Non-paritious Hilbert modular forms

The arithmetic of Hilbert modular forms has been extensively studied under the assumption that the forms concerned are"paritious"-- all the components of the weight are congruent modulo 2. In contrast, non-paritious Hilbert modular forms have been relatively little studied, both from a theoretical and a computational standpoint. In this article, we aim to redress the balance somewhat by studying the arithmetic of non-paritious Hilbert modular eigenforms. On the theoretical side, our starting point is a theorem of Patrikis, which associates projective l-adic Galois representations to these forms. We show that a general conjecture of Buzzard and Gee actually predicts that a strengthening of Patrikis' result should hold, giving Galois representations into certain groups intermediate between GL(2) and PGL(2), and we verify that the predicted Galois representations do indeed exist. On the computational side, we give an algorithm to compute non-paritious Hilbert modular forms using definite quaternion algebras. To our knowledge, this is the first time such a general method has been presented. We end the article with a selection of examples.


Introduction
Background. Let G be a reductive group over a number field F . One of the key themes of the Langlands programme is that "sufficiently nice" automorphic representations of G should give rise to -adic Galois representations, for any prime . However, translating this idea into a formal statement is surprisingly difficult, and a precise formulation of such a conjecture has only recently been given by Buzzard and Gee in [BG14].
In op.cit., they define a class of automorphic representations Π of G which are "L-algebraic"; and their conjecture predicts that if Π is L-algebraic, then for every prime (and isomorphism C ∼ = Q ), there should be a continuous representation of Gal(F /F ) with values in the Langlands L-group L G(Q ), whose restrictions to the decomposition groups at good primes v are determined by the corresponding local factors Π v of Π. (We shall recall the statement of this conjecture in more detail below.) One natural testing ground for this conjecture is provided by Hilbert modular forms. As noted in op.cit., if F is a totally real number field, and f is a Hilbert modular form for GL 2 /F , then the automorphic representation Π associated to f is L-algebraic (after a suitable twist) if and only if the weight of f is "paritious" (all of its components k σ are congruent modulo 2). It is well-known that paritious Hilbert eigenforms have associated 2-dimensional -adic Galois representations, confirming the Buzzard-Gee conjecture in this case.
However, what about non-paritious eigenforms? For quite a long time, there has been a common belief that these "do not have Galois representations", or even that they are "not arithmetical objects" (in particular, no effort has been spent in trying to compute them). However, this belief is incorrect! The first result in this direction is a result of Patrikis [Pat15], who proved that there are projective Galois representations attached to such forms. This has insipired us to take up a detailed study of non-paritious Hilbert modular forms, both from a theoretical and a computational standpoint.
Goals of this article. The goals of the present article are the following.
(1) For any Hilbert modular automorphic representation Π, we shall describe a subgroup of the restriction of scalars Res F/Q GL 2 , containing Res F/Q SL 2 , such that the restriction of Π to this group is L-algebraic after a suitable twist. This subgroup depends on "how close" the weight of f is to being paritious.
(2) We shall demonstrate that, as predicted by the Buzzard-Gee conjecture, we may associate -adic Galois representations to these automorphic representations, taking values in the Langlands L-groups of the subgroups in (1). Since our subgroup of Res F/Q GL 2 always strictly contains Res F/Q SL 2 , whose Langlands dual is Res F/Q PGL 2 , this result refines Patrikis' construction of projective Galois representations. (3) We describe algorithms for computing non-paritious Hilbert modular forms, via the Jacquet-Langlands correspondence between GL 2 and totally definite quaternion algebras. (4) We give explicit examples of non-paritious Hilbert modular forms computed using these algorithms, and describe the conjugacy classes of Frobenius elements in their associated Galois representations.
The article is organized as follows: in Section 1 we state Buzzard-Gee conjecture, and make a small detour through the concepts involved. Section 2 is about Hilbert modular forms: we recall their automorphic definition, and we prove that if a non-paritious Hilbert modular form is E-paritious (see Definition 2.2) then we can restrict it to an automorphic form of G * = G × (Res F /E GL1) GL 1 (as predicted by Buzzard-Gee). Section 3 contains the main theorem (Theorem 3.5), namely that non-paritious Hilbert modular forms, do have Galois representations attached to them, as predicted. Section 4 relates our construction with Patrikis' one. In Section 5 we focuss on real quadratic fields, where some exceptional isomorphism allows the Galois representation to land in GO 4 . In Section 6 we show how to use quaternion groups to compute Hilbert modular forms (paritious and non-paritious ones). In particular, Theorem 6.7 and Corollary 6.8 we prove how from automorphic forms for the quaternion group H we can construct forms in H * . This is the key result for computational purposes. In the same section we explain how to compute the Hecke action on such forms. We end the article with some particular examples.
Acknowledgements. It is a pleasure to thank the two authors of the conjecture we are studying: firstly, Kevin Buzzard for several helpful remarks, and in particular for pointing us towards the work of Blasius-Rogawski which is the key input to constructing the required Galois representations; and secondly, Toby Gee, for making us aware of the related work of Patrikis.

L-groups
In this section we'll recall from [BG14] the necessary notions to formulate their conjecture relating automorphic representations and Galois representations; and we will check the compatibility of their conjecture with restriction of scalars.
1.1. Global definitions. Let G be a connected reductive group over a number field F . The Langlands dualĜ is the connected reductive groupĜ over Q whose root datum is dual to that of G. The Galois group Γ F = Gal(F /F ) acts naturally onĜ, and the Langlands L-group L G is the pro-algebraic group over Q defined as the semidirect productĜ Γ F . See [BG14, §2.1] for details. If G is split over F (or is an inner form of a split group) the action of Γ F onĜ is trivial, so L G is a direct product.
We shall be interested in continuous homomorphisms ρ : Γ F → L G(M ), for various fields M , satisfying the following condition: the composite of ρ with the projection L G(M ) → Γ F is the identity map on Γ F . Such a morphism is called an admissible homomorphism, or sometimes L-homomorphism. More generally, if Γ ⊆ Γ F is a subgroup, we define a homomorphism Γ → L G(M ) to be admissible if its projection to Γ F is the inclusion map Γ → Γ F .
