1.1 The new ideas in this paper

When $m = 1$ , Theorems A, B and C were proved in [Reference Allen, Calegari, Caraiani, Gee, Helm, Hung, Newton, Scholze, Taylor and ThorneACC⁺23] (see [Reference Allen, Calegari, Caraiani, Gee, Helm, Hung, Newton, Scholze, Taylor and ThorneACC⁺23, Thm 1.01, Thm 1.0.2, Thm 7.1.14]). The deduction of Theorems A and B from Theorem C exactly parallels the arguments in [Reference Allen, Calegari, Caraiani, Gee, Helm, Hung, Newton, Scholze, Taylor and ThorneACC⁺23], so we now focus on explaining the proof of Theorem C.

Unsurprisingly, our arguments build on those of [Reference Allen, Calegari, Caraiani, Gee, Helm, Hung, Newton, Scholze, Taylor and ThorneACC⁺23]: in particular, we prove the potential automorphy of the compatible system of symmetric powers $\operatorname {\mathrm {Sym}}^{n-1}\mathcal {R} = \{\operatorname {\mathrm {Sym}}^{n-1} \rho _{\lambda }\}$ by checking the residual automorphy over some extension $F' / F$ and then applying an automorphy lifting theorem. We would like to highlight three new ingredients which appear here:

1. 1. A result on generic reducedness of special fibres of weight $0$ (local) crystalline deformation rings (see §1.2 for a further introductory discussion). Using the local-global compatibility result of [Reference Caraiani and NewtonCN23], this leads to a new automorphy lifting theorem in the setting of arbitrary ramification (Theorem 3.2.1).

2. 2. An application of a theorem of Drinfeld and Kedlaya [Reference Drinfeld and KedlayaDK17], showing generic ordinarity of families of Dwork motives (Proposition 4.2.6). This makes it possible to verify the potential residual automorphy of certain residual representations by an automorphic motive which is crystalline ordinary at some set of p-adic places.

3. 3. A ‘p-q-r’ switch including a version of the ‘Harris tensor product trick’ which incorporates an additional congruence between two tensor products of compatible families. One is a tensor product of $\operatorname {\mathrm {Sym}}^{n-1}\mathcal {R}$ with an induction of a character, as usual. The other is a tensor product of $\operatorname {\mathrm {Sym}}^{n-1}\mathcal {R}$ with a different auxiliary compatible family, which gives us more flexibility to realize different local properties at places related by complex conjugation. We discuss in the remainder of the introduction the need for this argument and give a more detailed sketch in §6.2 below.

To explain in more detail the need for these innovations, suppose we are given a compatible system $\mathcal {R} = \{\rho _{\lambda }\}$ as in the statement of Theorem C, therefore of Hodge–Tate weights $\{0, m \}$ for some $m \geq 1$ (and with $m \geq 2$ if we hope to go beyond the cases treated in [Reference Allen, Calegari, Caraiani, Gee, Helm, Hung, Newton, Scholze, Taylor and ThorneACC⁺23]). The general strategy for proving potential automorphy (the so-called ‘p-q switch’) is as follows:

1. 1. After making some CM base extension $H/F$ (depending on n), find an auxiliary n-dimensional compatible system $\mathcal {S} =\{ s_{\lambda }\}$ such that

   1. (a) For one prime $\lambda $ , the residual representations $\operatorname {\mathrm {Sym}}^{n-1} \overline {\rho }_{\lambda } |_{G_H}$ and $\overline {s}_{\lambda }$ coincide and moreover satisfy a number of standard ‘Taylor–Wiles’ conditions.

   2. (b) For a second prime $\lambda '$ , the residual representation $\overline {s}_{\lambda '}$ is induced from a character and is thus residually automorphic.

   3. (c) The Hodge–Tate weights of the compatible system $\mathcal {S}$ coincide with those of $\operatorname {\mathrm {Sym}}^{n-1}\mathcal {R} |_{G_H}$ .

2. 2. Apply an automorphy lifting theorem at $\lambda '$ to deduce that the compatible system $\mathcal {S}$ is automorphic. Then deduce that the residual representation $\overline {s}_{\lambda }$ is automorphic and use automorphy lifting theorems again to deduce that $\operatorname {\mathrm {Sym}}^{n-1}\mathcal {R} |_{G_H}$ is automorphic.

In our setting, both of these steps cause problems, but those affecting the second step are more serious.

