Nonparitious Hilbert modular forms
 44 Downloads
Abstract
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, nonparitious 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 nonparitious Hilbert modular eigenforms. On the theoretical side, our starting point is a theorem of Patrikis, which associates projective\(\ell \)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 \({{\,\mathrm{GL}\,}}_2\) and \({\text {PGL}}_2\); and we verify that the predicted Galois representations do indeed exist. On the computational side, we give an algorithm to compute nonparitious 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 an example.
Keywords
Hilbert modular forms Galois representationsMathematics Subject Classification
Primary 11F41 Secondary 11F801 Introduction
1.1 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 \(\ell \)adic Galois representations, for any prime \(\ell \). 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 [4].
In op.cit., they define a class of automorphic representations \(\Pi \) of G which are “Lalgebraic”; and their conjecture predicts that if \(\Pi \) is Lalgebraic, then for every prime \(\ell \) (and isomorphism \(\mathbf {C}\cong \overline{\mathbf {Q}}_\ell \)), there should be a continuous representation of \({{\,\mathrm{Gal}\,}}(\overline{F} / F)\) with values in the Langlands Lgroup \({}^L G(\overline{\mathbf {Q}}_\ell )\), whose restrictions to the decomposition groups at good primes v are determined by the corresponding local factors \(\Pi _v\) of \(\Pi \). (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 \({{\,\mathrm{GL}\,}}_2 / F\), then the automorphic representation \(\Pi \) associated to f is Lalgebraic (after a suitable twist) if and only if the weight of f is “paritious” (all of its components \(k_\sigma \) are congruent modulo 2). It is wellknown that paritious Hilbert eigenforms have associated 2dimensional \(\ell \)adic Galois representations, confirming the Buzzard–Gee conjecture in this case.
On the other hand, there are also eigenforms that are nonparitious. These do not have 2dimensional Galois representations; however, Patrikis [10] showed^{1} one can associate 2dimensional projective\(\ell \)adic Galois representations to such forms. This is wholly consistent with the Buzzard–Gee conjecture: the group \({\text {PGL}}_2\) is the Langlands dual of \({{\,\mathrm{SL}\,}}_2\), and one checks that nonparitious eigenforms give rise to automorphic representations of \({{\,\mathrm{GL}\,}}_2\) which are not Lalgebraic, but become Lalgebraic when restricted to \({{\,\mathrm{SL}\,}}_2\). This has inspired us to begin a more general study of nonparitious Hilbert modular forms, both from a theoretical and a computational viewpoint; as far as we are aware, the problem of computing nonparitious forms explicitly has not been considered before.
1.2 Goals of this article
 (1)
We introduce a hierarchy of conditions on the weight \((\underline{k}, \underline{t})\) of a Hilbert modular automorphic representation \(\Pi \) for \({{\,\mathrm{GL}\,}}_2 / F\), depending on a choice of a subfield \(E \subseteq F\); we call such weights “Eparitious”. (If \(E = F\), this is the usual parity condition that all the \(k_\sigma \) are congruent modulo 2. If \(E = \mathbf {Q}\) it is no condition at all, i.e. every \(\Pi \) is \(\mathbf {Q}\)paritious). We define a subgroup \(G^*\) of the restriction of scalars \(G :={{\,\mathrm{Res}\,}}_{F / E} {{\,\mathrm{GL}\,}}_2\), containing \({{\,\mathrm{Res}\,}}_{F / E} {{\,\mathrm{SL}\,}}_2\); and we show that if \(\Pi \) is Eparitious, the restriction of \(\Pi \) to \(G^*(\mathbf {A}_E)\)isLalgebraic after a suitable twist.
 (2)
We shall demonstrate that, as predicted by the Buzzard–Gee conjecture, we may associate \(\ell \)adic representations of \({\text {Gal}}(\overline{E} / E)\) to Eparitious automorphic representations of \({{\,\mathrm{GL}\,}}_2 / F\), taking values in the Langlands Lgroup of the group \(G^*\) defined in (1). Since our group \(G^*\) always strictly contains \({{\,\mathrm{Res}\,}}_{F / E}({{\,\mathrm{SL}\,}}_2)\), whose Langlands dual is \({{\,\mathrm{Res}\,}}_{F / E}({\text {PGL}}_2)\), this result refines Patrikis’ construction of projective Galois representations.
 (3)
We describe algorithms for computing nonparitious Hilbert modular forms, via the Jacquet–Langlands correspondence between \({{\,\mathrm{GL}\,}}_2\) and totally definite quaternion algebras.
 (4)
We give an explicit example of nonparitious Hilbert modular forms computed using these algorithms, and describe the conjugacy classes of Frobenius elements in their associated Galois representations.
1.3 Notation

F denotes a number field. (In Sect. 1F can be arbitrary, but from Sect. 2 onwards we shall assume F to be totally real.)

\(\mathcal {O}_F\) denotes the ring of integers of F, \(\mathcal {O}_F^\times \) the unit group, and \(\mathcal {O}_F^{\times +}\) the subgroup of totally positive units.

\(\mathbf {A}_F\) is the adèle ring of F.

\({{\,\mathrm{Cl}\,}}^+(F)\) denotes the narrow class group of F.

\(\Gamma _F\) denotes the Galois group \({{\,\mathrm{Gal}\,}}(\overline{F}/F)\).

E will denote a subfield of F, and the notations \(\mathcal {O}_E\), \(\Gamma _E\) etc have the same meanings as for F.
2 Lgroups
In this section we’ll recall from [4] 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.
2.1 Global definitions
Let G be a connected reductive group over a number field F. The Langlands dual\(\hat{G}\) is the connected reductive group \(\hat{G}\) over \(\overline{\mathbf {Q}}\) whose root datum is dual to that of G. The Galois group \(\Gamma _F = {{\,\mathrm{Gal}\,}}(\overline{F} / F)\) acts naturally on \(\hat{G}\), and the Langlands Lgroup\({}^L G\) is the proalgebraic group over \(\overline{\mathbf {Q}}\) defined as the semidirect product \(\hat{G} \rtimes \Gamma _F\). See [4, Sect. 2.1] for details. If G is split over F (or is an inner form of a split group) the action of \(\Gamma _F\) on \(\hat{G}\) is trivial, so \({}^L G\) is a direct product.
We shall be interested in continuous homomorphisms \(\rho : \Gamma _F \rightarrow {}^L G(M)\), for various fields M, satisfying the following condition: the composite of\(\rho \)with the projection\({}^L G(M) \rightarrow \Gamma _F\)is the identity map on\(\Gamma _F\). Such a morphism is called an admissible homomorphism, or sometimes Lhomomorphism. More generally, if \(\Gamma ' \subseteq \Gamma _F\) is a subgroup, we define a homomorphism \(\Gamma ' \rightarrow {}^L G(M)\) to be admissible if its projection to \(\Gamma _F\) is the inclusion map \(\Gamma ' \hookrightarrow \Gamma _F\).
Notation
If \(H_1\) and \(H_2\) are two reductive groups over F, then the Langlands Lgroup \({}^L(H_1 \times H_2)\) is the fibre product \({}^L H_1 \times _{\Gamma _F} {}^LH_2\); for \(r_1: \Gamma _F \rightarrow {}^L H_1\) and \(r_2: \Gamma _F \rightarrow {}^L H_2\) admissible homomorphisms, we write \(r_1 \times r_2: \Gamma _F \rightarrow {}^L(H_1 \times H_2)\) for their product.
2.2 Local theory
If \(\Gamma _{F_v}\) acts trivially on \(\hat{G}\)—equivalently, if G is split over \(F_v\)—then \(s_v\) is entirely determined by the conjugacy class of the projection to \(\hat{G}(\mathbf {C})\) of \(s_v({{\,\mathrm{Frob}\,}}_v)\). This semisimple conjugacy class in \(\hat{G}(\mathbf {C})\) is referred to simply as a Satake parameter.
As explained in [4, Sect. 2.2], there is a bijection between isomorphism classes of irreducible unramified representations of \(G(F_v)\), and Langlands–Satake parameters.
2.3 The Buzzard–Gee conjecture
Let \(\Pi = \bigotimes ' \Pi _v\) be an automorphic representation of \(G(\mathbf {A}_F)\). Then the local factor \(\Pi _v\) is unramified for almost all v, so we have a collection of Satake parameters \((s_v)_{v \notin \Sigma }\), where \(\Sigma \) is a finite set.
On the other hand, we also have a Harish–Chandra parameter for each infinite place \(\sigma \) of F, which is a Weyl group orbit^{2}\(\lambda _\sigma \in X_\bullet (\hat{T}) \otimes \mathbf {C}\), where \(\hat{T}\) is a maximal torus in \(\hat{G}\).
Definition 1.1
We say \(\Pi \) is Lalgebraic if \(\lambda _\sigma \in X_\bullet (\hat{T})\) for every infinite place \(\sigma \).
Conjecture 1.2
2.4 Weil restriction
We now check a compatibility property of the above conjecture. Let \(E \subseteq F\) be number fields. Let H be a reductive group over F, and let G be the Weil restriction \({{\,\mathrm{Res}\,}}_{F / E} H\), which is a reductive group over E. Then \(G(\mathbf {A}_E)\) is canonically isomorphic to \(H(\mathbf {A}_F)\), and this isomorphism sends G(E) to H(F); so automorphic representations of \(H(\mathbf {A}_F)\) and of \(G(\mathbf {A}_E)\) 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

The dual group \(\hat{G}\) is a product of [F : E] copies of \(\hat{H}\) indexed by the cosets \(\Gamma _E / \Gamma _F\); in particular the subgroup \(\Gamma _F\) preserves the first factor.

