1.1 The conjecture of Böckle–Juschka

Our second main result proves a generalisation of a conjecture of Böckle–Juschka concerning the irreducible components of $R^{\square }_{\overline {\rho }}$ . We assume that $G^0$ is split over $\mathscr O$ and $G/G^0$ is a constant group scheme. This can be achieved after replacing L by a finite unramified extension. Let $\Gamma _E$ be the kernel of the homomorphism $\Gamma _F\overset {\overline {\rho }}{\longrightarrow } G(k) \rightarrow (G/G^0)(k)$ . We assume that $\mathscr O$ contains $\mu _{p^{\infty }}(E)$ . One may further assume that this map is surjective, as replacing G with a subgroup does not change the deformation problem, so that $(G/G^0)(k)= {\mathrm {Gal}}(E/F)$ .

Let $G'$ be the derived group scheme of $G^0$ , let $G^{\prime }_{\operatorname {sc}}\rightarrow G'$ be the simply connected central cover of $G'$ and let $\pi _1(G')$ be the kernel of this map. Then $\pi _1(G')$ is a finite diagonalisable subgroup scheme of the centre of $G^{\prime }_{\operatorname {sc}}$ and is isomorphic to $\prod _{i=1}^r \mu _{n_i}$ for some integers $n_i$ . The group scheme $\pi _1(G')$ is étale if and only if p does not divide $n_i$ for all i.

Our assumptions on G imply that $G^0/G'$ is a split torus. The action of $G/G'$ on $G^0/G'$ by conjugation induces an action of $ {\mathrm {Gal}}(E/F)$ on its character lattice $M:=X^*(G^0/G')$ .

Let $\overline {\psi }: \Gamma _F \rightarrow (G/G')(k)$ be the representation $\overline {\psi }= \varphi \circ \overline {\rho }$ , where $\varphi : G\rightarrow G/G'$ is the quotient map. Let $R^{\square }_{\overline {\psi }}$ be the universal deformation ring of $\overline {\psi }$ .

In a companion paper [Reference Paškūnas and Quast48] we deal with the case, when $G^0$ is a torus. In particular, we show that the irreducible components of $R^{\square }_{\overline {\psi }}$ can be labelled by characters $\chi : (\mu _{p^{\infty }}(E)\otimes M)^{ {\mathrm {Gal}}(E/F)}\rightarrow \mathscr O^{\times }$ .

We assume that $\pi _1(G')$ is étale until the end of this subsection. Let $R^{\square , \chi }_{\overline {\rho }}$ be an irreducible component of $R^{\square }_{\overline {\rho }}$ corresponding to a character $\chi $ under Theorem 1.2.

Theorem 1.3 and a theorem of Grothendieck on factoriality of complete intersections imply:

If a further hypothesis on $\overline {\rho }$ and G is satisfied (for example, if $G/Z(G^0)$ does not have $ {\mathrm {PGL}}_2$ as a factor) then we show in Corollary 15.20 that $R^{\square , \chi }_{\overline {\rho }}$ and $R^{\square , \chi }_{\overline {\rho }}/\varpi $ are regular in codimension $2[F:\mathbb {Q}_p]-1$ and hence Corollary 1.4 holds if $[F:\mathbb {Q}_p]=2$ . If $F=\mathbb {Q}_p$ then [Reference Böckle, Iyengar and Paškūnas6, Remark 4.24, Corollary 4.25] provide examples, where Corollary 1.4 fails.

1.2 Deformation problems with fixed partial determinant

In fact, Theorems 1.1, 1.2, 1.3 and Corollary 1.4 hold in a more general setting, which might be called deformation problems with ‘fixed partial determinant’. Assume that we have a $ {\mathrm {Gal}}(E/F)$ -invariant decomposition $X^*(G^0/G')=M_1\oplus M_2$ and let $H_1$ be the quotient of $G/G'$ such that their component groups coincide and the character lattice of $H_1^0$ is equal to $M_1$ . We fix a continuous representation $\psi _1: \Gamma _F \rightarrow H_1(\mathscr O)$ , such that $\psi _1 \equiv \varphi _1\circ \overline {\rho } \pmod {\varpi }$ , where $\varphi _1: G\rightarrow H_1$ is the composition of $\varphi $ with the quotient map $G/G'\twoheadrightarrow H_1$ . Let $D^{\square , \psi _1}_{\overline {\rho }}$ be a subfunctor of $D^{\square }_{\overline {\rho }}$ , such that

$$ \begin{align*}D^{\square, \psi_1}_{\overline{\rho}}(A)=\{ \rho_{A}\in D^{\square}_{\overline{\rho}}(A): \varphi_1\circ \rho_{A}= \psi_1\otimes_{\mathscr{O}} A\}.\end{align*} $$

The functor $D^{\square , \psi _1}_{\overline {\rho }}$ is pro-represented by a quotient $R^{\square , \psi _1}_{\overline {\rho }}$ of $R^{\square }_{\overline {\rho }}$ . Then the above results hold with $R^{\square }_{\overline {\rho }}$ replaced with $R^{\square , \psi _1}_{\overline {\rho }}$ , see Corollary 15.14, Theorems 15.18, 15.19 and their Corollaries.

