ for all $g \in \mathcal{G}$ and $f(t_i) = 0$ for $1 \le i \le 3$ .

We have already recalled the R-basis of E given by $1, u, v,uv-vu$ . We set $w := uv-vu$ . The squares $u^2, v^2$ lie in R, the center of E.

Proposition 3.19. The following gives a free resolution of the left E-module ${{{\mathcal{O}}}}_{\mathbf{1}}$ :

(3.2) \begin{equation} 0 \to E \xrightarrow{(\begin{smallmatrix} v & u \end{smallmatrix})} E^{\oplus 2}\xrightarrow{\left(\begin{smallmatrix} vu & -u^2\\ -v^2 & uv \end{smallmatrix}\right)} E^{\oplus 2}\xrightarrow{\left(\begin{smallmatrix} u \\ v\end{smallmatrix}\right)} E \xrightarrow{f} {{{\mathcal{O}}}}_{\mathbf{1}} \to 0\end{equation}

with the matrices acting from the right on row vectors.

Proof. First we need to check that the left ideal generated by u, v coincides with the kernel of f. Since this left ideal contains Ru, Rv, and Rw, it suffices to show that it also contains the prime ideal $(t_1,t_2,t_3)$ . In fact, we have $(u^2,v^2,uv+vu) =(t_1,t_2,t_3)$ , which is useful later. This follows from the identities

\begin{align*}u^2 &= 2t_1 - t_1^2,\\ v^2 &= 2t_2 - t_2^2,\\ uv+vu &= 2(t_3-t_1-t_2-t_1t_2).\end{align*}

The first two of these identities are [Reference PaškūnasPaš13, Equation (159)]. The third can be checked by rewriting $uv+ vu - 2t_3$ using the identities $u = ({\gamma-\gamma^{-1}})/{2}, v =({\delta-\delta^{-1}})/{2}$ , and $2t_3 + 2 = ({T_{\rho}(\gamma\delta) +T_{\rho}(\delta\gamma) })/{2}$ .

Next we need to show that the kernel of ${(\begin{smallmatrix} u \\ v\end{smallmatrix})}$ is contained in the image of ${(\begin{smallmatrix} vu & -u^2\\-v^2 & uv \end{smallmatrix})}$ . Suppose $(\lambda_1 + \lambda_2 u + \lambda_3 v + \lambda_4w, \mu_1 + \mu_2 u + \mu_3 v + \mu_4 w) \in E^2$ is in the kernel. Applying the map ${(\begin{smallmatrix} u \\ v\end{smallmatrix})}$ and comparing coefficients tells us that this boilsdown to the following equalities in R:

(3.3) \begin{align}\lambda_1 &= -2v^2\mu_4 - \lambda_4(uv+vu),\end{align}

(3.4) \begin{align}\mu_1 &= 2u^2\lambda_4 + \mu_4(uv+vu),\end{align}

(3.5) \begin{align}\lambda_3 &= \mu _2 ,\end{align}

(3.6) \begin{align}0 &= \lambda_2u^2+\mu_3v^2 + \lambda_3(uv+vu).\end{align}

Translating by $(-2\lambda_4, 2\mu_4 ){(\begin{smallmatrix} vu & -u^2\\ -v^2 & uv \end{smallmatrix})} =(\lambda_1+\lambda_4 w, \mu_1+\mu_4 w)$ , we may assume that $\lambda_1 = \mu_1 = \lambda_4 = \mu_4 =0$ . Now we consider Equation (3.6). Since $u^2, v^2, (uv+vu)$ form a regular sequence in R, we can use the Koszul complex to write

\[ (\lambda_2,\mu_3,\lambda_3) = (x,y,z)\begin{pmatrix} 0 & -(uv+vu) & v^2\\ -(uv+vu) & 0 & u^2\\ -v^2& u^2 & 0\end{pmatrix} \]

for some $x,y,z \in R$ . Then, noting that $vuv = (uv+vu)v- v^2 u$ and $uvu = (uv+vu)u -u^2v$ , the reader can check that we have

\[(-yu, zu-xv)\begin{pmatrix}vu & -u^2\\ -v^2 & uv\end{pmatrix} = (\lambda_2 u + \lambda_3 v, \lambda_3u + \mu_3 v).\]