The Lgroup \({}^LG\) is isomorphic to the semidirect product \(\hat{G}\rtimes \Gamma _E\), with the natural action of \(\Gamma _E\) on \(\hat{G}\).
 If \(r: \Gamma _F \rightarrow {}^L H(\overline{\mathbf {Q}}_\ell )\) is an admissible homomorphism, there is an admissible homomorphism(uniquely determined up to conjugacy) such that the projection of \(\tilde{r} _{\Gamma _F}\) to the first factor of \(\hat{G}\) is r.$$\begin{aligned} \tilde{r} = {{\,\mathrm{Ind}\,}}_{F/E}(r): \Gamma _E \rightarrow {}^L G(\overline{\mathbf {Q}}_\ell ), \end{aligned}$$
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 \(\Gamma _F\) on \(\hat{H}\) is trivial, so \({}^L H\) is a direct product; and an admissible homomorphism \(\Gamma _F \rightarrow {}^L H (\overline{\mathbf {Q}}_\ell )\) is simply a homomorphism \(\Gamma _F \rightarrow \hat{H}(\overline{\mathbf {Q}}_\ell )\). Meanwhile, \(\hat{G} \cong \prod _{x \in \Gamma _E / \Gamma _F} \hat{H}\), with \(\Gamma _E\) acting by permuting the factors via its left action on \(\Gamma _E / \Gamma _F\).
In this situation, if r is an Lhomomorphism \(\Gamma _F \rightarrow {}^L H(\overline{\mathbf {Q}}_\ell )\), and \(\rho : \hat{H} \rightarrow {{\,\mathrm{GL}\,}}_m\) is a representation of \(\hat{H}\), then there is a natural representation \(\tilde{\rho }: {}^L G \rightarrow {{\,\mathrm{GL}\,}}_{[F : E] m}\) whose restriction to the identity component \(\hat{G}\) is given by \(\rho \times \dots \times \rho \); and the composite \(\tilde{\rho } \circ \tilde{r}\) is the induced representation \({{\,\mathrm{Ind}\,}}_{\Gamma _F}^{\Gamma _E} (\rho \circ r)\) in the usual sense. This justifies the notation “\({{\,\mathrm{Ind}\,}}_{F / E}(r)\)” for this homomorphism \(\tilde{r}\).
Proof of Proposition 1.3
The first two statements of the proposition are standard. We give an outline of the construction of the homomorphism \(\tilde{r}\).
It is convenient to work in a slightly more general setting: let V be an arbitrary group, and \(\rho : V \rightarrow H\) a homomorphism. Suppose \(U \ge V\) is an overgroup with \([U : V] = d < \infty \).
Let G be the group \(H^{U / V} \rtimes U\). Explicitly, an element of G is a pair (f, u) where f is a function \(U/V \rightarrow H\) and \(u \in U\), and the multiplication is given by \((f, u) (f', u) = (x \mapsto f(x) f'(u^{1} x), u u')\).
We define a map \(\tilde{\rho }: U \rightarrow G, u \mapsto (f_u, u)\), where \(f_u: U/V \rightarrow H\) is defined as follows. Choose a set of coset representatives \(U = \bigsqcup _{i = 1}^d u_i V\). We define \(f_u(u_i) = \rho (u_i^{1} u u_k)\), where \(k \in \{1, \dots , d\}\) is the unique index such that \(u_i^{1} u u_k \in V\). Then a routine but tedious check shows that \(\tilde{\rho }\) is a group homomorphism. \(\square \)
We now consider automorphic representations of G and H. Let \(\Pi \) be an automorphic representation of \(H(\mathbf {A}_F)\), and let \(\tilde{\Pi }\) denote the same space regarded as a representation of \(G(\mathbf {A}_E)\).
Proposition 1.5
 (i)
\(\Pi \) is Lalgebraic as a representation of \(G(\mathbf {A}_E)\) if and only if \(\tilde{\Pi }\) is Lalgebraic as a representation of \(H(\mathbf {A}_F)\) [4, Sect. 3.1].
 (ii)
If w is a finite place of E such that \(F_v / E_w\) is unramified for every \(v \mid w\), then \(\tilde{\Pi }_w = \bigotimes _{v \mid w} \Pi _v\) is unramified as a representation of \(G(E_w)\) if and only if each \(\Pi _v\) is unramified as a representation of \(H(F_v)\); and in this setting, the Langlands–Satake parameter \(\tilde{s}_w\) of \(\tilde{\Pi }_w\) is defined over a subfield E if and only if the same is true of each of the \(s_v\).
 (iii)
Let \(r: \Gamma _F \rightarrow {}^L H(\overline{\mathbf {Q}}_\ell )\) be an admissible homomorphism, and let \(\tilde{r}: \Gamma _E \rightarrow {}^L G(\overline{\mathbf {Q}}_\ell )\) be the induction of r described in Proposition 1.3. Then the restriction of \(\tilde{r}\) to \(W_{E_w}\) is \(\hat{G}\)conjugate to \(\iota ( \tilde{s}_w )\) if and only if the restriction of r to \(W_{F_v}\) is \(\hat{H}\)conjugate to \(\iota (s_v)\) for all \(v \mid w\).
Proof
Statements (i) and (ii) are proved in [4], in Sects. 3.1 and 3.2, respectively. So it remains to prove (iii), for which we need to make precise the relation between the Langlands–Satake parameters of \(\tilde{\Pi }_w\) and \(\Pi _v\).
There is a bijection between the orbits for the action of the Frobenius \(\sigma _w\) on the factors of \(\hat{G}(\mathbf {C})\), and the primes \(v \mid w\); so taking the fibre product (over \(\Gamma _{E_w}\)) of the representations \(\tilde{s}_v\) defines an admissible homomorphism \(\tilde{s}_w: W_{E_w} \rightarrow {}^L G(\mathbf {C})\). Since the Langlands–Satake parameter of a representation \(\Pi \otimes \Pi '\) of a product group \(U \times U'\) is the fibre product of the parameters of the factors, we see that \(\tilde{s}_w\) is exactly the Langlands–Satake parameter of \(\tilde{\Pi }_w\). On the other hand, since \(\tilde{s}_w\) is obtained from \((s_v)_{v \mid w}\) by induction, it is clear that \(\iota (\tilde{s}_w)\) is the restriction to \(W_{E_w}\) of a global homomorphism \(\tilde{r} = {{\,\mathrm{Ind}\,}}_{F / E}(r)\) if and only if \(\iota (s_v)\) is the restriction of r to \(W_{F_v}\) for all \(v \mid w\). \(\square \)
Corollary 1.6
The Buzzard–Gee conjecture is true for an automorphic representation \(\Pi \) of \(H(\mathbf {A}_F)\) if, and only if, it is true for the same representation regarded as a representation of \(G(\mathbf {A}_E)\). \(\square \)
3 Hilbert modular forms
3.1 Weights
Let F be a totally real field, and let \(\Sigma _F\) be the set of infinite places of F. By a weight for F, we mean a collection \(\underline{k}= (k_\sigma )_{\sigma \in \Sigma _F}\) of integers indexed by \(\Sigma _F\).
Notation
For \(x \in F^\times \) and \(\underline{k}\) a weight, we write \(x^{\underline{k}}\) for \(\prod _\sigma \sigma (x)^{k_\sigma } \in \mathbf {R}^\times \).
Thus weights are just the same thing as characters of the torus \({{\,\mathrm{Res}\,}}_{F / \mathbf {Q}} \mathbf {G}_m\).
Definition 2.1
We say k is paritious if the parity of \(k_\sigma \) is independent of \(\sigma \).
We also consider a slightly more general notion. For \(E \subseteq F\) a subfield and \(\underline{k}\) a weight of F, we define \(\underline{k}_E\) to be the weight for E defined by \((k_E)_\tau = \sum _{\sigma \mid \tau } k_\sigma \) (equivalently, the restriction of \(\underline{k}\) to \({{\,\mathrm{Res}\,}}_{E / \mathbf {Q}} \mathbf {G}_m \subset {{\,\mathrm{Res}\,}}_{F / \mathbf {Q}} \mathbf {G}_m\)).
Definition 2.2
We shall say \(\underline{k}\) is Eparitious if \(\underline{k}_E\) is paritious as a weight for E.
Thus being Eparitious is no condition at all if \(E = \mathbf {Q}\), and becomes more restrictive as E gets larger, with the opposite extreme \(E = F\) being the previous definition.
3.2 Adelic Hilbert modular forms
Notation
We say the pair \((\underline{k}, \underline{t})\) is reasonable if the quantity \(k_\sigma + 2 t_\sigma \) is independent of \(\sigma \), which is equivalent to requiring that \(\left( {\begin{matrix}x &{} 0 \\ 0 &{} x\end{matrix}}\right) \) acts trivially for all \(x \in \mathcal {O}_F^{\times +}\) (or just for all x in a finiteindex subgroup). We denote the common value of \(k_\sigma + 2t_\sigma \) by R.