If $\pi _1(G')$ is étale then the irreducible components of $R^{\square , \psi _1}_{\overline {\rho }}$ can be indexed by characters ${\chi : (\mu _{p^{\infty }}(E)\otimes M_2)^{ {\mathrm {Gal}}(E/F)} \rightarrow \mathscr O^{\times }}$ . There are always two interesting cases: if $M_1=0$ then we impose an empty condition and $R^{\square , \psi _1}_{\overline {\rho }}=R^{\square }_{\overline {\rho }}$ ; if $M_1=X^*(G^0/G')$ then $R^{\square , \psi _1}_{\overline {\rho }}$ parameterises deformations with ‘fixed determinant’ equal to $\psi _1$ and it follows from above that $R^{\square , \psi _1}_{\overline {\rho }}$ and $R^{\square , \psi _1}_{\overline {\rho }}/\varpi $ are integral domains. If $G= {\mathrm {GL}}_d$ then $X^*(G^0/G')=\mathbb Z$ and these are the only cases that can occur. However, in general there are further interesting cases naturally appearing in the Langlands program, as we discuss in Section 16.

If H is a connected reductive group over F (or more generally over a number field) which splits over a finite Galois extension E of F then it is expected in [Reference Buzzard and Gee13] that the Galois representations attached to C-algebraic automorphic forms on H take values in the C-group ${}^C H$ , which is a generalised reductive group scheme over $\mathscr O$ with component group the constant group scheme $ {\mathrm {Gal}}(E/F)$ . Moreover, they should satisfy the condition $d\circ \rho = \chi _{\mathrm {cyc}}$ , where $\chi _{\mathrm {cyc}}$ is the p-adic cyclotomic character and d is the canonical map $d:{}^C H\rightarrow \mathbb G_{m}$ . We explain in Theorem 16.5 that the functor $D^{\square ,\psi _1}_{\overline {\rho }}$ with $G={}^C H$ , $H_1= {\mathrm {Gal}}(E/F)\times \mathbb G_{m}$ and $\psi _1: \Gamma _F \rightarrow H_1(\mathscr O)$ , $\gamma \mapsto (\gamma |_E, \chi _{\mathrm {cyc}}(\gamma ))$ parameterises deformations $\rho _{A}$ of $\overline {\rho }: \Gamma _F \rightarrow {}^CH(k)$ satisfying $d\circ \rho _{A} = \chi _{\mathrm {cyc}}\otimes _{\mathscr {O}} A$ . We show that in this case, under the assumption that $\pi _1(\widehat {H}')$ is étale, the irreducible components are canonically in bijection with characters of p-power torsion subgroup of $Z(H)^0(F)$ , where $Z(H)^0$ is the neutral component of the centre of H.

1.3 Deformation space of Lafforgue’s pseudocharacters

We refer the reader to Section 7 for the definition of V. Lafforgue’s G-pseudocharacters. We single out some properties for the purpose of this introduction. To every representation $\rho : \Gamma _F \rightarrow G(A)$ one may attach an A-valued G-pseudocharacter $\Theta _{\rho }$ . Moreover, conjugation of $\rho $ with elements of $G^0(A)$ does not change the associated G-pseudocharacter. If $\Theta $ is a G-pseudocharacter with values in an algebraically closed field $\kappa $ then there exists a representation $\rho : \Gamma _F \rightarrow G(\kappa )$ , uniquely determined up to $G^0(\kappa )$ -conjugation, such that $\rho $ is G-semisimple (Definition 2.22) and $\Theta _{\rho }=\Theta $ . If A and $\kappa $ carry a topology then there is a notion of continuous G-pseudocharacters, which interacts well with continuous representations.

Let $\overline {\Theta }$ be the G-pseudocharacter attached to $\overline {\rho }$ . JQ has shown in [Reference Quast50] as part of his PhD thesis under the direction of Gebhard Böckle that the functor $D^{\mathrm {ps}}:{\mathfrak {A}}_{\mathscr {O}}\rightarrow \text {Set}$ , such that $D^{\mathrm {ps}}(A)$ is the set of continuous A-valued G-pseudocharacters deforming $\overline {\Theta }$ is pro-represented by a complete local noetherian $\mathscr O$ -algebra $R^{\mathrm {ps}}_G$ with residue field k. The proof that $R^{\mathrm {ps}}_G$ is noetherian uses the fact that $\Gamma _F$ is topologically finitely generated.

If $G= {\mathrm {GL}}_d$ then Emerson–Morel have shown in [Reference Emerson and Morel25] that there is a natural bijection between G-pseudocharacters and Chenevier’s d-dimensional determinant laws defined in [Reference Chenevier15]. Thus in this case the ring $R^{\mathrm {ps}}_{ {\mathrm {GL}}_d}$ coincides with the deformation ring studied by Böckle–Juschka in [Reference Böckle and Juschka7]. The results of that paper are an important input in [Reference Böckle, Iyengar and Paškūnas6]. In our paper we extend the main result [Reference Böckle and Juschka7, Theorem 5.5.1] to G-pseudocharacters, also obtaining a new proof of it, when $G= {\mathrm {GL}}_d$ .

Let $U^{\mathrm {ps}}:=( {\mathrm {Spec}} R^{\mathrm {ps}}_G)\setminus \{\text {the closed point}\}$ . The non-special absolutely irreducible locus $U^{ {\mathrm {n-spcl}}}$ is an open subscheme of $U^{\mathrm {ps}}$ such that a geometric point $x: {\mathrm {Spec}} \kappa \rightarrow U^{\mathrm {ps}}$ lies in $U^{ {\mathrm {n-spcl}}}$ if and only if the representation $\rho _x: \Gamma _F \rightarrow G(\kappa )$ associated to the specialisation of the universal G-pseudocharacter at x is G-irreducible (Definition 2.26) and $H^0(\Gamma _F, ( {\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})_{x}^*(1))=0$ , where $\Gamma _F$ acts on $( {\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})_x$ via $\rho _x$ composed with the adjoint action. If $\pi _1(G')$ is étale or $\mathrm {char}(\kappa )=0$ then $( {\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})_x=( {\mathrm {Lie}} G')_x$ and the last condition is equivalent to $H^0(\Gamma _F, ( {\mathrm {ad}}^0 \rho _x)^*(1))=0$ .