To check exactness at the next step of the sequence, we consider the condition that $(\lambda_1 + \lambda_2 u + \lambda_3 v + \lambda_4w, \mu_1 + \mu_2 u + \mu_3 v + \mu_4 w) \in E^2$ is in the kernel of ${(\begin{smallmatrix} vu \\ -v^2 \end{smallmatrix})}$ . Again, comparing coefficientsgives some equalities in R. One of them is

\[ -\lambda_2 u^2 = \mu_3 v^2, \]

which tells us that there is an $x \in R$ with $\mu_3 = xu^2$ and $\lambda_2 = -xv^2$ . Translating by

\[ (-xuv){\begin{pmatrix} vu \\ -v^2\end{pmatrix}} = (\lambda_2u, \mu_3 v - x(uv+vu)u), \]

we may assume that $\lambda_2 = \mu_3 = 0$ . Now the condition that $(\lambda_1 + \lambda_3 v + \lambda_4 w, \mu_1 + \mu_2 u + \mu_4 w)$ is in the kernel of ${(\begin{smallmatrix}vu \\ -v^2\end{smallmatrix})}$ boils down to the equalities

\begin{align*} \lambda_1 &= -2v^2\mu_4 + \lambda_4(uv+vu),\\ \mu_1 &= 2u^2\lambda_4 - \mu_4(uv+vu),\\ \lambda_3 &= \mu_2,\end{align*}

which means we have

\[ \lambda_1 + \lambda_3 v + \lambda_4 w = (2\lambda_4u-2\mu_4v+\lambda_3)v \]

and

\[ \mu_1+\mu_2u +\mu_4w = (2\lambda_4u-2\mu_4v+\lambda_3)u. \]

This shows that we do have something in the image of $(\begin{smallmatrix} v &u\end{smallmatrix})$ . Finally, the map $E \xrightarrow{(\begin{smallmatrix} v& u \end{smallmatrix})} E^{\oplus 2}$ is injective because $v^2$ is a non-zero divisor.

3.4 Non-generic case II

In this case, $\rho \simeq \chi_1 \oplus \chi_2 \cong \chi \otimes (\omega \oplus1)$ for some character $\chi : \Gamma \to{\mathbb{F}}^\times$ . Unlike all other cases, the moduli of representations $\mathfrak{X}$ is not smooth. Paškūnas has computed some deformation rings of representations with semisimplification isomorphic to $\rho$ (see [Reference PaškūnasPaš13, § B]), relying on a presentation due to Böckle [Reference BöckleBöc00]. We adapt these results to describe the entire moduli space $\mathfrak{X}$ .

Theorem 3.20. There is an isomorphism of graded rings

\[S \cong S' := \frac{{{{\mathcal{O}}}}[\![ a_0, a_1, b_0 c, b_1 c]\!] [b_0, b_1, c]}{(pb_0 + a_1 b_0 + a_0 b_1)} \]

where $b_i$ has degree 2 for $i=0,1$ , c has degree $-2$ ,and $a_i$ has degree 0 for $i=0,1$ . The isomorphism $S \cong S'$ induces an isomorphism of subrings of degree 0, $R = S_0 \cong R' =S'_0$ of degree 0,

\[R \cong R' := \frac{{{{\mathcal{O}}}}[\![ a_0, a_1, b_0 c, b_1 c]\!]}{(pb_0c + a_1 b_0 c + a_0 b_1 c)} \cong \frac{{{{\mathcal{O}}}}[\![ a_0,a_1, Y_0, Y_1]\!]}{(pY_0 + a_1 Y_0 + a_0Y_1)}. \]

The universal Cayley–Hamilton algebra E has R-GMA form

\[\begin{pmatrix}R & \frac{Rb_0 \oplus Rb_1}{\langle (p+a_1)b_0 + a_0 b_1\rangle}\\Rc & R\end{pmatrix} \]

with cross-diagonal multiplication given by

\[((x_0 b_0, x_1 b_1), yc) \mapsto x_0 yb_0c + x_1yb_1c. \]

The claim that S’ is a model for S is the main new statement and is developed in Theorem 3.24. For the moment,we deduce Theorem 3.20 from Theorem 3.24 using facts about the residually multiplicity-free case summarized in § 2.3.