The issue in the first step is the requirement (1)(c) on the Hodge–Tate weights. The most natural source of compatible systems $\mathcal {S}$ are those arising from motives, and a geometrically varying family of motives cannot have Hodge–Tate weights $0,m,\ldots ,m(n-1)$ with $m \ge 2$ by Griffiths transversality. (This difficulty is already present if $F={\mathbb Q}$ and one wants to prove the Sato–Tate conjecture for a classical modular form of weight greater than $2$ , such as $\Delta $ .) The now-standard resolution to this problem is to employ the ‘Harris tensor product trick’ [Reference HarrisHar09] and replace $\operatorname {\mathrm {Sym}}^{n-1} \mathcal {R}$ by $\operatorname {\mathrm {Sym}}^{n-1} \mathcal {R} \otimes \operatorname {\mathrm {Ind}}^{G_{F}}_{G_{L}} \mathcal {X}$ for some cyclic CM extension $L/F$ , where $\mathcal {X}$ is a compatible system of algebraic Hecke characters chosen sufficiently carefully so that this new compatible system has consecutive Hodge–Tate weights. Now in the second step, one wants to prove this new compatible system is potentially automorphic (using for $\mathcal {S}$ a compatible system coming from the cohomology of the Dwork family). The potential automorphy of $\operatorname {\mathrm {Sym}}^{n-1}\mathcal {R}$ can then be deduced using cyclic base change [Reference Arthur and ClozelAC89].

For the second step, applying an automorphy lifting theorem typically requires that the compatible systems $\mathcal {S}$ and $\operatorname {\mathrm {Sym}}^{n-1} \mathcal {R} \otimes \operatorname {\mathrm {Ind}}^{G_{F}}_{G_{L}} \mathcal {X}$ have ‘the same’ behaviour at places $v|p$ . There are two problems with this. First, we will need an automorphy lifting theorem that applies to arbitrarily ramified F, including nonordinary representations. Second, the compatible system of characters $\mathcal {X}$ has a restricted form, and in particular, its local behaviour cannot be chosen arbitrarily at a pair of conjugate places. We explain more about these difficulties and their resolution below.

For context, we first recall the situation when $F={\mathbb Q}$ . For example, one might try to demand that the p-adic representations in the compatible systems $\mathcal {S}$ and $\operatorname {\mathrm {Sym}}^{n-1} \mathcal {R} \otimes \operatorname {\mathrm {Ind}}^{G_{F}}_{G_{L}} \mathcal {X}$ are both ordinary, and indeed, it is straightforward (at least after a ramified base change) to find ordinary representations in the Dwork family and presumably difficult to understand the nonordinary representations in any generality. This means that one would like to show that many of the representations in the compatible system $\mathcal {R}$ are ordinary.

For a weight $2$ modular form, it is relatively easy to prove that there are infinitely many primes p for which the p-adic Galois representation is ordinary at p. However, the existence of infinitely many ordinary primes for $\Delta $ (or for any non-CM form of weight $k \ge 4$ ) remains an open question, so one also has to consider the possibility that the residual representation $\overline {r}_{\lambda }|_{G_{F_v}}$ is locally of the form $\omega ^m_2 \oplus \omega ^{mp}_2$ on inertia at p. This problem was resolved for classical modular forms in [Reference Barnet-Lamb, Geraghty, Harris and TaylorBLGHT11], via a further study of the Dwork family; in particular, showing that certain residual representations of the shape $\operatorname {\mathrm {Sym}}^{n-1} (\omega _2 \oplus \omega ^p_2)$ arise (locally on inertia) as residual representations in that family.

We now consider the case of an imaginary CM field F and explain why we need an automorphy lifting theorem allowing ramification at nonordinary places. Given the automorphy lifting theorems for CM fields proved in [Reference Allen, Calegari, Caraiani, Gee, Helm, Hung, Newton, Scholze, Taylor and ThorneACC⁺23], the most serious difficulty in adapting the strategy of [Reference Barnet-Lamb, Geraghty, Harris and TaylorBLGHT11] is that there is no way to avoid the possibility that a representation $\rho _{\lambda }$ can be simultaneously ordinary at one prime $v|p$ and nonordinary at the complex conjugate place $v^c$ . (One might hope to avoid this by considering places with $v=v^c$ , but then we would have to show that certain residual representations of $G_{{\mathbb Q}_{p^2}}$ occur in the Dwork family which seemed to us to be a difficult task.) This is a problem because the automorphy lifting theorems in [Reference Allen, Calegari, Caraiani, Gee, Helm, Hung, Newton, Scholze, Taylor and ThorneACC⁺23] for nonordinary representations require F to be unramified at our nonordinary prime $v^c$ , while at the ordinary prime v, we need to be able to make a highly ramified base change of (imaginary) CM fields $F'/F$ to find an appropriate representation in the Dwork family. It is, however, impossible to arrange that such an extension of CM fields is unramified at $v^c$ and ramified at v. One of the key innovations in this paper is to prove an automorphy lifting theorem that allows us to make a ramified base change at $v^c$ (Theorem 3.2.1). This was done in the two-dimensional case in [Reference Caraiani and NewtonCN23]; we discuss the difficulties in extending this result to higher dimensions and how we overcome them in Section 1.2 below. Note that, even when making a ramified base change, it is still important for us to keep track of the inertial type of residual representations in the Dwork family in order to show that the representations of interest are connected in the deformation space.

