Modularity of PGL2(𝔽p)-representations over totally real fields

Significance The connection between modular forms and Galois representations plays a significant role in modern algebraic number theory. J.-P. Serre made an influential conjecture relating mod p modular forms and mod p representations of the absolute Galois group of Q. Such a relationship has consequences for classical Diophantine questions, for example implying Fermat’s Last Theorem, and is also a mod p analogue of the Langlands program. It is thus important to study analogues of Serre’s conjecture in the broadest possible context. Serre’s modularity conjecture, definitively stated in 1986, was proved by Khare–Wintenberger in 2009. In this paper we prove new cases of extensions of Serre’s conjecture to mod p representations of absolute Galois groups of totally real number fields.

We study an analog of Serre's modularity conjecture for projective representations ρ : Gal(K/K) → PGL 2 (k), where K is a totally real number field. We prove cases of this conjecture when k = F 5 .
number theory | modular forms | Galois representations L et K be a number field, and consider a continuous representation where k is a finite field. (Here GK denotes the absolute Galois group of K ; for this and other notation, see 1.A. Notation below.) We say that ρ is of Serre type, or S type, if it is absolutely irreducible and totally odd, in the sense that for each real place v of K and each associated complex conjugation cv ∈ GK , det ρ(cv ) = −1.
Serre's conjecture and its generalizations assert that any ρ of S type should be automorphic (see for example refs. 1 and 2 in the case K = Q, ref. 3 when K is totally real, and ref. 4 for a general number field K ). The meaning of the word "automorphic" depends on the context but when K is totally real, for example, we can ask for ρ to be associated to a cuspidal automorphic representation π of GL2(AK ), which is regular algebraic of weight 0 (2.A. Automorphy of Linear and Projective Representations). Serre's conjecture is now a theorem when K = Q (5, 6). For a totally real field K , some results are available when k is "small." These are summarized in Theorem 1.1, which relies upon refs. 2 and 7-10: Theorem 1.1. Let K be a totally real number field, and let ρ : GK → GL2(k ) be a representation of S type. Then ρ is automorphic provided |k | ∈ {2, 3, 4, 5, 7, 9}.
One can equally consider continuous representations where again k is a finite field. We say that σ is of S type if it is absolutely irreducible and totally odd, in the sense that if k has odd characteristic, then for each real place v of K , σ(cv ) is nontrivial. One could formulate a projective analog of Serre's conjecture, asking that any representation σ of S type be automorphic. A theorem of Tate implies that σ lifts to a linear representation valued in GL2(k ) for some finite extension k /k , and by σ being automorphic we mean that a lift of it to a linear representation is automorphic (2.A. Automorphy of Linear and Projective Representations). Thus, if k is allowed to vary, this conjecture is equivalent to Serre's conjecture, since any representation ρ has an associated projective representation Proj(ρ), and any projective representation σ lifts to a representation valued in GL2(k ) for some finite extension k /k ; moreover, ρ is of S type if and only if Proj(ρ) is, and ρ is automorphic if and only if Proj(ρ) is. However, for fixed k the two conjectures are not equivalent: Certainly if ρ is valued in GL2(k ) then Proj(ρ) takes values in PGL2(k ), but it is not true that any representation σ : GK → PGL2(k ) admits a lift valued in GL2(k ), and in fact in general the determination of the minimal extension k /k such that there is a lift to GL2(k ) is somewhat subtle. It is therefore of interest to ask whether the consideration of projective representations allows one to expand the list of "known" cases of Serre's conjecture.
Our main theorem affirms that this is indeed the case. Before giving the statement we need to introduce one more piece of notation. We write ∆ : PGL2(k ) → k × /(k × ) 2 for the homomorphism induced by the determinant. We say that a homomorphism GK → k × /(k × ) 2 is totally even (resp. totally odd) if each complex conjugation in GK is a trivial (resp. nontrivial) image. Theorem 1.2. Let K be a totally real number field, and let σ : GK → PGL2(k ) be a representation of S type. Then σ is automorphic provided that one of the following conditions is satisfied: 2) |k | = 5, [K (ζ5) : K ] = 4, and ∆ • σ is totally even.