We also show in Corollary 15.25 that the reduced subscheme $({\mathrm {Spec}} R^{\mathrm {ps}}_G[1/p])^{\mathrm {red}}$ is normal. If $\pi _1(G')$ is étale then we compute the set of irreducible components of ${\mathrm {Spec}} R^{\mathrm {ps}}_G$ in Corollary 15.27, which together with Theorem 1.2 implies that ${\mathrm {Spec}} R^{\square }_{\overline {\rho }}\rightarrow {\mathrm {Spec}} R^{\mathrm {ps}}_G$ induces a bijection between the sets of irreducible components.

1.4 Complete intersection

We will now sketch the proof of Theorem 1.1. Let ${\mathrm {Rep}}^{\Gamma _F}_G: \mathscr O\text {-}\mathrm {alg} \rightarrow \text {Set}$ be the functor, which maps an $\mathscr O$ -algebra A to the set of group homomorphisms $\Gamma _F\rightarrow G(A)$ . The scheme $G^0$ acts on ${\mathrm {Rep}}^{\Gamma _F}_G$ by conjugation. We construct an affine scheme $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}={\mathrm {Spec}} A^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ over $\mathscr O$ , which is a $G^0$ -invariant subfunctor of ${\mathrm {Rep}}^{\Gamma _F}_G$ and the following desiderata hold:

1. (D1) the specialisation of the universal representation $\rho ^{\mathrm {univ}}: \Gamma _F \rightarrow G(A^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}})$ at $x\in X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}(\overline {k})$ induces a bijection $x\mapsto \rho _x$ between $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}(\overline {k})$ and the set of continuous representations $\rho : \Gamma _F \rightarrow G(\overline {k})$ such that the G-semisimplification (Definition 2.21) of $\rho $ is equal to $\overline {\rho }^{\mathrm {ss}}$ ;

2. (D2) the completion of a local ring at $x\in X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}(\overline {k})$ is isomorphic to $R^{\square }_{\rho _x}$ ;

3. (D3) the GIT quotient is the spectrum of a complete local noetherian $\mathscr O$ -algebra $R^{\mathrm {git}}_{G,\overline {\rho }^{\mathrm {ss}}}$ with residue field k.

We then bound the dimension of the special fibre $\overline {X}^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ from above

(2) $$ \begin{align} \dim \overline{X}^{\mathrm{gen}}_{G, \overline{\rho}^{\mathrm{ss}}}\le \dim G_k([F:\mathbb {Q}_p]+1). \end{align} $$

Once we can do this the first part of Theorem 1.1 follows immediately as (D2) implies that ${\dim R^{\square }_{\overline {\rho }}/\varpi \le \dim G_k([F:\mathbb {Q}_p]+1)}$ and the result follows from (1) and a little commutative algebra (Lemma 3.9). So the overall strategy is similar to the strategy in [Reference Böckle, Iyengar and Paškūnas6], however its execution requires substantially new ideas.

1.4.1 Definition

The difficulty in defining $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ stems from the fact that we would like to work with continuous representations of $\Gamma _F$ , but for arbitrary A there seems to be no sensible topology to put on $G(A)$ which allows us to express the desired continuity condition by simply asking the map $\Gamma _F \to G(A)$ to be continuous. The solution to this problem is inspired by the analogous definition by Fargues–Scholze [Reference Fargues and Scholze28] in the $\ell \neq p$ case, which uses condensed mathematics.

One may define $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ as a functor from $R^{\mathrm {ps}}_G\text {-}\mathrm {alg}\rightarrow \text {Set}$ such that $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}(A)$ is the set of representations $\rho : \Gamma _F \rightarrow G(A)$ , such that $\Theta _{\rho }$ is the specialisation of the universal G-pseudocharacter $\Theta ^u$ along $R^{\mathrm {ps}}_G\rightarrow A$ and $\rho $ extends to a homomorphism of condensed groups

$$ \begin{align*}\tilde{\rho}: \underline{\Gamma_F} \rightarrow G(A_{\operatorname{disc}}\otimes_{(R^{\mathrm{ps}}_G)_{\operatorname{disc}}} \underline{R^{\mathrm{ps}}_G}).\end{align*} $$

We refer the reader to Section 4.2 for the notation. In Lemma 4.8 we show that the condensed condition can be reformulated as follows: for some (equivalently any) closed immersion $\tau : G\hookrightarrow \mathbb A^n$ the image $\tau (\rho (\Gamma _F))$ is contained in a finitely generated $R^{\mathrm {ps}}_G$ -submodule $M \subseteq A^n = \mathbb A^n(A)$ such that the map $\tau \circ \rho : \Gamma _F \rightarrow M$ is continuous for the canonical topology on M as an $R^{\mathrm {ps}}_G$ -module. We call such representations $R^{\mathrm {ps}}_G$ -condensed. This definition of $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ is aesthetically pleasing and helpful to prove that $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ is functorial in G. However, it is not immediately clear (at least to the authors) from the definition that the functor is representable by a finite type $R^{\mathrm {ps}}_G$ -algebra.

If $G={\mathrm {GL}}_d$ then one may linearise representations and work instead with Cayley–Hamilton homomorphisms of $R^{\mathrm {ps}}_{{\mathrm {GL}}_d}$ -algebras $E^{u}\rightarrow M_d(A)$ , where $E^u$ is a universal Cayley–Hamilton quotient of the completed group algebra . We refer the reader to Section 5.1 for more details. Wang-Erickson has shown in [Reference Wang-Erickson62] that the algebra $E^u$ is a finitely generated $R^{\mathrm {ps}}_{{\mathrm {GL}}_d}$ -module. This result is the reason why $R^{\mathrm {ps}}_{{\mathrm {GL}}_d}$ -condensed representations do not explicitly appear in [Reference Böckle, Iyengar and Paškūnas6]. In Lemma 5.2 we show that $X^{\mathrm {gen}}_{{\mathrm {GL}}_d, \overline {\rho }^{\mathrm {ss}}}$ coincides with the scheme $X^{\mathrm {gen}}$ defined in [Reference Böckle, Iyengar and Paškūnas6]. In particular, it is represented by a finite type $R^{\mathrm {ps}}_{{\mathrm {GL}}_d}$ -algebra $A^{\mathrm {gen}}_{{\mathrm {GL}}_d}$ .