Proof of Theorem 3.20 given Theorem 3.24. We know that $R =S_0$ and $E = \mathrm{End}(\mathcal{V}) =M_2(S)^{{\mathbb{G}}_m}$ from Theorem 2.9. What remains is to deduce the claimed presentations of R by R’ and of E as above. This follows directly from Proposition 2.10, which implies that

\[E_{1,2} \cong S_2, \quad E_{2,1} \cong S_{-2} \]

and that the cross-diagonal multiplication map is compatible with the multiplication map $S_2 \times S_{-2} \to S_0 =R$ . Then the form of E given in Theorem 3.20 follows from straightforward calculations of $S_{\pm 2}$ given the isomorphism $S \cong S'$ proved in Theorem 3.24.

The proof that $S \cong S'$ is what remains. Without loss of generality, we writethis proof in the case that $\chi$ and $\psi$ are trivial and $\mathcal{O} = {\mathbb{Z}}_p$ ; the general case follows bytwisting. We begin this with Paškūnas’s description in [Reference PaškūnasPaš13, B] of a certain quotient group of $\Gamma$ . It requires the following data and notation.

• • Let $\mathcal{F}$ denote a free pro-p group on $p+1$ generators $x_0, \dotsc,x_p$ .

• • Given a profinite group H, let H(p) denote its maximal pro-p quotient.

• • Given a pro-p group H, there is a p-lower central series filtration defined inductively as

  \[H_1 = H, \quad H_{i+1} = H^p_i[H_i,H] \quad \text{for } i \in {\mathbb{Z}}_{\geq 1}. \]

• • Because $\Gamma_{{\mathbb{Q}}_p(\zeta_p)}(p)$ is a Demuškin group with invariants $n = p+1$ and $q = p$ (for a reference, see e.g. [Reference Neukirch, Schmidt and WingbergNSW08, 3.9]), there exists a surjection $\varphi : \mathcal{F} \twoheadrightarrow\Gamma_{{\mathbb{Q}}_p(\zeta_p)}(p)$ with kernel generated by a single element r.

We quote this lemma from [Reference PaškūnasPaš13, Appendix B].

Next we produce a representation of $\mathcal{F}$ with coefficients in the ring S’ of Theorem 3.20. Afterward we show that it factors through $\varphi$ and is universal, producing the isomorphism $S {\buildrel\sim\over\rightarrow}S'$ . This is a straightforward adaptation of the construction in [Reference PaškūnasPaš13, p.180] from a deformation ring to the whole moduli stack of representations.

Definition 3.22. Denote by $\alpha: \mathcal{F} \rtimes \mathrm{Gal}({\mathbb{Q}}_p(\zeta_p)/{\mathbb{Q}}_p) \to{\mathrm{GL}}_2(S')$ the homomorphism determined by

\begin{gather*}\mathrm{Gal}({\mathbb{Q}}_p(\zeta_p)/{\mathbb{Q}}_p) \ni g \mapsto {{{\left(\begin{smallmatrix}\tilde\omega(g) & 0 \\ 0 &1\end{smallmatrix}\right)}}} \\ \text{ for }i = 2,3,\dotsc,p-3 , \quad x_i \mapsto 1, \\x_{p-2} \mapsto \displaystyle \begin{pmatrix} 1 & 0 \\ c & 1\end{pmatrix}\!, \\ \text{for }j=0,1, \quad x_{1+j(p-1)} \mapsto \begin{pmatrix} 1 & b_j \\ 0 & 1 \end{pmatrix}\!, \\ \text{for }j=0,1, \quad x_{j(p-1)}\mapsto \begin{pmatrix} (1+a_j)^{-{1}/{2}} & 0 \\ 0 & (1+a_j)^{{1}/{2}}\end{pmatrix}\!,\end{gather*}

where the semidirect product structure is as in Lemma 3.21. The fact that these images of generators define a homomorphism can be read off from the semidirect product structure.