\(f(g, )\) is holomorphic on \(\mathfrak {H}_F\) for all \(g \in {{\,\mathrm{GL}\,}}_2(\mathbf {A}_{F, \mathrm {f}})\),

\(f(\gamma g, ) = f(g, ) \mid _{\underline{k}, \underline{t}} \gamma ^{1}\) for all \(\gamma \in {{\,\mathrm{GL}\,}}_2^+(F)\),

there exists an open compact subgroup U of \({{\,\mathrm{GL}\,}}_2(\mathbf {A}_{F, \mathrm {f}})\) such that \(f(gu, \tau ) = f(g, \tau )\) for all \(u \in U\) and \((g, \tau ) \in {{\,\mathrm{GL}\,}}_2(\mathbf {A}_{F, \mathrm {f}}) \times \mathfrak {H}_F\).
Remark 2.3
We have chosen to formulate the definition in terms of \({{\,\mathrm{GL}\,}}_2(\mathbf {A}_{F, \mathrm {f}}) \times \mathfrak {H}_F\) 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 \({{\,\mathrm{GL}\,}}_2(F) \backslash {{\,\mathrm{GL}\,}}_2(\mathbf {A}_F)\). Concretely, if f is a Hilbert modular form in the above sense, then the function \(\tilde{f}\) on \({{\,\mathrm{GL}\,}}_2(\mathbf {A}_F)\) given by \(\tilde{f}(g_\mathrm {fin}, g_\infty ) = \left( f(g_\mathrm {fin}, ) \mid _{\underline{k}, \underline{t}} g_\infty \right) (1 \otimes i)\) is left \({{\,\mathrm{GL}\,}}_2(F)\)invariant, and for each \(\sigma \in \Sigma _F\), it transforms by \(e^{i k_\sigma \theta }\) under right translation by \(\begin{pmatrix}\cos \theta &{} \sin \theta \\ \sin \theta &{} \cos \theta \end{pmatrix} \in \mathrm {SO}_2(F_\sigma )\). Conversely we can recover f from \(\tilde{f}\) via \(f(g, x + iy) = y^{(\underline{k}+ \underline{t} 1)} \tilde{f}\left( g, \left( {\begin{matrix}y &{} x \\ 0 &{} 1\end{matrix}}\right) \right) \).

The spaces \(M_{\underline{k}, \underline{t}}\) and \(S_{\underline{k}, \underline{t}}\) are admissible smooth representations of the group \({{\,\mathrm{GL}\,}}_2(\mathbf {A}_{F, \mathrm {f}})\), via the righttranslation action.
 If \(\underline{t}' = \underline{t}+ h \cdot \underline{1}\) for some \(h \in \mathbf {R}\), where \(\underline{1}\) is the weight all of whose components are 1, then the map \(f \mapsto f'\), \(f'(g, \tau ) = \Vert \det g\Vert ^h f(g, \tau )\), defines a bijection between \(M_{\underline{k}, \underline{t}}\) and \(M_{\underline{k}, \underline{t}'}\), and an isomorphism of \({{\,\mathrm{GL}\,}}_2(\mathbf {A}_{F, \mathrm {f}})\)representations.(Here \(\Vert x \Vert \) is the adèle norm map, sending a uniformiser at a prime \(\mathfrak {q}\) of F to the reciprocal of the size of its residue field.)$$\begin{aligned} M_{\underline{k}, \underline{t}'} = M_{\underline{k}, \underline{t}} \otimes \Vert \det \Vert ^h. \end{aligned}$$

For any \(f \in M_{\underline{k}, \underline{t}}\) there is a finiteindex subgroup of \(\mathbf {A}_{F, \mathrm {f}}^\times \), containing \(F^{\times +}\), such that for x in this subgroup, \(\left( {\begin{matrix}x &{} 0 \\ 0 &{} x\end{matrix}}\right) \in Z({{\,\mathrm{GL}\,}}_2(\mathbf {A}_{F, \mathrm {f}}))\) acts on f by \(\Vert x\Vert ^{R  2}\) where R is the common value of \(k_\sigma + 2t_\sigma \).