In the general case, we fix a closed immersion $\tau : G \hookrightarrow {\mathrm {GL}}_d$ of $\mathscr O$ -group schemes and define $X^{\mathrm {gen}, \tau }_G$ as a closed subscheme of $X^{\mathrm {gen}}_{{\mathrm {GL}}_d}:=X^{\mathrm {gen}}_{{\mathrm {GL}}_d, (\tau \circ \overline {\rho })^{\mathrm {ss}}}$ , such that $\rho \in X^{\mathrm {gen}}_{{\mathrm {GL}}_d}(A)$ lies in $X^{\mathrm {gen}, \tau }_G(A)$ if and only if $\rho (\Gamma _F)$ is contained in $\tau (G(A))$ . The functor $X^{\mathrm {gen}, \tau }_G$ is representable by a finite type $R^{\mathrm {ps}}_{{\mathrm {GL}}_d}$ -algebra $A^{\mathrm {gen}, \tau }_G$ . We show in Proposition 8.3 that the connected component of $X^{\mathrm {gen},\tau }_G$ containing the point corresponding to $\overline {\rho }:\Gamma _F \rightarrow G(k)$ is canonically isomorphic to the functor $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ defined via condensed mathematics as above. In particular, the connected component is independent of the chosen embedding $\tau $ . Using the construction via $X^{\mathrm {gen}, \tau }_G$ we show that desiderata (D1), (D2), (D3) hold. The condensed description allows to show in Proposition 8.5 that $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ is functorial in G.

A key input in the above argument is the following fundamental finiteness theorem proved by Vinberg [Reference Vinberg61], when S is a spectrum of a field of characteristic zero, by Martin [Reference Martin42], when S is a spectrum of a field of characteristic p, and by Cotner [Reference Cotner21] in general.

Theorem 1.7 (Cotner, [Reference Cotner21]).

Let S be a locally noetherian scheme and let $f: H\rightarrow G$ be a finite morphism of generalised reductive smooth affine S-group schemes. Then the induced map on the GIT quotients

is finite, where the action on both sides is given by conjugation.

The homomorphism $R^{\mathrm {ps}}_G \rightarrow A^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ is $G^0$ -equivariant for the trivial action on the source. Hence, its image is contained in $G^0$ -invariants $R^{\mathrm {git}}_{G, \overline {\rho }^{\mathrm {ss}}}$ , which represents the GIT quotient .

Proposition 1.8 (Proposition 7.4).

The morphism is a finite universal homeomorphism.

We expect that the morphism is adequate in the sense of Alper [Reference Alper1], that is, moreover ${R^{\mathrm {ps}}_G[1/p] \cong R^{\mathrm {git}}_{G, \overline {\rho }^{\mathrm {ss}}}[1/p]}$ , and hope to return to this question in future workFootnote ¹ . If this were true then one could show that ${\mathrm {Spec}} R^{\mathrm {ps}}_G[1/p]$ is reduced and hence normal, as explained in Section 1.3.

We define $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ for any profinite group $\Gamma $ satisfying Mazur’s p-finiteness condition. However, starting with Section 12 we use in an essential way that $\Gamma =\Gamma _F$ : besides Euler–Poincaré characteristic formula, it is crucial to our argument that using local Tate duality we may transfer obstructions to lifting from $H^2$ to $H^0$ .

1.4.2 Bounding the dimension

We will now sketch how we obtain the bound (2). Let $Y_{\overline {\rho }^{\mathrm {ss}}}$ be the preimage of the closed point of $X^{\mathrm {ps}}_G:= {\mathrm {Spec}} R^{\mathrm {ps}}_G$ in $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ . We bound the dimension of $Y_{\overline {\rho }^{\mathrm {ss}}}$ in Corollary 12.18. Its complement V and the special fibre $\overline {{V}}$ of V are Jacobson schemes and residue fields of closed points $x\in V$ are local fields. Moreover, in Lemma 5.17 we relate the completions of local rings $\mathscr O_{V, x}$ to the deformation rings $R^{\square }_{G, \rho _x}$ , and completions of local rings $\mathscr O_{\overline {{V}}, x}$ to $R^{\square }_{G, \rho _x}/\varpi $ . The upshot is that if we control the dimension of $R^{\square }_{G, \rho _x}/\varpi $ then we control the dimension of $\mathscr O_{\overline {{V}},x}$ and if we can do this at every closed point then we can bound the dimension of $\overline {{V}}$ and the dimension of its closure in $\overline {X}^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ . In Proposition 3.6 we establish a presentation

(3)

where $s=\dim _{\kappa (x)} H^2(\Gamma _F, {\mathrm {ad}}^0 \rho _x)$ and $r-s= \dim G^{\prime }_k ([F:\mathbb {Q}_p]+1)$ . We deduce from the results of [Reference Paškūnas and Quast48] that if $x\in \overline {{V}}$ is a closed point then $R^{\square }_{G/G', \varphi \circ \rho _x}/\varpi $ is formally smooth over the group algebra $\kappa (x)[\mu ]$ of dimension $\dim (G/G')_k ([F:\mathbb {Q}_p]+1)$ , where $\mu =(\mu _{p^{\infty }}(E)\otimes M)^{{\mathrm {Gal}}(E/F)}$ . This is easy if $G={\mathrm {GL}}_d$ or more generally if G is connected, but requires a non-trivial argument in the non-connected case, which we carry out in [Reference Paškūnas and Quast48]. This allows us to conclude that the closure in $\overline {X}^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ of the locus in $\overline {{V}}$ , where $H^2(\Gamma _F, {\mathrm {ad}}^0\rho _x)$ vanishes, has dimension $\dim G_k([F:\mathbb {Q}_p]+1)$ , if non-empty. This locus will be referred to as the $({\mathrm {Lie}} G^{\prime }_k)^*$ -non-special locus later on.