Notation. If H 1 and H 2 are two reductive groups over F , then the Langlands Lgroup L (H 1 × H 2 ) is the fibre product L H 1 × Γ F L H 2 ; for r 1 : Γ F → L H 1 and r 2 : Γ F → L H 2 admissible homomorphisms, we write r 1 × r 2 : Γ F → L (H 1 × H 2 ) for their product.

Local theory.
If v is a finite place of F at which G is unramified (i.e. G is quasi-split over F v and becomes split over an unramified extension of F v ), then there is a parametrisation of unramified representations of G(F v ) in terms of Langlands-Satake parameters. We choose an embedding F → F v , so we can identify Γ Fv with a subgroup of Γ F . Then a Langlands-Satake parameter is aĜ(C)-conjugacy class of admissible homomorphisms where I Fv is the inertia group, and satisfies a certain semisimplicity condition. (Note that this projection is well-defined, since the action of the inertia group I v onĜ(C) is trivial by assumption.) If Γ Fv acts trivially onĜ -equivalently, if G is split over F v -then s v is entirely determined by the conjugacy class of the projection toĜ(C) of s v (Frob v ). This semisimple conjugacy class inĜ(C) is referred to simply as a Satake parameter.
As explained in [BG14, §2.2], there is a bijection between isomorphism classes of irreducible unramified representations of G(F v ), and Langlands-Satake parameters.
1.3. The Buzzard-Gee conjecture. Let Π = Π v be an automorphic representation of G(A F ). Then the local factor Π v is unramified for almost all v, so we have a collection of Satake parameters (s v ) v / ∈Σ , where Σ is a finite set. On the other hand, we also have a Harish-Chandra parameter for each infinite place σ of F , which is a Weyl group orbit 1 λ σ ∈ X • (T ) ⊗ C, whereT is a maximal torus inĜ.
Then there is a finite extension E/Q such that the Satake parameters r(Π v ) are all defined over E; and for any prime and choice of embedding ι : E → Q , there exists an admissible homomorphism such that the restriction of r Π to Γ Fv is conjugate to ι(s v ) for every prime v / ∈ Σ such that v .
1.4. Weil restriction. We now check a compatibility property of the above conjecture. Let E ⊆ F be number fields. Let H be a reductive group over F , and let G be the Weil restriction Res F/E H, which is a reductive group over E. Then G(A E ) is canonically isomorphic to H(A F ), and this isomorphism sends G(E) to H(F ); so automorphic representations of H(A F ) and of G(A E ) are the same objects. However, the Buzzard-Gee conjecture for H over F , and for G over E, are apparently very different statements. In this section we shall check that the two statements are in fact equivalent. Proposition 1.3. Let E ⊆ F be number fields. Let H be a reductive group over F , and let G be the Weil restriction Res F/E H, which is a reductive group over E. Then: • The dual groupĜ is a product of [F : E] copies ofĤ indexed by the cosets Γ E /Γ F ; in particular the subgroup Γ F preserves the first factor. • The L-group L G is isomorphic to the semidirect productĜ Γ E , with the natural action of Γ E onĜ. • If r : Γ F → L H(Q ) is an admissible homomorphism, there is an admissible homomorphismr (uniquely determined up to conjugacy) such that the projection ofr| Γ F to the first factor ofĜ is r.
Remark 1.4. This proposition takes a particularly simple form if H is split over F (or is an inner form of a split group). In this case the action of Γ F onĤ is trivial, so L H is a direct product; and an admissible homomorphism Γ F → L H(Q ) is simply a homomorphism Γ F →Ĥ(Q ). Meanwhile,Ĝ ∼ = x∈Γ E /Γ FĤ , with Γ E acting by permuting the factors via its left action on Γ E /Γ F . In this situation, if r is an L-homomorphism Γ F → L H(Q ), and ρ :Ĥ → GL m is a representation ofĤ, then there is a natural representationρ : L G → GL [F :E]m whose restriction to the identity componentĜ is given by ρ × · · · × ρ; and the 1 If σ is a complex place then there is a small subtlety in that λσ actually depends not only on the place σ but also on a choice of isomorphism Fσ ∼ = C; but replacing this isomorphism with its conjugate changes λσ by an element of X•(T ), so the notion of L-algebraicity is well-defined. However, in this paper we shall mostly restrict to the case of totally real F where this subtlety does not arise. compositeρ •r is the induced representation Ind Γ E Γ F (ρ • r) in the usual sense. This justifies the notation "Ind F/E (r)" for this homomorphismr.
Proof of Proposition 1.3. The first two statements of the proposition are standard. We give an outline of the construction of the homomorphismr.
It is convenient to work in a slightly more general setting: let V be an arbitrary group, and ρ : Let G be the group H U/V U . Explicitly, an element of G is a pair (f, u) where f is a function U/V → H and u ∈ U , and the multiplication is given by We define a mapρ : . . , d} is the unique index such that u −1 i uu k ∈ V . Then a routine but tedious check shows thatρ is a group homomorphism.
We now consider automorphic representations of G and H. Let Π be an automorphic representation of H(A F ), and letΠ denote the same space regarded as a representation of G(A E ).
Proposition 1.5. We have the following compatibilities: Proof. Statements (i) and (ii) are proved in [BG14], in Section 3.1 and Section 3.2 respectively. So it remains to prove (iii), for which we need to make precise the relation between the Langlands-Satake parameters ofΠ w and Π v . Let H v denote the base extension of H to F v , and similarly for G w . Then we have G w = v|w Res Fv/Ew H v as algebraic groups over E w . For each v, we have a Langlands-Satake parameter s v : Applying exactly the same induction process as before, we obtain an admissible homomorphisms From the definition of the Langlands-Satake parameter, one sees thats v is exactly the Langlands-Satake parameter of Π v considered as a representation of the E wpoints of the algebraic group Res Fv/Ew H v over E w .
