2.1 A Question on Rational Points on Shimura Varieties

Let $\mathbb{S} $ denote the real torus $\mathrm {Res}_{\mathbb {C} / \mathbb {R}} (\mathbb {G}_m)$ , and let $\mathbb {A}_{\mathbb {Q}}^{f} $ be the finite adeles of ${\mathbb {Q}}$ . A Shimura datum is a pair $(G,X)$ where G is a reductive ${\mathbb {Q}}$ -algebraic group and X a $G(\mathbb {R})$ -orbit in the set of morphisms of $\mathbb {R}$ -algebraic groups $\mathrm {Hom}(\mathbb{S} , G_{\mathbb {R}})$ , satisfying the Shimura–Deligne axioms ([Reference Deligne15, Conditions 2.1.1(1-3)]); furthermore, in what follows, we also assume that G is the generic Mumford–Tate group on X. The Shimura–Deligne axioms imply that the connected components of X are hermitian symmetric domains and that faithful representations of G induce variations of polarisable ${\mathbb {Q}}$ -Hodge structures on X. Let $\widetilde {K}$ be a compact open subgroup of $G(\mathbb {A}_{\mathbb {Q}}^{f})$ and set

$$ \begin{align*} \mathrm{Sh}_{\widetilde{K}}(G,X) := G({\mathbb{Q}}) \backslash \big ( X \times G(\mathbb{A}_{\mathbb{Q}}^{f}) / \widetilde{K} \big). \end{align*} $$

Let $X^+$ be a connected component of X and let $G({\mathbb {Q}})^+$ be the stabiliser of $X^+$ in $G({\mathbb {Q}})$ . The above double coset set is a disjoint union of quotients of $X^+$ by the arithmetic groups $\Gamma _{g}:=G({\mathbb {Q}})^+ \cap gKg^{-1}$ , where g runs through a set of representatives for the finite double coset set $G({\mathbb {Q}})^+ \backslash G(\mathbb {A}_{\mathbb {Q}}^{f} )/K$ . Baily and Borel [Reference Baily and Borel2] proved that $\mathrm {Sh}_{\widetilde {K}}(G,X)$ has a unique structure of a quasi-projective complex algebraic variety. Thanks to the work of Borovoi, Deligne, Milne, and Milne–Shih, among others, the $\mathbb {C}$ -scheme

$$ \begin{align*} \mathrm{Sh} (G,X)= G({\mathbb{Q}}) \backslash \Big ( X \times G\big(\mathbb{A}_{\mathbb{Q}}^{f}\big) \Big), \end{align*} $$

together with its $G(\mathbb {A}_{\mathbb {Q}}^{f})$ -action, can be naturally defined over a number field $E:= E(G,X) \subset \mathbb {C}$ called the reflex field of $(G,X)$ . That is, there exists an E-scheme $\mathrm {Sh} (G,X)_{E}$ with an action of $G(\mathbb {A}_{\mathbb {Q}}^{f})$ whose base change to $\mathbb {C}$ gives $\mathrm {Sh} (G,X)$ with its $G(\mathbb {A}_{\mathbb {Q}}^{f})$ -action.

Let F be a finite extension of E such that there exists a point $x \in \mathrm {Sh}_{\widetilde {K}}(G,X)_{E}(F)$ . With such a point, we can naturally associate a continuous group homomorphism

$$ \begin{align*} \rho_{x} : G_{F}:=\mathrm{Gal}(\overline{F}/F) \longrightarrow \widetilde{K} \subset G(\mathbb{A}_{\mathbb{Q}}^{f}). \end{align*} $$

This paper is motivated by the following question.

When the Shimura variety has a natural interpretation as a moduli space of motives, the above question is naturally related to the Fontaine–Mazur conjecture [Reference Fontaine and Mazur19]. Indeed, both aim to predict when a Galois representation comes from the $\ell $ -adic (or adelic in our case) realisation of a motive. Even when they are not motivical (see [Reference Baldi4, Remark 1.6] for a discussion about this), representations arising in this way enjoy nice properties. For an example of geometric flavour, we refer the reader to [Reference Baldi4, Theorem 1.3].