One is left to understand the locus, where $H^2(\Gamma _F, {\mathrm {ad}}^0\rho _x)\neq 0$ . By local Tate duality this is equivalent to $H^0(\Gamma _F, ({\mathrm {ad}}^0 \rho _x)^*(1))\neq 0$ . Let us suppose that $\rho _x$ is absolutely G-irreducible. If $G={\mathrm {GL}}_d$ then Böckle–Juschka show in [Reference Böckle and Juschka7] that non-vanishing of $H^0(\Gamma _F, ({\mathrm {ad}}^0 \rho _x)^*(1))$ implies that $\rho _x$ is induced from $\Gamma _{F'}$ , where $F'$ is a proper abelian extension of F. We know the dimension of $\overline {X}^{\mathrm {ps}, \Gamma _{F'}}_{{\mathrm {GL}}_{d'}}$ , where $d'=\frac {d}{[F':F]}$ , inductively and using this it is shown in [Reference Böckle, Iyengar and Paškūnas6] that the closure of this locus has positive codimension. Such arguments might work for classical groups (JQ has studied the case $G={\mathrm {Sp}}_{2n}$ under the assumption $p>2$ in [Reference Quast50, Section 7]), but it seems impossible to get a characterisation like this for an arbitrary generalised reductive group.

Our key observation is that instead of having such a precise description of $\rho _x$ , it is enough to observe that $\rho _x$ has ‘small’ image. Namely, let H be the Zariski closure of $\rho _x(\Gamma _F)$ in $G(\kappa )$ , where $\kappa $ is the algebraic closure of $\kappa (x)$ . Since $\rho _x$ is absolutely G-irreducible, its image is not contained in any parabolic subgroup of $G_{\kappa }$ . A theorem of Martin [Reference Martin42] asserts that H is again generalised reductive. In Section 10 we prove two results:

Proposition 1.9 (Corollary 10.2).

If a generalised reductive subgroup H of $G_{\kappa }$ does not contain $G^{\prime }_{\kappa }$ then $\dim G_{\kappa } -\dim H\ge 2$ .

Our original proof of the above result used Matsushima’s theorem. The current simpler proof was suggested by both Sean Cotner and the anonymous referee.

Lemma 1.10 (Lemma 10.5).

The $G^{\prime }_{\kappa }$ -invariants in $({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^*_{\kappa }$ are zero.

We note that the above Lemma would be false in small characteristics if we did not take the dual. We first prove it for ${\mathrm {SL}}_2$ and then use this together with arguments with roots to prove it in the general case.

Let us go back to the situation at hand, where H is the Zariski closure of $\rho _x(\Gamma _F)$ in $G(\kappa )$ . If x lies in the special fibre, then the twisting operation by the cyclotomic character becomes trivial after the restriction to $\Gamma _{F(\zeta _p)}$ . Using this we show that non-vanishing of $H^0(\Gamma _F, ({\mathrm {ad}} \rho _x)^*(1))$ implies that the neutral component $H^0$ of H has non-zero invariants in $({\mathrm {Lie}} G')^*_{\kappa }$ . If $\pi _1(G')$ is étale or $\mathrm {char}(\kappa )=0$ then $({\mathrm {Lie}} G')^*_{\kappa } =({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^*_{\kappa }$ and Proposition 1.9 and Lemma 1.10 imply that $\dim G_{\kappa }-\dim H^0\ge 2$ and hence $\dim G_{\kappa }- \dim H \ge 2$ . The fact that the difference can be bounded below by $2$ (and not by $1$ ) plays an important role in verifying Serre’s criterion for normality in the proof of Theorem 1.3. As you can see there is an induction argument shaping up. To explain it we need two further ingredients.

Let W be an algebraic representation of G on a finite free $\mathscr O$ -module, and let $\rho : \Gamma _F \rightarrow G(A)$ be a representation, where A is a noetherian $\mathscr O$ -algebra. We show in Lemma 13.8 that there is a closed subscheme $X_{W,j}$ of $X={\mathrm {Spec}} A$ such that

$$ \begin{align*}x\in X_{W,j} \iff \dim_{\kappa(x)} H^0(\Gamma_F, W\otimes_{\mathscr{O}} \kappa(x)(1))\ge j+1.\end{align*} $$

We refer to $X_{W,j}$ as the W-special locus of level j and just the W-special locus when $j=0$ . The complement of the W-special locus is called W-non-special. The proof amounts to reformulating these conditions in terms of some finitely generated A-module and a little of commutative algebra. We apply this notion to $W=({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^*$ and if the image of $\rho $ is contained in $P(A)$ , where P is a parabolic subgroup scheme of G with unipotent radical U, then we apply it to $W=({\mathrm {Lie}} U)^*$ . When working with the special fibre we consider representations W over k instead of $\mathscr O$ .