There is a bijection between the orbits for the action of the Frobenius σ w on the factors ofĜ(C), and the primes v | w; so taking the fibre product (over Γ Ew ) of the representationss v defines an admissible homomorphisms w : W Ew → L G(C). Since the Langlands-Satake parameter of a representation Π ⊗ Π of a product group U × U is the fibre product of the parameters of the factors, we see thats w is exactly the Langlands-Satake parameter ofΠ w . On the other hand, sinces w is obtained from (s v ) v|w by induction, it is clear that ι(s w ) is the restriction to W Ew of a global homomorphismr = Ind F/E (r) if and only if ι(s v ) is the restriction of r to W Fv for all v | w.
Corollary 1.6. The Buzzard-Gee conjecture is true for an automorphic representation Π of H(A F ) if, and only if, it is true for the same representation regarded as a representation of G(A E ).
2. Hilbert modular forms 2.1. Weights. Let F be a totally real field, and let Σ F be the set of infinite places of F . By a weight for F , we mean a collection k = (k σ ) σ∈Σ F of integers indexed by Σ F .
Notation. For x ∈ F × and k a weight, we write x k for σ σ(x) kσ ∈ R × .
Thus weights are just the same thing as characters of the torus Res F/Q G m .
Definition 2.1. We say k is paritious if the parity of k σ is independent of σ.
We also consider a slightly more general notion. For E ⊆ F a subfield and k a weight of F , we define k E to be the weight for E defined by (k E ) τ = σ|τ k σ (equivalently, the restriction of k to Res E/Q G m ⊂ Res F/Q G m ).
Definition 2.2. We shall say k is E-paritious if k E is paritious as a weight for E.
Thus being E-paritious is no condition at all if E = Q, and becomes more restrictive as E gets larger, with the opposite extreme E = F being the previous definition.
2.2. Adelic Hilbert modular forms. Let H F be the set of elements of F ⊗ C of totally positive imaginary part, with its natural left action of GL + 2 (F ⊗ R). Let k = (k σ ) σ∈Σ F be a collection of integers, and t = (t σ ) σ∈Σ F a collection of real numbers. We can define the weight (k, t) right action of GL + 2 (F ⊗ R) on functions Notation. We say the pair (k, t) is reasonable if the quantity k σ +2t σ is independent of σ, which is equivalent to requiring that ( x 0 0 x ) acts trivially for all x ∈ O ×+ F (or just for all x in a finite-index subgroup). We denote the common value of k σ + 2t σ by R.
We define a Hilbert modular form of weight (k, t) to be a function we need an additional condition of holomorphy at the cusps, which is otherwise automatic by the Köcher principle.) We write M k,t for the space of such functions, and S k,t for the subspace of cusp forms. Both spaces are clearly zero unless (k, t) is reasonable.
Remark 2.3. We have chosen to formulate the definition in terms of GL 2 (A F,f )×H F since it makes the link to the classical theory slightly more direct. The alternative, more analytic, approach is to work with functions on the quotient GL 2 (F )\ GL 2 (A F ).
Concretely, if f is a Hilbert modular form in the above sense, then the function The following properties of M k,t and S k,t are well-known: • The spaces M k,t and S k,t are admissible smooth representations of the group GL 2 (A F,f ), via the right-translation action.
where 1 is the weight all of whose components are 1, then the map , defines a bijection between M k,t and M k,t , and an isomorphism of GL 2 (A F,f )-representations.
(Here x is the adèle norm map, sending a uniformiser at a prime q of F to the reciprocal of the size of its residue field.)

Hecke theory and Satake parameters. Let Π be an irreducible GL
where the product runs over finite primes of F , and each Π v is an irreducible smooth representation of GL 2 (F v ). All but finitely many of the Π v will be unramified, so we have a collection of Satake parameters s v . These s v can be described in terms of the action of Hecke operators. Let T (v) denote the double coset of 1 0 0 v , where v ∈ A F,f is a uniformiser at v; and let S(v) denote the double coset of v 0 0 v . If τ v and σ v denote the eigenvalues of these operators acting on the GL 2 (O F,v )-invariants of Π, then one has the following formula: Proposition 2.4. The Satake parameter s v is the semisimple conjugacy class such that We give a more explicit description of the s v if the prime v is narrowly principal, generated by a totally positive element ; compare [BG14, §3.3] for F = Q. Let f be the new vector of Π. Then the restriction of f to H F has a Fourier expansion There is a constant t( ), the "naive Hecke eigenvalue", such that c( α) = t( )c(α) if ( , αd F ) = 1. This is related to the "normalised Hecke eigenvalue" τ v above by Meanwhile, the quantity σ v is simply Nm(v) 2−R χ( ), where χ is the finite-order character by which the diamond operators act on F . It is shown in §3.2 of [BG14] that Π is L-algebraic if and only if t σ ∈ 1 2 +Z, for all σ ∈ Σ F . Notice that, for a given k, we can find t such that (k, t) is reasonable and t σ ∈ 1 2 + Z ∀σ if and only if k is paritious. Thus the automorphic representations of G arising from non-paritious Hilbert modular forms cannot be twisted to become L-algebraic.
It follows from Shimura's algebraicity theorem quoted above that if all t σ are in 1 2 + Z then the Satake parameters s v are all defined over a finite extension of Q (for all good primes v, not only those trivial in the narrow class group).
Remark 2.5. Buzzard and Gee define Π to be L-arithmetic if all the s v lie in a common finite extension. So Shimura's algebraicity theorem shows that if Π is L-algebraic, then it is L-arithmetic. If F = Q, the converse holds: L-arithmetic implies L-algebraic, as shown in [BG14]. The same holds over general fields F , as we will see in the next section.
