Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms

In this paper we study the compatible family of degree-4 Scholl representations $\rho_{\ell}$ associated with a space $S$ of weight $\kappa>2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It is shown that when either $K$ is totally real or $\kappa$ is odd, $\rho_\ell$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $GL_4(\mathbb Q)$. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $S$ and those of certain automorphic forms via the three-term Atkin and Swinnerton-Dyer congruences.


Introduction
To a d-dimensional space S κ (Γ) of cusp forms of weight κ > 2 for a noncongruence subgroup Γ under general assumptions, in [Sch85] Scholl attached a compatible family of 2d-dimensional ℓ-adic representations ρ ℓ of G := G Q = Gal(Q/Q). Due to the motivic nature of Scholl's construction, one expects these representations to be automorphic in the sense that they are related to automorphic forms as predicted by Langlands. For general Scholl representations, this is too much to hope for at present, owing to the ineffective Hecke operators on noncongruence modular forms and the currently available modularity techniques; but it is achievable if the space admits extra symmetries, 2000 Mathematics Subject Classification. Primary 11F11; secondary; 11F80 . 1 posthumous. The second author is supported in part by the NSF grant DMS-0801096, the third author by the NSF grant DMS-0901360 and the fourth author by the NSA grant H98230-08-1-0076 and the NSF grant DMS-1001332. Part of the paper was written when the fourth author was visiting the National Center for Theoretical Sciences in Hsinchu, Taiwan, and the University of California at Santa Cruz. She would like to thank both institutions for their hospitality.
1 as shown in [LLY05,ALL08,Lon08]. In each of these examples, the automorphy was established using the Faltings-Serre modularity technique, which becomes inefficient for general situation. On the other hand, tremendous progress in modularity has been made in recent years. In this paper, we prove the automorphy of degree-4 Scholl representations which admit quaternion multiplication by using modern technology. Moreover, the Atkin and Swinnerton-Dyer congruences for forms in the underlying space are also established. Now we outline our method and main results. The recent settlement of Serre's conjecture over Q (cf. Theorem 2.1.2) by Khare,Wintenberger and Kisin ([KW09], [Kis09a]) is useful for establishing the automorphy of 2-dimensional representations of G coming from geometry. In our situation, the Scholl representations ρ ℓ were constructed from geometry, as a variation of Deligne's construction of ℓ-adic Galois representations for congruence cusp forms. They have Hodge-Tate weights 0 and 1 − κ, each of multiplicity d, and all eigenvalues of the characteristic polynomial of a geometric Frobenius element are algebraic integers with the same complex absolute value. When d = 1, by appealing to the now established Serre's conjecture, one concludes that ρ ℓ arises from a congruence newform. When d > 1, all existing potential automorphy criteria, such as [BLTDR10], assume the regularity condition that the Hodge-Tate weights are distinct, hence they cannot be applied to Scholl representations directly. In this paper we consider the case d = 2 and the representation ρ ℓ has Quaternion Multiplication (QM) over a field K. We prove that when K is quadratic over Q, after extending the scalar field, ρ ℓ decomposes into the sum of two modular representations (Theorem 4.2.3). In the more interesting case that K is biquadratic over Q, we show that ρ ℓ is a tensor product of two 2-dimensional projective representationsρ andγ of G, in whichγ has finite image. By using Tate's vanishing theorem H 2 (G, C × ) = 0, we liftγ to an (ordinary) representation γ of G with finite image. Consequently,ρ can be lifted to a representation η of G such that ρ ℓ = η ⊗ γ is a tensor product of two degree 2 representations of G. Of these γ is induced from a finite character of the absolute Galois group G F of a quadratic field F contained in K, hence is automorphic. Moreover, it is odd if K is not totally real, and even otherwise (Theorem 3.2.1 and Proposition 3.2.3). The other component η is shown to be modular, arising from a weight κ newform of a congruence subgroup (Theorem 4.2.4). In fact, for each quadratic F contained in K, ρ ℓ is induced from a degree 2 representation of G F which comes from an automorphic form of GL 2 (A F ) (Theorem 4.3.1). Consequently, for K biquadratic over Q, the automorphy of ρ ℓ can be seen in many ways: in addition to what is described above, it also corresponds to an automorphic representation of GL 2 (A Q ) × GL 2 (A Q ), which also implies that it comes from an automorphic representation of GL 4 (A Q ) by a result of Ramakrishnan [Ram00] (Remark 4.2.5).
The automorphy of Scholl representations is useful for understanding arithmetic properties of noncongruence modular forms. An application of the above modularity/automorphy result is an intriguing link between the coefficients of noncongruence cusp forms and congruence automorphic forms via Atkin and Swinnerton-Dyer congruences. See §4.3 for details. Moreover, automorphy and Atkin and Swinnerton-Dyer congruences combined can be used to establish the d = 1 case of the unbounded denominator conjecture, a fundamental characterization for the Fourier coefficients of genuine noncongruence modular forms (cf. [LL10]). Our approach can be extended to handle other kinds of symmetries in a more general context, which will be dealt with in our future work. This paper is organized in the following manner. In §2, we obtain a simpler modularity criterion for 2-dimensional representation of G. §3 is devoted to the study of 4-dimensional Galois representations with QM over a biquadratic field. In §4, we prove the automorphy of Scholl representations of G for the cases d = 1 and d = 2 with QM, as well as the implications to Atkin and Swinnerton-Dyer congruences. The main results are recorded in Theorems 4.1.1, 4.2.4, 4.3.1 and 4.3.2. To illustrate our main results and methods, explicit examples are exhibited in §5. We recast the known results on automorphy and ASD congruences obtained in [LLY05,ALL08,Lon08] in the framework of QM. A new example of Scholl representations admitting QM over a biquadratic field is also given. The novelty of this example is the role played by the Atkin-Lehner involution. The automorphy of Galois representations and the conjectural ASD congruences are established.
The authors would like to thank Henri Darmon and Siu-Hung Ng for enlightening conversations, Luis Dieulefait for communicating to us modularity results.