Let $\Gamma'$ be the Galois group over ${\mathbb{Q}}_p$ of the maximal pro-p extension of ${\mathbb{Q}}_p(\zeta_p)$ . Let $\Gamma'_{{\mathbb{Q}}_p(\zeta_p)} \subset\Gamma'$ denote the subgroup fixing ${\mathbb{Q}}_p(\zeta_p)$ . Thus, we naturally have a quotient map $\pi : \Gamma \twoheadrightarrow \Gamma'$ , and the universal adapted representation $\rho_S : \Gamma \to {\mathrm{GL}}_2(S)$ factors through $\Gamma'$ .

Proposition 3.23 (Following [Reference PaškūnasPaš13, Proposition B.2]). There exists a continuous group homomorphism

\[\varphi' : \mathcal{F} \rtimes \mathrm{Gal}({\mathbb{Q}}_p(\zeta_p)/{\mathbb{Q}}_p) \twoheadrightarrow \Gamma' \]

such that $\varphi' \equiv \varphi\pmod{(\Gamma'_{{\mathbb{Q}}_p(\zeta_p)})_3}$ and there exists a factor $\tilde \rho$ of $\alpha$ producing the following commuting diagram.

In addition, there exists $r_1 \in \mathcal{F}$ such that $\mathrm{Gal}({\mathbb{Q}}_p(\zeta_p)/{\mathbb{Q}}_p)$ acts on $r_1$ by $\tilde\omega$ and $\ker \varphi'$ equals the closed normal subgroup of $\mathcal{F}$ generated by $r_1$ .

Proof. First we observe that $r' \in \ker \varphi'$ , where r’ was defined in Lemma 3.21. Indeed, for $j=0,1$ ,

\[[\varphi'(x_{1+(p-1)j}), \varphi'(x_{(p-1)(1-j)})] = \begin{pmatrix} 1 & a_{1-j}b_j \\ 0 & 1\end{pmatrix}\!, \]

whereas $[\varphi'(x_i), \varphi'(x_{p-i})] = 1$ for $i\not\equiv 1,0 \pmod{p-1}$ , and therefore

\[\varphi'(r) = \varphi'(x_1)^p \cdot \prod_{i=1}^{({p-1})/{2}} [\varphi'(x_i),\varphi'(x_{p-i})] \cdot [\varphi'(x_p),\varphi'(x_0)]=\begin{pmatrix}1 & pb_0 \\0 & 1\end{pmatrix}\begin{pmatrix}1 & a_1 b_0 \\0 & 1\end{pmatrix}\begin{pmatrix}1 & a_0 b_1 \\0 & 1\end{pmatrix}\!, \]

which vanishes in ${\mathrm{GL}}_2(S')$ due to the presence of the relation $pb_0 + a_1 b_0 + a_0 b_1$ .

By Lemma 3.21(2), $r \equiv r' \pmod{\mathcal{F}_3}$ , and therefore $\alpha(r) \in \alpha(\mathcal{F}_3)$ . The rest of the proof follows exactly as in [Reference PaškūnasPaš13, Proof ofProposition B.2], producing $r_1 \in \mathcal{F}$ , equivalent to r and r’ $\pmod{\mathcal{F}_3}$ , such that $r_1 \in \mathrm{Ker}\, \alpha$ and the conjugation action of $\mathrm{Gal}({\mathbb{Q}}_p(\zeta_p)/{\mathbb{Q}}_p)$ on $\mathcal{F}$ acts on $r_1$ by the character $\tilde\omega$ .