If the \(t_\sigma \) are all in \(\mathbf {Z}\), then \(M_{\underline{k}, \underline{t}}\) and \(S_{\underline{k}, \underline{t}}\) are the baseextensions to \(\mathbf {C}\) of \({{\,\mathrm{GL}\,}}_2(\mathbf {A}_{F, \mathrm {f}})\)representations defined over \(\tilde{F}\), the Galois closure^{3} of F in \(\mathbf {C}\) (see e.g. [12]).
3.3 Hecke theory and Satake parameters
Let \(\Pi \) be an irreducible \({{\,\mathrm{GL}\,}}_2(\mathbf {A}_{F, \mathrm {f}})\)subrepresentation of \(S_{\underline{k}, \underline{t}}\). Then we can write \(\Pi = \mathop {\bigotimes }_{v}^{'} \Pi _v\), where the product runs over finite primes of F, and each \(\Pi _v\) is an irreducible smooth representation of \({{\,\mathrm{GL}\,}}_2(F_v)\). All but finitely many of the \(\Pi _v\) 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 \(\mathcal {T}(v)\) denote the double coset of \(\left( {\begin{matrix}1 &{} 0 \\ 0 &{} \varpi _v\end{matrix}}\right) \), where \(\varpi _v \in \mathbf {A}_{F, \mathrm {f}}\) is a uniformiser at v; and let \(\mathcal {S}(v)\) denote the double coset of \(\left( {\begin{matrix}\varpi _v &{} 0 \\ 0 &{} \varpi _v\end{matrix}}\right) \). If \(\tau _v\) and \(\sigma _v\) denote the eigenvalues of these operators acting on the \({{\,\mathrm{GL}\,}}_2(\mathcal {O}_{F, v})\)invariants of \(\Pi \), then one has the following formula:
Proposition 2.4
It is shown in Sect. 3.2 of [4] that \(\Pi \) is Lalgebraic if and only if \(t_\sigma \in \tfrac{1}{2} + \mathbf {Z}\), for all \(\sigma \in \Sigma _F\). Notice that, for a given \(\underline{k}\), we can find \(\underline{t}\) such that \((\underline{k}, \underline{t})\) is reasonable and \(t_\sigma \in \tfrac{1}{2} + \mathbf {Z}\ \forall \sigma \) if and only if \(\underline{k}\) is paritious. Thus the automorphic representations of G arising from nonparitious Hilbert modular forms cannot be twisted to become Lalgebraic.
It follows from Shimura’s algebraicity theorem quoted above that if all \(t_\sigma \) are in \(\tfrac{1}{2} + \mathbf {Z}\) then the Satake parameters \(s_v\) are all defined over a finite extension of \(\mathbf {Q}\) (for all good primes v, not only those trivial in the narrow class group).
Remark 2.5
Buzzard and Gee define \(\Pi \) to be Larithmetic if all the \(s_v\) lie in a common finite extension. So Shimura’s algebraicity theorem shows that if \(\Pi \) is Lalgebraic, then it is Larithmetic. If \(F = \mathbf {Q}\), the converse holds: Larithmetic implies Lalgebraic, as shown in [4]. The same holds over general fields F, as we will see in the next section.
3.4 The group \(G^*\)
Proposition 2.6
Remark 2.7
The group \(\hat{G} = ({{\,\mathrm{GL}\,}}_2)^{\Gamma _E / \Gamma _F}\) has a \(2^d\)dimensional representation, where \(d = [F : E]\), given by the tensor product of the standard 2dimensional representations of the \({{\,\mathrm{GL}\,}}_2\) factors. This representation factors through \(\hat{G}^*\), 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 Lseries first appeared in the work of Asai [1]; see also Yoshida [13]. 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 3dimensional adjoint representation of \({}^L H\), where \(H = {{\,\mathrm{GL}\,}}_2 / F\).
The reason for introducing \(G^*\) is that it, so to speak, “makes more representations algebraic”. There is a natural quotient map \(X_\bullet (\hat{T})\) to \(X_\bullet (\hat{T}^*)\), where \(\hat{T}\) is the standard maximal torus of \(\hat{G}\). If \(\lambda \in X_\bullet (\hat{T})_\mathbf {C}\), and \(\lambda ^*\) is its image in \(X_\bullet (\hat{T}^*)_\mathbf {C}\), then it can occur that \(\lambda ^*\) is integral even if \(\lambda \) is not. In fact, we have the following result:
Proposition 2.8
Let \(\Pi \) be the automorphic representation of \(G(\mathbf {A}_E) = {{\,\mathrm{GL}\,}}_2(\mathbf {A}_F)\) given by a Hilbert modular form over F of weight \((\underline{k}, \underline{t})\); and for \(\tau \) a real place of E, let \(\lambda _\tau \) be the Harish–Chandra parameter of \(\Pi _\tau \).
Then the projection \(\lambda _\tau ^*\) lies in the integral cocharater lattice \(X_\bullet (\hat{T}^*)\) if, and only if, we have \(\sum _{\sigma \mid \tau } \left( t_\sigma  \tfrac{1}{2}\right) \in \mathbf {Z}\).
Proof
Proposition 2.9
If \(\underline{k}\) is Eparitious, then we may choose the \(t_\sigma \) such that \((\underline{k}, \underline{t})\) is reasonable and \(\lambda _\tau ^*\) is Lalgebraic for all real places \(\tau \) of E. Conversely, if \(\underline{k}\) is not Eparitious then no such \(\underline{t}\) exists.
Proof
Since \((\underline{k},\underline{t})\) is reasonable, the quantity \(k_\sigma + 2t_\sigma = R\) is independent of \(\sigma \). Then \(\sum _{\sigma \mid \tau } (t_\sigma \frac{1}{2}) = \frac{[F:E](R1)\sum _{\sigma \mid \tau }k_\sigma }{2}\). We can chose R so that this number is an integer if and only if the parity of \(\sum _{\sigma \mid \tau } k_\sigma \) is independent of \(\tau \). \(\square \)
3.5 Restriction of automorphic representations for G
Let \(\Pi \) be an irreducible \({{\,\mathrm{GL}\,}}_2(\mathbf {A}_{F, \mathrm {f}})\)subrepresentation of \(S_{\underline{k}, \underline{t}}\). Then we may consider the restriction of \(\Pi \) to the subgroup \(G^*(\mathbf {A}_{E, \mathrm {f}})\). This will usually not be irreducible. We denote by \(\Psi \) the set of irreducible constituents of \(\Pi \) as a \(G^*(\mathbf {A}_{E, \mathrm {f}})\)representation; this is (the finite part of) a global Lpacket for \(G^*\).
If \(\Pi \) is not of CM type (which we shall assume from now on), then all representations \(\Pi ^* \in \Psi \) are the finite parts of automorphic representations of \(G^*\), and they all have the same multiplicity in the spectrum of \(G^*\) [3, Sect. 3.2]. Moreover, any two representations \(\Pi _1^*, \Pi _2^* \in \Psi \) have the same Satake parameter at any prime where they are both unramified, and the same Harish–Chandra parameter at \(\infty \); these parameters are simply the images of the Satake and Harish–Chandra parameters of \(\Pi \) under the quotient map \({}^L G(\mathbf {C}) \rightarrow {}^L G^*(\mathbf {C})\).
In particular, the Buzzard–Gee conjecture is true for one \(\Pi ^* \in \Psi \) if and only if it holds for all of them, with the same representation \(r_{\Pi ^*, \iota }\). (That is, the Buzzard–Gee conjecture is really an assertion about automorphic Lpackets, not about individual automorphic representations.)
4 Galois representations
4.1 Setup
The following theorem, which establishes the Buzzard–Gee conjecture for automorphic representations of \({{\,\mathrm{GL}\,}}_2\) arising from paritious Hilbert modular forms, is well known:
Theorem 3.1
(For concreteness we take \({{\,\mathrm{Frob}\,}}_v\) to be the geometric Frobenius at v, inducing \(x \mapsto x^{1/{{\,\mathrm{Nm}\,}}(v)}\) on the residue field, although the validity of the above statement is obviously independent of the choice of geometric or arithmetic Frobenius.)
4.2 Representations over CM fields
Theorem 3.2
Proof
The existence of \(r_{\Pi ,\chi ,\iota }\) comes from [2, Theorem 2.6.1], while the irreducibility result is proved in the same way as [8, Theorem 4.14, Proposition 5.9] (using the fact that \(\Pi \) is assumed to be nonCM, so its basechange to M is cuspidal). \(\square \)
Corollary 3.3
The representation \(\Pi \) is Larithmetic if and only if it is Lalgebraic.
Proof
As mentioned in Remark 2.5, Shimura’s algebraicity results show that Lalgebraic implies Larithmetic. For the converse, the argument given in [4] generalizes as follows: by Theorem 3.2 there are infintely many principal primes v for which \(s_v\) is nonzero (look at the residual representation at a prime \(\ell \ne 2\) and primes mapping to the identity have this property). If \(\Pi \) is Larithmetic, by Shimura’s theorem the set \(\{v^{\underline{t}}{{\,\mathrm{Nm}\,}}(v)\}\) lies in a finite extension, so \(\underline{t}\in \frac{1}{2}+\mathbf {Z}\). \(\square \)
Before stating the main result, we need an auxiliary Lemma.
Lemma 3.4

it has big image, i.e. \(\{v \in V \; : \; v \psi (u) v^{1}=\psi (u) \, \forall u \in U'\}=Z(V)\).