1.1. Notation. Given a number field F , denote by O F its ring of integers and G F the absolute Galois group Gal(F /F ). For any finite prime l of O F dividing the rational prime ℓ, write F l for the completion of F at l and k l := O F /l its residue field. Its Galois group G F l := Gal(Q ℓ /F l ) is identified with a decomposition group of l in G F . Write I F l ⊂ G F l for the inertia subgroup over l and Fr l the arithmetic Frobenius over l, which is the usual topological generator of G F l /I F l ≃ Gal(k l /k l ) sending x ∈k l to x |k l | . The geometric Frobenius, which is Fr −1 l , is denoted by Frob l . We reserve G for G Q , G ℓ for G Q ℓ and I ℓ for I Q ℓ . For each ℓ we fix an embedding ι ℓ : Q ℓ ֒→ C. By an ℓ-adic Galois representation over E we mean a continuous representation ρ : where H is a subgroup of G and V a finite-dimensional vector space over E, a finite extension of Q ℓ . The reference to the ℓ-adic field E will be dropped when it is not important. We refer ρ to V if no confusions arise. Denote by ρ ss or V ss the semi-simplification of ρ. Let ρ be a Galois representation of H as above. If H is compact then there exists an Let m E be the maximal ideal of O E . We get a residual representationρ of H on the vector space T /m E T over the residue field k := O E /m E . Althoughρ depends on the choice of T , its semisimplificationρ ss does not. Two continuous linear representations ρ i , i = 1, 2, of the topological group H acting on finite-dimensional E i -vector spaces V i , where E 1 and E 2 are contained in a finite common extension E, are said to be equivalent, denoted

Modularity of degree two ℓ-adic Galois representations
In this section, we derive from known modularity results for 2-dimensional ℓ-adic Galois representations of G a useful modularity criterion for applications in later sections.
2.1. Modularity of 2-dimensional ℓ-adic Galois representations. Let κ ≥ 2 and N ≥ 1 be integers and S κ (Γ 1 (N ), C) be the space of cusp forms of weight κ and level N . Suppose that f = ∞ i=1 a n q n is a newform normalized with a 1 = 1. The following is a classical result proved by Deligne ([De]) when κ > 1, and Deligne and Serre when κ = 1 ( [DS75]).
Theorem 2.1.1. For a newform f as above, the field of coefficients E f = Q(a n , n ≥ 1) ⊂ C is a number field. Moreover, for any prime λ | ℓ of E f , there exists a continuous representation (1) ρ f,λ is odd and absolutely irreducible; (2) For any p ∤ N ℓ, ρ f,λ is unramified at p and Tr(ρ f,λ (Fr p )) = a p ; ( In what follows, an ℓ-adic representation ρ of G is said to be modular if there exists a modular form f ∈ S κ (Γ 1 (N ), C) and a prime λ | ℓ of E f such that ρ f,λ ∼ ρ or its dual ρ ∨ .
Recall that for any prime p, G p ⊂ G denotes a decomposition group at p and I p ⊂ G p is the inertia subgroup. Letρ : G → GL 2 (k) be a representation over a finite field k of characteristic ℓ. Callρ modular if there exists a modular form f ∈ S κ (Γ 1 (N ), C) and a prime λ | ℓ of E f such that the residual representationρ f,λ is equivalent toρ.
The following has been conjectured by Serre and proved by Khare, Wintenberger and Kisin ( [KW09,Kis09a]). We refer to [Kis05] for a nice review of Serre's conjecture.
Remark 2.1.3. The precise Serre's conjecture also predicts the (minimal) weight and the level of the modular form f , which are not needed here.
The aim of this subsection is to reprove the following result.
This theorem was essentially proved in [DM03], as explained in [Die08]. We sketch the proof below because some ingredients will be used later. We remark that the above modularity lifting theorem is an easy version (compared to that of [Kis09b]), however it is enough for the applications in this paper.
The proof will use the discussion of local representation at ℓ and several modularity lifting theorems. First we discuss the local representation ρ| G ℓ and its reduction. For i ≥ 1, the map Since ρ| G ℓ is crystalline with Hodge-Tate weights {0, r} and r ≤ ℓ − 2, there are two possibilities: Type I: ρ| G ℓ is absolutely reducible. In this case, ρ| I ℓ ∼ ǫ r ℓ * 0 1 and (ρ ⊗ kk )| I ℓ ≃ ω r 1 * 0 1 .
The reader is referred to §4.2.1 in [BM02] for the proof of the above statements. We also need the following result on Galois characters.
Lemma 2.1.5. Let χ : G → O × E be an ℓ-adic Galois character which is unramified outside a finite set S of finite primes excluding ℓ. Then χ is a finite character in the sense that the image of χ is a finite (cyclic) group.
Proof. Let p be a prime in S. It suffices to show that if χ is ramified at p then the image χ(I p ) is finite. Let I w p be the wild inertia subgroup of I p , I t p := I p /I w p the tame inertia group, and G w := χ(I w p ). We claim that the map χ : In fact, G w is a pro-p-group and ker(q) is a pro-ℓ-group. So they can only have trivial intersection. Since G w injects in k × , G w is a finite group. Replacing Q by a suitable finite extension, we may assume that χ factors though I t p . Let τ be a lift of Fr p and σ ∈ I t p . We have τ σ = σ p τ . Applying χ to this equation, we see that χ(σ) p−1 = 1. Thus χ(I t p ) is contained in the group of (p − 1)th-units in O × E , hence χ(I p ) is finite.
Corollary 2.1.6. Assume that χ : G → O × E is a Galois character such that χ is unramified almost everywhere and χ| G ℓ is crystalline. Then there exists a finite character ψ and an integer r such that χ = ψǫ r ℓ .
Proof. Using the p-adic Hodge theory on the classification of crystalline characters of G ℓ , we see that χ| G ℓ = ψ ℓ ǫ r ℓ with ψ ℓ a character unramified at ℓ. Applying the above lemma to χǫ −r ℓ , we prove the corollary.
Remark 2.1.7. The above corollary may fail if G is replaced by G K with K/Q a finite extension. The problem is that the classification of crystalline characters of G l is much more complicated if ℓ is inert in K. Here is a more concrete example: Let K be an imaginary quadratic extension of Q and consider an elliptic curve E defined over K with complex multiplication by the ring O K . Choose a prime ideal l of O K generated by a prime ℓ inert in K. Then the Tate module T ℓ (E) induces a Galois O K l -character χ : Note that χ has Hodge-Tate weights 0 and 1. So it can not be written as ψǫ r ℓ , which only has Hodge-Tate weight r.