There turns out to be one final wrinkle, where the second problem mentioned above arises. The p-adic representations in our compatible system ${\mathcal S}$ will satisfy one of two, mutually exclusive, local conditions at each p-adic place: they are crystalline at p and are either ordinary or are (on the same component of a local crystalline deformation ring as) a symmetric power of an induction of a Lubin–Tate character of $G_{{\mathbb Q}_{p^2}}$ . It turns out that we cannot always arrange for a tensor product $\operatorname {\mathrm {Sym}}^{n-1} \mathcal {R} \otimes \operatorname {\mathrm {Ind}}^{G_F}_{G_L} \mathcal {X}$ to have local p-adic representations of this shape. The problem is that algebraic Hecke characters have a very restricted form, and the fact that F is an imaginary CM field implies that a suitable choice of $\mathcal {X}$ will exist only if $\rho _{\lambda }$ is either both ordinary or nonordinary at each pair of places $\{v,v^c\}$ permuted by complex conjugation in $\operatorname {\mathrm {Gal}}(F/F^{+})$ .

Our solution is to instead consider tensor products of the form $(\operatorname {\mathrm {Sym}}^{n-1} {\mathcal R})\otimes {\mathcal R}_{\mathrm {aux}}$ , where ${\mathcal R}_{\mathrm {aux}}$ is a compatible system coming from (part of) the cohomology of the Dwork hypersurface [Reference QianQia23]. We will be able to choose ${\mathcal R}_{\mathrm {aux}}$ so that one of the local conditions mentioned in the previous paragraph will be satisfied by ${\mathcal S}_{\mathrm {aux}} = (\operatorname {\mathrm {Sym}}^{n-1} {\mathcal R}) \otimes {\mathcal R}_{\mathrm {aux}}$ at each p-adic place. It is now no longer possible to directly deduce the potential automorphy of $\operatorname {\mathrm {Sym}}^{n-1} \mathcal {R}$ from the potential automorphy of this product. This is not necessary to prove the Ramanujan conjecture – already the automorphy of this tensor product combined with the Jacquet–Shalika bounds (and the fact that $\mathcal {R}_{\mathrm {aux}}$ is pure) is enough to deduce purity – but it is necessary to prove the Sato–Tate conjecture. However, once the potential automorphy of ${\mathcal S}_{\mathrm {aux}}$ is established, we can (having chosen ${\mathcal R}_{\mathrm {aux}}$ carefully to begin with) find a third compatible system ${\mathcal R}_{\mathrm {CM}}$ such that ${\mathcal S}_{\mathrm {aux}} = (\operatorname {\mathrm {Sym}}^{n-1} {\mathcal R}) \otimes {\mathcal R}_{\mathrm {aux}}$ and ${\mathcal S}_{\mathrm {CM}} = (\operatorname {\mathrm {Sym}}^{n-1} {\mathcal R}) \otimes {\mathcal R}_{\mathrm {CM}}$ are residually the same at a third prime r, and ${\mathcal R}_{\mathrm {CM}}$ is induced from a character. Even though we do not have any control over the r-adic representation associated to $\operatorname {\mathrm {Sym}}^{n-1} {\mathcal R}$ locally at $v|r$ , the fact that it occurs as the same tensor factor in the r-adic representations of both ${\mathcal S}_{\mathrm {aux}}$ and ${\mathcal S}_{\mathrm {CM}}$ means we can still put ourselves in a situation where both r-adic Galois representations lie on the same component of a local deformation ring at $v|r$ . From this p-q-r switch, we can show that ${\mathcal S}_{\mathrm {CM}} = (\operatorname {\mathrm {Sym}}^{n-1} {\mathcal R}) \otimes {\mathcal R}_{\mathrm {CM}}$ is potentially automorphic, from which we deduce that $\operatorname {\mathrm {Sym}}^{n-1} {\mathcal R}$ is potentially automorphic.

One might also ask whether for general CM fields F, one can drop the hypothesis that ${\mathcal R}$ has parallel weight. The difficulty in doing so is as follows: in order to pass from $\operatorname {\mathrm {Sym}}^{n-1} \mathcal {R}$ to a compatible system with consecutive Hodge–Tate weights, one needs to tensor this compatible system with a second compatible system with certain prescribed local properties. If ${\mathcal R}$ does not have parallel weight, this auxiliary compatible system cannot have consecutive Hodge–Tate weights and, for reasons explained above, also cannot be induced from a compatible system of characters. It is very hard to construct such compatible systems because of the constraints on families of geometric local systems imposed by Griffiths transversality. The existence of even a single regular algebraic cuspidal automorphic representation for $\operatorname {\mathrm {GL}}_2(\mathbf {A}_F)$ for some CM field F which is neither of parallel weight $2$ , nor of CM type, nor arising from base change from the totally real subfield $F^{+}$ was only found (by a computation) in [Reference Calegari and MazurCM09, Lemma 8.11(2)] (see also [Reference Rahm and S¸engünRS13]).