2.4. The group G * . Now let E be a subfield of F , as before, and set G = Res F/E GL 2 . We are interested in subgroups of G defined by a condition on the determinant, as follows. The group GL 1 is a subgroup of Res F/E GL 1 in the obvious way. We define a group G * over E by given by the tensor product of the standard 2-dimensional representations of the GL 2 factors. This representation factors throughĜ * , and since it is invariant under permutation of the factors, it extends to a representation of L G * . We call this the Asai representation, as the corresponding L-series first appeared in the work of Asai [Asa77]; see also Yoshida [Yos94]. However, it is important to note that many other interesting algebraic representations of L G factor through L G * , such as the induction from L H of the 3-dimensional adjoint representation of L H, where H = GL 2 /F . The reason for introducing G * is that it, so to speak, "makes more representations algebraic". There is a natural quotient map X • (T ) to X • (T * ), whereT is the standard maximal torus ofĜ. If λ ∈ X • (T ) C , and λ * is its image in X • (T * ) C , then it can occur that λ * is integral even if λ is not. In fact, we have the following result: given by a Hilbert modular form over F of weight (k, t); and for τ a real place of E, let λ τ be the Harish-Chandra parameter of Π τ .
Proof. Using the basis of the Cartan subalgebra of gl 2 (C) described in [BG14, §3.3], we can identify X • (T ) with the abelian group (m σ , n σ ) σ|τ : m σ , n σ ∈ Z, m σ = n σ mod 2 , and in terms of this basis we have One has a similar description of X • (T * ); it is given by pairs ((m σ ) σ|τ , n), with m σ , n ∈ Z such that n = m σ mod 2. The quotient map is given by Proposition 2.9. If k is E-paritious, then we may choose the t σ such that (k, t) is reasonable and λ * τ is L-algebraic for all real places τ of E. Conversely, if k is not E-paritious then no such t exists.
Proof. Since (k, t) is reasonable, the quantity k σ + 2t σ = R is independent of σ.
. We can chose R so that this number is an integer if and only if the parity of σ|τ k σ is independent of τ .
2.5. Restriction of automorphic representations for G. Let Π be an irreducible GL 2 (A F,f )-subrepresentation of S k,t . Then we may consider the restriction of Π to the subgroup G * (A E,f ). This will usually not be irreducible. We denote by Ψ the set of irreducible constituents of Π as a G * (A E,f )-representation; this is (the finite part of) a global L-packet for G * .
If Π is not of CM type (which we shall assume from now on), then all representations Π * ∈ Ψ are the finite parts of automorphic representations of G * , and they all have the same multiplicity in the spectrum of G * [BL84, §3.2]. Moreover, any two representations Π * 1 , Π * 2 ∈ Ψ have the same Satake parameter at any prime where they are both unramified, and the same Harish-Chandra parameter at ∞; these parameters are simply the images of the Satake and Harish-Chandra parameters of Π under the quotient map L G(C) → L G * (C).
In particular, the Buzzard-Gee conjecture is true for one Π * ∈ Ψ if and only if it holds for all of them, with the same representation r Π * ,ι . (That is, the Buzzard-Gee conjecture is really an assertion about automorphic L-packets, not about individual automorphic representations.)

Galois representations
3.1. Setup. The following theorem, which establishes the Buzzard-Gee conjecture for automorphic representations of GL 2 arising from paritious Hilbert modular forms, is well known: where k σ ≥ 2 and t σ ∈ 1 2 + Z for all σ. Let be prime and let ι be an isomorphism C → Q . Then there exists a continuous Galois representation such that for all primes v at which the local factor Π v is unramified, the representation r Π,ι is also unramified, and the conjugacy class of r Π,ι (Frob v ) is ι(s v ).
(For concreteness we take Frob v to be the geometric Frobenius at v, inducing x → x 1/ Nm(v) on the residue field, although the validity of the above statement is obviously independent of the choice of geometric or arithmetic Frobenius.) Via the restriction-of-scalars compatibility above, the conjecture is true for the same representations Π regarded as automorphic representations of G = Res F/E GL 2 for any intermediate field E, giving admissible homomorphisms If k is not paritious, but is E-paritious for some subfield E (recall that this is always the case for E = Q), then the above theorem says nothing. However, as we have seen above, the restriction of Π to the group G * is L-algebraic for a suitable choice of t, and hence the Buzzard-Gee conjecture predicts Galois representations into L G * . The goal of this section will be to construct these "extra" Galois representations.
Theorem 3.2 (Blasius-Rogawski). Let Π be a non-CM irreducible subrepresentation of S k,t , where k σ ≥ 2 for all σ. Let K/Q be an imaginary quadratic extension and set M = F K. Then there exists a Hecke character χ of M , and a continuous Galois representation r Π,χ,ι : Γ M → GL 2 (Q ), with the following property: let v be a prime of F which splits in M/F and such that Π and χ are unramified at v. Then for each of the two primes w above v, the restriction of r Π,χ,ι to W Mw is conjugate to ι(s v ⊗ χ(w)). Furthermore, if Π E is not induced from a character of A × M , then r Π,χ,ι is irreducible. Proof. The existence of r Π,χ,ι comes from [BR93, Theorem 2.6.1], while the irreducibility result is proved in the same way as [Mok14, Theorem 4.14, Proposition 5.9] (using the fact that Π is assumed to be non-CM, so its base-change to M is cuspidal). Proof. As mentioned in Remark 2.5, Shimura's algebraicity results show that Lalgebraic implies L-arithmetic. For the converse, the argument given in [BG14] generalizes as follows: by Theorem 3.2 there are infintely many principal primes v for which s v is non-zero (look at the residual representation at a prime = 2 and primes mapping to the identity have this property). If Π is L-arithmetic, by Shimura's theorem the set {v t Nm(v)} lies in a finite extension, so t ∈ 1 2 + Z. Before stating the main result, we need an auxiliary Lemma.
Theorem 3.5. Let Π be a non-CM-type irreducible subrepresentation of S k,t , and E ⊂ F such that the restricted representation Π * is L-algebraic. Let ι : C → Q an isomorphism. Then there is a Galois representation r * Π,ι : Γ E → L G * (Q ), whose local factors at unramified places v are the ι(r * v ).
