A p$p$ ‐adic approach to the existence of level‐raising congruences

We construct level‐raising congruences between p$p$ ‐ordinary automorphic representations, and apply this to the problem of symmetric power functoriality for Hilbert modular forms. In particular, we prove the existence of the nth$n\text{th}$ symmetric power lift of a Hilbert modular eigenform of regular weight for each odd integer n=1,3,⋯,25$n = 1, 3, \dots, 25$ . In a future work with James Newton, we will use these results to establish the existence of the nth$n\text{th}$ symmetric power lift for all n⩾1$n \geqslant 1$ .


Introduction
Context. Let p be a prime number, and let ι : Q p → C be an isomorphism. Let F be a CM number field, and let π be a regular algebraic automorphic representation of GL n (A F ). Then (see [HLTT16,Sch15]) there exists a continuous semisimple representation r ι (π) : G F → GL n (Q p ), determined up to isomorphism by the requirement that for almost every finite place v ∤ p of F , we have the local-global compatibility relation WD(r ι (π)| GF v ) ∼ = rec T Fv (ι −1 π v ) (in other words, compatibility with the Tate-normalised local Langlands correspondence for GL n (F v )). By a level-raising congruence, we mean the data of another regular algebraic automorphic representation π ′ of GL n (A F ) such that there is an isomorphism of residual representations (1.1) r ι (π) ∼ = r ι (π ′ ) from G F to GL n (F p ) and a finite place v of F such that π ′ v is "more ramified" than π v (for example, such that π v is unramified but π ′ v is not, or such that the conductor ideal of π ′ v is strictly contained in the conductor ideal of π v ). The existence of an isomorphism (1.1) is equivalent to the eigenvalues of the unramified Hecke operators on π and π ′ , viewed as elements of Z p , being congruent modulo the maximal ideal of Z p , while for classical modular forms the conductor ideal corresponds to the level of the corresponding congruence subgroup of SL 2 (Z). This explains the terminology 'level-raising congruence', which goes back to Ribet's fundamental paper [Rib84].
Results. In this paper, we prove the existence of level-raising congruences for regular algebraic automorphic representations π of GL n (A F ) which are conjugate self-dual, and even 'θ-discrete', in the sense that they are of the form π = π 1 ⊞ · · · ⊞ π r , where π 1 , . . . , π r are conjugate self-dual, cuspidal (and therefore unitary) automorphic representations of lower rank general linear groups. This is a theme explored in our earlier paper [Tho14]. Here we use similar techniques to prove the following theorem, a special case of Theorem 6.11: Theorem A. Fix a partition n = n 1 + n 2 and let π 1 , π 2 be cuspidal, conjugate selfdual automorphic representations of GL n1 (A F ), GL n2 (A F ) such that π = π 1 ⊞π 2 is regular algebraic and ι-ordinary. Suppose that the following conditions are satisfied: (1) F contains an imaginary quadratic field F 0 in which p splits, and if w is a place of F such that π w is ramified, then the prime number lying below w splits in F 0 . The extension F/F + (where F + denotes the maximal totally real subfield of F ) is everywhere unramified.
Then there exists a regular algebraic, conjugate self-dual, cuspidal, ι-ordinary automorphic representation Π of GL n (A F ) such that r ι (Π) ∼ = r ι (π) and Π w0 is an unramified twist of St n .
(The case where π has the form π = π 1 ⊞ · · · ⊞ π r (and r > 2) may be treated by induction on r.) This strengthens [Tho14,Theorem 7.1], the main improvements being that the following two hypotheses are no longer required: (a) We no longer require that p is a banal characteristic for GL n (F w0 ), in the sense that p is prime to order of GL n (O Fw 0 /(̟ w0 )).
(b) We no longer require that the restriction r ι (π i | · | (ni−n)/2) )| GF w 0 of the residual representation to the decomposition group is as ramified as possible, in the sense that the image of a generator t w0 for the tame inertia group at w 0 has a single Jordan block.
These improvements are essential for the applications to symmetric power functoriality outlined below. Indeed, the possibility of such applications was the motivation in [Tho14], but the results proved there were too weak to be of any use. Before discussing these applications, we outline the main ideas behind the proof of Theorem A. The main principle is that if the situation modulo p is too degenerate, we should try to lift to a characteristic 0 (p-adic) situation where these degeneracies disappear. Here we achieve this by considering a twisted representation ρ ψ = r ι (π 1 | · | −n2/2 ) ⊗ ψ ⊕ r ι (π 2 | · | −n1/2 ) for some conjugate self-dual p-adic character ψ : G F → Q × p which is unramified at w 0 , and such that ρ ψ | GF w 0 satisfies the necessary condition for level-raising, i.e. that the characteristic polynomial of ρ ψ (Frob w0 ) has the form n i=1 (X − q i w0 α) for some α ∈ Q × p (as it would if ρ ψ | GF w 0 was in fact associated, under the local Langlands correspondence, to an unramified twist of the Steinberg representation). One now sees why the hypotheses (a) and (b) above fall away -characteristic 0 is certainly banal for GL n (F w0 ), and the condition on the Jordan blocks of ρ ψ (t w0 ) holds because it holds for r ι (π 1 |·| −n2/2 )| GF w 0 and r ι (π 2 |·| −n1/2 )| GF w 0 (by the strong form of local-global compatibility at w 0 , established in [TY07]).
The representation ρ ψ is not associated to a classical automorphic representation of GL n (A F ), but it can be seen as arising from a space of non-classical p-ordinary automorphic forms. Our hypotheses ensure that we can find such an automorphic form in a space H λ of p-ordinary weight λ automorphic forms on the A F + -points of a definite unitary group G over F + (for a suitable p-adic, non-classical weight λ). For us H λ is an admissible Z p [GL n (F w0 )]-module, flat over Z p , such that the irreducible subquotients of H λ [1/p] are isomorphic to subquotients of the induced representation (1.2) Ind GLn(Fw 0 ) Bn(Fw 0 ) Q p = C ∞ (B n (F w0 )\GL n (F w0 ), Q p ) from the Borel subgroup B n ⊂ GL n . We are thus in the situation of [Tho14,§4], which gives an abstract criterion on an admissible k[GL n (F w0 )]-module, where k is a field of banal characteristic, to admit level-raising (concretely, to admit the Steinberg representation as a quotient). The key property of H λ that we require in loc. cit. is that the homology groups (where GL n (F w0 ) 0 ⊂ GL n (F w0 ) is the subgroup of matrices whose determinant lies in O × Fw 0 ) are non-zero only in degree i = n − 1. To prove this, it is enough to show that the homology groups are concentrated in this degree. The difficulty faced in [Tho14] is that these groups are hard to compute in non-banal characteristic. To get around this here, we introduce the complex RΓ c (Ω n−1,ca Fw 0 , Z p ) studied in [Dat06] -the compactly supported etale cohomology of the Drinfeld upper half space Ω n−1 Fw 0 . (Following [Dat06], we use the superscript ca to denote base change to the completion of an algebraically closed extension.) It is a complex of smooth Z p [GL n (F w0 )]-modules, and according to [Dat06] there is an isomorphism RΓ c (Ω n−1,ca Fw 0 , Q p ) ∼ = ⊕ n−1 i=0 π i,1,...,1 [i − 2(n − 1)] in the derived category of smooth Q p [GL n (F w0 )]-modules (and where the representations π µ associated to an ordered partition µ of n are certain subquotients of the representation (1.2), whose definition we recall in §3). In particular, this complex is split, in the sense that it is isomorphic (in the derived category) to the direct sum of its cohomology groups. (Such a splitting exists also when Q p is replaced by a finite field of banal characteristic, but not in general, cf. [Dat14].) It is enough therefore for us to consider the derived tensor product RΓ c (Ω n−1,ca The cohomology of this complex will be concentrated in a single degree provided the same is true of the complex Now we use that H λ /m Zp H λ may be identified with a space of classical, p-ordinary automorphic forms on G with coefficients in F p . Using results of Rapoport-Zink [RZ96] on the p-adic uniformization of Shimura varieties, we identify the complex (1.3) with the F p -cohomology of a Shimura variety of Harris-Taylor type [HT01].
The results of Boyer [Boy19] and Caraiani-Scholze [CS17] imply that this cohomology vanishes outside degree n − 1 (after localisation a generic maximal ideal of the Hecke algebra). The reader familiar with existing level-raising arguments may wonder where "Ihara's lemma" (as formulated in e.g. [CHT08]) appears in the argument. In [Tho14], we deduced Ihara's lemma (in restricted circumstances) from the concentration of certain cohomology groups in a single degree. By contrast, here we deduce our level-raising result directly from this concentration of cohomology groups; it does not seem that this concentration alone is strong enough to imply Ihara's lemma in the generality required to then be able to deduce Theorem A as a consequence.
The appeal to the results of [Boy19,CS17] is the reason that Theorem A contains the genericity hypothesis (4) on the residual representation r ι (π). In fact, we work throughout with the p-ordinary part of cohomology, and it seems likely to us that the p-ordinary part of the cohomology of the Shimura varieties considered here is already concentrated in a single degree before localisation at a generic maximal ideal of the prime-to-p Hecke algebra. We don't know how to prove this, but such a result would make it possible to remove condition (4) from Theorem A, leading to a theorem which would be essentially optimal for the kinds of applications considered here.
Applications. Here is our first main application.
Theorem B. Let F + be a totally real number field, and let π be a cuspidal automorphic representation of GL 2 (A F + ) such that π ∞ is (essentially) square-integrable and which is not automorphically induced from a quadratic extension of F + . Then for each integer n = 1, . . . , 9, and for each odd integer n = 11, . . . , 25, the symmetric power Sym n π exists, in the sense that there is a cuspidal automorphic representation Π of GL n+1 (A F + ) such that for any place v of F + , we have . For a description of the importance of, and of previous progress on, the problem of symmetric power functoriality, we refer to the papers [CT14,NT21]. In our previous work joint work with Clozel [CT14], we outlined a programme to prove the existence of symmetric power functorial lifts of automorphic representations of GL 2 (A F + ) of the type considered in Theorem B, relying on two conjectures: one on the existence of certain level-raising congruences, and the second on the existence of a version of tensor product functoriality for the group GL 2 ×GL r . The results we establish in this paper, although not identical in strength to the level-raising conjecture stated in [CT14], are nevertheless sufficient to eliminate the need to assume this conjecture. Theorem B is then what one can prove unconditionally using the ideas in [CT14] by using known cases r = 1, 2, 3 of tensor product functoriality. We can also state a result conditional on the assumption of tensor product functoriality in all degrees: Theorem C. Let F + be a totally real number field. Assume the truth of [CT14, Conjecture 3.2]. Then for any cuspidal automorphic representation π of GL 2 (A F + ) such that π ∞ is (essentially) square-integrable and which is not automorphically induced from a quadratic extension of F + , and for any integer n ≥ 1, the symmetric power Sym n π exists, in the sense that there is a cuspidal automorphic representation Π of GL n+1 (A F + ) such that for any place v of F + , we have . After the first draft of this paper was completed, we further developed (together with James Newton) the strategy of [CT14] by proving enough cases of [CT14, Conjecture 3.2] to get the conclusion of Theorem C unconditionally. These results will appear elsewhere [NT22]. We have chosen to leave the statements and proofs of Theorem B and Theorem C in their current form.
Acknowledgements. The author's work received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 714405). We would like to thank Toby Gee and James Newton for useful comments on an earlier draft of this paper.
Notation. If F is a field of characteristic zero, we generally fix an algebraic closure F /F and write G F for the absolute Galois group of F with respect to this choice. If F is a number field, then we will also fix embeddings F → F v extending the map F → F v for each place v of F ; this choice determines a homomorphism G Fv → G F . When v is a finite place, we will write O Fv ⊂ F v for the valuation ring, ̟ v ∈ O Fv for a fixed choice of uniformizer, Frob v ∈ G Fv for a fixed choice of geometric Frobenius lift, k(v) = O Fv /(̟ v ) for the residue field, and q v = #k(v) for the cardinality of the residue field.