1.2 Ihara avoidance and the Emerton–Gee stack

There are two main difficulties in proving automorphy lifting liftings for p-adic representations with p ramified in F. One is having local–global compatibility theorems at the places dividing p; this was resolved in the recent work of Caraiani–Newton [Reference Caraiani and NewtonCN23]. The other difficulty was alluded to above: the usual Taylor–Wiles method for automorphy lifting only allows us to deduce the automorphy of a p-adic representation r from the automorphy of a congruent representation $r'$ if we know that for all finite places v, the representations $r|_{G_{F_v}}$ and $r'|_{G_{F_v}}$ are ‘connected’, in the sense that they lie on the same component of the appropriate local deformation ring.

As we have sketched above, in the particular cases that we consider in this paper, we have arranged this property at the places $v|p$ by considering the ordinary and nonordinary cases separately. (It was this construction that required us to pass to a situation where p is highly ramified in F.) We are not, however, able to arrange that our representations are connected at all the places $v\nmid p$ . Fortunately, Taylor [Reference TaylorTay08] found a way to prove automorphy lifting theorems when the representations fail to be connected at some places $v\nmid p$ , using his so-called ‘Ihara avoidance’ argument. This argument makes an ingenious use of two different local deformation problems at places $v\nmid p$ , which are congruent modulo p, and relates two corresponding patched modules of automorphic forms. The key point which makes this argument possible is to work with local deformation rings having the following ‘unique generalization’ property: any generic point of their special fibre has a unique generalization to the generic fibre. More geometrically, we need to avoid having two distinct irreducible components in characteristic zero which specialize to a common irreducible component in the special fibre.

In order to apply this argument, one also needs the unique generalization property for the deformation rings at the places $v|p$ . This was previously only known in the Fontaine–Laffaille and ordinary contexts, in which case the crystalline deformation rings can be understood completely explicitly (and in the former case, there is even a unique irreducible component). (This problem was sidestepped to some extent in [Reference Barnet-Lamb, Gee and GeraghtyBLGG11, Reference Barnet-Lamb, Gee, Geraghty and TaylorBLGGT14], but the approach there combines the Ihara avoidance argument with the Khare–Wintenberger lifting argument to produce characteristic zero lifts of residual representations of the prescribed weight and level. In our $\ell _0> 0$ situation (in the language of [Reference Calegari and GeraghtyCG18]), such lifts do not always exist.)

One way to establish the unique generalization property (when it holds) would be to explicitly compute the irreducible components of the generic fibres of the deformation rings, but this appears to be hopeless for crystalline deformation rings in any generality. However, as was already observed in [Reference TaylorTay08, §3] in the case $v\nmid p$ , an alternative approach is to consider an appropriate moduli stack of Galois representations, for which the deformation rings are versal rings at closed points. One shows that its special fibre is generically reduced (or even generically smooth) – for example, by showing that the deformation rings for generic choices of the residual Galois representation are formally smooth. It then follows that for an arbitrary residual representation, the special fibres of the ( ${\mathbb Z}_p$ -flat quotients of) the deformation rings are generically reduced, which implies the unique generalization property. (While [Reference TaylorTay08, §3] does not explicitly work with moduli stacks of Galois representations, [Reference TaylorTay08, Lem. 3.2] is easily reformulated in these terms, and while [Reference TaylorTay08, Prop. 3.1(3)] does not explicitly state that the ${\mathbb Z}_p$ -flat quotient of the deformation ring has generically reduced special fibre, this follows from the argument, as in [Reference Allen, Newton and ThorneANT20, Prop. 3.1].)

The unique generalization property has subsequently been used by Thorne in a context with $v\nmid p$ in the proof of [Reference ThorneTho12, Thm. 8.6] (in order to avoid any additional hypotheses when introducing an auxiliary prime to make the level structure sufficiently small), and in the case that $v|p$ by Caraiani–Newton [Reference Caraiani and NewtonCN23], who used the results of [Reference Caraiani, Emerton, Gee and SavittCEGS22a], which establish the generic reducedness of the special fibres of the crystalline deformation rings in the 2-dimensional (tamely potentially) Barsotti–Tate case, by an analysis of the corresponding Emerton–Gee stacks.

Unfortunately, an (unconditional) argument with the Breuil–Mézard conjecture shows that generic reducedness is extremely rare when $v|p$ (see Remark 2.5.6). We are, however, able to prove the following theorem.