Proof. As in Theorem 3.2, we choose an imaginary quadratic field K, and a character χ of A × M (where M = F K), such that there is a Galois representation r Π,χ,ι : Γ M → GL 2 (Q ) whose Satake parameters at the split primes are determined by Π and χ. Let L = KE. By Proposition 1.3 we can extend r Π,χ,ι to an admissible homomorphism Let us write r * Π,χ,ι for the projection ofr Π,χ,ι into the quotient L G * (Q ). Since Π is E-paritious, the Hecke character χ| GL1(A L ) is algebraic. Hence it has a Galois representation r χ,ι : Γ E → GL 1 (Q ) attached to it. We identify GL 1 (Q ) with the centre ofĜ * (Q ), and we consider the "tensor product" representation r * Π,K,ι := r * Π,χ,ι ⊗ r χ −1 ,ι : Γ EK → L G(Q ). where by "tensor product" we mean the component-wise product inĜ, which goes to the quotient (as it lies in the center).
Let us check that this morphism r * Π,K,ι is independent of the choice of the character χ. Because of the irreducibility of r Π,χ,ι , the centraliser of the image of r * Π,K,ι is the centre of L G * (Q ), which is just Q * and is thus certainly 2-divisible. So we are in a position to apply the preceding lemma. Let τ denote a lift to Γ E of the complex conjugation automorphism of K/Q. Since F is linearly disjoint from K (and K is Galois), we can and do assume that τ acts trivially on the dual groupĜ. Let (r * Π,χ,ι ) τ denote the morphism given by (r * Π,K,ι ) τ (σ) = r * Π,K,ι (τ στ −1 ). We claim that (r * Π,K,ι ) τ is conjugate to r * Π,K,ι . Tracing through the definitions, we find that (r * Π,K,ι ) τ is obtained by induction and twisting from the homomorphism (r Π,χ,ι ) τ : Γ M → GL 2 (Q ). Since the representations (r Π,χ,ι ) τ and r Π,τ (χ),ι are both irreducible and their traces agree on the Frobenii at split primes, they are conjugate by an element of GL 2 (Q ). Since the construction of r * Π,K,ι is independent of the choice of τ , as we have seen, this gives the required conjugacy between r * Π,K,ι and (r * Π,K,ι ) τ . Hence r * Π,K,ι extends to a representation of Γ E , uniquely determined up to twisting by the quadratic character associated to K/Q. By construction, r * Π,K,ι has the desired Satake parameters at all but finitely many primes split in L/E. It only remains to prove that the quadratic twists may be chosen in a uniform way, so that the morphisms obtained by extending r * Π,Kι for different choices of K coincide; this will imply that the resulting representation has the required Satake parameters at every prime (since for any given prime q, we may choose K such that q is split in K). This will be carried out in the next proposition.
Proposition 3.6. Let K i be an infinite list of imaginary quadratic fields, whose ramification set is pairwise disjoint and disjoint from the ramification set of F , and for each K i let r * Π,Ki,ι : Γ EKi → L G * (Q ) be the morphism constructed in the previous proof. Then there exists a morphism r * Π,ι : Γ E → L G * (Q ) whose restriction to Γ EKi is isomorphic to r * Π,Ki,ι for every i.
This completes the proof of the Buzzard-Gee conjecture for representations of G * arising from E-paritious Hilbert modular forms.
3.3. Realising the Asai representation geometrically. Composing the representation r * Π,ι constructed in the preceding subsection with the Asai representation L G * (Q ) → GL 2 d (Q ), we obtain a 2 d -dimensional -adic representation of Γ E , the Asai Galois representation associated to Π.
In the special case E = Q, this representation can be realised geometrically. Attached to the group G * is a compatible family of Shimura varieties (of varying levels), which are d-dimensional algebraic varieties defined over Q. The main result of [BL84] shows that if the level is taken small enough, the Asai Galois representation of Π is realised (up to semisimplification 3 ) as a direct summand of the middle-degree -adic intersection cohomology of this Shimura variety (with coefficients in some locally-constant sheaf determined by the weight k, t). Hence the content of Theorem 3.5 is to show that this representation factors naturally through the group L G * .
If Q E F then standard conjectures predict that the Asai Galois representation should still be realisable geometrically, via Shimura varieties attached to quaternion algebras. Let us suppose that at least one of the following conditions holds: (i) The degree d = [F : E] is even; (ii) The degree [E : Q] is odd; (iii) There is a finite place v of F at which the local factor Π v is in the discrete series. We then choose an infinite place τ of E, and a quaternion algebra B over F such that B ⊗ F,σ R is split for σ | τ and ramified for all other σ ∈ Σ F . If either (i) or (ii) holds there is a unique such B which is unramified at every finite place; if neither (i) nor (ii) holds, but (iii) does, then we can take B to ramify additionally at v. Then Π admits a Jacquet-Langlands transfer to B × , and the restriction of this representation to the group H * of elements of B × whose reduced norm is in Attached to H * , there is a Shimura variety X of dimension d, whose reflex field is E. It is expected that the Asai Galois representation of Π should appear in the middle-degree -adic cohomology of X , and a conditional proof of this has been given by Langlands [Lan79] modulo a conjecture describing the action of Frobenius on the special fibre.
3 If E = Q then the semisimplification can be dispensed with, since it has been shown by Nekovar [Nek16] that the -adic cohomology is semi-simple.

Relation to Patrikis' construction
In the above construction, we verified the Buzzard-Gee conjecture for the restriction of Π to the group G * ⊆ Res F/E GL 2 . One can also restrict further, all the way to the group G 0 = Res F/E SL 2 . This case has also been treated by Patrikis, who works more generally with essentially self-dual automorphic representations of GL N and SL N for general N [Pat15, Corollary 5.10].