The proof of Theorem 1.2 falls into three cases. The first one is when |k | is even or k = F3. When |k | is even, the homomorphism GL2(k ) → PGL2(k ) splits, so we reduce easily to Theorem 1.1. When k = F3, the homomorphism PGL2(Z[ √ −2]) → PGL2(F3) splits and we can use the Langlands-Tunnell theorem (7) to establish the automorphy of σ.
The second case is when |k | is odd and −1 is a square in k (resp. a nonsquare in k ) and ∆ • σ is totally even (resp. totally odd). In this case we are able to construct the following data: • A solvable totally real extension L/K and a representation ρ 1 : GL → GL2(k ) such that Proj(ρ 1 ) = σ|G L (by showing that L/K can be chosen to kill the Galois cohomological obstruction to lifting). • A representation ρ2 : GK → GL2(Q p ) such that Proj(ρ 2 ) and σ are conjugate in PGL2(Fp) (by choosing an arbitrary lift of σ to GL2(Fp) and applying the Khare-Wintenberger method).
We can then use Theorem 1.1 to verify the automorphy of ρ 1 and hence the residual automorphy of ρ 2 |G L . An automorphy lifting theorem then implies the automorphy of ρ2|G L , hence ρ2 itself by solvable descent, and hence finally of σ.
The final case is when k = F5 and ∆ • σ is totally odd. In this case there does not exist any totally real extension L/K such that σ|G L lifts to a representation valued in GL2(k ) (there is a local obstruction at the real places). However, it is possible to find a CM extension L/K such that σ|G L lifts to a representation valued in GL2(k ) with determinant the cyclotomic character. (By definition, a CM number field is a quadratic, totally imaginary extension of a totally real field.) When k = F5, such a representation necessarily appears in the group of 5-torsion points of an elliptic curve over L (8) and so we can use the automorphy results over CM fields established in ref. 11 together with a solvable descent argument to obtain the automorphy of σ. This "2-3 switch" strategy can also be used to prove the automorphy of representations σ : GK → PGL2(F3) with ∆ • σ totally odd using the 2-adic automorphy theorems proved in ref. 12; see Theorem 3.1. This class of representations includes the projective representations associated to the Galois action on the 3-torsion points of an elliptic curve over K . This gives a way to verify the modulo 3 residual automorphy of elliptic curves over K , which does not rely on the Langlands-Tunnell theorem (and in particular refs. 13 and 14) but only on the Saito-Shintani lifting for holomorphic Hilbert modular forms (15). (We note that we do need to use the Langlands-Tunnell theorem to prove the automorphy of representations σ : GK → PGL2(F3) with ∆ • σ totally even; cf. Theorem 2.12.) We now describe the structure of this paper. We begin in 2. Lifting Representations by studying the lifts of projective representations and collecting various results about the existence of characteristic 0 lifts of residual representations and their automorphy. We are then able to give the proofs of Theorem 1.1 and the first two cases in the proof of Theorem 1.

2.
A. Notation. If K is a perfect field, then we write GK = Gal(K /K ) for the Galois group of K with respect to a fixed choice of algebraic closure. If K is a number field and v is a place of K , then we write Kv for the completion of K at v and fix an embedding K → K v extending the natural embedding K → Kv ; this determines an injective homomorphism GK v → GK . If v is a finite place of K , then we write Frobv ∈ GK v for a lift of the geometric Frobenius, k (v ) for the residue field of Kv , and qv for the cardinality of Kv ; if v is a real place, then we write cv ∈ GK v for complex conjugation. Any homomorphism from a Galois group GK to another topological group will be assumed to be continuous.
If p is a prime and K is a field of characteristic 0, then we write : GK → Z × p for the p-adic cyclotomic character, : GK → F × p for its reduction modulo p, and ω : is a representation, then we write ρ : GK → GLn (Fp) for the associated semisimple residual representation (uniquely determined up to conjugation).
If k is a field, then we write Proj : GLn (k ) → PGLn (k ) for the natural projection and ∆ : PGLn (k ) → k × /(k × ) n for the character induced by the determinant. We use these maps only in the case n = 2.
If K is a field of characteristic 0, E is an elliptic curve over K , and p is a prime, then we write ρ E ,p : GK → GL2(Fp) for the representation associated to H 1 (E K , Fp) after a choice of basis. Thus det ρ E ,p = −1 .

Lifting Representations
In this section we study different kinds of liftings of representations: liftings to characteristic 0 (and the automorphy of such liftings) and liftings of projective representations to true (linear) representations. We begin by discussing what it means for a (projective or linear) representation to be automorphic.