We refer the reader to the introduction to Section 2 for a detailed overview of the proof of Theorem D, which, as above, relies on proving the corresponding property of the relevant Emerton–Gee stacks [Reference Emerton and GeeEG23] (whose versal rings are the crystalline lifting rings). The irreducible components of the special fibres of these stacks were described in [Reference Emerton and GeeEG23], and we prove our result by combining this description with a computation of extensions of rank $1$ Breuil modules. An amusing feature of this argument is that we prove a result about the deformation rings of arbitrary n-dimensional mod p representations by reducing to a calculation for reducible $2$ -dimensional representations.

1.3 Bianchi modular forms

Let us specialize to the case when $F/{\mathbb Q}$ is an imaginary quadratic field. Let $\pi $ be a regular algebraic cuspidal automorphic representation of $\operatorname {\mathrm {GL}}_2(\mathbf {A}_F)$ . Let $\chi $ be the central character of $\pi $ . By definition, the representation $\pi $ occurs in $L^2_{\mathrm {cusp}}(\operatorname {\mathrm {GL}}_2(F) \backslash \operatorname {\mathrm {GL}}_2(\mathbf {A}_F))$ . Let $\mathfrak {g}$ be the Lie algebra of $\operatorname {\mathrm {GL}}_2({\mathbb C})$ as a real group. The assumption that $\pi $ is regular algebraic is equivalent to the condition that the infinitesimal character of $\pi _{\infty }$ is the same as $V^{\vee }$ for an algebraic representation V of $\mathrm {Res}_{F/{\mathbb Q}} \operatorname {\mathrm {GL}}_2$ . The assumption that $\pi $ is cuspidal places a restriction on V corresponding to the fact (noted earlier) that such $\pi $ has parallel weight; the corresponding representations are parametrized (up to twist) by an integer $k \ge 2$ , where $k=2$ corresponds to the case when V is trivial. This choice of k determines the action of $Z(\mathfrak {g})$ on $\pi _{\infty }$ , and by taking functions which are suitable eigenvectors under $Z(\mathfrak {g})$ , we may arrive at certain vector valued Hecke eigenfunctions $\Phi $ on $\operatorname {\mathrm {GL}}_2(\mathbf {A}_F)$ with Fourier expansions ([Reference WilliamsWil17, §1.2], [Reference HidaHid94, §6])

$$ \begin{align*}f \left[ \left( \begin{matrix} t & z \\ 0 & 1 \end{matrix} \right) \right] = |t|_F \sum_{\alpha \in F^{\times}} c(\alpha t \delta_F,f) W(\alpha t_{\infty}) e_F(\alpha z),\end{align*} $$

where $\delta = \delta _F$ is the different, $\alpha t \delta $ can be interpreted as a fractional ideal of $\mathcal {O}_F$ , $c(I,f)$ is a Fourier coefficient which vanishes unless $I \subset \mathcal {O}_F$ and which we may assume is normalized so that $c(\mathcal {O}_F,f) = 1$ , W is an explicit Whittaker function which is vector valued in some explicit representation of $\operatorname {\mathrm {SU}}(2)$ depending on k, and $e_F$ is an explicit additive character of $F \backslash \mathbf {A}_F$ . This has a direct translation into more classical language and can be interpreted as a collection of $h_F$ functions on a finite union of hyperbolic spaces $\mathbf {H}^3$ . The explicit functions f (either adelically or classically) are known as Bianchi modular forms. For a normalized Bianchi eigenform f of weight k and level prime to $\mathfrak {p}$ , Theorem 7.1.1 implies the following bound:

Theorem E. Let f be a cuspidal Bianchi modular eigenform of level $\mathfrak {n}$ and weight k. Let $\mathfrak {p}$ be a prime ideal of $\mathcal {O}_F$ not dividing $\mathfrak {n}$ . Then

(1.3.0) $$ \begin{align} |c(\mathfrak{p},f)| \le 2 N(\mathfrak{p})^{(k-1)/2}. \end{align} $$

This connects our theorem with the more classical version of the Ramanujan conjecture for modular forms [Reference DeligneDel71] as discussed earlier in the introduction.

The eigenvalues $c(\mathfrak {p},f)$ associated to f have a second interpretation in terms of the cohomology of arithmetic groups and arithmetic hyperbolic $3$ -manifolds. The algebraic representations V of $\operatorname {Res}_{F/{\mathbb Q}}\operatorname {\mathrm {GL}}_2$ are all, up to twist, given on real points $\operatorname {Res}_{F/{\mathbb Q}}\operatorname {\mathrm {GL}}_2({\mathbb R}) = \operatorname {\mathrm {GL}}_2({\mathbb C})$ by the representations $ \operatorname {\mathrm {Sym}}^{k-2} {\mathbb C}^2 \otimes \overline {\operatorname {\mathrm {Sym}}^{l-2} {\mathbb C}^2}$ for a pair of integers $k, l \geq 2$ . Let $\mathfrak {n} \leq \mathcal {O}_F$ be a nonzero ideal. Having fixed k and l, we can form the group cohomology