Patrikis' result in the case of GL 2 amounts to the construction of an admissible homomorphism Γ F → PGL 2 (Q ), or (equivalently, via the restriction-of-scalars formalism of Corollary 1.6) an admissible homomorphism Γ E → L G 0 , with the appropriate Satake parameters. This can be seen as a consequence of Theorem 3.5 by composing with the quotient map Remark 4.1. Patrikis' work suggests that a generalisation of Theorem 3.5 should hold for any mixed-parity, regular, essentially self-dual, cuspidal automorphic representation Π of GL n /F . This could potentially be proved, by essentially the same method as above, if one knew that for sufficiently many CM extensions M of F , the representations Γ M → GL n (Q ) associated to L-algebraic twists of the base change of Π to M were irreducible.

The case [F : E] = 2
If F/E is a quadratic extension, then the L-group L G * has a particularly simple description. In this case,Ĝ * is the quotient of GL 2 × GL 2 by the subgroup of elements of the form An explicit model for the Asai representation ofĜ = GL 2 × GL 2 is given by the action on 2 × 2 matrices, via (g 1 , g 2 )(m) = g 1 · m · g t 2 . This factors throughĜ * , and is a faithful representation ofĜ * . We may extend this to a representation of L G * , factoring through the quotientĜ * Gal(F/E), by letting the non-trivial element σ ∈ Gal(F/E) act as m → m t .
This representation preserves the quadratic form q(m) = det m up to scalar multiplication, with the multiplier character given by (g 1 , g 2 ) → det(g 1 ) det(g 2 ). Thus we may regard this representation as a homomorphismĜ * Gal(F/E) → GO 4 . In fact it is an isomorphism between these groups [Ram02,§1]. The identity component GSO 4 thus corresponds toĜ * . We thus obtain the following result: Theorem 5.1. Let F/E be a quadratic extension of totally real fields, and Π a non-CM Hilbert modular automorphic representation of GL 2 /F whose restriction to G * is L-algebraic. Then, for every embedding ι : Q → Q , there exists a Galois representation r * Π,ι : Γ E → GO 4 (Q ) such that for primes w = w 1 w 2 of E split in F , r * Π,ι (Frob w ) is conjugate to the image of (s w1 (Π), s w2 (Π)) under the map GL 2 × GL 2 → GO 4 .
Let ν denote the orthogonal multiplier GO 4 → G m . Then ν • r * Π,ι is the -adic Galois character corresponding (via ι) to the algebraic Grössencharacter where ω : F × \A × F → C × is the central character of Π. (Note that ω will not generally be algebraic as a Grössencharacter of F , but its restriction to E will be.) The determinant of the standard 4-dimensional representation of GO 4 agrees with ν 2 on GSO 4 , but not on GO 4 ; the determinant of r * Π,ι is therefore given by ω 2 | A × E · χ F/E , where χ F/E is the character associated to our quadratic extension.
Remark 5.2. For d > 2 we do not know of a simple description of the image of L G * in GL 2 d .

Computing Hilbert modular forms and Quaternion groups
We now explain how these non-paritious Hilbert modular forms can be computed explicitly. For computational purposes, it is better to work with a definite quaternion algebra, rather than with the Hilbert modular variety; so we need to explain how to explicitly compute examples of non-paritious automorphic forms for definite quaternion algebras over F , extending the algorithms explained in [DV13] for the paritious case.
6.1. Groups. Let B be a totally definite quaternion algebra over F , of discriminant d B , and let O B be a maximal order in B. Then H = Res F/E B × is an algebraic group over E; it is an inner form of G = Res F/E GL 2 , and in particular it has the same L-group as G.
Let H * be the fibre product of H with GL 1 over Res F/E GL 1 (with respect to the reduced norm map H → Res F/E GL 1 ); this is an inner form of G * . The Eparitious Hilbert modular forms will give rise to automorphic forms for H which are not algebraic, but become algebraic while restricted to H * . These are exactly the automorphic forms we shall compute. As is well known, B × \(B ⊗ A F,f ) × /U is finite. If C U denotes a set of representatives for this set, and for x ∈ C U we write Γ x = B × ∩ xU x −1 , then the map f → (f (x)) x∈C U gives an isomorphism In We shall now analyse this map more closely, under the following hypothesis: the image of U under the reduced norm map nrd : or if U is one of the subgroups U 1 (N) or U 0 (N) to be introduced below. In this case, all three maps induced by the reduced norm are surjective. We thus obtain a surjection from which is the narrow class group Cl + (F ); and this fits into a commutative diagram where the vertical arrows are natural surjections. Proof. It is clear from the commutativity of the diagram that the image of ψ cannot be any larger than this. Conversely, let x ∈ H(A E,f ) be such that the class of nrd(x) is in the image of Cl + (E). Since the maps (3) are surjective, there exist γ ∈ H(E) and u ∈ U such that nrd(γxu) ∈ A × E,f . That is, γxu ∈ H * (A E,f ), and γxu lies in the same double coset as x.
We now study the fibres of ψ. We will need the following definition: Definition 6.4. The capitulation group is the group Clearly, if a ∈ F ×+ represents a class in the capitulation group, then the ideal aO F is the base-extension to O F of an ideal of O E , whose narrow ideal class is independent of the representative a and is in the kernel of the natural map Cl + (E) → Cl + (F ) (the capitulation kernel ). This gives an exact sequence Definition 6.5. We define an action of K F/E on H * (E)\H * (A E,f )/U * as follows. Given a ∈ F ×+ representing a class in K F/E , there exists γ ∈ H(E) such that nrd(γ) = a, and u ∈ U such that a nrd(u) which clearly is independent of the choice of γ and u, and preserves the fibres of ψ.
Remark 6.6. If a ∈ (O × F ) 2 then the action is trivial, since for such a we may choose γ to be in Z(B) ∩ U and u = γ −1 . Thus the action of K F/E factors through the quotient of K F/E by the image of (O × F ) 2 , which is a finite group.