The homomorphism \(\psi ^\mu : U' \rightarrow V\) defined by \(\psi ^\mu (u)=\psi (\mu u \mu ^{1})\) is conjugate in V to \(\psi \).
Proof
Theorem 3.5
Proof
Let us check that this morphism \(r^*_{\Pi ,K,\iota }\) is independent of the choice of the character \(\chi \). If we multiply \(\chi \) by an algebraic character \(\psi \) of \(\mathbf {A}_M^\times \), then \(\psi \) has an associated Galois representation \(\Gamma _M \rightarrow {{\,\mathrm{GL}\,}}_1(\overline{\mathbf {Q}}_\ell )\), and we may induce this to a homomorphism \(\Gamma _L \rightarrow ({{\,\mathrm{GL}\,}}_1)^{[M : L]} \rtimes {{\,\mathrm{Gal}\,}}(M / L)\). If we compose this homomorphism with the product map \(({{\,\mathrm{GL}\,}}_1)^{[M : L]} \rightarrow {{\,\mathrm{GL}\,}}_1\), then the action of \({{\,\mathrm{Gal}\,}}(M / L)\) becomes trivial, and one checks easily that the result is exactly the Galois representation \(\Gamma _L \rightarrow {{\,\mathrm{GL}\,}}_1(\overline{\mathbf {Q}}_\ell )\) associated to \(\psi _{\mathbf {A}_L^\times }\). Hence the twists cancel out, showing that the representation \(r^*_{\Pi ,K,\iota }\) is independent of the choice.
Because of the irreducibility of \(r_{\Pi , \chi , \iota }\), the centraliser of the image of \(r^*_{\Pi ,K,\iota }\) is the centre of \({}^L G^*(\overline{\mathbf {Q}}_\ell )\), which is just \(\overline{\mathbf {Q}}_\ell ^*\) and is thus certainly 2divisible. So we are in a position to apply the preceding lemma.
Let \(\tau \) denote a lift to \(\Gamma _E\) of the complex conjugation automorphism of \(K / \mathbf {Q}\). Since F is linearly disjoint from K (and K is Galois), we can and do assume that \(\tau \) acts trivially on the dual group \(\hat{G}\). Let \((r^*_{\Pi ,\chi , \iota })^\tau \) denote the morphism given by \((r^*_{\Pi ,K,\iota })^\tau (\sigma )=r^*_{\Pi ,K,\iota }(\tau \sigma \tau ^{1})\). We claim that \((r^*_{\Pi ,K,\iota })^\tau \) is conjugate to \(r^*_{\Pi ,K,\iota }\).
Tracing through the definitions, we find that \((r^*_{\Pi ,K,\iota })^\tau \) is obtained by induction and twisting from the homomorphism \((r_{\Pi , \chi , \iota })^\tau : \Gamma _M \rightarrow {{\,\mathrm{GL}\,}}_2(\overline{\mathbf {Q}}_\ell )\). Since the representations \((r_{\Pi , \chi , \iota })^\tau \) and \(r_{\Pi , \tau (\chi ), \iota }\) are both irreducible and their traces agree on the Frobenii at split primes, they are conjugate by an element of \({{\,\mathrm{GL}\,}}_2(\overline{\mathbf {Q}}_\ell )\). Since the construction of \(r_{\Pi , K, \iota }^*\) is independent of the choice of \(\tau \), as we have seen, this gives the required conjugacy between \(r_{\Pi , K, \iota }^*\) and \((r_{\Pi , K, \iota }^*)^\tau \). Hence \(r_{\Pi , K, \iota }^*\) extends to a representation of \(\Gamma _E\), uniquely determined up to twisting by the quadratic character associated to \(K / \mathbf {Q}\).
By construction, \(r_{\Pi , K, \iota }^*\) 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 \(r^*_{\Pi ,K\iota }\) 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. \(\square \)
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 \(r^*_{\Pi ,K_i,\iota }:\Gamma _{EK_i} \rightarrow {}^LG^*(\mathbf {Q}_\ell )\) be the morphism constructed in the previous proof. Then there exists a morphism \(r_{\Pi ,\iota }^* : \Gamma _E \rightarrow {}^L{G^*}(\overline{\mathbf {Q}_\ell })\) whose restriction to \(\Gamma _{EK_i}\) is isomorphic to \(r^*_{\Pi ,K_i,\iota }\) for every i.
Proof
This completes the proof of the Buzzard–Gee conjecture for representations of \(G^*\) arising from Eparitious Hilbert modular forms.
4.3 Realising the Asai representation geometrically
Composing the representation \(r^*_{\Pi , \iota }\) constructed in the preceding subsection with the Asai representation \({}^L G^*(\overline{\mathbf {Q}}_\ell ) \rightarrow {{\,\mathrm{GL}\,}}_{2^d}(\overline{\mathbf {Q}}_\ell )\), we obtain a \(2^d\)dimensional \(\ell \)adic representation of \(\Gamma _E\), the Asai Galois representation associated to \(\Pi \).
In the special case \(E = \mathbf {Q}\), this representation can be realised geometrically. Attached to the group \(G^*\) is a compatible family of Shimura varieties (of varying levels), which are ddimensional algebraic varieties defined over \(\mathbf {Q}\). The main result of [3] shows that if the level is taken small enough, the Asai Galois representation of \(\Pi \) is realised (up to semisimplification^{4}) as a direct summand of the middledegree \(\ell \)adic intersection cohomology of this Shimura variety (with coefficients in some locallyconstant sheaf determined by the weight \(\underline{k}, \underline{t}\)). Hence the content of Theorem 3.5 is to show that this representation factors naturally through the group \({}^L G^*\).
 (i)
The degree \(d = [F : E]\) is even;
 (ii)
The degree \([E : \mathbf {Q}]\) is odd;
 (iii)
There is a finite place v of F at which the local factor \(\Pi _v\) is in the discrete series.
Attached to \(H^*\), there is a Shimura variety \(\mathcal {X}\) of dimension d, whose reflex field is E. It is expected that the Asai Galois representation of \(\Pi \) should appear in the middledegree \(\ell \)adic cohomology of \(\mathcal {X}\), and a conditional proof of this has been given by Langlands [6] modulo a conjecture describing the action of Frobenius on the special fibre.
5 Relation to Patrikis’ construction
In the above construction, we verified the Buzzard–Gee conjecture for the restriction of \(\Pi \) to the group \(G^* \subseteq {{\,\mathrm{Res}\,}}_{F/E} {{\,\mathrm{GL}\,}}_2\). One can also restrict further, all the way to the group \(G^0 = {{\,\mathrm{Res}\,}}_{F/E} {{\,\mathrm{SL}\,}}_2\). This case has also been treated by Patrikis, who works more generally with essentially selfdual automorphic representations of \({{\,\mathrm{GL}\,}}_n\) and \({{\,\mathrm{SL}\,}}_n\) for general n [10, Corollary 5.10].
For a Hilbert modular automorphic representation \(\Pi \), it follows from the \(n = 2\) case of Patrikis’ result that there is an admissible homomorphism \(\Gamma _F \rightarrow {\text {PGL}}_2(\overline{\mathbf {Q}}_\ell )\), or (equivalently, via the restrictionofscalars formalism of Corollary 1.6) an admissible homomorphism \(\Gamma _E \rightarrow {}^L G_0\), with the appropriate Satake parameters. This can be seen as a consequence of Theorem 3.5 by composing with the quotient map \({}^L G^* \rightarrow \dfrac{{}^L G^*}{Z(\hat{G}^*)} = {}^L G^0\).
Remark 4.1
Patrikis’ work suggests that a generalisation of Theorem 3.5 should hold for any mixedparity, regular, essentially selfdual, cuspidal automorphic representation \(\Pi \) of \({{\,\mathrm{GL}\,}}_n / F\). 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 \(\Gamma _M \rightarrow {\text {GL}}_n(\overline{\mathbf {Q}}_\ell )\) associated to Lalgebraic twists of the base change of \(\Pi \) to M were irreducible.
6 The case \([F : E] = 2\)
If F / E is a quadratic extension, then the Lgroup \({}^L G^*\) has a particularly simple description. In this case, \(\hat{G}^*\) is the quotient of \({{\,\mathrm{GL}\,}}_2 \times {{\,\mathrm{GL}\,}}_2\) by the subgroup of elements of the form \(\left( \left( {\begin{matrix}z &{} 0 \\ 0 &{} z\end{matrix}}\right) , \left( {\begin{matrix}z &{} 0 \\ 0 &{} z\end{matrix}}\right) ^{1}\right) \).
An explicit model for the Asai representation of \(\hat{G} = {{\,\mathrm{GL}\,}}_2 \times {{\,\mathrm{GL}\,}}_2\) is given by the action on \(2 \times 2\) matrices, via \((g_1, g_2)( m) = g_1 \cdot m \cdot g_2^t\). This factors through \(\hat{G}^*\), and is a faithful representation of \(\hat{G}^*\). We may extend this to a representation of \({}^L G^*\), factoring through the quotient \(\hat{G}^* \rtimes {{\,\mathrm{Gal}\,}}(F / E)\), by letting the nontrivial element \(\sigma \in {{\,\mathrm{Gal}\,}}(F / E)\) act as \(m \mapsto m^t\).
This representation preserves the quadratic form \(q(m) = \det m\) up to scalar multiplication, with the multiplier character given by \((g_1, g_2) \mapsto \det (g_1) \det (g_2)\). Thus we may regard this representation as a homomorphism \(\hat{G}^* \rtimes {{\,\mathrm{Gal}\,}}(F / E) \rightarrow {{\,\mathrm{GO}\,}}_4\). In fact it is an isomorphism between these groups [11, Sect. 1]. The identity component \({{\,\mathrm{GSO}\,}}_4\) thus corresponds to \(\hat{G}^*\). We thus obtain the following result:
Theorem 5.1
Let \(\nu \) denote the orthogonal multiplier \({{\,\mathrm{GO}\,}}_4 \rightarrow \mathbf {G}_m\). Then \(\nu \circ r^*_{\Pi , \iota }\) is the \(\ell \)adic Galois character corresponding (via \(\iota \)) to the algebraic Grössencharacter \(\omega _{\mathbf {A}_E^\times }\), where \(\omega : F^\times \backslash \mathbf {A}_F^\times \rightarrow \mathbf {C}^\times \) is the central character of \(\Pi \). (Note that \(\omega \) will not generally be algebraic as a Grössencharacter of F, but its restriction to E will be.)