If F is a CM field (totally imaginary extension of a totally real field F + ), then we write c ∈ Gal(F/F + ) for the non-trivial element. If S is a finite set of finite places of F + which are unramified in F , then we write F S /F for the maximal subextension of F which is unramified over F + , and G F,S = Gal(F S /F ).
If p is a prime, then we call a coefficient field a finite extension E/Q p contained inside our fixed algebraic closure Q p , and write O for the valuation ring of E, ̟ ∈ O for a fixed choice of uniformizer, and k = O/(̟) for the residue field.
If G is a locally profinite group and U ⊂ G is an open compact subgroup, then we write H(G, U ) for the set of compactly supported, U -biinvariant functions f : G → Z. It is a Z-algebra, where convolution is defined using the left-invariant Haar measure normalized to give U measure 1; see [NT16,§2.2]. It is free as a Z-module, with basis given by the characteristic functions [U gU ] of double cosets.
Let K be a non-archimedean characteristic 0 local field of normaliser ̟ K and residue field k K , and let | · | K : K × → R >0 be the absolute value satisfying |̟ K | K = |k K | −1 . We write W K ⊂ G K for the Weil group of K and I K ⊂ W K for the inertia subgroup. If ρ : G K → GL n (Q p ) is a continuous representation (assumed to be de Rham if p equals the residue characteristic of K), then we write WD(ρ) = (r, N ) for the associated Weil-Deligne representation, and WD(ρ) F −ss for its Frobenius semisimplification. We use the cohomological normalisation of class field theory: it is the isomorphism Art K : K × → W ab K which sends uniformizers to geometric Frobenius elements. The local Langlands correspondence rec K is a bijection between the set of isomorphism classes of irreducible admissible C[GL n (K)]-modules and the set of isomorphism classes of Frobenius-semisimple Weil-Deligne representations (r, N ) of W K of rank n over C. The Tate normalisation rec T K is defined by rec T K (π) = rec K (π| · | (1−n)/2 ); it respects the action of Aut(C) on the set of isomorphism classes, so makes sense over any field Ω which is abstractly isomorphic to C (such as Q p ).
If P ⊂ GL n is a parabolic subgroup, Ω is a field, and π is a smooth Ω[GL n (K)]module, then we write Ind π for the smooth induction: the set of locally constant functions f : GL n (K) → π such that for all p ∈ P (K), g ∈ GL n (K), f (pg) = π(p)f (g). If Ω = C then we write i GLn(K) P (K) π = Ind GLn(K) P (K) (π ⊗ δ 1/2 P ) for the normalised induction, where δ P (p) = | det(Ad(p) : Lie N → Lie N )| K . If σ is a smooth C[GL n (K)]-module, then we write r P (σ) = (σ) N (K) ⊗ δ −1/2 P for the normalised restriction (twist of N (K)-coinvariants, where N ⊂ P is the unipotent radical).
By definition, the Steinberg representation St n of GL n (K) is the unique irreducible quotient of Ind GLn(K) Bn(K) C, where B n ⊂ GL n is the upper-triangular Borel subgroup. It satisfies rec K (St n ) = Sp n , where Sp n = (r, N ) is the Weil-Deligne representation on C n = ⊕ n i=1 Ce i defined by r(Art K (x))(e i ) = |x| (n+1−2i)/2 e i , N e i = e i−1 for each i = 1, . . . , n (and where by convention e 0 = 0). Now let K be a finite extension of R. We again write W K for the Weil group of K. In this case the local Langlands correspondence rec K is a bijection between the set of isomorphism classes of infinitesimal equivalence classes of irreducible admissible representations of GL n (K) over C and the set of isomorphism classes of semisimple representations of W K of rank n over C. We again define rec T K (π) = rec K (π| · | (1−n)/2 ). These notions are reviewed in more detail in [CT14, §2.1].
If F is a number field and π is an automorphic representation of GL n (A F ), we say that π is regular algebraic if π ∞ has the same infinitesimal character as an irreducible algebraic representation of Res F/Q GL n . Let Z n + ⊂ Z n denote the set of tuples (λ 1 , . . . , λ n ) such that λ 1 ≥ · · · ≥ λ n . We identify Z n + with the set of characters of the diagonal maximal torus T n ⊂ GL n which are dominant with respect to the upper-triangular Borel subgroup B n ⊂ GL n . If π is regular algebraic, we say that it is of weight λ = (λ τ ) τ ∈Hom(F,C) ∈ (Z n + ) Hom(F,C) if for each place v|∞ of F , π v has the same infinitesimal character as the dual of the tensor product of the representations of GL n (F v ) of highest weights λ τ (τ ∈ Hom(F v , C)).
If χ : F × \A × F → C × is a Hecke character which is regular algebraic (equivalently: algebraic), then for any isomorphism ι : Q p → C there is a continuous character p is a continuous character which is de Rham and unramified at all but finitely many places, then there exists an algebraic Hecke character χ : Let F + be a totally real field. By definition, a RAESDC automorphic representation of GL n (A F + ) is a pair (π, χ), where π is a regular algebraic cuspidal automorphic representation of GL n (A F + ) and χ : (F + ) × \(A F + ) × → C × is a continuous algebraic character such that χ v (−1) is independent of the choice of place v|∞ of F + and π ∼ = π ∨ ⊗ (χ • det). Now let F be a CM field. By definition, a RAECSDC automorphic representation of GL n (A F ) is a pair (π, χ), where π is a regular algebraic cuspidal automorphic representation of GL n (A F ) and χ : (F + ) × \(A F + ) × → C × is a continuous algebraic character such that χ v (−1) = (−1) n for each place v|∞ of F + and π ∼ = π c,∨ ⊗ (χ • N F/F + • det). A RACSDC automorphic representation π of GL n (A F ) is a regular algebraic cuspidal automorphic representation π such that π ∼ = π c,∨ .
Let p be a prime number and let ι : Q p → C be an isomorphism. Any RACSDC or RAECSDC (resp. RAESDC) automorphic representation π has an associated Galois representation r ι (π) : G F → GL n (Q p ) (resp. r ι (π) : G F + → GL n (Q p )) which satisfies WD(r ι (π)| GF v ) F −ss ∼ = rec T Fv (π v ) for any place v ∤ p∞ of F (resp. similarly with F replaced by F + ). This notation can be applied more generally to regular algebraic automorphic representations of GL n (A F ) of the form π = π 1 ⊞· · ·⊞π r , where each π i is cuspidal and conjugate self-dual. See [CT14, Theorem 2.2] for more details.

Homology of smooth representations
Let p be a prime, let G be a topological group containing an open compact subgroup which is a pro-p group containing only countably many open compact subgroups, and let R be a Z[1/p]-algebra. We write Mod sm (R[G]) for the category of smooth (left) R[G]-modules. Our main examples will be G = GL n (F v ), where F v /Q p is a finite extension, G = Γ, where Γ is a discrete group, and products of these.
If X is a totally disconnected, locally compact space (in other words, X is a Hausdorff topological space such that each point has a basis of open compact neighbourhoods), we write C ∞ c (X, R) for the set of locally constant and compactly supported functions f : X → R. If G acts on X (and the action map ). (In this paper we will consider only left actions on spaces and modules; thus an element g ∈ G acts . When we wish to consider only one of these actions, we write ) to indicate that we are considering only the action of G either by left or by right translation. ) and there is, by adjunction, an embedding M → Ind G 1 N . We next show that the category has enough projectives. Let G r denote G, with G acting on itself by right translation. We claim that C ∞ c (G r , R) is a projective object of Mod sm (R[G]). To see this, fix a decreasing sequence denote the R-linear map which, for any g ∈ G and open compact subgroup K ≤ K 1 , sends the indicator function 1 gK to [K : this is integration against a Haar measure -see [Vig96].) We make C ∞ c (G, R) into an R-algebra by the formula The algebra C ∞ c (G, R) contains the idempotent elements e Ki = [K 1 : There are compact induction functors , sending a smooth K i -module M to the set of compactly supported and locally constant functions f : G → M satisfying f (kg) = kf (g) for all k ∈ K i , g ∈ G. This functor has an exact right adjoint, namely restriction to K i . In particu- This map has a splitting given by the formula is a direct summand of a projective object, and is therefore itself projective. To see that our category has enough projectives, take M ∈ Mod sm (R[G]) and choose a free R-module F and a surjection p : F → M of R-modules. Then the map ). The proof of Proposition 2.1 also shows that C ∞ c (G r , R) is a direct summand of ⊕ i∈I C ∞ c (K i \G, R) for some index set I and collection (K i ) i∈I of open, compact, pro-p subgroups of G. It's therefore enough to show that C ∞ c (K\G, R) is projective in Mod sm (R[H]) whenever K is an open, compact, pro-p subgroup of G.
Let (g j ) j∈J be a set of representatives for the double cosets K\G/H. We claim that there is an isomorphism If G 1 , G 2 , G 3 are groups satisfying our conditions, then we define a bifunctor by the formula

Given complexes
. This defines the functor of total tensor product: may be identified with the left-derived functor of the functor is naturally isomorphic to the identity functor. In particular for any M ∈ Mod sm (R[G 1 × G 2 ]), there are natural isomorphisms Proof. This is all standard when the category of smooth R[G 1 × G 2 ]-modules is replaced by the category of all R[G 1 × G 2 ]-modules. The same arguments apply equally in our case. We give the details, following [Wei94, • is a quasi-isomorphism from a bounded above complex of projective objects, and by the general machinery of derived functors, this is canonically independent of the choice of Q • . If It is easy to check that this map is well-defined and satisfies p( . We need to check that p is an isomorphism. It is surjective since if K is an open compact pro-p subgroup of G 1 which fixes m, then p(e K ⊗ m) = m. To see that it is injective, suppose that p( i f i ⊗ v i ) = 0. Choose a sufficiently small open compact pro-p subgroup K of G 1 so that each f i is left K-invariant; then we can rewrite i f i ⊗ v i = j 1 Kgj ⊗ m j for some elements g j ∈ G 1 and m j ∈ M . Choose an open normal subgroup K 1 ≤ K such that for each j, g j m j is fixed by K 1 , and choose coset representatives k l so that K = ⊔ k K 1 k l . Then we can further rewrite j 1 Kgj ⊗m j = j,l 1 K1k l gj ⊗m j . Since K 1 is normal in K, K 1 fixes each vector k l g j m j , and so we have p(1 K1k l gj ⊗m j ) = vol(K 1 )k l g j m j , and therefore l,j k l g j m j = 0, and

This vector is zero in
M , as required.
One important special case is when the groups G 1 , G 3 are trivial, and G 2 = G (say), in which case we have a functor . This gives the homology groups used in the introduction to this paper (for the remainder of the paper, we stick to cohomology groups only).
The total tensor product is associative: Proposition 2.4. If G 1 , G 2 , G 3 , G 4 are groups satisfying our conditions then for Proof. After replacing A • , B • , and C • by complexes of projective objects, we can apply [Sta21, Remark 08BI].
We also have a change of ring functor. If G is a group satisfying our conditions, S is a (commutative) R-algebra, and M ∈ Mod sm (R[G]), then M ⊗ R S ∈ Mod sm (S[G]). We write − ⊗ L R S for the left-derived functor of − ⊗ R S. Proposition 2.5. If G 1 , G 2 , G 3 are groups satisfying our assumptions and S is an R-algebra, then for any Proof. After replacing A • , B • by complexes of projective objects, it is enough to check that if P ∈ Mod sm (R[G 1 × G 2 ]) and Q ∈ Mod sm (R[G 2 × G 3 ]) are projective then P ⊗ R[G2] Q is a projective object of Mod sm (R[G 1 × G 3 ]) and moreover that there is a natural isomorphism The existence of the latter isomorphism follows from the right-exactness of − ⊗ R S; to show that P ⊗ R[G2] Q is projective, we can assume that We can also use the derived tensor product to construct an internal tensor product, i.e. a functor The following lemma concerns a variant of this construction and will be used in the proof of Theorem 5.2.
and it suffices to show that these functors are naturally isomorphic, i.e. that if , where (g 1 , g 2 ) acts on the source by (g 1 , g 2 ) · (m ⊗ f ) = g 1 m ⊗ L g1 R g2 f and on the target by (g 1 , where the two actions are now given respectively by ((g 1 , g 2 ) · f )(g) = g 1 f (g −1 1 gg 2 ) and ((g 1 , g 2 ) · f )(g) = g 2 f (g −1 1 gg 2 ). These two actions are intertwined by the map f → (F f (g) = g −1 f (g)). This is the desired natural isomorphism.