Clearly the quotient K F/E /O ×+ F permutes different fibers, so the stabilizer is and u depending on f as above, and suppose that there existsγ ∈ H * (E) andũ ∈ U * such that γxu =γxũ. Taking implies that the element on the right belongs to Γ x and has norm equal to nrd(γ), up to O ×+ E . If there is no such element, the orbits cannot be equivalent, while if such an element ξ exists,γ = γξ −1 ∈ H * (E) andũ = x −1 ξxu ∈ U * gives the required equivalence.
Corollary 6.8. There exist an algorithm to compute the space M W (H * ; U * ).
Proof. The action of K E/F on the above double quotients translates readily into an action on the space M W (H * ; U * ). For a ∈ F ×+ representing a class in K E/F , and γ, u as before, and f ∈ M W (H * ; U * ), , we define From Theorem 6.7, we see that the image of the pullback map ψ * consists of exactly those forms in M W (H * ; U * ) which are invariant under the action of K F/E . Therefore, provided we have determined the image of Cl + (E) inside Cl + (F ) and the capitulation group K F/E , the algorithms described in [DV13] can be readily adapted to work with ψ * (M W (H; U )).
6.4. Weights. We now define the specific modules W in which we are interested.
Definition 6.9. For (k, t) a weight, with all k σ ≥ 2, we define the weight module of weight (k, t) to be the C-linear representation W (k, t) of B × given by (The appearance of nrd 2−kσ−tσ is needed in order for our parametrisation of the weights to be consistent with automorphic forms for GL 2 via the Jacquet-Langlands correspondence.) Here the action of B × on the first factor is given by choosing splittings B ⊗ F,σ C ∼ = M 2×2 (C), for each σ ∈ Σ F . This representation is, of course, not algebraic unless the t σ are all in Z.
Notation. We write M k,t (H; U ) for M W (k,t) (H; U ) and similarly for H * .
The restriction map ψ * is clearly compatible with taking direct limits as U shrinks. So we have a well defined map where M k,t (H) := lim − →U M k,t (H; U ) and likewise for H * . We now recall the precise statement of the Jacquet-Langlands correspondence. Let S k,t (H) = M k,t (H) if k = (2, . . . , 2), and if k = (2, . . . , 2) let it be the quotient of M k,t (H) by its unique one-dimensional subrepresentation.
Theorem 6.10 (Jacquet-Langlands). There is a bijection between the H(A E,f )subrepresentations of S k,t (H), and the GL 2 (A F,f )-subrepresentations of the space S k,t of holomorphic Hilbert modular forms whose local factors at the primes dividing d B are discrete series; and this bijection preserves Satake parameters at the unramified primes.
Let Π H * be an automorphic representation of H * of weight (k, t) which arises from ψ * (S k,t (H)). Then Π H * is a constituent of some automorphic representation Π H of H, which is the Jacquet-Langlands correspondent of an automorphic representation Π G of G arising in S k,t . If Π G * is any G * -constituent of Π G , then the Satake parameters of Π G * at unramified primes are the same as those of Π H * ; and we can compute these using the action of Hecke operators on M k,t (H * ). This gives an explicit approach to computing with automorphic representations arising from (possibly non-paritious) Hilbert modular forms. 6.5. Induction and Shapiro's lemma. We shall also need to consider some more general modules incorporating some finite-order character. Let N be an ideal of O F coprime to d B . For each q | N we fix an isomorphism , and the quotient is isomorphic to (O F /N) × .
Definition 6.11. Let ε be a character of (O F /N) × . The weight module for (N, where the action on C[ is given by induction from the character ε : The module V (N, k, t, ε) is not a representation of B × , but only of the subgroup consisting of elements that are units locally at the primes dividing N. However, by weak approximation, an automorphic form for H or H * (of any level) is uniquely determined by its values on elements of H(A E,f ) or H * (A E,f ) that are units at N. Thus we may make the following definition: Definition 6.12. We define the space of quaternionic Hilbert modular forms of weight (k, t), level N and character ε by . We define similarly a space M * k,t (N, ε) of automorphic forms on H * . From Shapiro's lemma, one sees readily that there is an isomorphism between M k,t (N, ε) and the subspace of M W (k,t) (H; U 1 (N)) where the quotient U 0 (N)/U 1 (N) acts via the character ε. However, the former interpretation is more convenient for computations, since for U = O × B the double cosets C U have an interpretation as equivalence classes of right O B -ideals in B, and there are robust algorithms available for computing with them, as explained in [DV13].
acts via a character on V (N, k, t, ε), and this character is trivial if and only if (k, t) is reasonable and ε(u) = σ sign σ(u) kσ for all u ∈ O × F . Remark 6.14. The conditions of the lemma are equivalent to ε being the finite part of a Hecke character of conductor N, whose signs at the infinite places are determined by the k σ .
For U = O × B , each of the groups Γ x appearing in (1) will contain O × F as a finite-index subgroup; so M k,t (N, ε) is zero unless the conditions of Lemma 6.13 are satisfied. If these conditions do hold, then M k,t (N, ε) can be decomposed into a direct sum of eigenspaces for the action of Z(H)(A E,f ), corresponding to the set of Grössencharacters of F extending ε. For M * k,t (N, ε), the action of Hecke operators is more restricted. We obtain Hecke operators T (m) and S(m) for any ideal m of O E (rather than O F ) coprime to Nd, and these are compatible with the corresponding operators for H via the map ψ. More generally, we can descend to H * those Hecke operators for H corresponding to double cosets with a natural choice of representative lying in H * . For instance, if p is a prime of F , then the operator S(p) −1 T (p 2 ) is well-defined as a Hecke operator for H * , although S(p) and T (p 2 ) themselves are not, since in the spherical Hecke algebra of GL 2 (F q ) we have for a uniformizer at q, and the double-coset representatives on the left are in SL 2 (F q ) and thus a fortiori in H * (A E,f ).