Galois representations endowed with quaternion multiplication
In this section we show that if a 4-dimensional ℓ-adic representation of G is endowed with quaternion multiplication over a quadratic or biquadratic field, then either it decomposes into the sum of two degree 2 representations, or it is induced from a degree 2 representation of an index 2 subgroup G K of G. In the latter case, we show that this degree 2 representation of G K , after twisting by a character, can be extended to a representation of G.
3.1. Quaternion multiplication. If there is a quaternionic action on the space of a 4-dimensional ℓ-adic representation in the following sense, then a lot more can be said. In this section, F is always assumed to be a finite extension of Q ℓ .
Definition 3.1.1. Let ρ ℓ be an ℓ-adic representation of G acting on a 4-dimensional F -vector space W ℓ . It is said to have quaternion multiplication (QM) if there are linear operators J s and J t on W ℓ , parametrized by two distinct non-square integers s and t, satisfying (a) J 2 s = J 2 t = −id, J s J t = −J t J s ; (b) For u ∈ {s, t} and g ∈ G, we have J u ρ ℓ (g) = ±ρ ℓ (g)J u with + sign if and only if g ∈ G Q( √ u) .
In this case, we say that the representation has QM over Q( √ s, √ t).
Theorem 3.1.2. Let ρ ℓ be a 4-dimensional ℓ-adic representation over F with QM over Q( √ s, √ t). Then the following two statements hold.
There is a vast literature on abelian varieties attached to weight 2 congruence Hecke eigenform or more general modular motives possessing quaternion multiplication (QM) due to the existence of extra twists given by operators like Atkin-Lehner involutions (see [Rib80], [Mom81], [BG03], [GGJQ05], [Die03] etc). Compared to these cases our situation distinguishes itself by the specific condition on the field of definition for each J operator.
Proof. We follow the proof of Theorem 6 in [ALL08].
and write ρ l for ρ ℓ ⊗ F E for simplicity. Given a nonsquare u ∈ {s, t, st}, we first show that there exists an ℓ- , ρ l (g)v is an eigenvector of J u with the opposite eigenvalue −i. Thus J u has eigenvalues ±i with 2-dimensional ±i-eigenspace. Since J u commutes with the action of G Q( √ u) , each ±i-eigenspace affords a G Q( √ u) -action, denoted by σ u , σ − u , respectively. When u is a nonsquare, by property (b), any τ ∈ G \ G Q( √ u) gives rise to an E-isomorphism from the (−i)-eigenspace to the i-eigenspace. On the (−i)-eigenspace, for all Let {v 1 , v 2 } be a basis of the i-eigenspace of J u and choose a nonsquare v ∈ {s, t, st} not equal to u. Then v 3 := J v v 1 and v 4 := J v v 2 form a basis of the −i-eigenspace of J u . With respect to the ordered basis {v 1 , v 2 , v 3 , v 4 } we may express the operators by the matrices This gives a representation γ of the quaternion group generated by J s and J t .
It follows from (b) that with respect to the same basis, ρ l (g) is represented by This shows that the map sending g ∈ G Q( √ u) to the matrix P (g) is the 2-dimensional representation σ u of G Q( √ u) and the representation σ − u is given by . In this case ; denote the 2-dimensional representations on W and W − by σ and σ − , respectively. It is straightforward to check that with respect to the above bases of W and W − , we have σ(g) = P (g) = σ − (g) for g ∈ N and σ(g) is the product of the characteristic polynomials of σ(Frob p ) and (σ ⊗ θ)(Frob p ). We shall see that there is also a natural factorization for As stated in the theorem above, for any Thus for any prime p = ℓ splitting into two places p ± in Q( √ u), τ permutes the two places p ± ; and if ρ ℓ is unramified at p, the characteristic polynomial of The statement (a) below on the factorization of the characteristic polynomial of ρ ℓ (Frob p ) at an unramified place p follows immediately from Theorem 3.1.2 and the remark above.
(a) Let p be a prime different from ℓ at which ρ ℓ is unramified. Then the degree-4 characteristic polynomial H p (x) of ρ ℓ (Frob p ) and the degree-2 characteristic polynomial H p,u (x) of σ u (Frob p ) at a place p of Q( √ u) dividing p are related as follows: is totally real, and Trσ u (c) = −Trσ τ u (c) otherwise. Proof. It remains to prove part (b). This follows from the fact that σ τ Consequently, in case (a2) above, H p (x) is also the characteristic polynomial of ρ ℓ (Frob p ± ). Since a prime p splits in at least one of the quadratic fields contained in the biquadratic field Q( √ s, √ t), when the representation ρ ℓ admits QM over Q( √ s, √ t), we obtain a natural factorization of the characteristic polynomial of ρ ℓ (Frob p ) as a product of two quadratic polynomials.
Proposition 3.1.5. Keep the same notation and assumptions as in Theorem 3.1.2. Assume that −1 is not a square in F . Let j denote a square root of −1 and set E = F (j).
(a) For any g ∈ G Q( √ u) , let H g,+ (x) and H g,− (x) be the characteristic polynomials of σ u (g) and σ − u (g) respectively. Then H g,+ (x) and H g,− (x) are conjugate under the map j → −j.
For u, c and τ as in Corollary 3.1.4 (b), we have Consequently, if Trρ ℓ (c) = 0 at the complex conjugation c, then σ is odd when Q( √ s, √ t) is quadratic, and σ u is odd for positive u ∈ {s, t, st} when Q( √ s, √ t) is biquadratic.
Proof. (a) Let w be a nonzero vector in the representation space W ℓ of ρ ℓ . Then w and J u w are linearly independent over F since the eigenvalues of J u are outside F by assumption. Let w ′ be a vector in W ℓ and not in the F -span w, J u w . Claim that w, w ′ , J u w, J u w ′ are linearly independent over F . If not, then J u w ′ = αw + βJ u w + γw ′ for some α, β, γ ∈ F not all zero. Apply J u to the above relation to get another relation γJ u w ′ = βw − αJ u w − w ′ . Comparing both relations yields γ 2 = −1, a contradiction. Therefore {w, w ′ , J u w, J u w ′ } forms an F -basis of W ℓ . With respect to this ordered basis, J u is represented by the block matrix On the extension W ℓ ⊗ F E the actions of ρ ℓ (G) and J u are E-linear. Recall that σ u and σ − u are ρ ℓ restricted to the ±j-eigenspaces of J u on W ℓ ⊗ F E. It is easy to check that {w − jJ u w, w ′ − jJ u w ′ } forms an E-basis for the space of σ u and {w + jJ u w, w ′ + jJ u w ′ } forms a basis for the space of σ − u . In other words, for some 2 × 2 matrices P and R with entries in F . One checks that This shows that the action of ρ ℓ (g) on the ±j-eigenspaces of J u are represented by matrices conjugate under j → −j, hence the characteristic polynomials H g,± (x) are as asserted.