Finally, we establish some useful finiteness properties.
Lemma 2.7. Suppose that R is a local ring of residue field k = R/m R , and let V be an admissible R[G]-module.
Proof. The second part follows from the first (take W to be the submodule generated by the lifts of a generating set for V /m R V ). We prove the first. If K ⊂ G is an open compact subgroup of pro-p order, then Since V is the union of the submodules V K (as K ranges over all open compact pro-p-subgroups of G), we find V = W .
Lemma 2.8. Suppose that R is Noetherian, and let V, W ∈ Mod sm (R[G]) be such that V is finitely generated as an R[G]-module and W is admissible. Then V ⊗ R[G] W is a finitely generated R-module.
Proof. Let v 1 , . . . , v n ∈ V be generators for V as R[G]-module and let K ⊂ G be an open compact pro-p subgroup such that v 1 , . . . , v n ∈ V K . We claim that the Since W is admissible and R is Noetherian, this will imply the truth of the lemma.
It is easy to see that V ⊗ R[G] W is generated as an R-module by elements of the form v i ⊗ w, with w ∈ W . Take such an element, and choose an open subgroup Since tr K/K ′ (w) ∈ W K , we're done.
Proposition 2.9. Suppose that R is Noetherian, let K ⊂ G be an open compact subgroup, and let W ∈ Mod sm (R[K]) be finitely generated as is a finitely generated R-module. The same holds more generally if V is replaced by a bounded above complex A • ∈ D − (R[G]) whose cohomology groups are admissible R[G]-modules.
V (as this compact induction is projective, cf. the proof of Proposition 2.1). Since c-Ind G K W is finitely generated, Lemma 2.8 implies that c- If K is not pro-p, then we can find a resolution P • → W by projective R[K]modules which are finitely generated as R-modules. Then V and the result follows from the fact that each c- V is a finitely generated R-module and our assumption that R is Noetherian.
Theorem 2.10. Suppose that R is Noetherian and that G = GL n (F v ) for number field F and p-adic A similar statement in the case that R is a field appears in [Vig97].
Proof. The second part follows from the first by a spectral sequence argument, so we prove just the first. Since V is finitely generated, we can find m ≥ 2 such that, taking K to be the principal congruence subgroup of level m, V is generated by V K . Since V is assumed admissible, V K is a finitely generated R-module. We now appeal to [GK05, Theorem 3.2]; this states that there is an isomorphism • X is the Bruhat-Tits building of PGL n (F v ), considered as simplicial complex, each simplex σ corresponding to a homothety class of lattice chains L 0 ⊃ L 1 ⊃ · · · ⊃ L k ⊃ ̟ v L 0 being endowed with the cyclic orientation (L 0 , . . . , L k ). • V is the coefficient system on X which assigns to each simplex σ as above the finitely generated given by the formula (g · (v σ ) σ ) τ = gv g −1 τ . Let σ 0 , . . . , σ r be representatives of the finitely many G-orbits of simplices of dimension −q. Let G j = Stab G (σ j ). Then G j contains K σj as a normal subgroup and is the pre-image in G of an open compact subgroup of PGL n (F v ), and there is an isomorphism is a finitely generated R-module. This implies the statement of the theorem, by a spectral sequence argument. It suffices to show that for each W ) is a finitely generated R-module. This is the content of Proposition 2.9.
We will apply this theorem in conjunction with the following proposition.
Proposition 2.11. Suppose that R is Noetherian and that G = GL n (F v ) for number field F and p-adic Proof. Since the centre of G acts trivially on M , and G 0 has finite index in G mod centre, it suffices to show that M is finitely generated as an R[G]-module. Guillem Garcia Tarrach has shown us an elementary proof of this statement. It also follows from the deeper statement that parabolic induction from a Levi subgroup of GL n (F v ) preserves the property of being finitely generated (see [Dat09, Corollaire 3.9, Proposition 7.5]).
We have the following observations.
(2) Each representation π λ has the same cuspidal support as π n . The Jordan-Hölder factors of the representation i
(3) The normalised Jacquet module r Bn (π λ ) may be described as follows: it is a direct sum of the characters where σ ranges over the set of bijective functions Proof. According to the construction of the local Langlands correspondence rec Fv using the Bernstein-Zelevinsky classification (as explained e.g. in [Wed08]), π λ is the unique irreducible quotient of the induced representation i GLn(Fv )

This representation is in turn a quotient of the induced representation
By [Zel80, Proposition 2.10], this induction from B n (F v ) in fact has π λ as its unique irreducible quotient. The same result contains a description of r Bn (π λ ). The identification of π n and π 1,...,1 in the first part of the proposition is a very special case of this. Finally, [Zel80, Proposition 2.1] shows that the induced representation decomposes with multiplicity one.
Corollary 3.2. Let n = n 1 + n 2 be a partition with n 1 , n 2 ≥ 1, and let λ : n = λ 1 +· · ·+λ k be any partition such that λ = n, (n 1 , n 2 ). Then there exists a character which is a subquotient of the normalised Jacquet module r Bn (π λ ) and such that there is no increasing filtration 0 ⊂ Fil 1 ⊂ Fil 2 ⊂ · · · ⊂ Fil n = C n by sub-Weil-Deligne representations of rec Fv (π n1,n2 ) such that for each i = 1, . . . , n, Proof. Using the third part of Proposition 3.1, we see that r Bn (π λ ) contains the character We will show that the corollary holds with this choice of χ. In order to see this, we split into cases. The existence of a filtration Fil • as in the statement of the corollary would imply that rec Fv (π n1,n2 ) contains an invariant line on which W Fv acts by the character |·| (2λ k −(n+1))/2 . However, there are exactly two invariant lines in rec Fv (π n1,n2 ), on which W Fv acts by the characters | · | (n−1)/2 and | · | (n−1−2n1)/2 . If λ k = n 2 , then neither of these equal | · | (2λ k −(n+1))/2 , leading to the desired contradiction. Suppose then that λ k = n 2 . Then λ k−1 < n 1 and we look at Fil λ k +1 = Fil n2+1 , which would be the sum of those 1-dimensional subspaces of rec Fv (π n1,n2 ) where W Fv acts by the characters This subspace of rec Fv (π n1,n2 ) is not in fact a sub-Weil-Deligne representation (as it is not invariant by N ), so we obtain the desired contradiction in this case also.
We now compute some derived tensor products of these representations. We need some notation.
We identify {1, . . . , n − 1} with the set of simple roots of the maximal torus T n ⊂ GL n , sending i to (t 1 , . . . , t n ) ∈ T n → t i /t i+1 . If λ is a partition of n, let P λ denote the associated standard parabolic subgroup of GL n , M λ its standard Levi subgroup, and let I λ ⊂ {1, . . . , n − 1} denote the set of simple roots which are roots of M λ . Identifying S n with the Weyl group of T n in GL n , we write w 0 ∈ S n for the longest element of S n with respect to the Borel subgroup B n and, for subsets I, J ⊂ {1, . . . , n − 1}, δ(I, J) for the cardinality of the symmetric difference (I ∪ J) − (I ∩ J). We write Theorem 3.3. Let λ be a partition of n and let π be an irreducible admissible C[G]-module of trivial central character. If π λ ⊗ L C[PGLn(Fv )] π = 0, then there is a partition µ such that π ∼ = π µ . In this case, Proof. Tensor-hom adjunction gives for any i ∈ Z an isomorphism where the two ∨'s denote vector space dual and smooth dual, respectively). For a subset J ⊂ {1, . . . , n − 1}, let π J denote the representation defined in [Dat06]; it is the unique irreducible quotient of the unnormalised induction Ind where P J is the standard parabolic subgroup generated by B n and the negative roots corresponding to elements of J.
To complete the proof of the theorem, we need to show that if π is an irreducible admissible representation of trivial central character that is not of the form π µ for any partition µ, then , then the two actions of α on Ext * Modsm(C[PGLn(Fv )]) (π λ , π ∨ ), one induced by the endomorphism α π λ : π λ → π λ , the other induced by the endomorphism α π ∨ : π ∨ → π ∨ , are the same (as follows e.g. from the fact that Since π is not of the form π µ , the representations π and π ∨ have distinct cuspidal supports. Bernstein's description of the centre of Mod sm (C[PGL n (F v )]) implies that we can choose α so that α π λ = 0 but α π ∨ is the identity. This implies that The definition of the representations π J appearing in the proof makes sense over any field K of characteristic 0. We extend the definition of π λ to any such field by setting π λ = Ind , where we note that the definition of the representation on the right-hand side makes sense over any such field because only integer powers of the norm character | · | appear in the definition.
Here is a variant of Theorem 3.3.
Theorem 3.4. Let λ be a partition of n and let π be an irreducible admissible , then there is a partition µ and an unramified character ψ : is non-zero if and only if i = δ(I λ , −w 0 (I µ )), in which case it is a 1-dimensional C-vector space.
Proof. Let Z denote the centre of G and let Z 0 = Z ∩ G 0 . Let ω π : Z → C × denote the central character of π. If ω π | Z 0 = 1, then π λ ⊗ L C[G 0 ] π = 0 and π is not an unramified twist of any π µ . If ω π | Z 0 = 1, then we can assume without loss of generality (after replacing π by an unramified twist) that in fact ω π = 1 and π is a smooth C[PGL n (F v )]-module. Let , and hence such that Ext i C[PGLn(Fv )] (π λ , π ⊗ ψ) = 0. Theorem 3.3 implies that π ⊗ ψ ∼ = π µ for some µ. If ψ ′ is another character which appears in this way then we similarly obtain π ⊗ ψ ′ ∼ = π µ ′ for some µ ′ . Looking at cuspidal supports, we see that this is possible only if ψ = ψ ′ , and moreover that there is an isomorphism . The desired statement now follows from Theorem 3.3.
The representations π λ are of interest to us for two reasons. The first is that they show up in spaces of automorphic representations when the level-raising congruence is satisfied (because of the second part of Proposition 3.1). The second is that they appear in the cohomology of the Drinfeld upper half-space. Indeed, let p be a prime not dividing q v and let E/Q p be a finite extension. Let R be one of E, O E , or O E /(̟ c ) for some c ≥ 1. The Drinfeld upper half-space is the rigid analytic space Ω n Fv which is the admissible open subspace of P n−1 Fv obtained by deleting all F v -rational hyperplanes. The group PGL n (F v ) acts on it and, according to [Dat06, §4.1.3], its compactly supported cohomology is computed by a complex (Here the superscript ca, following the notation of [Dat06], denotes base extension to the completion F v of the fixed algebraic closure of F v .) Formation of this complex is compatible with change of coefficients R → R ′ (where R ′ is also of our allowed form). When R = E, we have the following explicit description of this complex, which is part of [Dat06, Proposition 4.2.2]: For general R, we can at least prove: Proof. This follows from Theorem 2.10, provided we can show that each group , R) is a quotient of a parabolic induction).