The second ingredient, which uses Theorem 1.7 as an input, is the following finiteness result proved in Proposition 8.6. If $H\rightarrow G_k$ is a finite morphism of generalised reductive k-group schemes which maps an H-pseudocharacter $\overline {\Psi }$ to $\overline {\Theta }$ then the natural map $\overline {X}^{\mathrm {ps}}_{H, \overline {\Psi }} \rightarrow \overline {X}^{\mathrm {ps}}_G$ is finite. Moreover, there are only finitely many H-pseudocharacters $\overline {\Psi }$ mapping to $\overline {\Theta }$ . In particular, the dimension of the union $\overline {X}^{\mathrm {ps}}_{HG}$ of the scheme theoretic images can be bounded by the largest $\dim \overline {X}^{\mathrm {ps}}_{H,\overline {\Psi }}$ .

We are now in a position to sketch the proof of the bound in (2) when $\overline {\rho }$ is absolutely G-irreducible. From (3) and the relation between $R^{\square }_{G, \rho _x}/\varpi $ and the completions of $\mathscr O_{\overline {{V}},x}$ we obtain a lower bound

(4) $$ \begin{align} \dim \overline{X}^{\mathrm{gen}}_{G, \overline{\rho}^{\mathrm{ss}}}\ge \dim G_k ([F:\mathbb {Q}_p]+1). \end{align} $$

We then construct a pair $(G_1, \overline {\rho }_1)$ such that $\dim G_1\le \dim G$ , $\pi _1(G^{\prime }_1)$ is étale and if (2) holds for $(G_1, \overline {\rho }_1)$ then it also holds for $(G, \overline {\rho })$ . For example, if $G={\mathrm {PGL}}_d$ then $G_1= {\mathrm {SL}}_d/\mu _m$ , where $d=p^e m$ and $p \nmid m$ , and $\overline {\rho }_1=\overline {\rho }$ as $G(k)=G_1(k)$ . In general, the presence of the component group of G complicates the construction of $G_1$ , see Section 2.3. This reduction step allows us to assume that $\pi _1(G')$ is étale, which implies that ${\mathrm {Lie}} G^{\prime }_{\operatorname {sc}}={\mathrm {Lie}} G'$ . We then show that there are finitely many generalised reductive subgroup schemes $H_i$ of $G_k$ , defined over some finite extension of k, such that $\dim G_k - \dim H_i \ge 2$ and the $({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^*_k$ -special locus in $\overline {X}^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ is contained in the preimage of the union of $\overline {X}^{\mathrm {ps}}_{H_i G}$ . The assumption that $\overline {\rho }$ is absolutely G-irreducible implies that every fibre $X^{\mathrm {gen}}_{G,\overline {\rho }^{\mathrm {ss}}}\rightarrow X^{\mathrm {ps}}_G$ has dimension $\dim G_k - \dim Z(G)_k$ . Inductively, we may bound the dimension of $\overline {X}^{\mathrm {ps}}_{H_iG}$ by $\dim H_i [F:\mathbb {Q}_p] +\dim Z(H_i)$ . Using this we show that the codimension of the $({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^*_k$ -special locus in $\overline {X}^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ is at least $2[F:\mathbb {Q}_p]$ . It follows from (4) that the $({\mathrm {Lie}} G')^*_k$ -non-special locus in $\overline {X}^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ is non-empty and as explained above its closure has dimension $\dim G_k ([F:\mathbb {Q}_p]+1)$ . This lets us conclude that (4) is an equality.

Let us briefly indicate how we find the subgroup schemes $H_i$ in the above argument. The $({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^*_k$ -special locus of level j is non-empty only for finitely many j. The group schemes $H_i$ correspond to the generic points of these loci for $j\ge 0$ . Given such a generic point $\eta $ we let H be the Zariski closure of $\rho _{\eta }(\Gamma _F)$ in $G(\overline {\kappa (\eta )})$ . Since $\overline {\rho }^{\mathrm {ss}}$ is assumed to be absolutely G-irreducible H is not contained in any parabolic subgroup of $G_{\overline {\kappa (\eta )}}$ and hence is generalised reductive. A conjugate of H by an element of $G(\overline {\kappa (\eta )})$ can be defined over $\overline {k}$ and hence over a finite extension of k. Our arguments with generalised reductive groups over algebraically closed fields use results of Martin [Reference Martin42] and Bate–Martin–Röhrle [Reference Bate, Martin and Röhrle3] in an essential way.

The general case is an elaboration of this inductive argument. If $\overline {\rho }^{\mathrm {ss}}$ is not absolutely G-irreducible then the dimensions of the fibres $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}\rightarrow X^{\mathrm {ps}}_G$ can vary. We bound their dimensions in Section 12. We stratify $\overline {X}^{\mathrm {ps}}_G$ by $\overline {X}^{\mathrm {ps}}_{LG}$ , where L runs over the Levi subgroup schemes of G containing a fixed maximal split torus. If $y\in \overline {X}^{\mathrm {ps}}_{LG}$ , but does not lie in any $\overline {X}^{\mathrm {ps}}_{L'G}$ for a proper Levi $L'\subsetneq L$ then for typical y the fibre will have dimension $\dim G_k - \dim Z(L)_k +\dim U_k [F:\mathbb {Q}_p]$ , where U is the unipotent radical of any parabolic subgroup with Levi L. By induction we know that $\dim \overline {X}^{\mathrm {ps}}_{LG}$ is at most $\dim L_k [F:\mathbb {Q}_p]+ \dim Z(L)_k$ . The sum of these numbers does not exceed $\dim G_k ([F:\mathbb {Q}_p]+1)$ . In fact, the difference is equal to $\dim U_k [F:\mathbb {Q}_p]$ . We deal with the fibres, which have dimension exceeding $\dim G_k - \dim Z(L)_k +\dim U_k [F:\mathbb {Q}_p]$ by considering the $({\mathrm {Lie}} U_k)^{\ast }$ -special locus inside $X^{\mathrm {ps}}_L$ . The inductive argument is carried out in Section 13.