A. Automorphy of Linear and Projective Representations. Let K be a CM or totally real number field. If π is a cuspidal, regular algebraic automorphic representation of GL2(AK ), then (16,17) for any isomorphism ι : Q p → C, there exists a semisimple representation rι(π) : GK → GL2(Q p ) satisfying the following condition, which determines rι(π) uniquely up to conjugation: For all but finitely many finite places v of K such that πv is unramified, rι(π)|G Kv is unramified and rι(π)| ss G Kv is related to the representation ι −1 πv under the Tate-normalized unramified local Langlands correspondence. (See ref. 18, section 2 for an explanation of how the characteristic polynomial of rι(π)|G Kv may be expressed in terms of the eigenvalues' explicit unramified Hecke operators.) In this paper we need only to consider automorphic representations that are of regular algebraic automorphic representations π that are of weight 0, in the sense that for each place v |∞ of K , πv has the same infinitesimal character as the trivial representation.
Let k be a finite field of characteristic p, viewed inside its algebraic closure Fp. In this paper, we say that a representation ρ : GK → GL2(k ) is automorphic if it is GL2(Fp) conjugate to a representation of the form rι(π), where π is a cuspidal, regular algebraic automorphic representation of GL2(AK ) of weight 0. We say that a representation σ : conjugate to a representation of the form Proj(rι(π)), where π is a cuspidal, regular algebraic automorphic representation of GL2(AK ) of weight 0.
We say that a representation ρ : GK → GL2(Q p ) is automorphic if it is conjugate to a representation of the form rι(π), where π is a cuspidal, regular algebraic automorphic representation of GL2(AK ) of weight 0. We say that an elliptic curve E over K is modular if the representation of GK afforded by H 1 (E K , Qp) is automorphic in this sense.
Lemma 2.1. Let K be a CM or totally real number field, let ρ : GK → GL2(k ) be a representation, and let σ = Proj(ρ). Then, 1) Let χ : GK → k × be a character. Then ρ is automorphic if and only if ρ ⊗ χ is automorphic.

2) σ is automorphic if and only if ρ is automorphic.
Proof: If χ : GK → k × is a character, then its Teichmüller lift X : GK → Q × p is associated, by class field theory, to a finite-order Hecke character Ξ : If π is a cuspidal automorphic representation that is regular algebraic of weight 0 and rι(π) is conjugate to ρ, then π ⊗ (Ξ • det) is also cuspidal and regular algebraic of weight 0 and rι(π ⊗ (Ξ • det)) is conjugate to ρ ⊗ χ.
It is clear from the definition that if ρ is automorphic, then so is σ. Conversely, if σ is automorphic, then there is a cuspidal, regular algebraic automorphic representation π of GL2(AK ) and isomorphism ι : Q p → C such that Proj(rι(π)) = Proj(ρ). It follows that there exists a character χ : GK → F × p such that ρ is conjugate to rι(π) ⊗ χ. The automorphy of ρ follows from the first part of Lemma 2.1.
B. Lifting to Characteristic 0. We recall a result on the existence of liftings with prescribed properties. We first need to say what it means for a representation to be exceptional. If K is a number field and σ : GK → PGL2(k ) is a projective representation, we say that σ is exceptional if it is PGL2(Fp) conjugate to a representation σ : GK → PGL2(F5) such that σ (GK ) contains PSL2(F5) and the character (−1) ∆•σ is trivial. [Here we write (−1) ∆•σ for the composition of ∆ • σ with the unique isomorphism F × 5 /(F × 5 ) 2 ∼ = {±1}.] We say that a representation ρ : GK → GL2(k ) is exceptional if Proj(ρ) is exceptional. If K is totally real, then this is equivalent to the definition given in ref. 19, section 3. The exceptional case is often excluded in the statements of automorphy lifting theorems (the root cause being the nontriviality of the group H 1 (σ(GK ), Ad 0 ρ(1))).
Theorem 2.2. Let K be a totally real field, let ρ : GK → GL2(k ) be a representation of S type, and let ψ : GK → Z × p be a continuous character lifting det ρ such that ψ is of finite order. Suppose that the following conditions are satisfied: Then ρ lifts to a continuous representation ρ : GK → GL2(Zp) satisfying the following conditions: We now combine the previous two theorems to obtain a "solvable descent of automorphy" theorem for residual representations, along similar lines to refs. 23 and 24.