Although we have fewer Hecke operators to consider when working with H * , we have potentially gained an algebraicity property. If k is not F -paritious, but is E-paritious, then we can choose t such that (k, t) is reasonable and W is algebraic as a representation of H * (although we cannot, of course, make it algebraic as a representation of H). In this case, we can find a finite extension L/Q to which V (N, k, t, ε) descends, and hence M * k,t (N, ε) is the base-extension to C of an Lvector space which is preserved by the action of the Hecke operators for H * .
Remark 6.15. We can re-introduce some of the "missing" Hecke action using a trick due to Shimura (cf. [LLZ, Definition 2.2.4]). Let H denote the subgroup of (B ⊗ A F,f ) × consisting of the elements whose reduced norms are in F ×+ · A × E,f ⊂ A × F,f . Then the double quotient H(E)\H /U * bijects with H * (E)\H * (A E,f )/U * , so we can interpret M * k,t (N, ε) as a space of functions on H /U * . Thus we may define a Hecke operator for any double U * -coset in H . In particular, we can use this to make sense of T (p) as an operator on M * k,t (N, ε) for any prime p Nd B of F whose ideal class lies in the image of Cl + (E) in Cl + (F ); however, this will only be well-defined modulo the action of the capitulation group K E/F . Note that the Hecke operators associated to double cosets in H make sense even if (k, t) is not "reasonable" in the sense of §2. There is a 6-dimensional C representation of the group H = Res F/Q B × corresponding to k = (4, 3) and t = (− 7 4 , − 5 4 ), given by where V σi is the 2-dimensional representation of H coming from a splitting of B ⊗ F,σi C. This representation is, of course, not algebraic, but its restriction to H * is algebraic and can be descended to any finite extension K/F over which B splits, such as the cyclotomic field Q(ζ 8 ).
In order to obtain non-zero Hilbert modular forms, we need to take a non-trivial character. Let N be the ideal generated by 5 − 3 √ 2 (so N is one of the two prime ideals above 7). There is a unique non-trivial quadratic character ε : (O F /N) × → ±1, and one checks that for u ∈ O F we have ε(u) = sign σ 2 (u), where σ 2 is the embedding F → R mapping √ 2 to − √ 2; in particular, the restriction of ε to O × F is the inverse of the central character of V , a necessary condition for Hilbert modular forms of weight V and character ε to exist.
With this choice we compute that the space M k,t (N, ε) is 2-dimensional. Since F has narrow class number one, and O ×+ F = (O × F ) 2 , this is isomorphic (via the pullback map ψ) to the space M * k,t (N, ε). 7.2. Hecke operators. If m is an ideal of F coprime to n, then we have two related definitions of a Hecke operator at m: • A normalized Hecke operator T (m), defined as in §6.6 above.
• A naive Hecke operator T ( ), depending on a choice of totally-positive generator of m. This is given by identifying W as an H * -representation with the representation W (k, t ) = Sym 2 V σ1 ⊗Sym 1 V σ2 , where t = 2−k = (−2, −1); and treating T ( ) as a double coset in the group H of Remark 6.15. The normalisation of the "naive Hecke operator" is chosen in such a way that its eigenvalue corresponds to the "naive Hecke eigenvalue" defined above in the complex-analytic theory. The two operators are related by the formula In particular, if m is the base-extension to F of an ideal of Z, and is the positive integer generating m, then T (m) and T ( ) agree. The normalised Hecke operator T (m) is canonically defined, but it does not preserve the natural K-structure on the space, so the collection of eigenvalues of these operators (for varying m) do not all lie in a finite extension of Q. On the other hand, the naive Hecke operator T ( ) preserves the K-structure, but it will depend on the the choice of generator .
From equation (4), it is clear that if p is a prime inert in F and m = (p), then T (m) = T (p); whereas if p = p 1 p 2 is a prime split or ramified in F , and 1 , 2 are totally positive generators of these ideals such that 1 2 = p, then T (p 1 )T (p 2 ) = T ( 1 )T ( 2 ) = T (p). So in either case we do have a canonical operator T (p), which is both independent of choices and has eigenvalues defined over a finite extension, which is the Hecke operator of H * and can be computed with either definition.
Similarly we can define a normalized operator S(m) for any ideal m, and a naive operator S( ) for ∈ O F , via the action of ( 0 0 ). Note that if p is a split prime and 1 2 = p, the operators T ( 2 1 )S( 2 ) and T ( 2 2 )S( 1 ) are well defined and are independent of the choice of generators with either (but consistent) definition. Clearly the action of S(p) is given by p 3 ε(p).
7.3. Hecke eigenvalues. Our space M k,t (N, ε) is an irreducible module for the Hecke algebra with coefficients in F ; it decomposes over the CM field L = F [b], where b 2 = −3 √ 2 − 8. (We note that L is not Galois over Q.) In Table 7.1, we display the Hecke eigenvalues for all primes of F of norm up to 200. For an inert prime p, we list the eigenvalue t(p) of the Hecke operator T (p) = T (p). For a split prime, we choose arbitrary totally-positive generators 1 and 2 of the two primes above p such that 1 2 = p, and we list the eigenvalues t( i ) of the naive Hecke operators T ( 1 ) and T ( 2 ).
7.4. Satake parameters. Let Π = Π 0 ⊗ nrd −1/2 , where Π 0 is the automorphic representation of H arising from the system of eigenvalues described above (and tabulated in Table 7.1). The shift by nrd −1/2 is included in order to give a slightly more pleasant normalisation of the Satake parameters. If s p denotes the Satake parameter of Π at a finite prime p, then s p is the conjugacy class of matrices with characteristic polynomial H p (X) = X 2 − τ (p)X + Nm(p) 5/2 ε(p), where τ (p) denotes the T (p)-eigenvalue. On the other hand, we may consider the "naive Satake parameter" where is a choice of totally-positive generator of p. Then the characteristic polynomial of s is the polynomial H (X) = X 2 − t( )X + σ 1 ( ) 3 σ 2 ( ) 2 ε(p) where as above t( ) is the eigenvalue of T ( ); and these polynomials all have coefficients in the finite extension L = F [b].  Table 7.1).