If $X^{\mathrm {ps}}_{LG}$ is non-empty for a Levi subgroup L satisfying $\dim G -\dim L=2$ then the bound ${\dim U_k [F:\mathbb {Q}_p]}$ is not good enough to apply the Serre’s criterion for normality, when $F=\mathbb {Q}_p$ , as the assumption on L forces $\dim U_k = 1$ . We analyse the $({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^{\ast }$ -non-special locus in the preimage of $X^{\mathrm {ps}}_{LG}$ in $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ in this case in detail in Section 11. In Section 14 we show that the $({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^{\ast }$ -special locus in $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ and also in $\overline {X}^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ has codimension of at least $1+[F:\mathbb {Q}_p]$ and of at least $2[F:\mathbb {Q}_p]$ if $X^{\mathrm {ps}}_{LG}$ is empty for all Levi subgroups L of codimension $2$ .

1.5 Irreducible components

We will sketch the proof of Theorem 1.3. Using the results of [Reference Paškūnas and Quast48] we show that

(5)

where $l= \dim (G/G')_k - \dim Z(G/G')_k$ , $m+l= \dim (G/G')_k ([F:\mathbb {Q}_p]+1)$ and $\mu ={(\mu _{p^{\infty }}(E)\otimes M)^{{\mathrm {Gal}}(E/F)}}$ . Let $\mathrm {X}(\mu )$ be the group of characters $\chi : \mu \rightarrow \mathscr O^{\times }$ . If A is an $A^{\mathrm {gen}}_{G/G', \overline {\psi }^{\mathrm {ss}}}$ -algebra then we use (5) to define $A^{\chi }:= A\otimes _{\mathscr O[\mu ], \chi } \mathscr O$ and if $X={\mathrm {Spec}} A$ we write $X^{\chi }:={\mathrm {Spec}} A^{\chi }$ . It follows from (5) that the irreducible components of $X^{\mathrm {gen}}_{G/G', \overline {\psi }^{\mathrm {ss}}}$ are given by $X^{\mathrm {gen}, \chi }_{G/G', \overline {\psi }^{\mathrm {ss}}}$ for $\chi \in \mathrm X(\mu )$ and are regular. In particular, the completions of local rings of $X^{\mathrm {gen}, \chi }_{G/G', \overline {\psi }^{\mathrm {ss}}}$ are also regular. Thus the presentation (3) gives us a presentation

(6)

where $R^{\square ,\chi }_{G/G', \varphi \circ \rho _x}$ is a regular ring. Further, if x is in the $({\mathrm {Lie}} G')^{\ast }$ -non-special locus then s in the above presentation is $0$ and we can conclude that $R^{\square ,\chi }_{G, \rho _x}$ is a regular ring. We deduce that the $({\mathrm {Lie}} G')^{\ast }$ -non-special locus in $X^{\mathrm {gen},\chi }_{G, \overline {\rho }^{\mathrm {ss}}}$ is contained in the regular locus. In general, the $({\mathrm {Lie}} G')^{\ast }$ -non-special locus might be empty, for example this happens if $G={\mathrm {PGL}}_p$ . On the other hand the complement of $({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^{\ast }$ -non-special locus has codimension of at least $1+[F:\mathbb {Q}_p]$ in $X^{\mathrm {gen}, \chi }_{G, \overline {\rho }^{\mathrm {ss}}}$ . If $\pi _1(G')$ is étale then these loci coincide and this implies that $X^{\mathrm {gen},\chi }_{G, \overline {\rho }^{\mathrm {ss}}}$ is regular in codimension $[F:\mathbb {Q}_p]$ . Since $X^{\mathrm {gen},\chi }_{G, \overline {\rho }^{\mathrm {ss}}}$ is excellent the same applies to the completions of its local rings, and we may deduce that the deformation rings $R^{\square ,\chi }_{G, \rho _x}$ are regular in codimension $[F:\mathbb {Q}_p]$ . To show that they are complete intersection it is enough to bound their dimension from above and then use (6). Since $k[\mu ]$ is a local artinian algebra the underlying reduced subschemes of the special fibres of $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ and $X^{\mathrm {gen}, \chi }_{G, \overline {\rho }^{\mathrm {ss}}}$ coincide and hence they have the same dimension. Serre’s criterion for normality implies that $R^{\square ,\chi }_{G, \rho _x}$ is normal. Since the ring is local, it is a domain.

Since $({\mathrm {Lie}} G')_L= ({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})_L$ the $({\mathrm {Lie}} G')^{\ast }$ -non-special and $({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^{\ast }$ -non-special loci in the generic fibre coincide. This allows us to prove that $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}[1/p]$ is normal without any assumption on $\pi _1(G')$ .

A similar argument works in the ‘fixed partial determinant’ setting described in Section 1.2. However, there is one new idea not present in [Reference Böckle, Iyengar and Paškūnas6]. To bound the dimension of $X^{\mathrm {gen}, \psi _1}_{G, \overline {\rho }^{\mathrm {ss}}}$ we show that there is a finite morphism $X^{\mathrm {gen}, \psi _1}_{G, \overline {\rho }^{\mathrm {ss}}}\rightarrow X^{\mathrm {gen}}_{G/Z_1, \overline {\rho }^{\mathrm {ss}}}$ , where $Z_1$ is a central subtorus of $G^0$ such that the composition $Z_1\rightarrow G \rightarrow H_1$ is an isogeny. We can then bound the dimension of various subschemes of $X^{\mathrm {gen}, \psi _1}_{G, \overline {\rho }^{\mathrm {ss}}}$ by dimensions of the corresponding subschemes in $X^{\mathrm {gen}}_{G/Z_1, \overline {\rho }^{\mathrm {ss}}}$ . So for example, to prove the results in the fixed determinant case, when $G={\mathrm {GL}}_d$ , we use the results proved for $G={\mathrm {PGL}}_d$ . In [Reference Böckle, Iyengar and Paškūnas6] one twists by characters to unfix the determinant instead. This argument can be made to work, when G is connected, but will not work in general.