The definite unitary group
The proof of our level-raising theorem will take place on a definite unitary group. We introduce the notation and assumptions we need here. Let F 0 be an imaginary quadratic field and let F be a CM number field of the form F = F + F 0 , such that F/F + is everywhere unramified. Let n ≥ 1 is an integer, and let G be the reductive group over O F + whose functor of points is We write U n for the reductive group over O F + whose functor of points is The group U n is quasi-split (with Borel subgroup B and maximal torus T defined over O F + respectively by the uppertriangular matrices and diagonal matrices) and admits an F -splitting (given by the standard matrices X i = E i,i+1 for i = 1, . . . , n − 1). For any finite place v of F + , the tuple (B, T, We can identify We define an inner twist ξ : U n,F → G F by the formula ξ(g 1 , g 2 ) = (g 1 , Φ n g 2 Φ −1 n ). By definition, this means that ξ −1c ξ is an inner automorphism of U n,F . We can lift this to a pure inner twist. We recall that a pure inner twist (see [Kal11,§2] Two pure inner twists are isomorphic if and only if the cocycles x, x ′ have the same image in H 1 (F + , U n ). Similar definitions and remarks apply with F + replaced by any completion F + v .
To lift ξ to the data of pure inner twist, it suffices to give z ∈ U n (F ) such that z c z = 1 and ξ −1c ξ = Ad(z). If n is odd, then (Φ n , Φ n ) ∈ U n (F + ) and we take z = (Φ n , Φ n ). If n is even, then we choose ζ ∈ F × 0 such that c ζ = −ζ and take z = (ζΦ n , −ζΦ n ). Our constructions will depend on this choice but the precise choice is unimportant. We do make use of the observation that since F 0 ⊂ F , the group G can be naturally defined over Q, and moreover that the pure inner twist is also defined over Q.
If v is a finite place of F + , then the cocycle in . It follows that (ξ, z) becomes isomorphic to the trivial pure inner twist (U n , 1, 1) over F + v ; such an isomorphism is determined uniquely up to automorphisms of the trivial pure inner twist, or in other words up to conjugation by U n (F + v ). We thus have for each finite place v of We can choose the isomorphisms ι v so that for all but finitely many places v of F + , and for every v which splits in F , The claim is vacuous at split places v (as then there is a unique conjugacy class of hyperspecial maximal compact subgroups). It is also vacuous at inert places when n is odd, for the same reason, although we don't need this. In general, let v be any finite place of F + such that, if w denotes a place of F lying above v, then z lies in U n (O Fw ). Let F u w denote the ring of integers of the maximal unramified extension of F w . Lang's theorem implies that , and we can take ι −1 v to be the map obtained after passage to F + v -points. Henceforth we fix a system of isomorphisms ι v satisfying the conclusion of the lemma. If v is a finite place of F + which splits v = ww c in F , then we also write ι w for the isomorphism G(F + v ) → GL n (F w ) given by canonical inclusion Theorem 4.2. Let σ be an automorphic representation of G(A F + ). Then we can find a partition n = n 1 + · · · + n k and discrete, conjugate self-dual automorphic representations π i of GL ni (A F ) (i = 1, . . . , k) with the following properties: (1) Let π = π 1 ⊞ · · · ⊞ π k . Then for any finite place v of F + such that σ v is unramified, and any place w|v of F , π w is unramified and is related to σ v by unramified base change.
(3) For each place v|∞ of F , π v has the same infinitesimal character as the algebraic representation ⊗ τ :Fv→C W τ , where W τ is the unique algebraic rep- Corollary 4.3. Let σ be an automorphic representation of G(A F + ), let p be a prime number, and let ι : Q p → C be an isomorphism. Then there exists a continuous semisimple representation r σ,ι : G F → GL n (Q p ) satisfying the following conditions: (1) For any finite, prime-to-p place v of F + such that σ v is unramified, and for any place w|v of F , r σ,ι | GF w is unramified.
In the situation of Corollary 4.3, it is useful to note that if there is a split place v = ww c such that σ w is not generic (in the sense of having a Whittaker model), then r σ,ι has two irreducible subquotients which differ by a twist of the cyclotomic character (see the formula for r σ,ι given in the proof of [AT21, Corollary 3.4] -this happens precisely when one of the constituents π i appearing in the statement of Theorem 4.2 is not cuspidal). Conversely, if the residual representation r σ,ι satisfies the genericity condition (3) of Theorem A, then σ w must be generic at every split place w.
Theorem 4.4. Let π be a RACSDC automorphic representation of GL n (A F ). Suppose that for each place w of F such that π w is ramified, w is split over F + . Then there exists an automorphic representation σ of G(A F + ) satisfying the following conditions: ) v = 0 and if w denotes the unique place of F lying above v, then π w is related to σ v by unramified base change. ( For each place v|∞ of F , π v has the same infinitesimal character as the algebraic representation ⊗ τ : The rest of this section is devoted to an analogue of Theorem 4.4 for conjugate self-dual automorphic representations of GL n (A F ) which are not cuspidal. It is a special case of the endoscopic classification of automorphic representations of G(A F + ). It would follow from the results of [KMSW14]. Here it is relatively simple to deduce what we need from the results of [Lab11], following similar lines to the arguments given in [NT21].
Theorem 4.5. Fix a partition n = n 1 +n 2 and let π 1 , π 2 be cuspidal, conjugate selfdual automorphic representations of GL n1 (A F ), GL n2 (A F ) such that π = π 1 ⊞ π 2 is regular algebraic. Suppose that the following conditions are satisfied: (1) Let λ i = (λ i,τ ) ∈ (Z ni + ) Hom(F,C) be the weight of π i |·| (ni−n)/2 . Then for each i = 1, 2, for any embedding τ 0 : F 0 → C and any (µ 1 , µ 2 ) ∈ Z n1 + × Z n2 + , the number of τ ∈ Hom(F, C) such that τ | F0 = τ 0 and (λ 1,τ , λ 2,τ ) = (µ 1 , µ 2 ) is even. (For example, this condition holds if π 1 , π 2 arise by base change from a quadratic CM extension F/F ′ .) (2) If w is a finite place of F such that π w is ramified, then w is split over F + . Then there exists an automorphic representation σ of G(A F + ) with the following properties: ) v = 0 and if w denotes the unique place of F lying above v, then π w is related to σ v by unramified base change. ( The theorem is proved using a comparison of trace formulae, using the results of [Lab11]. Most of the the necessary set-up is detailed in [NT21,§1], see especially [NT21, §1.5] for a description of the equivalence classes of endoscopic data for G and a normalisation of local transfer factors under the assumption that n is odd. We first describe a normalisation of local transfer factors also in the case that n is even. As outlined in [NT21, §1.5], the choice of pinning determines the Langlands-Shelstad [LS87] normalisation of local transfer factors for the quasi-split group U n which can be transferred, using the fixed structure of pure inner twist, to a normalisation of local transfer factors ∆ E v for the group G (with respect to any given endoscopic datum E). It follows from [Kal18, Proposition 4.4.1] that this normalisation of local transfer factors satisfies the adelic product formula. More precisely, if a choice of non-trivial character ψ : F + \A F + → C is fixed, one can define the associated Whittaker normalisation of the local transfer factors for U n , hence for G, which differs from ∆ E v by a local root number ǫ v , a sign which is equal to 1 for all but finitely many places v of F + , and [Kal18, Proposition 4.4.1] shows that the local transfer factors ǫ v ∆ E v satisfy the adelic product formula. However, the product v ǫ v is 1 (as it is the root number of a real representation -this observation appears also in the proof of the cited proposition).
We can then state the following result, which is a generalisation of [NT21, Proposition 1.7] to include the case where n is even: Proposition 4.6. Let v be a finite place of F + , and let f v ∈ C ∞ c (G(F + v )). Suppose given an extended endoscopic triple E = (H, s, η).
(1) Suppose that v is inert in F and that f v is ι −1 v (U n (O F + v ))-biinvariant. Suppose given an unramified Langlands parameter ϕ H : Then there is an identity where the twisted trace is Whittaker normalised.
The proof is essentially the same as the proof of [NT21, Proposition 1.7], which relies only on the properties of the local transfer factors associated to the F + vpinning of the quasi-split group U n . Here is another result valid at the infinite places, which is a generalisation of [NT21, Proposition 1.6] that includes the case that n is even: Let v be an infinite place of F + . Suppose given an extended endoscopic triple E = (H, s, η) for G and a Langlands parameter ϕ H : The proof is again the same as the proof of the proof of [NT21, Proposition 1.6], taking into account our normalisation of local transfer factors in the case n even. Repeating the argument of [NT21, Proposition 4.6] (if n 1 = n 2 ) or [AT21, §3.5] (if n 1 = n 2 ) and using the above two propositions, we see that the multiplicity of σ as an automorphic representation of G(A F + ) is 1 2 (1 + v|∞ ǫ(v, E, ϕ H )), where E is the endoscopic triple associated to the group H = U n1 × U n2 . We claim that the product of signs v|∞ ǫ(v, E, ϕ H ) equals 1. Indeed, we are assuming that F has the form F + F 0 and that our choice of pure inner twist is defined over F 0 . It follows that if v, v ′ are infinite places of F then there is a canonical isomorphism Our claim therefore follows from assumption 1 of Theorem 4.5.

Analysis of a derived tensor product
We continue with the set-up of the previous section. Thus we let F 0 be an imaginary quadratic field and let F be a CM number field of the form F = F + F 0 , such that F/F + is everywhere unramified, and take G to be the reductive group over O F + defined by (4.1). We need to set up some more notation. Recall that we have fixed for each finite place v of F + an isomorphism ι v : . Let p be a prime number, and let E/Q p be a coefficient field. Let S p denote the set of p-adic places of F + . If S is a finite set of places of F + , then we write J S for the set of open compact subgroups which are sufficiently small, in the sense that such that if g ∈ G(A ∞ F + ) then G(F + ) ∩ gKg −1 is torsion-free (hence trivial), and moroever such that for each v / ∈ S, which are sufficiently small, in the sense that for each g ∈ G(A ∞,v0 F + ) the group G(F + ) ∩ gKg −1 (intersection in G(A ∞,v0 F + )) is torsion-free, and such that if v / ∈ S, is an open compact subgroup such that K v0 ∈ J v0 S , then K ∈ J S . (The finite O F + -congruence subgroups of K are contained in the torsion-free Here T i w denotes the usual unramified Hecke operator We define as usual the polynomial (characteristic polynomial of Frobenius under rec T Fw ). Now fix a prime number q = p which splits q = u 0 u c 0 in F 0 , let v 0 be a place of F + dividing q, and let w 0 be the unique place of F lying above both u 0 and v 0 . Let S be a finite set of places of F + containing S p and such that S ∩ S q = {v 0 }.
is admissible and there is a natural map Proposition 5.1. Let notation and assumptions be as above. Then: (1) If m ⊂ T S is a maximal ideal in the support of A, then the localisation A m is naturally isomorphic to lim In particular, m occurs in the support of A.