Because $\tilde \rho \circ \pi : \Gamma \to {\mathrm{GL}}_2(S')$ hasresidual pseudorepresentation $\psi(\omega \oplus 1)$ , the universal property of the universalCayley–Hamilton algebra $(E,R,D_E : E \to R)$ (see Definition 2.3) produces:

• a ring homomorphism $R \to S'_0 = R' \subset S'$ ;

• an R-algebra homomorphism $\eta : E \to M_2(S')$ suchthat

  • * $(E,R,D_E) \to (M_2(S'),S',\det : M_2(S') \to S')$ is a morphism of Cayley–Hamilton algebras and

  • * $\tilde \rho = \eta \circ \rho^u$ .

We impose the R-GMA structure on E arising from the idempotents arising from pullback over $\eta$ ,

\[(\eta^{-1}({{{(\begin{smallmatrix}1 & 0 \\ 0 & 0\end{smallmatrix})}}}), \ \eta^{-1}({{{(\begin{smallmatrix}0 & 0 \\ 0 & 1\end{smallmatrix})}}}).\]

(These idempotents lie in $\eta(E)$ because they are ${\mathbb{Z}}_p$ -linear combinations of the image of $\mathrm{Gal}({\mathbb{Q}}_p(\zeta_p)/{\mathbb{Q}}_p)$ specified in Definition 3.22.) Now that E has been endowed with an R-GMA structure, we write S for the graded R-algebra representing its adapted representation moduli functor. Thus, its universal property along with $\eta$ induce a graded R-algebra homomorphism

\[\phi : S \to S' \quad \text{with } 0\text{-degree part } R=S_0 \to R'=S'_0, \]

where R is the pseudodeformation ring.

To prove the theorem, we import a description of $\hat S/p \hat S$ from [Reference Wang-EricksonWE20]. We remind the reader that the completion $\hat S$ of S is not local: see Remark 3.5. Because $\rho \simeq \omega \oplus 1$ , we can apply the decomposition $\mathrm{Ad}^0 \rho \cong \omega \oplus 1 \oplus \omega^{-1}$ .

Proposition 3.25 [Reference Wang-EricksonWE20]. There exists a presentation of $\hat S/p\hat S$ of the form

\[\left[\frac{(\mathrm{Sym}^*_{{\mathbb{F}}_p} H^1(\Gamma, \mathrm{Ad}^0 \rho)^*)}{(m^*(H^2(\Gamma, \mathrm{Ad}^0 \rho)^*)}\right]^\wedge{\buildrel\sim\over\rightarrow} \hat S/p\hat S \]

where we have the following.

1. (1) The completion denoted $[\dotsm]^\wedge$ is at the ideal generated by $H^1(\Gamma,1)^*$ , $H^1(\Gamma, \omega)^* \otimes_{{\mathbb{F}}_p}H^1(\Gamma, \omega^{-1})^*$ .

2. (2) The presentation is ${\mathbb{G}}_m$ -equivariant, as expressed by a ${\mathbb{Z}}$ -grading (of each constituent of the limit that is the completion) where the degrees of the modules of generators and relations are given by:

   • – $\deg H^i(\Gamma, \omega)^* =2$ for $i =1,2$ ;

   • – $\deg H^1(\Gamma, 1)^* = 0$ ;

   • – $\deg H^1(\Gamma, \omega^{-1})^* = -2.$

3. (3) The $m^*$ is ${\mathbb{G}}_m$ -equivariant.

4. (4) The image of $m^*$ lies in the ideal $\mathrm{Sym}^{\geq 2}$ , and the quadratic term (modulo $\mathrm{Sym}^{\geq3}$ ) $m_2^*$ is the ${\mathbb{F}}_p$ -linear dual of the composite of the cup product and the Lie bracket map

   \[H^1(\Gamma, \mathrm{Ad}^0 \rho) \otimes_{{\mathbb{F}}_p} H^1(\Gamma, \mathrm{Ad}^0 \rho) \to H^2(\Gamma, \mathrm{Ad}^0 \rho\otimes_{{\mathbb{F}}_p} \mathrm{Ad}^0 \rho) \buildrel{[\cdot,\cdot]}\over\longrightarrow H^2(\Gamma, \mathrm{Ad}^0 \rho). \]