(b) For u > 0, a complex conjugation c lies in G Q( √ u) . By applying (a) to g = c, we get that Trσ u (c) and Trσ τ u (c) are conjugate under j → −j. As c 2 = id, the possible values for Trσ u (c) are ±2 and 0, which all lie in F . Hence Trσ u (c) = Trσ τ u (c). When Q( √ s, √ t) is not totally real, it follows from Corollary 3.1.4 (b) that the two traces are opposite, therefore they are equal to zero, and hence σ u is odd.
(c) In this case, choose u = st, which is a square. The operator J u = J st := J s J t commutes with ρ ℓ (G). The representation σ u is called σ and σ − u called σ ⊗ θ in Theorem 3.1.2. Apply (a) to g = c, the complex conjugation in G. By the same argument as in (b) we conclude Trσ(c) = Trσ(c)θ(c).
Finally, using 0 = Trρ is biquadratic, we conclude the oddness of σ and σ u respectively.
3.2. Quaternion multiplication and extension of representations. Let K be a finite extension of Q and V be an ℓ-adic representation of G K . We say that V can be extended to an ℓ-adic We shall show that, for each u ∈ {s, t, st}, there exists a finite character χ u of G Q( and a 2-dimensional representation η u of G such that σ u ⊗ χ u ∼ η u | G Q( √ u) . In other words, σ u ⊗ χ u can be extended to η u .
To see this, we return to the proof of Theorem 3.1.2, (1) with a chosen u ∈ {s, t, st}. The argument there shows that given g, h ∈ G, there exists a constant α 0 (g, h) ∈ {±1} such that P (g)P (h) = α 0 (g, h)P (gh). Hence g → P (g) defines a degree 2 irreducible projective representatioñ ρ u of G whose restriction to G Q( √ u) is the representation σ u . Letγ u be the map sending gN for g ∈ N, G Q( respectively. Note thatγ u is a degree 2 irreducible projective representation of G trivial on N and By Tate's vanishing theorem H 2 (G, C × ) = 0 ([Ta62]), there is a representation γ u : G → GL 2 (C) with finite image which liftsγ u . Then the kernel H of γ u is a normal subgroup of N with quotient N/H finite cyclic. Embed the finite image γ u (G) into GL 2 (Q ℓ ) and regard γ u as an ℓ-adic representation.
For any g ∈ G, there exists s u (g) ∈ Q × ℓ such thatγ u (g) = s u (g)γ u (g). It follows from ρ ℓ = ρ u ⊗γ u =ρ u · s u ⊗ s −1 u ·γ u =ρ u · s u ⊗ γ u that η u :=ρ u · s u is also an ordinary representation of G. Sinceρ u restricted to G Q( √ u) is equal to the representation σ u as noted above, s u restricted to G Q( √ u) is a character, denoted by χ u , with kernel containing H. Hence η u is an extension to G of the representation σ u ⊗ χ u of G Q( √ u) . Note that γ u and Ind G G Q( √ u) χ −1 u have the same restrictions to G Q( √ u) , so they differ at most by the quadratic character θ u of G with kernel G Q( √ u) . Replacing γ u by γ u ⊗ θ u and η u by η u ⊗ θ u if necessary, we may assume that γ u = Ind G For u ∈ {s, t, st}, let σ u be as in Theorem 3.1.2. Then there exists a finite character χ u of G Q( √ u) such that σ u ⊗ χ u extends to a degree 2 representation η u of G and Remark 3.2.2. As irreducible projective representations of Gal(Q( √ s, √ t)/Q), theγ u 's are in fact equivalent. This can be shown directly (for instance,γ u conjugated by 1 i i 1 is equivalent tõ γ v ) or seen from the fact that the Schur multiplier of Gal(Q( √ s, √ t)/Q) has only one nontrivial element. Hence the η u 's are also projectively equivalent. In particular, they have the same parity, equal to that of η u with u > 0.
We close this section by discussing the oddness of the representations occurred.
Proposition 3.2.3. With the same notation and assumption as in Theorem 3.2.1, we have (a) for positive u ∈ {s, t, st}, η u is odd if and only if σ u is odd at both real places of Q( √ u); is not totally real and even otherwise.
Proof. Denote by c the complex conjugation in G.
As det(η u )(g −1 cg) remains the same for all g in G, we see that σ u and η u have the same parity.
(b) The first statement follows from base change [AC89]. Write γ u for Ind G G Q( Let Γ ⊂ SL 2 (Z) be a noncongruence subgroup, that is, Γ is a finite index subgroup of SL 2 (Z) not containing any principal congruence subgroup Γ(N ). For any integer κ ≥ 2, the space S κ (Γ) of weight κ cusp forms for Γ is finitedimensional; denote by d = d(Γ, κ) its dimension. Assume that the compactified modular curve (Γ\H) * is defined over Q and the cusp at infinity is Q-rational. For even κ ≥ 4 and any prime ℓ, in [Sch85] Scholl constructed an ℓ-adic Galois representation ρ ℓ : G → GL 2d (Q ℓ ) attached to S κ (Γ). The representations ρ ℓ form a compatible system in the sense that there exists a finite set S of finite primes of Q such that for any prime p ∈ S and primes ℓ and ℓ ′ different from p, the representations ρ ℓ and ρ ℓ ′ are unramified at p and the characteristic polynomials of Frob p under ρ ℓ and ρ ℓ ′ have coefficients in Z and agree. Scholl also showed that all the roots of the characteristic polynomial of Frob p have the same complex absolute value p (κ−1)/2 (cf. §5.3 in [Sch85]). Scholl's results can be extended to odd weights under some extra hypotheses (e.g., ±(Γ ∩ Γ(N )) = ±(Γ) ∩ ±(Γ(N )), where ± : SL 2 (Z) → PSL 2 (Z) is the projection). The readers are referred to the end of [Sch85] for more details. In this subsection we always assume that ρ ℓ does exist. Our main concern is Whether ρ ℓ or its dual ρ ∨ ℓ is automorphic as predicted by the Langlands conjecture? If so, then the associated L-function coincides with the L-function of an automorphic representation of some adelic reductive group. When the reductive group is GL 2 over Q, the representation is called "modular" in §2.