Proof. We have A = lim − →Kv 0 A Kv 0 as T S -modules, where the direct limit runs over the set of open compact subgroups K v0 ⊂ G(F + v0 ). Since localisation commutes with direct limits, this shows that for any maximal ideal m ⊂ T S , we have In particular, A m is non-zero if and only if some (A Kv 0 ) m is non-zero. Each A Kv 0 is finite-dimensional as k-vector space, which shows that if A m is non-zero then the residue field of m is a finite extension of k and that (A Kv 0 ) m may be identified with the submodule A Kv 0 [m ∞ ] ⊂ A Kv 0 of m ∞ -torsion. By passage to the direct limit, we find that A m may be identified with the submodule of A of m ∞ -torsion. This shows the first and second parts of the proposition. For the third, we note that there is a T S -invariant direct sum decomposition It suffices therefore to show that if H * (RΓ c (Ω n−1,ca is nonzero, then m is the only maximal ideal of T S in its support. It even suffices to show that the natural map is an isomorphism. This is a consequence of the fact that if A standard consequence of Corollary 4.3 and the theory of algebraic modular forms states that if m occurs in the support of A Kv 0 , then there is a continuous semisimple representation ρ m : G F,S → GL n (T S /m) such that for each finite place w ∤ S of F which splits over F + , the characteristic polynomial det(X − ρ m (Frob w )) equals P w (X). The goal of this section is to prove the following theorem concerning the finite-dimensional k-vector spaces H j (RΓ c (Ω n−1,ca Theorem 5.2. With notation and assumptions as above, let m ⊂ T S be a maximal ideal of residue field k with the following property: (1) There exists a finite place w ∤ S of F such that ρ m | ss Then H j (RΓ c (Ω n−1,ca We will prove this theorem by comparing the cohomology groups in the statement of the theorem with the cohomology groups of a q-adically uniformized Shimura variety using [RZ96], and then using the vanishing theorems proved in [Boy19] or [CS17]. We begin by finding PEL data that will allow a comparison with the group G. Let D be a central simple algebra over F of rank n and let * be a positive involution of D (therefore of the second kind). Let V = D, considered as left D-module, and let ψ : V × V → Q be a non-degenerate alternating bilinear form such that for all d ∈ D, v, w ∈ V , we have ψ(dv, w) = ψ(v, d * w). Then the tuple (D, * , V, ψ) is a PEL datum and gives rise to a Shimura datum ( H, X), where H is the reductive group over Q whose functor of points is Lemma 5.3. Fix an embedding τ : F → C. Then we can find data (D, * , V, ψ) such that the following conditions are satisfied: (1) inv D w0 = 1/n, inv D w c 0 = −1/n, and inv D w = 0 for every place w = w 0 , w c 0 of F . (2) For each prime r = q, the group H Qr is quasi-split.
We henceforth fix a choice of τ : F → C and data (D, * , V, ψ) as in the statement of the lemma. Then the reflex field of the Shimura datum ( H, X) equals τ (F ) and for any sufficiently small open compact subgroup K ⊂ H(A ∞ Q ), there is an associated Shimura variety Sh K over τ (F ). Let ι : Q p → C be an isomorphism such that ι −1 • τ induces the place w 0 of F . We write G for the unitary similitude group associated to G. More precisely, G is the reductive group over Q whose functor of points is given by In particular, viewing Q q as an F 0 -algebra via the isomorphism F 0,u0 ∼ = Q q , we can identify This identification appears in the statement of the following theorem.
Theorem 5.4. Let K q ⊂ G(A ∞,q Q ) be a sufficiently small open compact subgroup, and let an open (but not compact) subgroup of G(Q q ). Then we can find an isomorphism f : G(A ∞,q Q ) → H(A ∞,q Q ) and an H( G(A ∞,q Q ), K q )-equivariant isomorphism of rigid analytic spaces over F w0 : Let G ad denote the adjoint group of G, and let I denote the unitary group over F + defined by (B, †). Since I Qr is quasi-split for every prime r, I F + v is quasi-split for every finite place v of F + . It follows that for every place v of F + , the groups G F + v and I F + v are isomorphic (if v is finite, these groups are both quasi-split; if v is infinite, then they are both the compact form of GL n,F + v ). Choose g ∈ GL n (F ) such that the involution b → b † of B is identified with the involution X → g t X c g −1 .
Then g t g c is a scalar matrix and g defines a 1-cocycle in Z 1 (F/F + , G ad ). For any place v of F + , the image of this 1-cocycle in H 1 (F + v , Aut(G)) is trivial. The map H 1 (F + v , G ad ) → H 1 (F + v , Aut(G)) has trivial pointed kernel, because the map Aut(G)(F + v ) → Out(G)(F + v ) is surjective. It follows that the image of our 1cocycle in H 1 (F + v , G ad ) is trivial. By the Hasse principle for H 1 (F + , G ad ) (proved using the description of ker 1 given in [Kot84, §4]), we conclude that our 1-cocycle in Z 1 (F/F + , G ad ) is in fact a coboundary and that we can in fact choose the isomorphism B ∼ = M n (F ) so that † is identified with the involution X → t X c and I = G. This leads to the claimed statement.
Before stating the result we need from [CS17], we need to introduce a bit more notation. Suppose given a finite set T of prime numbers containing p, q, and all the primes which are ramified in F , and let K H ⊂ H(A ∞ Q ) be a sufficiently small open compact subgroup of the form K H = l K H,l such that for each l ∈ T , K H,l ⊂ H(Q l ) is a hyperspecial maximal compact subgroup. Let Spl F0/Q denote the set of prime numbers which split in F 0 . We define to be the O-subalgebra generated by all the Hecke operators T i w and (T n w ) −1 , as w ranges over places of F lying above a prime l ∈ Spl F0/Q −T , and where these operators can be considered as elements of H( H(Q l ), K H,l ) using the analogue of the isomorphism (5.1) for the group H.
Theorem 5.5. Let T , K H be as in the previous paragraph. Let T F + denote the set of places of F + lying above an element of T . Suppose given a maximal ideal m ⊂ T T which is in the support of H * (Sh rig,ca K H , k). Then: (1) There exists a continuous semisimple representation such that for each prime number l ∈ Spl F0/Q −T and each place w|l of F , det(X − ρ m (Frob w )) = P w (X) mod m.
(2) Suppose further that there exists a finite place w ∤ T of F such that ρ m | ss Then [CS17, Theorem 6.3.1(1)] states that for any maximal ideal n ⊂ H T that occurs in the support of H * (Sh rig,ca there exists a continuous representation ρ n : G F,T F + → GL n (H T / n) such that for each prime number l ∈ Spl F0/Q −T and each place w|l of F , det(X − ρ n (Frob w )) = P w (X) mod n. If m ⊂ T S is a maximal ideal that is in the support of the cohomology, then m = T S ∩ n for some maximal ideal n ⊂ H T that is in the support, and we can take ρ m to be a conjugate of ρ n that is defined over T S / m. Such a conjugate exists because the conjugacy classes of elements Frob w (w a place of F lying above Spl F0/Q −T ) are dense in G F,T F + . This shows the first part.
For the second, we can decompose

Using [Dat06, Proposition B.3.1], we see that it is enough to construct a T Sequivariant isomorphism
RΓ c (Ω n−1,ca . The right hand-side here may be further decomposed as Since there is an isomorphism we see that it is even enough to construct a T S -equivariant isomorphism f on the left-hand side and by (g 1 , g 2 ) · (ω ⊗ f ) = g 2 ω ⊗ L g1 R g2 f on the right-hand side (restricting these actions to the subgroup G(F + )×GL n (F w0 ) 0 and passing to derived coinvariants then recovers the complexes whose cohomology groups appear in (5.3)). The existence of such an isomorphism is the content of Lemma 2.6. We next note that we are free to enlarge S if necessary (replacing m by its intersection with the subalgebra T S ′ ⊂ T S ) and also to replace K v0 by an open normal subgroup K v0 1 . Indeed, the Hochschild-Serre spectral sequence implies that if the Theorem holds for K v0 1 then the groups H j (RΓ c (Ω n−1,ca is the rigid space associated to a smooth proper scheme, which therefore satisfies Poincaré duality forétale cohomology with k-coefficients, then implies that the truth of the Theorem for K v0 .
We can therefore assume that the following additional conditions hold: • There is a sufficiently small open compact subgroup K q ⊂ G(A ∞,q Q ) such that K q ∩ G(A ∞,q Q ) = K q . • Defining K q as in the statement of Theorem 5.2, and writing µ : G → G m for the similitude character, we have µ( K) ∩ Q × = {1}.
Let T be a finite set of rational primes including p, q, and all those primes l such that K l is not a hyperspecial maximal compact subgroup of G(Q l ) or lying below an element of S. These conditions imply that the induced map , k) m , the desired result follows in view of the isomorphism (5.3).

Raising the level
We continue with the set-up of the previous section. Thus we let F 0 be an imaginary quadratic field, F = F + F 0 a CM field such that F/F + is everywhere unramified, and take G to be the reductive group over O F + whose functor of points is given by (4.1), with its associated isomorphisms ι v : for places w of F split over a place v of F + . We fix a prime number p, an isomorphism ι : Q p → C, a coefficient field E/Q p , and a finite set S of places of F + , each of which splits in F , and containing the set S p of p-adic places. We assume that E is large enough to contain the image of any embedding F → Q p .
We also fix for each v ∈ S a place v of F lying above v, and set S = { v | v ∈ S}. We write I p for the set of embeddings τ : F → Q p which induce a place of S p . We have defined J S to be the set of sufficiently small open compact ). We define J p S to be the set of sufficiently small open compact ) (a slight variation of the notation J v0 S also established in the previous section).
If w is a place of F , then we define Iw w ⊂ GL n (O Fw ) to be the standard Iwahori subgroup (pre-image of B n (k(w)) under reduction modulo ̟ w ) and for d ≥ c ≥ 1, Iw w (c, d) to be the intersection of the pre-image of B n (O Fw /̟ d w ) under reduction modulo ̟ d w and the pre-image of N n (O Fw /̟ c w ) under reduction modulo ̟ c w (recall B n ⊃ N n denote the standard Borel subgroup of GL n and its unipotent radical, respectively).
If K p ∈ J p S , then we have defined the Hecke algebra . In this case we define an enlarged (and still commutative) algebra . We define T S v to be the algebra generated by T S and the image of O[Λ v ]. We define unramified characters (1, . . . , 1, ̟ v , 1, . . . , 1) ∈ Λ v (where ̟ v sits in the i th position). We can define a group determinant D v of W F v of rank n with coefficients in T S v as the one associated to the representation ⊕ n i=1 χ v,i • Art −1 F v . Then [ACC + 18, Proposition 2.2.8] shows that D v "is" the semi-simplified Tate-normalised local Langlands correspondence rec T F v for Iwahori-spherical representations of GL n (F v ). We define ∆ v ∈ T S v to be the discriminant of the characteristic polynomial D v (X − Frob v ) of Frobenius.
We need to define spaces of ordinary algebraic modular forms. We first define what we mean by ordinary parts. If M is an O-module equipped with an endomorphism U : M → M , then the ordinary part M ord ⊂ M is the O-submodule consisting of elements f ∈ M such that there is a polynomial P (X) ∈ O[X] such that P (0) ∈ O × and P (U )f = 0. Let P (X) ∈ k[X] denote the reduction of P (X) modulo ̟ and factor P (X) = P 1 (X)P 2 (X), where P 1 is a power of X and P 2 (0) = 0. Hensel's lemma implies that this lifts uniquely to a factorisation P (X) = P 1 (X)P 2 (X) in O[X] where P 1 , P 2 are also monic. Then P 1 (U )P 2 (U ) = 0 in A. The image of P 1 (U ) in A/m A is a power of U , showing that P 1 (U ) ∈ A × and therefore that P 2 (U ) = 0 in A. This implies that M = M ord . For the third part, suppose that M is a finite-dimensional E-vector space, and let P (X) = det(X − U : M → M ). We can assume that all of the roots of P (X) in and and an element , . . . , 1)) v∈Sp ∈ G(F + p ). Observe that t p normalises n p . We define an endomorphism U p of A(n p , R) by the formula U p (f )(g) = n∈np/ηpnpη −1 p f (gnη p ). Then the action of U p on A(n p , R) commutes with the action of the group G(A p,∞ F + ) × t p . We define A(n p , R) ord to be the ordinary part of A(n p , R) with respect to the action of U p . Thus is an R-submodule which is invariant both under U p and the right translation action We now explain how H λ (K p ) may be compared with spaces of classical algebraic modular forms.
]submodule denoted by M µ in loc. cit., which is an O-lattice. There is an embedding which sends µ to the character µ : (ι −1 v (t v )) v∈Sp → τ ∈ Ip τ (µ(t v(τ ) )) (here v(τ ) denotes the place of F induced by τ ). We say that a character λ ∈ Hom cts (t p , O × ) is locally algebraic if it has the form λ = µψ for some µ ∈ (Z n ) Ip and finite order character ψ : t p → O × , and locally dominant algebraic if further µ ∈ (Z n + ) Ip . If K p ∈ J p S and c ≥ b ≥ 1, then we define and (Iw v (b, c)).