1) For all but finitely many places
Proposition 2.4. Let K be a totally real number field and let ρ : GK → GL2(k ) be a representation of S type. Suppose that there exists a solvable totally real extension L/K such that the following conditions are satisfied: 1) p > 2 and ρ|G L(ζp ) is absolutely irreducible. If p = 5, then ρ is nonexceptional.
Then ρ is automorphic. Proof: Let ψ : GK → Z × p be the character such that ψ is the Teichmüller lift of (det ρ) , and let ρ : GK → GL2(Zp) be the lift of ρ whose existence is asserted by Theorem 2.2. Then Theorem 2.3 implies the automorphy of ρ|G L , and the automorphy of ρ itself and hence of ρ follows by cyclic descent, using the results of Langlands (13).
We can now give the proof of Theorem 1.1, which we restate here for the convenience of the reader: Theorem 2.5. Let K be a totally real field and let ρ : GK → GL2(k ) be a representation of S type. Suppose that |k | ∈ {2, 3, 4, 5, 7, 9}. Then ρ is automorphic. Proof: Many of the results we quote here are stated in the case of K = Q but hold more generally for totally real fields with minor modification. We apply them in the more general setting without further comment. If ρ is dihedral, then this is a consequence of results of Hecke (ref. 2, section 5.1). If k = F3, it is a consequence of the Langlands-Tunnell theorem (7) (see the discussion following theorem 5.1 in ref. 25, chap. 5). We may thus assume for the remainder of the proof that |k | > 3. We may also assume that for any abelian extension L/K , the restriction ρ|G L(ζp ) is absolutely irreducible (as otherwise ρ would be dihedral).
Next suppose that k = F5. We note that ρ is not exceptional, by ref. Moreover, if K is a CM field, we can choose L also to be a CM field. Proof: Let H denote the 2-Sylow subgroup of k × , of order 2 m , and let H ≤ k × denote its prime-to-2 complement. If 0 ≤ k ≤ m, we write G k = GL2(k )/(2 m−k H × H ), which is an extension We show by induction on k ≥ 0 that we can find a solvable, S -split extension L k /K and a homomorphism ρ k : GL k → G k lifting σ|G L k and such that for each v ∈ S and each place w |v of L k , ρ k |G L k ,w = ρv mod For the induction step, suppose the induction hypothesis holds for a fixed value of k . We consider the obstruction to lifting ρ k to a homomorphism ρ k +1 : GL k → G k +1 . This defines an element of H 2 (GL k , Z/2Z), which is locally trivial at the places of L k lying above S . We can therefore find an extension of the form L k +1 = L k · E k +1 , where E k +1 /K is a solvable S -split extension, such that the image of this obstruction class in H 2 (GL k +1 , Z/2Z) vanishes and so there is a homomorphism ρ k +1 : GL k +1 → G k +1 lifting ρ k |G L k +1 .
It remains to explain why we can choose K to be CM if L is. Since the extensions E k in the proof are required only to satisfy some local conditions, which are vacuous if K is CM, we can choose the fields E k to be of the form KE k , where E k is a totally real extension, in which case the field L constructed in the proof is seen to be CM. Remark 2.9: We remark that if v is a real place of K and σ(cv ) = 1, then there exists a lift of σ|G Kv to GL2(k ) if and only if either −1 is a square in k × and ∆ • σ(cv ) = 1 or −1 is not a square in k × and ∆ • σ(cv ) = 1. We also note the utility of the "S -split" condition: We can add any set of places at which σ is unramified to S and in this way ensure that the S -split extension L/K is linearly disjoint from any other fixed finite extension of K .
Here is a variant.
Lemma 2.10. Suppose that p > 2. Let K be a number field, let σ : GK → PGL2(k ) be a homomorphism, and let χ : GK → k × be a character. Suppose that the following conditions are satisfied: 2) For each finite place v of K , σ|G Kv and χ|G Kv are unramified.
We now prove an analog of Proposition 2.4 for projective representations.
Proposition 2.11. Let K be a totally real number field and let σ : GK → PGL2(k ) be a representation of S type. Suppose that there exists a solvable totally real extension L/K satisfying the following conditions: 1) p > 2 and σ|G L(ζp ) is absolutely irreducible. If p = 5, then σ is nonexceptional.