We provide an affirmative answer to the case d = 1.
Proof. In fact, it has been known for a long time (see [Ser87] (4.8)) that the strong form of Serre's conjecture implies the modularity of 2-dimensional motives over Q, which can be directly applied to our theorem. Here we use a slightly different method which will be useful later.
Since {ρ ℓ } forms a compatible system, by Chebotarev Density Theorem, it suffices to show that there exists an ℓ such that ρ ∨ ℓ is modular. The plan is to conclude this by applying Theorem 2.1.4 to ρ ∨ ℓ for a large ℓ. Hence we have to show the existence of an ℓ such that ρ ∨ ℓ satisfies the three hypotheses in Theorem 2.1.4. In [Sch96], Scholl proved that, a general Scholl representation ρ ℓ as above is the ℓ-adic realization of a certain motive over Q in the sense of Grothendieck. 3 In particular, there exists a smooth projective Q-scheme X such that ρ ℓ ≃ H κ−1 et (X) as G-representations. Furthermore, he also proved that the Hodge type of H κ−1 et (X) is (κ − 1, 0) d and (0, κ − 1) d (Theorem 2.12 in [Sch85]). Consequently, we know the Hodge-Tate weights of ρ ℓ are 0 and −(κ − 1) and those of ρ ∨ ℓ are 0 and κ − 1. Pick ℓ > 2κ − 2 so that X has a smooth model over Z (ℓ) , the localization of Z with respect to the prime ℓ. Then ρ ℓ | G ℓ is crystalline by Faltings' comparison theorem ( [Fa89]). To complete the proof, it remains to show that ρ ℓ is odd and absolutely irreducible.
Contained in the proof above are the following two useful results.

4.2.
Automorphy of certain Scholl representations with d ≥ 2. In the remainder of this paper, we consider the automorphy of certain Scholl representations for the case d ≥ 2.
In this subsection, we fix a degree d cyclic extension K/Q. Let ρ ℓ be a 2d-dimensional subrepresentation of the Scholl representation of G attached to a space of weight κ cusp forms for a noncongruence subgroup Γ. We further assume that the Hodge-Tate weights of ρ ℓ are {0, −(κ−1)}, each with multiplicity d. For an ℓ-adic Galois representation V of G K and a prime l of O K above ℓ, denote by HT l (V ) the set of Hodge-Tate weights of the local Galois representation V | G K l and by HT(V ) the union of HT l (V ) for all primes l above ℓ.
We begin by proving a lemma dealing with Hodge-Tate weights: Lemma 4.2.2. Let E/Q ℓ be a finite Galois extension and V be a finite dimensional Hodge-Tate representation of G E := Gal(Q ℓ /E) over Q ℓ . Then for any σ ∈ Gal(Q ℓ /Q ℓ ), the Hodge-Tate weights of V σ are the same as those of V .

Proof. Set Fil
is the Hodge-Tate ring. One can find a construction and discussion of B HT in [Con09] (see Definition 2.3.6). Both of them are E-vector spaces. It suffices to show that they have the same dimension over E. For any j v j ⊗ a j ∈ Fil i D, where v j ∈ V and a j ∈ Fil i B HT , it can be easily checked that j v j ⊗ σ(a j ) ∈ Fil i D σ . Hence σ induces a Q ℓ -linear isomorphism between Fil i D and Fil i D σ . Thus Fil i D and Fil i D σ have the same E-dimension.
Proof of Proposition 4.2.1. Since Hodge-Tate weights are stable when restricted to a Galois subgroup of finite index, the representationsρ, ρ, and ρ ⊗ χ have the same Hodge-Tate weights. More precisely, suppose that V is an ℓ-adic Galois representation of G and F is a finite extension of Q. If V | G ℓ is Hodge-Tate then HT ℓ (V | G ℓ ) = HT l (V | G F l ) for each prime l of O K above ℓ. Now it suffices to show that HT( ρ) = {0, −(κ − 1)}. Select σ ∈ G such that its image in Gal(K/Q) is a generator. Note that ρ ℓ | G K ≃ d−1 i=0 ρ σ i . We claim that HT( ρ σ i ) are the same for all i. If so, since ρ ℓ has Hodge-Tate weights {0, −(κ − 1)} and each weight appears with multiplicity d, we get that HT( ρ) = {0, −(κ − 1)}. To prove the claim, it suffices to show that for any prime l of O K we have HT l ( ρ) = HT σ i (l) ( ρ σ i ). We should be careful that we can not directly use Lemma 4.2.2 here because it is not clear that σ i (l) = l.
Observe that ρ| G K l is isomorphic to ρ σ i restricted to σ i G K l σ −i . Let ̟ be the maximal ideal of O Q above l which gives rise to the decomposition group G K l . Then there exists τ ∈ G such that So without loss of generality, we may assume that σ i (̟) = τ (̟). Thus σ i τ −1 is in G ℓ but may not be in G K l . Identify The identity map Id : ρ σ i → ρ τ is an isomorphism of G K -modules. Now we need to show that ρ τ and ρ σ i have the same Hodge-Tate weights on G K σ i (l) . This follows from Lemma 4.2.2 and the fact that ρ τ = ( ρ σ i ) τ σ −i .
We are ready to prove the automorphy of certain Scholl representations. First suppose that the Scholl representation ρ ℓ has degree 4 and admits QM over a quadratic field Q( √ s). By Theorem 3.1.2, (2), we have, over Q ℓ ( √ −1), for a degree 2 representation σ of G and a quadratic character θ of G with kernel G Q( √ s) . So σ and σ ⊗ θ restricted to the decomposition group G ℓ are crystalline, and they have the same Hodge-Tate weights {0, −(1 − κ)} by Proposition 4.2.1. We conclude their absolute irreducibility from Corollary 4.1.3. Proposition 4.1.2 says that ρ ℓ has trace zero at the complex conjugation, hence by Proposition 3.1.5, both σ and σ ⊗ θ are odd. It then follows from Theorem 2.1.4 that σ and σ ⊗ θ are modular for large ℓ, and thus for all ℓ by compatibility. We record this in Theorem 4.2.3. Suppose that the Scholl representation ρ ℓ has degree 4 and admits QM over a quadratic field Q( √ s). Then over Q ℓ ( √ −1) it decomposes as a direct sum of two modular representations σ and σ ⊗ θ. Here θ is the quadratic character of Gal(Q( √ s)/Q).