Choose an integer c ≥ 1 such that ψ| tp(c,c) is trivial. We can then view ψ as a character of the group K p (1, c) (by projection to the factor at p). If 1 ≤ b ≤ c and λ = µψ is locally dominant algebraic then we define M λ (K p (b, c), R) to be the set of functions f : c), R) by the formula (u · f )(g) = uf (gu). By definition, the invariants of this action are M λ (K p (b, c), R). It follows from [Ger19, Lemma 2.6] and the fact that K p is sufficiently small that M λ (K p (b ′ , c), R) is in fact a free R[t p (b)/t p (b ′ )]module and that there is an isomorphism of T S -modules, induced by the trace. We define the U p -operator on M λ (K p (b, c), R) as in [Ger19, Definition 2.8], as a Hecke operator at level K p (b, c) (normalized by a scalar if µ = 0). Then [Ger19, Lemma 2.10] shows that the natural inclusion M λ (K p (b, c) Lemma 2.19] shows that this inclusion induces an isomorphism on ordinary parts. We define S λ (K p , R) = M λ (K p (1, c), R) ord , safe in the knowledge that this is independent of the choice of integer c ≥ 1 such that ψ| tp∩K p (c,c) is trivial. The description of S λ (K p , O) ⊗ O,ι C in terms of automorphic representations of G(A F + ) is given by [Ger19, Lemma 2.5]. If λ is the trivial character, we omit it from the notation. It follows from [Ger19, Proposition 2.22] that for any locally dominant algebraic character there is a T S [U p ]-equivariant isomorphism The following proposition includes the comparison between H λ (K p ) and S λ (K p , O).
Proof. Let λ ∈ Hom cts (t p , O × ). There is an isomorphism and hence an embedding We will first show that for each c ≥ 1, A(K p × n p , O/(̟ c )) ord,tp=w0λ −1 is a finite free O/(̟ c )-module, and the natural map is an isomorphism. This will imply that H λ (K p ) is a finite free O-module and that for each c ≥ 1, there is an isomorphism Since the right-hand side here depends only on the reduction of λ mod (̟ c ), this will imply the truth of the first two parts of the proposition. If d ≥ 1 is such that λ mod (̟ c ) is trivial on t p (d), then λ mod (̟ c ) extends to a character of K p (1, d). We can compute where the maps are the natural inclusions (which respect the action of the U p operator, by [Ger19, Lemma 2.10]). Then [Ger19, Lemma 2.19] shows that these inclusions become isomorphisms after passage to ordinary part, and so we have simply To complete the proof of the first two parts of the proposition, it is enough to observe that if d ≥ 1 is such that λ mod (̟ c+1 ) is trivial on t p (d), then M (K p (d, d), O/(̟ c+1 )) ord,tp=w0λ −1 is a finite free O/(̟ c+1 )-module and the natural map is an isomorphism (both claims being true before passage to ordinary direct summands and as a consequence of K p being sufficiently small, cf. [Ger19, Lemma 2.6]).
To prove the third part of the lemma, it is enough to construct a compatible sequence of isomorphisms or equivalently a compatible sequence of isomorphisms The existence of such a sequence is the content of [Ger19, Proposition 2.22].
To formulate a corollary, we introduce some notation. Let λ ∈ Hom cts (t p , O × ) and let m ⊂ T S be a maximal ideal which is in the support of S(K p , k). Fix a place v 0 ∈ S − S p of F + which splits v 0 = w 0 w c 0 in F , and define the limit in each case running over the directed system of open compact subgroups V v0 ⊂ G(F + v0 ). Then: Corollary 6.3.
(2) There is a T S -equivariant isomorphism We recall that the representations π ν associated to an ordered partition ν of n have been defined in §3.
Proof. The O[GL n (F w0 )]-module H λ is a saturated submodule of A(K p,v0 × n p , O) (i.e. the cokernel of the inclusion map is ̟-torsion-free). In particular, it is ̟- Since H λ is smooth by definition, and H λ (K p,v0 V v0 ) m is a finite free O-module, this shows that H λ is even admissible. For any open compact subgroup V v0 , there is an isomorphism v0 V v0 , k) m , and the second part of the corollary follows by passage to the limit.
For the last part, let is a finitely generated O-module, by Theorem 2.10. Invoking Proposition 2.5 and our assumption, we find that Applying a version of Nakayama's lemma for complexes (e.g. [Sta21, Lemma 0G1U]) we find that C • = 0, and therefore that The final claim in the corollary follows from Theorem 3.5 (namely, the decomposition of RΓ c (Ω n−1,ca Fw 0 , E)).
In the next proposition, we introduce some new notation: if M is a T S -module, then T S (M ) denotes the image of the map T S → End O (M ) (and similarly for T S v0 ).
Proposition 6.4. Let λ ∈ Hom cts (t p , O × ) and K p ∈ J p S . Then: (1) There exists an n-dimensional group determinant D of G F,S with coefficients in T S (H λ (K p )) such that for any place w ∤ S of F which splits over F + , D(X − Frob w ) equals the image of P w (X) in T S (H λ (K p )).
(2) Let v 0 ∈ S − S p and suppose that K v0 = ι −1 v0 (Iw v0 ). Then the pushforward of D| WF v 0 under the inclusion ) be the trace associated to the group determinant D. Then for all σ ∈ G F,S and for all τ 1 , . . . , τ n ∈ W F v 0 , we have the relation ). Proof. The first part is a standard consequence of Corollary 4.3. Indeed, we have while for any c ≥ 1 and any d ≥ c such that λ mod (̟ c ) is trivial on there is a surjective homomorphism of T S -algebras the direct summands being indexed by the finitely many systems of T S -eigenvalues that appear in S (K p (d, d), O) ⊗ O Q p . The group determinant D exists as a consequence of existence of the family of Q p -valued determinants indexed by these summands and glueing for group determinants ([Che14, Example 2.32]). The second part of the lemma may be proved similarly. It is enough to show that for any d ≥ 1, the group determinant D d of G F,S valued in T S v0 (S (K p (d, d), O)) has the property that D d | WF v 0 equals the pushforward of D v0 . This can be checked after extending scalars to Q p , in which case it follows from [ACC + 18, Proposition 2.2.8].
The third part of the lemma is slightly more subtle, since we need to take into account Hecke operators which do not act as scalars on automorphic representations (in other words, which do not lie in the centre of the Iwahori-Hecke algebra at the place v 0 ). What we need to check is that if d ≥ 1 and T d : ) is the trace associated to the group determinant D d , then for all σ ∈ G F,S and for all τ 1 , . . . , τ n ∈ W F v 0 , we have the relation ). This can be checked in T S v0 (S (K p (d, d), O)) ⊗ O Q p , which now splits as a sum of its localisations at the finitely many systems of T S v0 -eigenvalues which appear in S (K p (d, d), Let q ⊂ T S v0 (S (K p (d, d), O)) be a minimal prime ideal, kernel of a homomorphism T S v0 (S (K p (d, d), O)) → Q p . If ∆ v0 ∈ q, then ∆ n! v0 = 0 in T S v0 (S (K p (d, d), O)) (q) , because this Artinian local ring has length at most n! (cf. [Tho,Proposition 3.11]). The required relation thus certainly holds in this localisation. If ∆ v0 ∈ q, then T S v0 (S(K p (d, d), O)) (q) is a field, the map T S v0 (S(K p (d, d), O)) (q) → Q p is an isomorphism, and we need to check that if σ is an automorphic representation of G(A F + ) which contributes to S(K p (d, d), O) and ρ = r ι (σ), then the relation (ρ(τ 1 )−(χ v0,1 mod q)(τ 1 ))(ρ(τ 2 )−(χ v0,2 mod q)(τ 2 )) . . . (ρ(τ n )−(χ v0,n mod q)(τ n )) = 0 holds in M n (Q p ). This is a consequence of local-global compatibility at the place v 0 and [NT21, Lemma 2.18].
We next record some consequences of the results of §4.
Corollary 6.6. With assumptions as in Proposition 6.5(2), suppose given a continuous character ψ : G F,S → O × of finite order such that ψψ c = 1 and for each Then there exists a unique O-algebra homomorphism f π ψ : T S (S λπ ψ (K p , O)) → Q p such that for each place w ∤ S of F which is split over F + , the image of P w (X) ∈ T S [X] equals det(X − (ρ 1 ⊗ ψ)(Frob w )) det(X − ρ 2 (Frob w )).
Proof. We need only observe that if π satisfies the hypotheses of Proposition 6.5(2), then so does π ψ .
We can describe λ π ψ in terms of λ π . Indeed, π ψ and π have the same weight µ. In the notation of Proposition 6.5(1), we can define for each v ∈ S p subsets We need a version of Corollary 6.6 "in families". With assumptions as in Proposition 6.5(2), fix an integer d ≥ 1 such that α π | tp(d) is trivial, and let F d /F be the largest abelian extension of F with the following properties: • F d is unramified outside p.
• For each place v ∈ S p , the conductor of F d, v /F v divides (̟ d v ). • The extension F d /F + is Galois and c ∈ Gal(F/F + ) acts on F d by −1.
Then ∆ d = Gal(F d /F ) is finite and any character ψ : ∆ d → O × satisfies the conditions of Corollary 6.6. The map β π , together with the Artin map, defines a homomorphism t p → ∆ d .
Assume (as we may, after possibly enlarging E) that ρ 1 , ρ 2 take values in GL n1 (O), GL n2 (O). We claim that there is an O t p -algebra homomorphism . To see this, we consider the natural embedding Taking the product over the maps f π ψ coming from Corollary 6.6 determines a map T S tp (M λπ (K p (d, d), The next proposition shows what happens when we allow the character ψ to have infinite order (the proof is by reduction to the finite order case): Proposition 6.7. With assumptions as in Proposition 6.5(2), suppose given a continuous character ψ : G F,Sp → O × such that ψψ c = 1. Let λ π ψ ∈ Hom cts (t p , O × ) be the character defind by the formula (6.2). Then, after possibly enlarging O, there exists a homomorphism T S (H λπ ψ (K p )) → O of O-algebras such that for each fi- (1, 1), k) ord . Let I γ denote the kernel of the homomorphism O t p → O determined by w 0 (γ) −1 .
Fix c ≥ 1, and let d ≥ c be an integer such that γ mod (̟ c ) is trivial on t p (d). By Proposition 6.2, there is a T S [U p ]-equivariant isomorphism Equation (6.1)). There is also an isomorphism Since M λπ (K p (d, d), O/(̟ c )) ord,tp=w0(γ) −1 is generated as an O/(̟ c )-module, hence as R-module, by m elements, it follows that which for each split place w ∤ S of F , sends P w (X) to the image of det(X − ((ρ 1 ⊗ ψ) ⊕ ρ 2 )(Frob w )) in O/(̟ ⌊c/m⌋ )[X]. Using the identification of H λπ ψ (K p )/(̟ c ), we see that there is a homomorphism with the same property with respect to the polynomials P w (X). Since m was fixed and c was arbitrary, we can pass to the limit to obtain the desired homomorphism T S (H λπ ψ (K p )) → O.
Proof. Let L/F denote the maximal abelian pro-p extension of F which is unramified away from S p , and let ∆ = Gal(L/F ). We need to show that, after possibly enlarging E, there is a homomorphism ∆/(c + 1) → 1 + ̟O sending Frob v0 to α. This will be the case if Frob v0 does not have finite order in the finitely generated Z pmodule ∆/(c+ 1), or if Frob v0 /Frob v c 0 does not have finite order in ∆. By class field theory, it suffices to show that if β ∈ O F generates a (positive) power of the ideal corresponding to the place v 0 , then β/β c has infinite order in F + , so after increasing N we can assume that (β/β c ) N is fixed by c, therefore that (β) 2N = (β c ) 2N . This is a contradiction since these ideals are distinct (v 0 splits in F ).
We can now prove our main level-raising theorem. For the reader's convenience we first state our assumptions: • F is a CM field of the form F = F + F 0 , where F 0 is an imaginary quadratic field and F/F + is everywhere unramified. • p, q are primes which split in F 0 .