The determinant of the standard 4dimensional representation of \({{\,\mathrm{GO}\,}}_4\) agrees with \(\nu ^2\) on \({{\,\mathrm{GSO}\,}}_4\), but not on \({{\,\mathrm{GO}\,}}_4\); the determinant of \(r^*_{\Pi , \iota }\) is therefore given by \(\omega ^2 _{\mathbf {A}_E^\times } \cdot \chi _{F / E}\), where \(\chi _{F / E}\) 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 \({{\,\mathrm{GL}\,}}_{2^d}\).
7 Computing Hilbert modular forms and quaternion groups
We now explain how these nonparitious 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 nonparitious automorphic forms for definite quaternion algebras over F, extending the algorithms explained in [5] for the paritious case.
7.1 Groups
Let B be a totally definite quaternion algebra over F, of discriminant \(\mathfrak {d}_B\), and let \(\mathcal {O}_B\) be a maximal order in B. Then \(H = {{\,\mathrm{Res}\,}}_{F / E} B^\times \) is an algebraic group over E; it is an inner form of \(G = {{\,\mathrm{Res}\,}}_{F/E} {{\,\mathrm{GL}\,}}_2\), and in particular it has the same Lgroup as G.
Let \(H^*\) be the fibre product of H with \({{\,\mathrm{GL}\,}}_1\) over \({{\,\mathrm{Res}\,}}_{F/ E} {{\,\mathrm{GL}\,}}_1\) (with respect to the reduced norm map \(H \rightarrow {{\,\mathrm{Res}\,}}_{F / E} {{\,\mathrm{GL}\,}}_1\)); 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.
7.2 Automorphic forms for H and \(H^*\)
The following definition is standard:
Definition 6.1
Similarly, if \(U^*\) is an open compact subgroup of \(H^*(\mathbf {A}_{F, \mathrm {f}})\), and W a representation of \(H^*(E)\), we can define a space \(M_W(H^*; U^*)\) of automorphic forms for \(H^*\) of weight W and level \(U^*\).
7.3 Pullback from H to \(H^*\)
Definition 6.2
Lemma 6.3
The image of \(\psi \) consists of those elements of \(H(E) \backslash H(\mathbf {A}_{E, \mathrm {f}}) / U\) whose reduced norm lies in the image of \({{\,\mathrm{Cl}\,}}^+(E)\) in \({{\,\mathrm{Cl}\,}}^+(F)\).
Proof
It is clear from the commutativity of the diagram that the image of \(\psi \) cannot be any larger than this. Conversely, let \(x \in H(\mathbf {A}_{E, \mathrm {f}})\) be such that the class of \({{\,\mathrm{nrd}\,}}(x)\) is in the image of \({{\,\mathrm{Cl}\,}}^+(E)\). Since the maps (3) are surjective, there exist \(\gamma \in H(E)\) and \(u \in U\) such that \({{\,\mathrm{nrd}\,}}(\gamma x u) \in \mathbf {A}_{E, \mathrm {f}}^\times \). That is, \(\gamma x u \in H^*(\mathbf {A}_{E, \mathrm {f}})\), and \(\gamma x u\) lies in the same double coset as x. \(\square \)
We now study the fibres of \(\psi \). We will need the following definition:
Definition 6.4
Definition 6.5
Remark 6.6
If \(a \in (\mathcal {O}_F^\times )^2\) then the action is trivial, since for such a we may choose \(\gamma \) to be in \(Z(B) \cap U\) and \(u = \gamma ^{1}\). Thus the action of \(K_{F/E}\) factors through the quotient of \(K_{F/E}\) by the image of \((\mathcal {O}_F^\times )^2\), which is a finite group.
For \(x \in H(\mathbf {A}_{E, \mathrm {f}})\), let \(\Gamma _x\) denote the group \(B^\times \cap x Ux^{1}\), as above. Let \(\mathcal {O}_x = \{{{\,\mathrm{nrd}\,}}(\nu ) \; : \; \nu \in \Gamma _x \} \subset \mathcal {O}_F^{\times +}\). As \((\mathcal {O}_F^\times )^2 \subset \mathcal {O}_x\), the quotient \(\mathcal {O}_F^{\times +}/\mathcal {O}_x\) is finite.
Theorem 6.7
Let \(x \in H^*(\mathbf {A}_{E, \mathrm {f}})\). Then \(K_{F/E}\) acts transitively on \(\psi ^{1}(\psi (x))\), and the stabiliser of x is \(\mathcal {O}_x\); i.e. the fiber at \(\psi (x)\) is an homogeneous space for \(K_{F/E}/\mathcal {O}_x\).
Proof
Let \(x,y \in H^*(\mathbf {A}_{E, \mathrm {f}})\) be such that \(\psi ([x]) = \psi ([y])\). Then there exists \(\gamma \in H(E)\) and \(u \in U\) such that \(\gamma x u = y\), so \({{\,\mathrm{nrd}\,}}(\gamma ) \in K_{F/E}\) and \([y] = {{\,\mathrm{nrd}\,}}(\gamma ) \cdot [x]\), proving that the action is transitive.
Corollary 6.8
There exist an algorithm to compute the space \(M_W(H^*; U^*)\).
Proof
7.4 Weights
We now define the specific modules W in which we are interested.
Definition 6.9
(The appearance of \({{\,\mathrm{nrd}\,}}^{2k_\sigma t_\sigma }\) is needed in order for our parametrisation of the weights to be consistent with automorphic forms for \({{\,\mathrm{GL}\,}}_2\) via the Jacquet–Langlands correspondence.) Here the action of \(B^\times \) on the first factor is given by choosing splittings \(B \otimes _{F, \sigma } \mathbf {C}\cong M_{2\times 2}(\mathbf {C})\), for each \(\sigma \in \Sigma _F\). This representation is, of course, not algebraic unless the \(t_\sigma \) are all in \(\mathbf {Z}\).
Notation
We write \(M_{\underline{k}, \underline{t}}(H; U)\) for \(M_{W(\underline{k}, \underline{t})}(H; U)\) and similarly for \(H^*\).
We now recall the precise statement of the Jacquet–Langlands correspondence. Let \(S_{\underline{k}, \underline{t}}(H) = M_{\underline{k}, \underline{t}}(H)\) if \(\underline{k}\ne (2, \dots , 2)\), and if \(\underline{k}= (2, \dots , 2)\) let it be the quotient of \(M_{\underline{k}, \underline{t}}(H)\) by its unique onedimensional subrepresentation.
Theorem 6.10
(Jacquet–Langlands) There is a bijection between the \(H(\mathbf {A}_{E, \mathrm {f}})\)subrepresentations of \(S_{\underline{k}, \underline{t}}(H)\), and the \({{\,\mathrm{GL}\,}}_2(\mathbf {A}_{F, \mathrm {f}})\)subrepresentations of the space \(S_{\underline{k}, \underline{t}}\) of holomorphic Hilbert modular forms whose local factors at the primes dividing \(\mathfrak {d}_B\) are discrete series; and this bijection preserves Satake parameters at the unramified primes.
Let \(\Pi _{H^*}\) be an automorphic representation of \(H^*\) of weight \((\underline{k}, \underline{t})\) which arises from \(\psi ^*(S_{\underline{k}, \underline{t}}(H))\). Then \(\Pi _{H^*}\) is a constituent of some automorphic representation \(\Pi _{H}\) of H, which is the Jacquet–Langlands correspondent of an automorphic representation \(\Pi _G\) of G arising in \(S_{\underline{k}, \underline{t}}\). If \(\Pi _{G^*}\) is any \(G^*\)constituent of \(\Pi _G\), then the Satake parameters of \(\Pi _{G^*}\) at unramified primes are the same as those of \(\Pi _{H^*}\); and we can compute these using the action of Hecke operators on \(M_{\underline{k}, \underline{t}}(H^*)\). This gives an explicit approach to computing with automorphic representations arising from (possibly nonparitious) Hilbert modular forms.
7.5 Induction and Shapiro’s lemma
Definition 6.11
The module \(V(\mathfrak {N},\underline{k},\underline{t}, \varepsilon )\) is not a representation of \(B^\times \), but only of the subgroup consisting of elements that are units locally at the primes dividing \(\mathfrak {N}\). However, by weak approximation, an automorphic form for H or \(H^*\) (of any level) is uniquely determined by its values on elements of \(H(\mathbf {A}_{E, \mathrm {f}})\) or \(H^*(\mathbf {A}_{E, \mathrm {f}})\) that are units at \(\mathfrak {N}\). Thus we may make the following definition:
Definition 6.12
From Shapiro’s lemma, one sees readily that there is an isomorphism between \(M_{\underline{k}, \underline{t}}(\mathfrak {N}, \varepsilon )\) and the subspace of \(M_{W(\underline{k}, \underline{t})}(H; U_1(\mathfrak {N}))\) where the quotient \(U_0(\mathfrak {N}) / U_1(\mathfrak {N})\) acts via the character \(\varepsilon \). However, the former interpretation is more convenient for computations, since for \(U = \widehat{\mathcal {O}}_B^\times \) the double cosets \(C_U\) have an interpretation as equivalence classes of right \(\mathcal {O}_B\)ideals in B, and there are robust algorithms available for computing with them, as explained in [5].
Lemma 6.13
The group \(\mathcal {O}_F^\times \subseteq \mathcal {O}_B^\times \) acts via a character on \(V(\mathfrak {N},\underline{k},\underline{t}, \varepsilon )\), and this character is trivial if and only if \((\underline{k}, \underline{t})\) is reasonable and \(\varepsilon (u) = \prod _{\sigma } {\text {sign}} \sigma (u)^{k_\sigma }\) for all \(u \in \mathcal {O}_F^\times \). \(\square \)
Remark 6.14
The conditions of the lemma are equivalent to \(\varepsilon \) being the finite part of a Hecke character of conductor \(\mathfrak {N}\), whose signs at the infinite places are determined by the \(k_\sigma \).
For \(U = \widehat{\mathcal {O}}_B^\times \), each of the groups \(\Gamma _x\) appearing in (1) will contain \(\mathcal {O}_F^\times \) as a finiteindex subgroup; so \(M_{\underline{k}, \underline{t}}(\mathfrak {N}, \varepsilon )\) is zero unless the conditions of Lemma 6.13 are satisfied. If these conditions do hold, then \(M_{\underline{k}, \underline{t}}(\mathfrak {N}, \varepsilon )\) can be decomposed into a direct sum of eigenspaces for the action of \(Z(H)(\mathbf {A}_{E, \mathrm {f}})\), corresponding to the set of Grössencharacters of F extending \(\varepsilon \).
7.6 Hecke operators