1.6 Overview by section

In Section 2 we review some facts about the generalised reductive $\mathscr O$ -group schemes and their parabolic subgroup schemes, which we call R-parabolic. We also review G-semisimplification and G-irreducibility. In Section 3 we recall deformation theory of representations of a profinite group satisfying Mazur’s p-finiteness condition. Proposition 3.6 is a key result of that section. In Section 4 we discuss and compare notions of continuity via algebra and via condensed sets. As explained in Section 1.4.1 this is used to define $X^{\mathrm {gen}}_{G,\overline {\rho }^{\mathrm {ss}}}$ later on. In Section 5 we construct the scheme $X^{\mathrm {gen},\tau }_G$ and study the relation between the completion of its local rings and deformation rings of Galois representations. In Section 6 we study the GIT quotient and define $X^{\mathrm {gen},\tau }_{G,\overline {\rho }^{\mathrm {ss}}}$ and its GIT quotient $X^{\mathrm {git}}_{G,\overline {\rho }^{\mathrm {ss}}}$ . In Section 7 we introduce Lafforgue’s G-pseudocharacters and their deformation spaces $X^{\mathrm {ps}}_G$ and show that the natural map $X^{\mathrm {git}}_{G,\overline {\rho }^{\mathrm {ss}}}\rightarrow X^{\mathrm {ps}}_G$ is a finite universal homeomorphism. In Section 8 we give a condensed description of $X^{\mathrm {gen},\tau }_{G,\overline {\rho }^{\mathrm {ss}}}$ as explained in Section 1.4.1. This allows us to conclude that $X^{\mathrm {gen},\tau }_{G,\overline {\rho }^{\mathrm {ss}}}$ is independent of $\tau $ , which we then omit from notation, and is functorial in G. In Section 9 we show that without loss of generality one may assume that the map $\Gamma _F \overset {\overline {\rho }}{\longrightarrow } G(k) \rightarrow (G/G^0)(\overline {k})$ is surjective. In this case we define the absolutely G-irreducible locus in $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}$ and show that if it is non-empty then the difference between its dimension and the dimension of its GIT quotient is $\dim G_k - \dim Z(G)_k$ . In Section 10 we bound the codimension of generalised reductive subgroups of generalised reductive groups over algebraically closed fields. The results of this section are used to bound the dimensions of $({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^{\ast }$ -special and $({\mathrm {Lie}} U)^{\ast }$ -special loci, as explained in Section 1.4.2. In Section 11 we show that if G is a generalised reductive group over an algebraically closed field with an R-Levi L of codimension $2$ such that $L/L^0\rightarrow G/G^0$ is an isomorphism then $G/Z(G^0)\cong G_1\times {\mathrm {PGL}}_2$ and $L/Z(G)\cong G_1\times \mathbb G_{m}$ . Proposition 11.5 is used to bound the dimension of the $({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^{\ast }$ -special locus in Section 14. This section can be omitted if one is only interested in the complete intersection property and does not care for normality and irreducible components. In Section 12 we bound the dimensions of fibres of $X^{\mathrm {gen}}_{G, \overline {\rho }^{\mathrm {ss}}}\rightarrow X^{\mathrm {ps}}_G$ . In Section 13, which is the core of the paper, we carry out the induction argument described in Section 1.4.2. In Section 14 we bound the dimension of the $({\mathrm {Lie}} G^{\prime }_{\operatorname {sc}})^*$ -special locus. In Section 15 we study the irreducible components of $X^{\mathrm {gen},\psi _1}_{G, \overline {\rho }^{\mathrm {ss}}}$ . In Section 16 we explain that our results apply, when G is an L-group or a C-group, for the convenience of the reader. The appendix recalls some background on condensed mathematics, which is used in Section 4.

1.7 Notation

Let F be a finite extension of $\mathbb {Q}_p$ . We fix an algebraic closure $\overline {F}$ over F. Let $\Gamma _F:={\mathrm {Gal}}(\overline {F}/F)$ be the absolute Galois group of F. We will denote by $\zeta _p$ a primitive p-th root of unity in $\overline {F}$ .

Let L be another finite extension of $\mathbb Q_p$ , such that ${\mathrm {Hom}}_{\mathbb {Q}_p\text {-}\mathrm {alg}}(F, L)$ has cardinality $[F:\mathbb {Q}_p]$ . Let $\mathscr O$ be the ring of integers in L, $\varpi $ a uniformiser, and k the residue field. All our schemes are defined over $\mathscr O$ (unless stated otherwise). We omit ${\mathrm {Spec}} \mathscr O$ from the notation, when we take fibre products over it, so that $X\times Y:= X\times _{{\mathrm {Spec}} \mathscr O} Y$ .

If A is a commutative ring and $\rho : \Gamma \rightarrow {\mathrm {GL}}_d(A)$ is a representation, we let

$$ \begin{align*}\rho^{\mathrm{lin}}: A[\Gamma]\rightarrow M_d(A), ~\sum_{\gamma\in \Gamma} a_{\gamma} \gamma \mapsto \sum_{\gamma\in \Gamma} a_{\gamma} \rho(\gamma)\end{align*} $$

be its linearisation, where $M_d(A)$ is the set of $d \times d$ matrices with entries in A.

We denote by $A\text {-}\mathrm {alg}$ the category of commutative A-algebras.