• v 0 is a q-adic place of F + and u 0 is a q-adic place of F 0 . We take v 0 to be the unique place of F lying above both v 0 and u 0 . • ι : Q p → C is an isomorphism.
Theorem 6.11. With the above assumptions, fix a partition n = n 1 + n 2 and let π 1 , π 2 be cuspidal, conjugate self-dual automorphic representations of GL n1 (A F ), GL n2 (A F ) such that π = π 1 ⊞ π 2 is regular algebraic and ι-ordinary. Suppose that the following conditions are satisfied: (1) If w is a place of F such that π w is ramified, then w is split over F + .
(3) There exists a finite place w ∤ S of F , split over F + , such that there is an isomorphism r ι (π)| ss GF w ∼ = ⊕ n i=1 χ i , where χ 1 , . . . , χ n are unramified characters such that if i = j then χ i /χ j = ǫ.
(5) If w is a q-adic place of F not lying above v 0 , then π w is unramified. Then there exists a RACSDC ι-ordinary automorphic representation Π of GL n (A F ) such that r ι (Π) ∼ = r ι (π) and Π v0 is an unramified twist of St n . Moreover, if w is a q-adic place of F not lying above v 0 , then Π w is unramified; and if w is any place of F such that Π w is ramified, then w is split over F + .
Proof. We can choose S, S, and K p ∈ J p S so that the following conditions are satisfied: • S contains S p , v 0 , and each place below which π is ramified. S does not contain any q-adic places of Suppose for contradiction that the conclusion of the theorem does not hold. By Proposition 6.5, we can find (after possibly enlarging E) a locally dominant algebraic character λ π ∈ Hom cts (t p , O × ) and a maximal ideal m ⊂ T S in the support of S λπ (K p , k) ∼ = S(K p , k) such that for each place w ∤ S of F which is split over F + , P w (X) mod m = det(X − r ι (π)(Frob w )). Thus S, H λ are defined for any λ ∈ Hom cts (t p , O × ), and Corollary 6.3 shows that H λ ⊗ O k ∼ = S.
We claim that Using Lemma 6.10 and after possibly enlarging O, we can find a character ψ : G F,Sp → O × with the following properties: • ψψ c = 1.

Application to symmetric power functoriality
The following conjecture is [CT14, Conjecture 3.2].
The following conjecture is [CT14,Conjecture 3.3] in the most general case K = ∅.
The main result we will prove here is the following (cf. [CT14, Theorem 3.4]: Theorem 7.1. Let p ≥ 5 be a prime, and let 0 < r < p be an integer. Then the implication SP p−r + SP r + TP r ⇒ SP p+r holds.
Before proceeding to the proof of Theorem 7.1 we give a corollary.
(2) Assume TP r for all r ≥ 1. Then SP n holds for all n ≥ 1.
Proof. The proof is the same as the proofs of [CT14, Corollary 3.5, Corollary 3.6]. We give the details again here. We know that SP n is true for n = 1, 2, 3, 4, 5 and TP r is true for r = 1, 2, 3 (see [CT14,§3] for detailed references). Therefore we have the unconditional implications, valid for any prime number p ≥ 5: It is now elementary to check that this implies the first part of the corollary, using p = 5, . . . , 23. For the second, we use induction on n. Take an integer n > 5. Bertrand's postulate implies that there is a prime number p such that n < p < 2n. Writing n = p + r, we have 0 < r < p and we deduce SP n = SP p+r by Theorem 7.1 and induction.
We now get everything in place to prove Theorem 7.1, following the lines of [CT14].
Lemma 7.3. Let p ≥ 5 be a prime and let ϕ ∈ G Qp be a lift of arithmetic Frobenius. There exists an integer a 0 = a 0 (p) ≥ 3 such that if a ≥ a 0 and G ≤ GL 2 (F p ) is a finite subgroup which contains a conjugate of SL 2 (F p a ), then G has the following properties: (1) For each 0 < r < p, the image of the homomorphism Sym r−1 : G → GL r (F p ) is adequate, in the sense of [Tho12, Definition 2.3].
(3) If H ≤ G is a subgroup of index [G : H] < 2p, then H contains a conjugate of SL 2 (F p a ).
Proof. If a ≥ 3 then the classification of finite subgroups of PGL 2 (F p ) shows that the projective image of G is conjugate to PSL 2 (F p b ) or PGL 2 (F p b ) for some b ≥ a, and therefore that there is (after replacing G by a conjugate) a sandwich for some finite extension k/F p b .
The classification of finite subgroups of PSL 2 (F p b ) shows that as b ≥ 3, H ∩ PSL 2 (F p b ) must equal PSL 2 (F p b ) (because all the other subgroups of PGL 2 (F p b ) have index greater than 2p). In particular, H must contain SL 2 (F p b ).
Proposition 7.4. Let F be an imaginary CM field and let π be a RACSDC automorphic representation of GL 2 (A F ) of weight 0. Let p ≥ 5 be a prime and let ι : Q p → C be an isomorphism. Fix an integer 0 < r < p, and suppose that the following conditions are satisfied: (1) The extension F/F + is everywhere unramified and linearly disjoint from F + (ζ p )/F + . Moreover, F contains an imaginary quadratic field F 0 in which p splits.
(2) If v is a place of F such that π v is ramified, then v is split over F + .
(4) There exists an integer a ≥ a 0 (p) such that r ι (π)(G F ) contains a conjugate of SL 2 (F p a ). (5) There exist RACSDC automorphic representations Π 1 , Π 2 of GL r (A F ), GL p−r (A F ), respectively, such that r ι (Π 1 ) ∼ = Sym r−1 r ι (π) and r ι (Π 2 ) ∼ = Sym p−r−1 r ι (π). (6) There exists a place w 0 ∤ p of F such that π w0 is an unramified twist of the Steinberg representation and q w0 ≡ 1 mod p. The residue characteristic q of w 0 splits in F 0 , and π w is unramified for each q-adic place of F such that w = w 0 , w c 0 . Moreover r ι (π)| GF w 0 is trivial. Suppose finally that Conjecture TP r holds. Then there exists a RACSDC automorphic representation Π of GL p+r (A F ) with the the following properties: (1) Π is ι-ordinary.
(3) Π w0 is an unramified twist of the Steinberg representation.
The conclusion of the proposition now follows from Theorem 6.11 applied to Σ, provided we can check hypothesis (3) of Theorem 6.11, i.e. the existence of γ ∈ G F such that r ι (Σ)(γ), or equivalently (Sym p+r−1 r ι (π))(γ), has p + r distinct eigenvalues, no two of which are in ratio ǫ(γ). Since r ι (π)(G F (ζp) ) contains a conjugate of SL 2 (F p a ), we can choose γ ∈ G F (ζp) with image a conjugate of diag(t, t −1 ) for any t ∈ F p a such that 1, t 2 , . . . , t 2(p+r−1) are all distinct. Such elements exist because of our assumption that a ≥ 3.
We next want to state a slightly strengthened version of the main result of [ANT20]: Theorem 7.5. Let F be a CM number field, let n ≥ 1, let p ≥ 5 be a prime not dividing n, and let ρ : G F → GL n (Q p ) be a continuous representation satisfying the following conditions: (1) ρ c ∼ = ρ ∨ ǫ 1−n .
(3) There exists λ ∈ (Z n + ) Hom(F,Q p ) such that ρ is ordinary of weight λ. (4) There is an isomorphism ρ ∼ = ⊕ r i=1 ρ i , where each representation ρ i is absolutely irreducible and satisfies ρ c (5) There is a place w 0 ∤ p of F such that ρ| ss is an unramified twist of There exists an ι-ordinary RACSDC automorphic representation Π of GL n (A F ) such that r ι (Π) ∼ = ρ and Π w0 is an unramified twist of the Steinberg representation. (7) There exists a place w ∤ p of F such that ρ| GF w is unramified and H 0 (F w , ad ρ(1)) = 0. We have Moreover, ρ is primitive and ρ(G F ) has no quotient of order p.
Proof. This theorem is identical in statement to [ANT20, Theorem 6.1], except that the condition "F (ζ p ) not contained in F ker ad ρ " has been replaced by the condition "there exists a place w ∤ p of F such that ρ| GF w is unramified and H 0 (F w , ad ρ(1)) = 0". The former condition is used only to show the existence, by the Chebotarev density theorem, of a place w ∤ p of F such that ρ| GF w is unramified, q w ≡ 1 mod p, and ρ| GF w is scalar. This implies that H 0 (F w , ad ρ(1)) = 0 and therefore that any lifting of ρ| GF w is unramified. On the automorphic side, this means that we can choose a sufficiently small level subgroup at the place w without introducing any new ramification. However, we can in fact take w to be any place such that ρ| GF w is unramified and H 0 (F w , ad ρ(1)) = 0 and the rest of the argument remains unchanged.
We can use Theorem 7.5 to get the following consequence of Proposition 7.4. In contrast to the results of [CT14,CT17], there are no conditions here on the presence (or otherwise) of roots of unity in the base field F + .
Theorem 7.6. Let p ≥ 5 be a prime, let 0 < r < p be an integer, and assume Conjectures SP p−r , SP r , and TP r . Let F + be a totally real field and let (π, χ) be a RAESDC automorphic representation of GL 2 (A F + ) satisfying the following conditions: (1) There exists an isomorphism ι : Q p → C such that π is ι-ordinary and r ι (π)(G F ) contains a conjugate of SL 2 (F p a ), for some a > a 0 (p).
(3) There exists a place v 0 ∤ p such that π v0 is an unramified twist of the Steinberg representation. Then the (p + r − 1) th symmetric power of π exists: there is an ι-ordinary cuspidal RAESDC automorphic representation Π of GL p+r (A F + ) such that Sym p+r−1 r ι (π) ∼ = r ι (Π).
Proof. We are free, by soluble base change, to replace F + by any soluble totally real extension. We can therefore assume that q v0 ≡ 1 mod p and that there exists a CM quadratic extension F/F + with the following properties: • The extension F/F + is everywhere unramified and linearly disjoint from the extension of F + (ζ p ) cut out by r ι (π)| G F + (ζp ) . In particular, F ⊂ F + (ζ p ) and r ι (π)(G F ) = r ι (π)(G F + ). • F contains an imaginary quadratic field F 0 in which p and the residue characteristic q of the place v 0 split. • For each q-adic place w of F , r ι (π)| GF w is trivial.
• Each place v of F + such that π v is ramified splits in F . • There exists an everywhere unramified Hecke character ψ : Increasing a, we assume also that the projective image of r ι (π) is conjugate either to PSL 2 (F p a ) or PGL 2 (F p a ). Let π F denote the base change of π to F , and let ρ = r ι (π F ⊗ ψ −1 ), r = Sym p+r−1 ρ. The representation π F ⊗ ψ −1 is RACSDC. Let w 0 be a place of F lying above v 0 . Using [Tho12, Theorem 9.1, Theorem 10.2] again we can find another RACSDC ι-ordinary automorphic representation π ′ of GL 2 (A F ) such that r ι (π ′ ) ∼ = ρ, π ′ w0 is an unramified twist of the Steinberg representation, and such that for each q-adic place w = w 0 , w c 0 of F , π ′ w is unramified.
We can then apply Theorem 7.5 to deduce the automorphy of Sym p+r−1 ρ, using Proposition 7.4 applied to π ′ to verify the residual automorphy hypothesis, provided we can check the conditions on the residual representation r. Before we carry out these checks, we note that this will conclude the proof: the automorphy of Sym p+r−1 r ι (π) follows by quadratic descent.
The representation r has two irreducible constituents r 1 = ϕ ρ ⊗ Sym r−1 ρ, r 2 = det ρ r ⊗ Sym p−r−1 ρ of distinct dimensions, which remain irreducible on restriction to G F (ζp) , so the conditions that require an argument to check are as follows: • r is primitive.