The operator \(\mathcal {T}(\mathfrak {m})\), given by the double Ucoset of elements of \(\widehat{\mathcal {O}}_B\) whose norms generate the ideal \(\mathfrak {m}\widehat{\mathcal {O}}_F\);

the operator \(\mathcal {S}(\mathfrak {m})\), given by the double Ucoset generated by the element \(x \in Z(H)(\mathbf {A}_{E, \mathrm {f}})\), for any \(x \in \widehat{\mathcal {O}}_F\) generating the ideal \(\mathfrak {m}\widehat{\mathcal {O}}_F\).
Although we have fewer Hecke operators to consider when working with \(H^*\), we have potentially gained an algebraicity property. If \(\underline{k}\) is not Fparitious, but is Eparitious, then we can choose \(\underline{t}\) such that \((\underline{k}, \underline{t})\) 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 \(L / \mathbf {Q}\) to which \(V(\mathfrak {N}, \underline{k}, \underline{t}, \varepsilon )\) descends, and hence \(M^*_{\underline{k}, \underline{t}}(\mathfrak {N}, \varepsilon )\) is the baseextension to \(\mathbf {C}\) of an Lvector space which is preserved by the action of the Hecke operators for \(H^*\).
Remark 6.15
We can reintroduce some of the “missing” Hecke action using a trick due to Shimura (cf. [7, Definition 2.2.4]). Let \(\mathscr {H}\) denote the subgroup of \((B \otimes \mathbf {A}_{F, \mathrm {f}})^\times \) consisting of the elements whose reduced norms are in \(F^{\times +} \cdot \mathbf {A}_{E, \mathrm {f}}^\times \subset \mathbf {A}_{F, \mathrm {f}}^\times \). Then the double quotient \(H(E) \backslash \mathscr {H} / U^*\) bijects with \(H^*(E) \backslash H^*(\mathbf {A}_{E, \mathrm {f}}) / U^*\), so we can interpret \(M_{\underline{k}, \underline{t}}^*(\mathfrak {N}, \varepsilon )\) as a space of functions on \(\mathscr {H} / U^*\). Thus we may define a Hecke operator for any double \(U^*\)coset in \(\mathscr {H}\). In particular, we can use this to make sense of \(\mathcal {T}(\mathfrak {p})\) as an operator on \(M^*_{\underline{k}, \underline{t}}(\mathfrak {N}, \varepsilon )\) for any prime \(\mathfrak {p}\not \mid \mathfrak {N}\mathfrak {d}_B\) of F whose ideal class lies in the image of \({{\,\mathrm{Cl}\,}}^+(E)\) in \({{\,\mathrm{Cl}\,}}^+(F)\); however, this will only be welldefined modulo the action of the capitulation group \(K_{E/F}\).
Note that the Hecke operators associated to double cosets in \(\mathscr {H}\) make sense even if \((\underline{k}, \underline{t})\) is not “reasonable” in the sense of Sect. 2.2, since we only need \(\mathcal {O}_E^\times \) to act trivially, not \(\mathcal {O}_F^\times \). We shall see an application of this in the next section.
8 An explicit example of a nonparitious Hilbert eigenform
8.1 Setup
Let \(F = \mathbf {Q}(\sqrt{2})\), and let \(\sigma _1, \sigma _2\) denote the two embeddings \(F \hookrightarrow \mathbf {R}\) (mapping \(\sqrt{2}\) to \(\sqrt{2}\) and \(\sqrt{2}\) respectively). Let \(B = \left( \frac{1, 1}{F} \right) \) be the Hamilton quaternions over F, so that B is the unique quaternion algebra over F unramified at all finite places; and let \(\mathcal {O}_B\) be a maximal order in B, so that \(\widehat{\mathcal {O}}_B^\times \) is a maximal compact subgroup of \(H(\mathbf {A}_{F, \mathrm {f}})\). The class number of \(\mathcal {O}_B\) is one.
With this choice we compute that the space \(M_{\underline{k}, \underline{t}}(\mathfrak {N}, \varepsilon )\) is 2dimensional. Since F has narrow class number one, and \(\mathcal {O}_F^{\times +} = (\mathcal {O}_F^\times )^2\), this is isomorphic (via the pullback map \(\psi \)) to the space \(M^*_{\underline{k}, \underline{t}}(\mathfrak {N}, \varepsilon )\).
8.2 Hecke operators

A normalized Hecke operator\(\mathcal {T}(\mathfrak {m})\), defined as in Sect. 6.6 above.

A naive Hecke operator\(T(\varpi )\), depending on a choice of totallypositive generator \(\varpi \) of \(\mathfrak {m}\). This is given by identifying W as an \(H^*\)representation with the representation \(W(\underline{k}, \underline{t}') = {\text {Sym}}^2 V_{\sigma _1} \otimes {\text {Sym}}^1 V_{\sigma _2}\), where \(\underline{t}' = 2  \underline{k}= (2, 1)\); and treating \(T(\varpi )\) as a double coset in the group \(\mathscr {H}\) of Remark 6.15.
The normalised Hecke operator \(\mathcal {T}(\mathfrak {m})\) is canonically defined, but it does not preserve the natural Kstructure on the space, so the collection of eigenvalues of these operators (for varying \(\mathfrak {m}\)) do not all lie in a finite extension of \(\mathbf {Q}\). On the other hand, the naive Hecke operator \(T(\varpi )\) preserves the Kstructure, but it will depend on the the choice of generator \(\varpi \).
From Eq. (4), it is clear that if p is a prime inert in F and \(\mathfrak {m} = (p)\), then \(\mathcal {T}(\mathfrak {m}) = T(p)\); whereas if \(p = \mathfrak {p}_1 \mathfrak {p}_2\) is a prime split or ramified in F, and \(\varpi _1, \varpi _2\) are totally positive generators of these ideals such that \(\varpi _1 \varpi _2 = p\), then \(\mathcal {T}(\mathfrak {p}_1) \mathcal {T}(\mathfrak {p}_2) = T(\varpi _1) T(\varpi _2) = T(p)\). 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 \(\mathcal {S}(\mathfrak {m})\) for any ideal \(\mathfrak {m}\), and a naive operator \(S(\varpi )\) for \(\varpi \in \mathcal {O}_F\), via the action of \(\left( {\begin{matrix}\varpi &{} 0 \\ 0 &{} \varpi \end{matrix}}\right) \). Note that if p is a split prime and \(\varpi _1 \varpi _2 = p\), the operators \(T(\varpi _1^2)S(\varpi _2)\) and \(T(\varpi _2^2)S(\varpi _1)\) 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 \(p^3\varepsilon (p)\).
8.3 Hecke eigenvalues
Our space \(M_{\underline{k},\underline{t}}(\mathfrak {N},\varepsilon )\) is an irreducible module for the Hecke algebra with coefficients in F; it decomposes over the CM field \(L = F[b]\), where \(b^2 = 3\sqrt{2}8\). (We note that L is not Galois over \(\mathbf {Q}\).)
In Table 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 \(T(p) = \mathcal {T}(p)\). For a split prime, we choose arbitrary totallypositive generators \(\varpi _1\) and \(\varpi _2\) of the two primes above p such that \(\varpi _1 \varpi _2 = p\), and we list the eigenvalues \(t(\varpi _i)\) of the naive Hecke operators \(T(\varpi _1)\) and \(T(\varpi _2)\).
The eigenvalues displayed show many of the interesting features we expect for such an eigensystem. For example, we see that the eigenvalue \(t(\varpi )\) lies in F when \(\varepsilon (\varpi ) = 1\), and in \(b\cdot F\) when \(\varepsilon (\varpi ) = 1\). In particular, when p is totally split in \(\mathbf {Q}(\sqrt{2}, \sqrt{7})\), such as \(p = 23\), then we see that \(t(\varpi _1)\) and \(t(\varpi _2)\) are both in F.
The smallest rational prime which is inert in F is \(p = 3\). In that case, we have \(\varepsilon (3) = 1\), and \(t(3) = (7\sqrt{2}  4)b\).
The smallest rational prime which splits in F is \(p = 17\): we have \(17 = \varpi _1 \varpi _2\) where \(\varpi _1 = 2\sqrt{2} + 5\). Note that \(\varepsilon (\varpi _1) = 1\), but \(\varepsilon (\varpi _2) = +1\), so \(t(\varpi _2)\) is in F but \(t(\varpi _1)\) is not, and nor is the product \(t(p) = t(\varpi _1) t(\varpi _2) = (150\sqrt{2} + 264)b\) is not in F.
Naive Hecke eigenvalues at level \((53\sqrt{2})\) and weight (4, 3) over \(\mathbf {Q}(\sqrt{2})\), for primes of norm \(< 200\)
\({{\,\mathrm{Nm}\,}}(p)\)  \(\varpi _1\)  \(t(\varpi _1)\)  \(t(\varpi _2)\) 