$$\begin{align*}H = H^1(\Gamma_1(\mathfrak{n}), \operatorname{\mathrm{Sym}}^{k-2} {\mathbb C}^2 \otimes \overline{\operatorname{\mathrm{Sym}}^{l-2} {\mathbb C}^2}) \end{align*}$$

of the standard congruence subgroup $\Gamma _1(\mathfrak {n}) \leq \operatorname {\mathrm {GL}}_2(\mathcal {O}_F)$ . Then H is a finite-dimensional ${\mathbb C}$ -vector space. Let $H_{\mathrm {par}} \subset H$ denote the subgroup consisting of classes which vanish under the restriction of H to $H^1(P, \operatorname {\mathrm {Sym}}^{k-2} {\mathbb C}^2 \otimes \overline {\operatorname {\mathrm {Sym}}^{l-2} {\mathbb C}^2})$ for any parabolic subgroup $P \subset \Gamma _1(\mathfrak {n})$ . More geometrically, one can interpret H as the cohomology of a local system on the Bianchi manifold $Y_1(\mathfrak {n}) = \mathbf {H}^3/\Gamma _1(\mathfrak {n})$ . If $X_1(\mathfrak {n})$ is the Borel–Serre compactification of $Y_1(\mathfrak {n})$ , then parabolic cohomology consists of classes which are trivial on the boundary $X_1(\mathfrak {n}) \smallsetminus Y_1(\mathfrak {n})$ ; this boundary may be identified (when $\Gamma _1(\mathfrak {n})$ is torsion free) with a finite disjoint union of complex tori. The spaces H and $H_{\mathrm {par}}$ are equipped with a commuting family of linear operators, the unramified Hecke operators $T_{\mathfrak {p}}$ , indexed by the principal ideals $\mathfrak {p} \leq \mathcal {O}_F$ not dividing $\mathfrak {n}$ . More precisely, if one writes $\mathfrak {p} = (\pi )$ , then the group $A \Gamma _1(\mathfrak {n}) A^{-1} \cap \Gamma _1(\mathfrak {n}) = \Gamma _1(\mathfrak {n},\mathfrak {p})$ has finite index in $\Gamma _1(\mathfrak {n})$ , where

$$ \begin{align*}A = \left(\begin{matrix} \pi & 0 \\ 0 & 1 \end{matrix} \right);\end{align*} $$

the map $T_{\mathfrak {p}}$ is induced by composing (in a suitable order) a restriction map, a conjugation by A map, and a trace map, respectively. Note that the existence of (a large family of) such operators comes from the fact that $\Gamma = \operatorname {\mathrm {GL}}_2(\mathcal {O}_F)$ has infinite index inside its commensurator in $\operatorname {\mathrm {GL}}_2(F)$ ; as shown by Margulis [Reference MargulisMar91, Thm. IX.1.13], this characterizes the arithmeticity of $\Gamma $ .

In order to obtain an action of $T_{\mathfrak {p}}$ for more general prime ideals $\mathfrak {p}$ , one needs to replace $Y_1(\mathfrak {n})$ by a disconnected union of $h_F$ commensurable arithmetic hyperbolic manifolds $Y_1(\mathfrak {n};\mathfrak {a}) = \mathbf {H}/\Gamma _1(\mathfrak {n};\mathfrak {a})$ indexed by ideals $\mathfrak {a}$ in the class group $\mathrm {Cl}(\mathcal {O}_F)$ of F prime to $\mathfrak {n}$ , and where $Y_1(\mathfrak {n};\mathcal {O}_F) = Y_1(\mathfrak {n})$ . The group $\Gamma (\mathcal {O}_F;\mathfrak {a})$ is the automorphism group of the $\mathcal {O}_F$ -module $\mathcal {O}_F \oplus \mathfrak {a}$ , which consists explicitly of matrices of the form

$$ \begin{align*}\left( \begin{matrix} \mathcal{O}_F & \mathfrak{a}^{-1} \\ \mathfrak{a} & \mathcal{O}_F \end{matrix} \right) \cap \operatorname{\mathrm{GL}}_2(F)\end{align*} $$

with determinant in $\mathcal {O}^{\times }_F$ . The space $H_{\mathrm {par}}$ vanishes unless $k = l$ ([Reference HarderHar87, 3.6.1]), which we now assume. If $h_F = 1$ , the Eichler–Shimura isomorphism ([Reference HarderHar87, §3.6]) gives a map from Bianchi cuspidal modular eigenforms f of weight k as described above and cohomology classes $\eta _f \in H_{\mathrm {par}}$ which are simultaneous eigenforms for all the Hecke operators. Moreover, the eigenvalues of $T_{\mathfrak {p}}$ on $\eta _f$ are given exactly by $c(\mathfrak {p},f)$ . If $h_F> 1$ , one must replace H and $H_{\mathrm {par}}$ by the direct sum of the corresponding cohomology groups over $Y_1(\mathfrak {n};\mathfrak {a})$ for $\mathfrak {a} \in \mathrm {Cl}(\mathcal {O}_F)$ . Theorem E now implies the following:

These explicit formulations of our theorems can be generalized in a number of ways. Remaining in the setting of arithmetic hyperbolic $3$ -manifolds (or orbifolds), we can replace $\operatorname {\mathrm {GL}}_2(\mathcal {O}_F)$ by a congruence subgroup $\Gamma $ of the norm one units in a maximal order $\mathcal {O}$ of a division algebra $D/F$ , where $F \hookrightarrow {\mathbb C}$ is a number field with one complex place and D is definite at all real places of F. When $[F:{\mathbb Q}] = 2$ , we obtain the Bianchi manifolds as above (when $D/F$ is split) but also certain compact hyperbolic arithmetic three manifolds; our theorem applies equally well in the latter case (note that $H=H_{\mathrm {par}}$ in this setting). On the other hand, suppose that F has at least one real place; for example, take $F = \mathbf {Q}[\theta ]/(\theta ^3 - \theta + 1)$ , let $k = 2$ , and let $D/F$ be ramified at the real place and the unique prime of norm $5$ . Now $H = H_{\mathrm {par}}$ is the first cohomology group of a congruence cover of the Weeks manifold. The generalized Ramanujan conjecture still predicts a bound of the shape $|a_{\mathfrak {p}}| \le 2 N(\mathfrak {p})^{1/2}$ for the eigenvalues of the Hecke operators $T_{\mathfrak {p}}$ . However, our methods do not apply in this situation, and the best current bounds remain those of the form $|a_{\mathfrak {p}}| \le 2 N(\mathfrak {p})^{1/2 + 7/64}$ proved using analytic methods (see [Reference SarnakSar05]).

We finish with an application of a different sort. Let $\Gamma = \operatorname {\mathrm {SL}}_2(\mathcal {O}_F)$ . The quotients $\Gamma \backslash \mathbf {H}^3$ were first investigated by Bianchi [Reference BianchiBia1892], and for that reason, they are known as Bianchi orbifolds. For a Bianchi modular form f of level one, one may [Reference MarshallMar11, §3] associate to f a normalized measure $\mu _f$ on $\Gamma \backslash \mathbf {H}^3$ . One then has the following [Reference MarshallMar11, Cor 3]:

1.4 Recent work of Matsumoto

A few months after the first preprint version of this work was circulated, a remarkable new work by Matsumoto appeared [Reference MatsumotoMat24] which proves Theorems A and B with no parallel weight condition. Matsumoto’s approach introduces several new ideas of a global nature, while our approach here is based on refining our understanding of the local ingredients in the potential automorphy argument. For this reason, one might hope that the two approaches could be profitably combined in the future.

1.5 Notation

Let $K/{\mathbb Q}_p$ be a finite extension. If $\sigma :K \hookrightarrow \overline {{\mathbb Q}}_p$ is a continuous embedding of fields, then we will write $\mathrm {HT}_\sigma (\rho )$ for the multiset of Hodge–Tate numbers of $\rho $ with respect to $\sigma $ , which by definition contains i with multiplicity $\dim _{\overline {{\mathbb Q}}_p} (W \otimes _{\sigma ,K} \widehat {\overline {{K}}}(i))^{G_K} $ . We write $\varepsilon $ for the p-adic cyclotomic character, which is a crystalline representation with $\mathrm {HT}_\sigma (\varepsilon )=\{ -1\}$ for each $\sigma $ .

We say that $\rho $ has weight $0$ if for each $\sigma :K\hookrightarrow \overline {{\mathbb Q}}_p$ , we have $\mathrm {HT}_\sigma (\rho )=\{0,1,\dots ,d-1\}$ . We often somewhat abusively write that a representation $\rho :G_K\to \operatorname {\mathrm {GL}}_d(\overline {{\mathbb Z}}_p)$ is crystalline of weight $0$ if the corresponding representation $\rho :G_K\to \operatorname {\mathrm {GL}}_d(\overline {{\mathbb Q}}_p)$ is crystalline of weight $0$ .

Let ${\mathcal O}$ be the ring of integers in some finite extension $E/{\mathbb Q}_p$ , and suppose that E is large enough that it contains the images of all embeddings $\sigma :K\hookrightarrow \overline {{\mathbb Q}}_p$ . Write $\varpi $ for a uniformizer of ${\mathcal O}$ , and ${\mathcal O}/\varpi ={\mathbb F}$ for its residue field. We write $\operatorname {Art}_K: K^\times \to W_K^{\operatorname {ab}}$ for the isomorphism of local class field theory, normalized so that uniformizers correspond to geometric Frobenius elements.