Next we consider a more general situation.
Theorem 4.2.4. Let K be a degree d cyclic extension of Q and ρ ℓ be a 2d-dimensional ℓ-adic subrepresentation of a Scholl representation of G as above. Assume that (a) ρ ℓ is induced from a 2-dimensional representation ρ of G K ; (b) There exists a character χ of G K of finite order such that ρ ⊗ χ can be extended to an ℓ-adic representationρ of G; (c) ρ ℓ | G ℓ is crystalline and χ is unramified at any prime above ℓ.
Thenρ is absolutely irreducible. If we further assume that (d) K is unramified over ℓ and ℓ > 2κ − 2; (e)ρ is odd, then the dual ofρ is isomorphic to a modular representation ρ g,λ of G attached to a weight κ cuspidal newform g, and the following relations on L-series hold: Remark 4.2.5. The character χ of G K corresponds to an automorphic form π χ of GL d (A Q ) by base change [AC89]. The above expression shows that the semi-simplification of ρ ∨ ℓ is automorphic, arising from the form g × π χ of the group GL 2 (A Q ) × GL d (A Q ). This is because When d = 2, it follows from the work of Ramakrishnan [Ram00] that an automorphic form for GL 2 (A Q ) × GL 2 (A Q ) corresponds to an automorphic form for GL 4 (A Q ). In this case the L-function attached to the degree 4 representation ρ ∨ ℓ is an automorphic L-function for GL 4 (A Q ). This fact also holds for general d by base change [AC89, §3.6]. More precisely, thatρ ∨ comes from an automorphic form of GL 2 over Q implies thatρ ∨ comes from an automorphic form of GL 2 over K, which in turn implies that ρ ∨ ℓ corresponds to an automorphic form of GL 2d over Q. Proof of Theorem 4.2.4. We first show thatρ is absolutely irreducible. Suppose otherwise. Enlarging the field of coefficients if necessary, we may assume that (ρ) ss = η 1 ⊕ η 2 for some characters η 1 and η 2 of G. Note that (ρ| G K ) ss = ( ρ ⊗ χ) ss . Since ρ ℓ is crystalline at ℓ, ρ is crystalline at any prime above ℓ, which in turn implies that ρ ⊗ χ is crystalline at any prime above ℓ because χ is unramified at any prime dividing ℓ. Consequently, each η i is crystalline at ℓ. Hence η i = µ i ǫ κ i ℓ with µ i a finite character and κ i the Hodge-Tate weight of η i by Corollary 2.1.6. Sinceρ has Hodge-Tate weights 0 and −(κ − 1) by Proposition 4.2.1, we see that each η i has Hodge-Tate weight either 0 or −(κ − 1). Therefore ρ ss = (µ 1 ǫ 1 ⊕ µ 2 ǫ −(κ−1) ℓ )| G K . This contradicts the fact that the roots of the characteristic polynomial of Frob p under the representation ρ ℓ = Ind G G K ρ have the same complex absolute value. Soρ cannot be absolutely reducible. Now we prove the remaining assertions under the additional assumptions. As shown above,ρ is absolutely irreducible and has Hodge-Tate weights {0, −(κ − 1)}. Further, since K is unramified above ℓ,ρ| G ℓ is crystalline. Finally, sinceρ is odd by assumption, we conclude from Theorem 2.1.4 thatρ ∨ comes from a weight κ cuspidal newform g as described in Theorem 2.1.1. Hence ρ = ρ ∨ g,λ | G K ⊗ χ −1 and the relations on L-functions hold. Remark 4.2.6. Since we have a compatible family of Scholl representations ρ ℓ constructed from geometry, there always exists a prime ℓ large enough such that the conditions (c) and (d) are satisfied. However, the oddness ofρ is not automatic.

Degree 4 Scholl representations with QM and Atkin-Swinnerton-Dyer conjecture.
Hecke operators played a fundamental role in the arithmetic of congruence modular forms. It was shown in [Th89,Be94,Sch97] that similarly defined Hecke operators yielded little information on genuine noncongruence forms. When the modular curve of a noncongruence subgroup Γ has a model over Q, Atkin and Swinnerton-Dyer in [ASD71] predicted a "p-adic" Hecke theory in terms of 3-term congruence relations on Fourier coefficients of weight κ cusp forms in S κ (Γ) as follows. Suppose the cusp at ∞ is a Q-rational point with cusp width µ. The d-dimensional space S κ (Γ) admits a basis whose Fourier coefficients are in a finite extension of Q; moreover, there is an integer M such that for each prime p ∤ M , these Fourier coefficients are p-adically integral, that is, integral over Z p . Atkin and Swinnerton-Dyer conjectured in [ASD71] that, given κ ≥ 2 even, for almost all primes p, S κ (Γ) has a p-adically integral basis f j (z) = n≥1 a j (n)q n/µ , where q = e 2πiz and 1 ≤ j ≤ d, depending on p, whose Fourier coefficients satisfy the 3-term congruences (4.1) a j (pn) − A j (p)a j (n) + p (κ−1)/2 a j (n/p) ≡ 0 mod p (κ−1)(1+ordpn) ∀n ≥ 1 for some algebraic integers A j (p) satisfying the Ramanujan bound. The congruence (4.1) is carefully explained in [ASD71] and [Sch85]. It can be summed up as the quotient (a j (pn) − A j (p)a j (n) + p (κ−1)/2 a j (n/p))/p (κ−1)(1+ordpn) is integral over Z p , as in [ALL08]. In what follows, this is the interpretation of ASD congruences we shall adopt. The κ = 2 case of this conjecture was verified in [ASD71] for d = 1 and discussed in [Car71] using formal group language. Scholl showed in [Sch85] that, for κ ≥ 4 even and p large enough, all p-adically integral cusp forms f = n≥1 a(n)q n/µ in S κ (Γ) satisfy (2d + 1)-term congruences a(p d n) + C 1 (p)a(p d−1 n) + · · · + C 2d (p)a(n/p d ) ≡ 0 mod p (κ−1)(1+ordpn) ∀n ≥ 1, is the characteristic polynomial of the associated Scholl representation ρ ℓ at Frob p . Similar statement holds for κ odd. Hence the ASD conjecture holds for d = 1. For arbitrary