5. (5) The universal representation $\rho_S : \Gamma \to M_2(S)$ has the following form modulo $(p, \mathrm{Sym}^{\geq 2} H^1(-)^*)$ :

   \[\begin{pmatrix}\omega(1+ \tilde A)& \tilde B\\ \omega \tilde C & 1 - \tilde A\end{pmatrix}\!, \]
   where
   \begin{align*}\tilde B &\in Z^1(\Gamma, \omega) \otimes H^1(\Gamma, \omega)^*,\\ \tilde A &\in Z^1(\Gamma, {\mathbb{F}}_p) \otimes H^1(\Gamma,{\mathbb{F}}_p)^*, \\ \tilde C &\in Z^1(\Gamma, \omega^{-1}) \otimes H^1(\Gamma, \omega^{-1})^*\end{align*}
   are choices of lift of $\mathrm{id} \in \mathrm{End}_{{\mathbb{F}}_p}(H^1(-)) \cong H^1(-) \otimes_{{\mathbb{F}}_p}H^1(-)^*$ under the natural projection
   \[Z^1(-) \otimes H^1(-)^* \twoheadrightarrow H^1(-) \otimes H^1(-)^*. \]

In order to work explicitly with this presentation of $S/pS$ , we use the following choice of basis of $H^1(\Gamma, \mathrm{Ad}^0 \rho)$ . To specify this basis, we use generators $x_i \in \Gamma'$ , $0 \leq i \leq p$ , produced by Definition 3.22 and Proposition 3.23. (This is a slight abuse of notation, since these generators are actually in $\mathcal{F}$ and we use $x_i$ to refer to $\varphi'(x_i) \in \Gamma'$ .) The basis is labeled so that it matches the deformations to ${\mathbb{F}}_p[\varepsilon]/(\varepsilon^2)$ of $\rho$ that arise from using each non-identity matrix listed in Definition 3.22, as follows.

• • $\{\bar b_0^*, \bar b_1^*\} \subset H^1(\Gamma,\omega)$ ;

• • $\{\bar a_0^*, \bar a_1^*\} \subset H^1(\Gamma, {\mathbb{F}}_p)$ ;

• • $\{\bar c^*\} \subset H^1(\Gamma, \omega^{-1})$ ;

characterized by the property that for each $y \in Y = \{b_0, b_1, a_0, a_1, c\} \subset S'$ ,the lift $\rho_y : \Gamma \to{\mathrm{GL}}_2({\mathbb{F}}_p[\epsilon]/(\epsilon^2))$ of $\rho$ given by specializing the coefficients of $\tilde \rho \circ \pi: \Gamma \to {\mathrm{GL}}_2(S')$ along themap $\nu_y : S \to {\mathbb{F}}_p[\epsilon]/(\epsilon^2)$ given by

\[y \mapsto \epsilon, \quad z \mapsto 0 \quad \text{for all } z \in Y \smallsetminus \{y\} \]

realizes the cohomology class $\bar y^*$ under the standard bijection between lifts of $\rho$ to ${\mathbb{F}}_p[\epsilon]/(\epsilon^2)$ and $Z^1(\Gamma, \mathrm{Ad}^0\rho)$ .

• • concentrated in the summand of $Z^1(\mathrm{Ad}^0\rho)$ named in the lemma (e.g. $\rho_{b_0} \in Z^1(\Gamma,\omega)$ ) under the standard decomposition $\mathrm{Ad}^0 \rho \simeq \omega \oplus 1 \oplus\omega^{-1}$ ; and

• • linearly independent after projection to $H^1(\mathcal{F} \rtimes \mathrm{Gal}({\mathbb{Q}}_p(\zeta_p)/{\mathbb{Q}}_p),-)$ , and therefore also linearly independent subsets of the cohomology groups $H^1(\Gamma,-)$ named in the lemma.