Let $\overline {\rho }:G_K\to \operatorname {\mathrm {GL}}_d(\overline {{\mathbb F}}_p)$ be a continuous representation. Then after enlarging E and thus ${\mathbb F}$ if necessary, we may assume that the image of $\overline {\rho }$ is contained in $\operatorname {\mathrm {GL}}_d({\mathbb F})$ . We write $R^{\square ,{\mathcal O}}_{\overline {\rho }}$ for the universal lifting ${\mathcal O}$ -algebra of $\overline {\rho }$ ; by definition, this (pro-)represents the functor ${\mathcal D}^{\square ,{\mathcal O}}_{\overline {\rho }}$ given by lifts of $\overline {\rho }$ to representations $\rho : G_K \to \operatorname {\mathrm {GL}}_d(A)$ , for A an Artin local ${\mathcal O}$ -algebra with residue field ${\mathbb F}$ . The precise choice of E is unimportant, in the sense that if ${\mathcal O}'$ is the ring of integers in a finite extension $E'/E$ , then by [Reference Barnet-Lamb, Gee, Geraghty and TaylorBLGGT14, Lem. 1.2.1], we have $R^{\square ,{\mathcal O}'}_{\overline {\rho }}=R^{\square ,{\mathcal O}}_{\overline {\rho }}\otimes _{{\mathcal O}}{\mathcal O}'$ .

We write $R^{\mathrm {crys},\underline {0},{\mathcal O}}_{\overline {\rho }}$ for the unique ${\mathcal O}$ -flat quotient of $R_{\overline {\rho }}^{\square ,{\mathcal O}}$ with the property that if B is a finite flat E-algebra, then an ${\mathcal O}$ -algebra homomorphism $R_{\overline {\rho }}^{\square ,{\mathcal O}}\to B$ factors through $R^{\mathrm {crys},0,{\mathcal O}}_{\overline {\rho }}$ if and only if the corresponding representation of $G_K$ is crystalline of weight $0$ .

We will let $\operatorname {rec}_K$ be the local Langlands correspondence of [Reference Harris and TaylorHT01], so that if $\pi $ is an irreducible complex admissible representation of $\operatorname {\mathrm {GL}}_n(K)$ , then $\operatorname {rec}_K(\pi )$ is a Frobenius semisimple Weil–Deligne representation of the Weil group $W_K$ . We write $\operatorname {rec}^T_K$ for the arithmetic normalization of the local Langlands correspondence, as defined in, for example, [Reference Clozel and ThorneCT14, §2.1]; it is defined on irreducible admissible representations of $\operatorname {\mathrm {GL}}_n(K)$ defined over any field which is abstractly isomorphic to ${\mathbb C}$ (e.g., $\overline {{\mathbb Q}}_l$ ).

Let F be a number field. If v is a finite place of F, then we write $k(v)$ for the residue field of $F_v$ . We identify dominant weights $\lambda $ of $\operatorname {Res}_{F/{\mathbb Q}}\operatorname {\mathrm {GL}}_n$ with sets of tuples of integers $(\lambda _{\tau ,1}\ge \lambda _{\tau ,2} \ge \cdots \ge \lambda _{\tau ,n})_{{\tau : F\hookrightarrow {\mathbb C}}}$ indexed by complex embeddings of F (cf. [Reference Allen, Calegari, Caraiani, Gee, Helm, Hung, Newton, Scholze, Taylor and ThorneACC⁺23, §2.2.1]). If $\pi $ is an irreducible admissible representation of $\operatorname {\mathrm {GL}}_n(\mathbf {A}_F)$ and $\lambda $ is a dominant weight, we say that $\pi $ is regular algebraic of weight $\lambda $ if the infinitesimal character of $\pi _\infty $ is the same as that of $V_\lambda ^\vee $ , where $V_\lambda $ is the algebraic representation of $\operatorname {Res}_{F/{\mathbb Q}}\operatorname {\mathrm {GL}}_n$ of highest weight $\lambda $ . We say that $\pi $ is regular algebraic if it is regular algebraic of some weight.

Let F be an imaginary CM field, and let $\pi $ be a cuspidal, regular algebraic weight $\lambda $ automorphic representation of $\operatorname {\mathrm {GL}}_2(\mathbf {A}_F)$ . The weight ${\lambda } = (\lambda _{\tau , 1}, \lambda _{\tau , 2})_\tau \in ({\mathbb Z}^2)^{\operatorname {\mathrm {Hom}}(F, {\mathbb C})}$ satisfies the following:

• • There is an integer $w \in {\mathbb Z}$ such that for all $\tau $ , we have $\lambda _{\tau , 1} + \lambda _{\tau c, 2} = w$ . In particular, for all $\tau $ , we have $\lambda _{\tau , 1} - \lambda _{\tau , 2} = \lambda _{\tau c, 1} - \lambda _{\tau c, 2}$ .

This is a consequence of Clozel’s purity lemma [Reference ClozelClo90, Lemma 4.9]. In particular, if F is imaginary quadratic, $\pi $ is necessarily of parallel weight.