d, proving the ASD conjecture amounts to factoring H p (x) as a product of d quadratic polynomials x 2 − A j (p)x + B j (p), 1 ≤ j ≤ d, and finding a basis f j = n≥1 a j (n)q n/µ of S κ (Γ) with p-adically integral coefficients such that the 3-term congruences Assume that S κ (Γ) has a 2-dimensional subspace S to which one can associate a family of compatible degree-4 Scholl representations {ρ ℓ } of G over Q ℓ . Suppose that there is a finite extension F of Q and there are operators J s and J t acting on extensions over fields in Q ℓ ⊗ F of the representation spaces of all ρ ℓ that satisfy the QM conditions in Definition 3.1.1 over a biquadratic Q( √ s, √ t). Then, as stated in Corollary 3.1.4, for almost all primes p they give rise to a factorization of the characteristic polynomial H p (x) of ρ ℓ (Frob p ) into a product of two quadratic polynomials H p ± ,u (x) = x 2 − A ±,u (p)x + B u (p) by choosing a u ∈ {s, t, st} such that p splits into two places p ± in Q( √ u). Recall that H p ± ,u (x) are the respective characteristic polynomial of σ u (Frob p + ) and σ u (Frob p − ) = (σ u ⊗ δ u )(Frob p + ), where σ u and δ u are as in Theorem 3.1.2. For p ∤ M and unramified in Q( √ s, √ t), let V p be the 4-dimensional p-adic Scholl space containing the space S p of p-adically integral forms in S as a subspace and its dual (S p ) ∨ as a quotient (cf. [Sch85]). On V p there is the action of F p , the Frobenius at p. Scholl proved in [Sch85] that, for ℓ = p and ρ ℓ unramified at p, ρ ℓ (Frob p ) and F p have the same characteristic polynomial.
Assume that the operators J s and J t also act on S, preserving forms whose Fourier coefficients are in a suitable number field K which are p-adically integral for almost all p. For such a p, their actions on S p extend to V p , again denoted by J s and J t . Suppose that they satisfy the condition (a) in Definition 3.1.1 and (b)' for u ∈ {s, t, st}, the operator J u commutes with F p for p split in Q( √ u), and J u F p = −F p J u for p inert in Q( √ u). Here J st = J s J t . We say that V p admits QM over Q( √ s, √ t) for simplicity. The same argument as in the proof of Theorem 3.1.2 shows that over any field in Q p ⊗ K( √ −1), for every u ∈ {s, t, st}, each ±ieigenspace V p,±,u of J u is 2-dimensional, containing a form in S p , and J v , where v ∈ {s, t, st} and v = u, permutes the two eigenspaces. If p splits completely in Q( √ s, √ t), then V p,±,u are invariant under F p and J v commutes with F p . Therefore the actions of F p on V p,±,u are isomorphic and hence have the same characteristic polynomial, equal to H p ± ,u (x). If p does not split in Q( √ s, √ t), choose an element u ∈ {s, t, st} such that p splits in Q( √ u). Then V p,±,u are invariant under F p .
On the ℓ-adic side, the representation space W ℓ of ρ ℓ decomposes similarly into two ±i-eigenspaces W ℓ,p,±,u of J u , each is invariant under ρ ℓ (Frob p ). Moreover, viewed as J u ρ ℓ (Frob p ) invariant spaces, they are intertwined by J v because of condition (b) in Definition 3.1.1. Therefore J u ρ ℓ (Frob p ) has the same characteristic polynomial on W ℓ,p,±,u . A normalizer of Γ acts on the space S and hence V p . Further, its action on the modular curve X Γ induces an action on the ℓ-adic cohomology group, which is the representation space of ρ ℓ . When J s and J t arise from normalizers of Γ, or algebraic real linear combinations of such, we conclude from the argument in [Sch85,Prop. 4.4 and proof] that J u F p on V p and J u ρ ℓ (Frob p ) on W ℓ have the same characteristic polynomials. Combined with the fact that ρ ℓ (Frob p ) and F p have the same characteristic polynomials, we get, by using the argument of Lemma 8 of [ALL08], that when p splits in Q( √ u), F p on V p,±,u and ρ ℓ (Frob p ) on W ℓ,p,±,u have the same characteristic polynomials As observed above, for almost all primes p, V p,±,u contain a nonzero f ±,u = n≥1 a ±,u (n)q n/µ ∈ S with p-adically integral Fourier coefficients. They form a basis of S. It follows from [Sch85,FHL+] that the ASD congruences hold, namely, Now we turn to the modularity of the degree 4 representations ρ ℓ with QM over Q( √ s, √ t) as above. Theorem 3.2.1 says for each u ∈ {s, t, st}, there is a finite ℓ-adic characters χ u of G Q( √ u) such that σ u ⊗ χ u extends to a representation η u of G. We know from Corollary 4.1.2 that all ρ ℓ have trace 0 at the complex conjugation. If there is a large ℓ such that the representation ρ ℓ is over a field not containing a square root of −1, then by combining Propositions 3.1.5 and 3.2.3, we get that η u is odd for this and hence all ℓ because of compatibility. We then conclude from Theorem 4.2.4 that the dual of ρ ℓ is automorphic. In this case the dual of σ u corresponds to an automorphic form h u of GL 2 over Q( √ u). As σ s , σ t , σ st have the same restrictions to G Q( √ s, √ t) , the three automorphic forms h u of GL 2 over Q( √ u) for u ∈ {s, t, st} base change to the same automorphic form h of GL 2 over Q( √ s, √ t). We summarize the above discussions in the theorems below. If ρ ℓ is unramified at a prime p which splits in Q( √ u), then h u is an eigenfunction of the Hecke operators at the two places p ± of Q( √ u) above p, and the product of the local L-factors attached to h u at these two places is where H p ± ,u (x) = x 2 − A ±,u (p)x + B u (p) are the characteristic polynomials of σ u (Frob p ± ).