• There exists a place w ∤ p of F such that r| GF w is unramified and H 0 (F w , ad r(1)) = 0.
We treat these in turn. If r is not primitive, then there is an isomorphism r ∼ = Ind GF GL σ for some finite extension L/F of degree > 1, hence an embedding σ ֒→ r| GL . The extension of F cut out by r has Galois group isomorphic to a subgroup of GL 2 (F p ) whose image in PGL 2 (F p ) is a conjugate of PSL 2 (F p a ) or PGL 2 (F p a ). Since we assume a ≥ a 0 (p), and [L : F ] ≤ dim r < 2p, it follows from Lemma 7.3 that r(G L ) contains a conjugate of SL 2 (F p a ), and therefore that the irreducible constituents of r remain irreducible on restriction to G L . We find that σ is isomorphic to one of r 1 | GL or r 2 | GL . In particular, σ extends to G F and we can take it outside of the induction to get an isomorphism Then for any g ∈ G F , the eigenvalues of g on Ind GF GL F p are among the roots of unity of order at most d, and 1 is an eigenvalue. The above isomorphism implies that for any g ∈ G F , r(g) has two eigenvalues whose ratio is a root of unity of order at most d. We will show that this leads to a contradiction. Indeed, since the image of ρ contains a conjugate of SL 2 (F p a ), we can find an element in the image of r which has eigenvalues t p+r−1 , t p+r−3 , . . . , t −(p+r−1) , there t ∈ F × p a is an element of order p a −1. The ratios of these eigenvalues are t 2 , . . . , t 2(p+r−1) . We'll be done therefore it none of these ratios can have order at most d. This would imply that t itself has order at most 2d(p + r − 1) < 8p 2 . As p ≥ 5 and a > 3, we have p a − 1 > 8p 2 , so this is the desired contradiction.
Since Ind GF GL F p decomposes as a direct sum of characters and r 1 , r 2 are irreducible and of distinct dimensions, this leads to a contradiction.
We now show that there exists a place w ∤ p of F such that r| GF w is unramified and H 0 (F w , ad r(1)) = 0. It suffices to find σ ∈ G F such that ǫ(σ) = −1 and if α, β ∈ F p are the eigenvalues of ρ(σ), then (α/β) i = −1 for each i = 1, . . . , p + r − 1. Since F/F + is linearly disjoint from the extension of F + (ζ p ) cut out by r ι (π)| G F + (ζp ) , it is even enough to find σ ∈ G F + such that ǫ(σ) = −1 and r ι (π) satisfies the analogous condition on eigenvalues.
Consider the homomorphism ad r ι (π) × ǫ : G F + → PGL 2 (F p ) × F × p . We can assume that the projective image of ad r ι (π) is equal to one of PSL 2 (F p a ) or PGL 2 (F p a ). Since PSL 2 (F p a ) is a simple group, the image of ad r ι (π) × ǫ contains PSL 2 (F p a ) × {1}. This image also contains (g, −1), where g ∈ PGL 2 (F p a ) has order 2 (the image of complex conjugation). The image of ad r ι (π) × ǫ therefore contains all elements of the form (gh, −1), where h ∈ PSL 2 (F p a ). We see finally that the proof will be complete if we can show that for any g ∈ PGL 2 (F p a ) of order 2, there exists h ∈ PSL 2 (F p a ) such that the eigenvalues α, β ∈ F p of gh (defined only up to scalar multiplication) satisfy (α/β) i = −1 for each i = 1, . . . , p + r − 1. This is elementary (for example, consider elements h which commute with g and use the conditions p ≥ 5, a ≥ 3).
Theorem 7.1 now follows on combining Theorem 7.6 and the following proposition.
Proposition 7.7. Let F + be a totally real number field and let (π, χ) be a RAESDC automorphic representation of GL 2 (A F + ) without CM. Let p ≥ 5 be a prime and let 0 < r < p be an integer. Then we can find a soluble totally real extension E + /F + and a RAESDC automorphic representation (π ′ , χ ′ ) of GL 2 (A E + ) without CM satisfying the following conditions: (1) Sym p+r−1 r ι (π) is automorphic, associated to a RAESDC automorphic representation of GL p+r (A F + ), if and only if Sym p+r−1 r ι (π ′ ) is automorphic.
(3) For each isomorphism ι : Q p → C, π ′ is ι-ordinary and r ι (π ′ )(G E + ) contains a conjugate of SL 2 (F p a ) for some a > a 0 (p). (4) There exists a place v 0 ∤ p of E + such that π ′ v0 is an unramified twist of the Steinberg representation.

The mixed parity case
Let F + be a totally real number field. If F + = Q, then not every cuspidal Hilbert modular eigenform of regular weight over F + admits an associated Galois representation. This is related to the fact that not every cuspidal automorphic representation π of GL 2 (A F + ) such that π ∞ is essentially square-integrable admits a twist which is algebraic. In order to deduce Theorems B and C of the introduction from the results of the previous section, we therefore need an additional argument. As a warm-up, we first review the description of those π which are indeed algebraic. The arguments that follow will extend those given in [CT17,§7].
The local Langlands correspondence rec R for GL 2 (R) gives a bijection between the set of infinitesimal equivalence clases of essentially square-integrable irreducible admissible representations of GL 2 (R) and the set of conjugacy classes of continuous irreducible representations W R → GL 2 (C). Each such representation of W R is isomorphic to a unique one of the form ρ s,a = Ind WR C × χ s,a for (s, a) ∈ C × Z ≥1 , and where χ s,a (re iθ ) = r s e iaθ . The corresponding representation of GL 2 (R) is the unique essentially square-integrable subquotient of the unitary induction of the character (t 1 , t 2 ) → |t 1 | (s+a)/2 sgn(t 1 ) a+1 |t 2 | (s−a)/2 of the maximal torus T 2 (R) = R × × R × of GL 2 (R).
Suppose given integers k v ∈ Z ≥0 for each place v|∞ of F + and w ∈ R. Cuspidal Hilbert modular forms of weights ((k v + 2) v|∞ , w) may be lifted to cuspidal automorphic forms. In particular, [Oht83, Proposition 5.2.3] explains how a cuspidal Hilbert modular eigenform of weights ((k v + 2) v|∞ , w) determines a cuspidal automorphic representation π of GL 2 (A F + ) such that for each place v|∞ of F + , rec Fv (π v ) = ρ −s,kv +1 (and conversely). Repeating the computations of [Clo90, we see that π is regular algebraic if and only if w is an integer and k v ≡ w mod 2 for each place v|∞; and that π admits a character twist which is regular algebraic if and only if the parity of k v is independent of v|∞.
Here then is the formulation of symmetric power functoriality for a more general class of automorphic representations of GL 2 (A F + ) than the RAESDC ones considered in the previous section.
Conjecture (SP ′ n ). Let n ≥ 1. Let F + be a totally real field, and let π be a cuspidal automorphic representation of GL 2 (A F + ) without CM, such that π ∞ is essentially square-integrable. Then there exists a cuspidal automorphic representation Π of GL n (A F + ) such that for each place v of F + , we have . Our goal will be to prove the following result.
Theorem 8.1. Conjecture SP n implies Conjecture SP ′ n . Together with Corollary 7.2, this theorem implies Theorems B and C of the introduction. The rest of this section is devoted to its proof. Let π be a cuspidal automorphic representation of GL 2 (A F + ) without CM such that π ∞ is essentially square-integrable, and fix n ≥ 1. Assuming Conjecture SP n , we will show Sym n−1 π exists in the sense of Conjecture SP ′ n . We first quote what is proved in [CT17,§7], assuming SP n .
Lemma 8.2. Let F/F + be a quadratic CM extension. Let π F denote the base change of π to F .
Proposition 8.3. There exists a cuspidal automorphic representation Π of GL n (A F + ) satisfying the following properties: (1) For any finite place v of F + , The existence of a Π having the first property is part of the content of [CT17, Theorem 7.1]; the proof of this in loc. cit. appears incomplete, so we give a more detailed explanation here as well as a description of the lifting at the infinite place.
Proof. We use an automorphic version of the patching argument of [BR93, Proposition 4.3.1]. Let F/F + be a quadratic CM extension and let ψ, Π 1 be as in the statement of Lemma 8.2. Then we have rec Fw (Π 1,w ) ∼ = Sym n−1 rec Fw ((π F ⊗ ψ −1 ) w ) for every place w of F . Indeed, if w is a finite place, then this follows from localglobal compatibility for the representations r ι (Π 1 ), r ι (π F ⊗ ψ −1 ). If w is an infinite place then rec Fw (Π 1,w ) can be read off from the Hodge-Tate weights of r ι (Π 1 ) (use the recipe of [CT14, Theorem 2.2] -here we are using the fact that Π 1,w is essentially tempered), so the desired result holds also at the infinite places. Twisting by a character, we find that we also have rec Fw ((Π 1 ⊗ ψ n−1 ) w ) ∼ = Sym n−1 rec Fw (π F,w ) for every place w of F . In particular, Π 1 ⊗ ψ n−1 is stable under the action of Gal(F/F + ) so by [AC89, Ch. 3, Theorem 4.2] there exists a cuspidal automorphic representation Π(F ) of GL n (A F + ) with base change Π 1 ⊗ ψ n−1 , determined up to twist by the quadratic Hecke character associated to the extension F/F + . For any place w of F lying above a place v of F + , we have (8.1) rec Fw (Π(F ) F,w ) ∼ = Sym n−1 rec F + v (π v )| WF w . In particular, if v splits in F then we have . We now choose an infinite sequence F 0 , F 1 , F 2 . . . of CM quadratic extensions of F + with the following properties: • For each 0 ≤ i < j < k, [F i F j F k : F + ] = 8.
• For each finite place v of F + , there exists i > 0 such that v splits in F i . We can construct such a sequence inductively by enumerating the finite places v 1 , v 2 , . . . of F + , choosing F 0 arbitrarily, and in general choosing F n so that v n splits in F n and also so that for each quadratic subfield M of F 0 . . . F n−1 /F + , there exist a place v of F + which is inert in M and split in F n . For any i ≥ 0, we write ǫ i for the quadratic Hecke character associated to the extension F i /F + .
For each i ≥ 0 we are given a cuspidal automorphic representation Π(F i ) of GL n (A F + ). For any i < j, the base change representations Π(F i ) FiFj and Π(F j ) FiFj are cuspidal and isomorphic. Indeed, they are isomorphic by (8.1). To show they are cuspidal, it's enough to show that (Π(F i ) Fi ⊗ ψ 1−n ) FiFj is cuspidal. However, Π(F i ) Fi ⊗ ψ 1−n is RACSDC and r ι (Π(F i ) Fi ⊗ ψ 1−n )| GF i F j is irreducible (as the Zariski closure of its image contains a conjugate of the principal SL 2 , by [CT17,Theorem 7.3] and [NT, Example 2.34]). It is therefore enough to know that if σ is a RACSDC automorphic representation of GL n (A Fi ) and r ι (σ)| GF i F j is irreducible, then σ FiFj is cuspidal. This is true, because otherwise [AC89, Ch. 3, Theorem 4.2] would imply that σ is isomorphic to its twist by the quadratic Hecke character associated to the extension F i F j /F i , and therefore that r ι (σ) is induced from G FiFj and reducible on restriction to this subgroup, a contradiction.
The base change of Σ 0i to F i F j F k is cuspidal (by the same argument as before), so by [AC89, Ch. 3, Theorem 4.2] the character ǫ n0 0 ǫ ni i ǫ nj j must be trivial. By hypothesis the map is an isomorphism, so this is possible only if n 0 = n i = n j = 0 and the representations Σ 0i (for any i > 0) are all isomorphic. We set Π = Σ 01 = Σ 0i .
To complete the proof, we need to explain why Π has the required properties. If v is a finite place, we choose F i such that v splits in F i . Then (8.3) shows that Π v has the expected form. If v is an infinite place, then the desired property follows already from (8.1).
This completes the proof of Theorem 8.1.