9  3  \((7w  4)b\)  
17  \(2w + 5\)  \((3w + 12)b\)  \(8w  18\) 
23  \(w + 5\)  \(22w + 14\)  \(26w + 36\) 
25  5  \((16w + 18)b\)  
31  \(3w + 7\)  \((13w  18)b\)  \(30w + 34\) 
41  \(2w + 7\)  \(16w  106\)  \((32w + 26)b\) 
47  \(w + 7\)  \(76w + 46\)  \((7w  70)b\) 
71  \(5w + 11\)  \((74w  6)b\)  \((3w  32)b\) 
73  \(2w + 9\)  \((27w + 18)b\)  \(168w + 14\) 
79  \(w + 9\)  \((46w + 60)b\)  \((7w + 40)b\) 
89  \(4w + 11\)  \((65w + 64)b\)  \(206w + 30\) 
97  \(6w + 13\)  \(272w + 38\)  \((83w  32)b\) 
103  \(3w + 11\)  \(78w + 228\)  \((8w + 122)b\) 
113  \(2w + 11\)  \((46w  56)b\)  \((18w + 8)b\) 
121  11  \(170w + 366\)  
127  \(9w + 17\)  \(50w + 46\)  \(272w + 372\) 
137  \(14w + 23\)  \(10\)  \(74w + 114\) 
151  \(3w + 13\)  \(282w  168\)  \(172w  318\) 
167  \(w + 13\)  \((172w  166)b\)  \(398w  24\) 
169  13  \((84w + 62)b\)  
191  \(7w + 17\)  \((11w + 12)b\)  \((114w + 184)b\) 
193  \(4w + 15\)  \((129w + 162)b\)  \((185w  486)b\) 
199  \(11w + 21\)  \(250w  188\)  \((288w + 430)b\) 
8.4 Satake parameters
Let \(\Pi = \Pi _0 \otimes \Vert {{\,\mathrm{nrd}\,}}\Vert ^{1/2}\), where \(\Pi _0\) is the automorphic representation of H arising from the system of eigenvalues described above (and tabulated in Table 1). The shift by \(\Vert {{\,\mathrm{nrd}\,}}\Vert ^{1/2}\) is included in order to give a slightly more pleasant normalisation of the Satake parameters.
Characteristic polynomials of \({{\,\mathrm{Frob}\,}}_p\) in the standard representation of \({{\,\mathrm{GO}\,}}_4\) (notations as in Table 1)
p  \(H_p(X)\) 

3  \(X^4 + (7w + 4)bX^3 + (1701w + 972)bX 3^{10}\) 
5  \(X^4 + (16w  18)bX^3 + (50000w  56250)bX 5^{10}\) 
11  \(X^4 + (170w  366)X^3 + (27378670w + 58944666)X 11^{10}\) 
17  \(X^4 + (150w + 264)bX^3 + (1213222w + 584358)X^2\) 
\(+ (212978550w  374842248)bX + 17^{10}\)  
23  \(X^4 + (428w + 640)X^3 + (4107156w  157642)X^2\) 
\(+ (2754754804w + 4119259520)X + 23^{10}\)  
31  \(X^4 + (982w + 1392)bX^3 + (24199902w + 22262526)X^2\) 
\(+ (28113826282w  39851778192)bX + 31^{10}\) 
Footnotes
 1.
Patrikis’ result is actually considerably more general, applying to regular algebraic, essentially selfdual cuspidal automorphic representations of \({{\,\mathrm{GL}\,}}_n\) over totally real fields. However, we shall consider only the \(n = 2\) case in the present paper.
 2.
If \(\sigma \) is a complex place then there is a small subtlety in that \(\lambda _\sigma \) actually depends not only on the place \(\sigma \) but also on a choice of isomorphism \(F_\sigma \cong \mathbf {C}\); but replacing this isomorphism with its conjugate changes \(\lambda _\sigma \) by an element of \(X_\bullet (\hat{T})\), so the notion of Lalgebraicity is welldefined. However, in this paper we shall mostly restrict to the case of totally real F where this subtlety does not arise.
 3.
Actually a somewhat smaller space suffices: one can take here the fixed field of the largest subgroup of \({\text {Gal}}(\tilde{F} / \mathbf {Q})\) whose permutation action on \(\Sigma _F\) stabilises the weight \(\underline{k}\).
 4.
If \(E = \mathbf {Q}\) then the semisimplification can be dispensed with, since it has been shown by Nekovar [9] that the \(\ell \)adic cohomology is semisimple.
 5.
Note that for the \(X^2\) coefficient we need to compute the Hecke operators \(T(p)^2\) and \(T(p^2)\); these can be calculated directly as double cosets, but it is quicker computationally to express these operators as polynomials in \(T(\varpi _1)\) and \(T(\varpi _2)\), since evaluating these nonnormalised operators involves summing over fewer double coset representatives.
Notes
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. We are also grateful to Stefan Patrikis for his comments on an earlier version of this paper.
References
 1.Asai, T.: On certain Dirichlet series associated with Hilbert modular forms and Rankin’s method. Math. Ann. 226(1), 81–94 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
 2.Blasius, D., Rogawski, J.D.: Motives for Hilbert modular forms. Invent. Math. 114(1), 55–87 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 3.Brylinski, J.L., Labesse, J.P.: Cohomologie d’intersection et fonctions \(L\) de certaines variétés de Shimura. Ann. Sci. École Norm. Sup. (4) 17(3), 361–412 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
 4.Buzzard, K., Gee, T.: The conjectural connections between automorphic representations and Galois representations, Automorphic forms and Galois representations. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 414, Cambridge Univ. Press, Cambridge, pp. 135–187 (2014)Google Scholar
 5.Dembélé, L., Voight, J.: Explicit methods for Hilbert modular forms, Elliptic curves, Hilbert modular forms and Galois deformations, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Basel, pp. 135–198 (2013)Google Scholar
 6.Langlands, R.P.: On the zeta functions of some simple Shimura varieties. Canad. J. Math. 31(6), 1121–1216 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
 7.Lei, A., Loeffler, D., Zerbes, S.L.: Euler systems for Hilbert modular surfaces. Forum Math. Sigma 6, e23 (2017)Google Scholar
 8.Mok, C.P.: Galois representations attached to automorphic forms on \(\operatorname{GL}_2\) over CM fields. Compos. Math. 150(4), 523–567 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
 9.Nekovář, J.: Eichler–Shimura relations and semisimplicity of étale cohomology of quaternionic Shimura varieties. Ann. Sci. École Norm. Sup. 51(5), 1179–1252 (2018)Google Scholar
 10.Patrikis, S.: On the sign of regular algebraic polarizable automorphic representations. Math. Ann. 362(1–2), 147–171 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 11.Ramakrishnan, D.: Modularity of solvable Artin representations of \(\text{ GO }(4)\)type. Int. Math. Res. Not. 2002(1), 1–54 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
 12.Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J. 45(3), 637–679 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
 13.Yoshida, H.: On the zeta functions of Shimura varieties and periods of Hilbert modular forms. Duke Math. J. 75(1), 121–191 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
Copyright information
OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.