Theorem 4.3.2. Keep the same notation and hypotheses as in Theorem 4.3.1. Assume the modular curve X Γ has a model defined over Q such that the cusp at ∞ is a Q-rational point with cusp width µ. Further, assume that J s and J t arise from algebraic real linear combinations of normalizers of Γ whose actions on V p admit QM over Q( √ s, √ t) for almost all p. Then there exists a finite set of primes T , including ramified primes, primes < 2κ − 2, and primes where X Γ has bad reductions, such that for each p / ∈ T , S has a p-adically integral basis f ±,p = n≥1 a ±,u (n)q n/µ which are ±ieigenfunctions of J u for some u ∈ {s, t, st} such that p splits in Q( √ u). The Atkin and Swinnerton-Dyer congruences (4.2) hold with A ±,u (p) and B u (p) coming from the characteristic polynomials are the local factors at the two places p ± of Q( √ u) above p of an automorphic form h u for GL 2 over Q( √ u).
Observe that the basis for which the ASD conjecture holds depends on p modulo the discriminant of Q( √ s, √ t). Further, the ASD congruences (4.2) describe congruence relations between Fourier coefficients of noncongruence forms f ±,p and those of congruence forms h u .
Remark 4.3.3. Under the same assumptions as above except that the field Q( √ s, √ t) is a quadratic extension of Q, the same argument shows that, for almost all primes p, if each eigenspace of J s contains a nonzero form in S, then the ASD conjecture on S holds at such p.

Applications
As applications, we exhibit some examples of Scholl representations which admit QM. We use Theorem 4.3.1 to conclude the automorphy of the representation and Theorem 4.3.2 to conclude the ASD congruences. We expect that there is a wide variety of cases to which Theorem 4.2.4 could be applied.

Old cases.
We first re-establish several known automorphy results, previously proved using Faltings-Serre modularity criterion. While the new argument is more conceptual, no information on the congruence modular form giving rise to the Galois representation is revealed.
In a series of papers [LLY05, ALL08, Lon08], a sequence of genus 0 normal subgroups, denoted by Γ n , of Γ 1 (5) are considered. Their modular curves are n-fold covers of the modular curve for Γ 1 (5), ramified only at two cusps ∞ and −2 of Γ 1 (5) with ramification degree n. The group Γ n is noncongruence when n = 1, 5, its modular curve is defined over Q with the cusp at ∞ a Q-rational point, and an explicit basis for S 3 (Γ n ) with rational coefficients was constructed (cf. [ALL08, Prop. 1]). To the (n − 1)-dimensional space S 3 (Γ n ) Scholl has attached a compatible family of 2(n − 1)-dimensional ℓ-adic Galois representations of G. The functions in S 3 (Γ n ) belonging to a nontrivial supergroup Γ m of Γ n are called "old" forms; denote by S 3 (Γ n ) new the subspace of forms in S 3 (Γ n ) orthogonal to the "old" forms. Accordingly, there is a compatible family of degree 2ϕ(n)-dimensional ℓ-adic representations ρ new 3,l,n of G attached to S 3 (Γ n ) new . Here ϕ(n) stands for the Euler phi-function (cf. [Lon08,§3]). For almost all ℓ, ρ new 3,l,n | G ℓ is crystalline with Hodge Tate weights {0, −2}, each with multiplicity ϕ(n), and the action of complex conjugation has trace zero.
Remark 5.1.2. The above automorphy argument does not work for n > 6. However, we conjecture that when n = p is an odd prime, the representation ρ 3,l,p is related to a Hilbert modular form.
Another known example of 4-dimensional Scholl representation with QM by a biquadratic field has been investigated in [HLV10].
It is straightforward to check that where the ζ on the left acts on E W −1 2 ΓW 2 and the one on the right acts on E Γ . The composition T −2 = ζ • W 2 is an isogeny defined over Q ℓ ( √ −2). Let T −2 denote its dual isogeny.
We first note that the above map T −2 induces a natural homomorphism g 1 from the parabolic cohomology H 1 (X Γ ⊗ Q, i * R 1 h 0 * Q ℓ ) to that of X W −1 2 ΓW 2 . Here i : H/Γ → X Γ = (H/Γ) * is the inclusion map and h : E Γ → (H/Γ) * is the natural projection and h 0 is the restriction of h to H/Γ. We further let w 2 : X Γ → X W −1 2 ΓW 2 denote the isomorphism which sends t to 6 √ −8/t. Compared with T −2 , this map is on the base curve level. Using w 2 , we have a natural isomorphism g 2 : H 1 (X Γ ⊗ Q, i * R 1 h 0 * Q ℓ ) ∼ = H 1 (X W −1 2 ΓW 2 ⊗ Q, (w 2 ) * i * R 1 h 0 * Q ℓ ), and the cohomology on the right is isomorphic to the parabolic cohomology attached to weight 3 cusp forms of W −1 2 ΓW 2 . The composition g −1 2 • g 1 is an endomorphism on H 1 (X Γ ⊗ Q, i * R 1 h 0 * Q ℓ ) and is denoted by T −2 again. On H 1 (X Γ ⊗ Q, i * R 1 h 0 * Q ℓ ) the map (T −2 ) 2 = −2 id is induced by the fiber-wise multiplication by 2 map T −2 • T −2 .
Denote by ρ new ℓ the 4-dimensional ℓ-adic Scholl representation of G corresponding to the "new" forms of S 3 (Γ) (cf. [Lon08]). Let J −2 = 1 √ 2 T −2 and let J −3 denote 1 √ 3 (2ζ − 1) * on ρ new ℓ . As linear operators on the representation space of ρ new ℓ , J −2 and J −3 generate a quaternion group. We know from the above discussions that the actions of J δ are defined over Q( √ δ) for δ = −2, −3. Choose the field F in Theorem 4.3.1 to be Q( √ 2, √ 3). The space of ρ new ℓ over fields in Q ℓ ⊗ F is endowed with QM over Q( √ −2, √ −3). By Theorem 4.3.1, we conclude the automorphy of ρ new ℓ . We have also seen that J −2 and J −3 act on the space F 1 , F 5 and the space V p admits QM over Q( √ −2, √ −3). Since J −2 and J −3 arise from normalizers of Γ 1 (6), by Theorems 4.3.1 and 4.3.2, we have Theorem 5.2.1. The representation ρ new ℓ is automorphic, and the ASD congruences hold on the space F 1 , F 5 .
Choosing different signs of i, j, k gives rise to 8 eigenforms that are conjugate to each other under G.