Then σ is automorphic. Proof: By Lemma 2.7, we can lift σ to a representation ρ : GK → GL2(k ). Then ρ|G L is automorphic and we can apply Proposition 2.4 to conclude that ρ is automorphic and hence that σ is automorphic.
We are now in a position to establish a large part of Theorem 1.2.
Theorem 2.12. Let K be a totally real number field and let σ : GK → PGL2(k ) be a representation of S type. If one of the following conditions holds, then σ is automorphic: 2) |k | = 5 or 9 and ∆ • σ is totally even. If |k | = 5, then σ is nonexceptional.
Proof: When k = F2 or F4, the map SL2(k ) → PGL2(k ) is an isomorphism, so σ trivially lifts to a GL2(k ) representation and we can apply Theorem 2.5. The case when |k | = 3 follows from ref. 7. In the other cases, we can assume that σ|G K (ζp ) is absolutely irreducible (as otherwise σ lifts to a dihedral representation). Let S∞ be the set of infinite places of K and choose a finite set S of finite places of K at which σ is unramified such that Gal(K ker(σ| G K (ζp ) ) /K ) is generated by {Frobv } v ∈S . We can apply Lemma 2.8, see also Remark 2.9, with S = S∞ ∪ S to find a solvable, totally real extension L/K such that σ lifts to a representation ρ : GL → GL2(k ) such that ρ|G L(ζp ) is absolutely irreducible and ρ is not exceptional if p = 5. Then Theorem 2.5 implies the automorphy of ρ and Proposition 2.11 implies the automorphy of σ, as desired.

Modularity of Mod 3 Representations
In this section, which is a warmup for the next one, we give a proof of the following theorem that does not depend on the Langlands-Tunnell theorem: Theorem 3.1. Let K be a totally real number field, and let σ : GK → PGL2(F3) be a representation of S type such that ∆ • σ is totally odd. Then σ is automorphic. Proof: We can assume that σ is not dihedral; by the classification of finite subgroups of PGL2(F3), we can therefore assume that σ(GK ) contains PSL2(F3). By Proposition 2.11, we can moreover assume, after replacing K by a solvable totally real extension, that σ is everywhere unramified and that for each place v |2 of K , qv ≡ 1 mod 3 and σ|G Kv is trivial. Lemma 3.2. There exists a solvable totally real extension L/K and a modular elliptic curve E over L satisfying the following conditions: 1) σ(GL) contains PSL2(F3). In particular, σ|G L is of S type.
We can then apply ref. 11, lemma 9.7 to conclude that there exists an elliptic curve E /L satisfying the following conditions: • There is an isomorphism ρ E ,3 ∼ = ρ.
• For each place v |2 of L, E has multiplicative reduction at v and the valuation at v of the minimal discriminant of E is 3.
Then ref. 12, p. 1237, corollary implies that E is modular, proving the lemma. We see that σ|G L is automorphic. We can then apply Proposition 2.11 to conclude that σ itself is automorphic, as required.
• For each place v |5 of K , ζ5 ∈ Kv , ρ|G Kv is trivial, and ρ|G Kv is ordinary, in the sense of ref. 18, section 5.1.
• Let χ = det ρ. Then χ has finite-order prime to 5 and for each finite place v of K , χ |G Kv is unramified. In particular, χ is everywhere unramified.
Proof: The representation τ |G K (ζ 5 ) is absolutely irreducible because its projective image contains PSL2(F5). If ζ5 ∈ K , then √ 5 ∈ K and so K = K (∆ • σ) = K (ζ5); this possibility is ruled out because σ is nonexceptional. It follows that τ is nonexceptional. The representation τ is decomposed generic because ρ|G K is (and this condition depends only on the associated projective representation).
Due to Lemma 4.3, we can apply ref. 11, lemma 9.7 and ref. 11, corollary 9.13 to conclude the existence of a modular elliptic curve E over K such that ρ E ,5 ∼ = τ and for each place v |5 of K , E has multiplicative reduction at the place v . We can then apply the automorphy lifting theorem (ref. 11, theorem 8.1) to conclude that ρ|G K ⊗ ψ is automorphic and hence that ρ|G K is automorphic. It follows by cyclic descent (13) that ρ and hence σ are also automorphic, and this completes the proof.
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