2.2 Hilbert Modular Varieties

Let $F/{\mathbb {Q}}$ be a totally real extension of degree $n_{F}$ and fix $\{\sigma _{i}\}_{i=1}^{n_{F}}$ the set of real embeddings of F into $\mathbb {C}$ . We let G be the ${\mathbb {Q}}$ -algebraic group obtained as the Weil restriction of $\mathrm {GL}_{2}$ from F to ${\mathbb {Q}}$ and let X be $n_{F}$ copies of $\mathcal {H}^{\pm }=\{\tau \in \mathbb {C}: \mathrm {Im}(\tau )\neq 0\}$ , on which $G({\mathbb {Q}})=\mathrm {GL}_{2}(F)$ acts on the i-th component via $\sigma _{i}$ . That is,

$$ \begin{align*} \bigg( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot (\tau_{1},\ldots,\tau_{n_{F}})\bigg)_{i} = \frac{\sigma_{i}(a) \tau_{i}+\sigma_{i}(b)}{\sigma_{i}(c)\tau_{i}+\sigma_{i}(d)}. \end{align*} $$

In this case, the reflex field of $(G,X)$ is ${\mathbb {Q}}$ , and the set of geometrically connected components of the Shimura variety $S:=\mathrm {Sh}_{G(\widehat {\mathbb {Z}})}(G,X)$ is $\mathrm {Pic}(\mathcal {O}_{F})^+$ , where $\widehat {\mathbb {Z}}$ denotes the profinite completion of $\mathbb {Z}$ (different choices of level structure will appear later).

It is interesting to notice here a first difference between modular curves (i.e., when $F={\mathbb {Q}}$ ) and higher dimensional Hilbert modular varieties. We recall the following folklore result (see [Reference Blasius6, Section 2.3.2.]) to see how it follows from Matsushima’s formula [Reference Borel and Wallach8, Theorem VII.5.2].

To have a better interpretation as moduli space, we actually consider the subgroup $G^*$ of G given by its elements whose determinant is in ${\mathbb {Q}}$ . More precisely, we let $G^*$ be the pull-back of

$$ \begin{align*} \det: G \longrightarrow \mathrm{Res} _{\mathcal{O}_{F} /\mathbb{Z}}\mathbb{G}_m \end{align*} $$

to $\mathbb {G}_m/{\mathbb {Q}}$ . The Shimura variety $Y_{F}:=\mathrm {Sh}_{G^*(\widehat {\mathbb {Z}})}(G^*,X^*)(\mathbb {C})$ is connected and comes with a finite map to $S/\mathbb {C}$ . It is a quasi projective $n_{F}$ -dimensional ${\mathbb {Q}}$ -scheme.

In the next section, we present the moduli problem solved by $Y_{F}$ . It will be also clear from such moduli interpretation that the reflex field of $Y_{F}$ is the field of rational numbers.

2.2.1 Hilbert Modular Varieties as Moduli Spaces

As explained, for example, in [Reference Edixhoven17, Section 3], the Shimura variety $Y_{F}$ represents the (coarse) moduli space for triplets $(A, \alpha , \lambda )$ where:

• • A is a complex abelian variety of dimension $n_{F}$ ;

• • $\alpha : \mathcal {O}_{F} \hookrightarrow \mathrm {End} (A)$ is a morphism of rings;

• • $\lambda : A \to A^*$ is a principal $\mathcal {O}_{F}$ -polarisation.

By $A^*,$ we denoted the $\mathcal {O}_{F}$ -dual abelian variety of A, i.e,. it is defined as $\mathrm {Ext}^{1}(A,\mathcal {O}_{F}\otimes \mathbb {G}_m)$ . Otherwise, one can obtain such abelian variety considering $A^\vee $ (the dual of A, in the standard sense) and tensoring it over $\mathcal {O}_{F}$ with the different ideal of the extension $F/{\mathbb {Q}}$ . By principal $\mathcal {O}_{F}$ -polarisation we mean an isomorphism $\lambda : A \to A^*$ , such that the induced map $A\to A^\vee $ is a polarisation.

Furthermore, the Shimura variety of level

$$ \begin{align*} U_{0}(\mathfrak{N}):= \big\lbrace \gamma\in G(\widehat{\mathbb{Z}}): \gamma \equiv \bigg(\begin{matrix}* &*\\0& 1\end{matrix}\bigg) \text{ mod } \mathfrak{N}\big\rbrace, \end{align*} $$

where $\mathfrak {N}$ is an integral ideal of $\mathcal {O}_{F}$ , parametrises triplets as above, equipped with a $\mathfrak {N}$ -level structure as follows. We fix an isomorphism of $\mathcal {O}_{F}$ -modules

$$ \begin{align*} \big(\mathcal{O}_{F}/\mathfrak{N} \mathcal{O}_{F} \big)^{2} \to A[\mathfrak{N}] \end{align*} $$

making the following diagram commutative:

where $(N)=\mathbb {Z}\cap \mathfrak {N}$ , $\psi _{\mathfrak {N}}$ is the pairing given by $\bigg(\begin{matrix}0 &1\\-1&0\end{matrix}\bigg)$ , $e_{\lambda ,\mathfrak {N}}$ is the perfect Weil pairing on $A[\mathfrak {N}]$ induced by the $\mathcal {O}_{F}$ -polarisation $\lambda ,$ and $\mathcal {O}_{F}\otimes \mathbb {Z}/N\mathbb {Z}\to \mathcal {O}_{F} \otimes \mu _{N}$ is an arbitrarily chosen isomorphism. When a level structure is needed, we always assume that $N>3$ in order to have a fine moduli space. A rational point of $Y_{F}(\mathfrak {N}):=\mathrm {Sh}_{U_{0}(\mathfrak {N})\cap G^*}(G^*,X)$ then represents a triple, as above, together with such level structure. In Section 4.2, we will see a similar description for $\mathcal {O}_{L,S}$ -points of an integral model of $Y_{F}(\mathfrak {N})$ .

2.3 Eichler–Shimura Theory

We discuss Eichler–Shimura theory for classical and Hilbert modular forms, reviewing results that attach opportune abelian varieties to modular forms.

2.3.1 Classical Modular Eigenform

The following is [Reference Shimura37, Theorem 7.14, p. 183 and Theorem 7.24, p. 194].

Theorem 2.3 (Shimura)

Let f be a holomorphic newform of weight $2$ with rational Fourier coefficients $(a_{n}(f))_{n}$ . There exists an elliptic curve $E/{\mathbb {Q}}$ such that, for all primes p at which E has good reduction, one has

$$ \begin{align*} a_{p}(f) = 1- N_{p}(E) + p, \end{align*} $$

where $N_{p}(E)$ denotes the number of points of the reduction mod p of E, over the field with p-elements. In other words, up to a finite number of Euler factors, $L(s,E/{\mathbb {Q}})$ and $L(s,f)$ coincide.

More generally, let $K(f)$ be the subfield of $\mathbb {C}$ generated over ${\mathbb {Q}}$ by $(a_{n}(f))_{n}$ for all n. Then there exists an abelian variety $A/{\mathbb {Q}}$ and an isomorphism $K(f) \cong \mathrm {End}^{0}(A)$ with the following properties:

• • $\dim (A)=[K(f):{\mathbb {Q}}]$ ;

• • up to a finite number of Euler factors at primes at which A has good reduction, $L(s,A/{\mathbb {Q}},K(f))$ coincides with $L(s,f)$ ;

where the L-function $L(s,A/{\mathbb {Q}},K(f))$ is defined by the product of the following local factors where v is a prime of $K(f)$ not dividing $\ell $

$$ \begin{align*} \det\big(1-\ell^{-s}\operatorname{Frob}_{\ell}|T_{v}(A)\big). \end{align*} $$

Shimura considers the Jacobian of the modular curve of level N and takes the quotient by the kernel of the homomorphism giving the Hecke action on f. What happens if we want to produce an abelian variety with such properties, defined over our totally real field F, when ${\mathbb {Q}}\subsetneq F$ ? Here is where Hilbert modular forms come into play. In the next section, we recall what we need from such theory, and explain Blasius’ generalisation of Theorem 2.3 and why the absolute Hodge conjecture is needed.

2.3.2 Hilbert Modular Forms for F

Let $\mathcal {H}_{F}$ denote $n_{F}$ copies of the upper half plane $\mathcal {H}^+$ . We consider subgroups $\Gamma \subset \mathrm {GL}_{2}(\mathcal {O}_{F})$ of the form $U_{0}(\mathfrak {N}) \cap G({\mathbb {Q}})^+$ . Moreover, for $\lambda \in F$ and $\underline {r}=(r_{1},\ldots , r_{n_{F}})\in \mathbb {Z}^{n_{F}}$ , we write $\lambda ^{\underline {r}}=\lambda _{1}^{r_{1}}\cdots \lambda _{n_{F}}^{r_{n_{F}}}$ , where $\lambda _{i}=\sigma _{i}(\lambda )$ . Similarly, if $\underline {\tau }=(\tau _{1},\ldots ,\tau _{n_{F}})\in \mathcal {H}_{F}$ , we write $\underline {\tau }^{\lambda }=\tau _{1}^{r_{1}}\cdots \tau _{n_{F}}^{r_{n_{F}}}$ .

Definition 2.1 A Hilbert modular form of level $\mathfrak {N}$ and weight $(\underline {r},w)\in \mathbb {Z}^{n_{F}}\times \mathbb {Z}$ , with $r_{i}\equiv w \mod 2$ (and trivial nebentype character) is a holomorphic function $f: \mathcal {H}_{F} \rightarrow \mathbb {C}$ such that

$$ \begin{align*} f(\gamma \cdot \underline{\tau})= (\det\gamma)^{-\underline{r}/2}(c\underline{\tau}+d)^{\underline{r}}f(\underline{\tau}), \end{align*} $$

for every $\gamma =\bigg(\begin{matrix}a & b \\ c & d\end{matrix}\bigg) \in U_{0}(\mathfrak {N}) \cap G({\mathbb {Q}})^+$ and for every $\underline {\tau }=(\tau _{1},\ldots ,\tau _{n_{F}})\in \mathcal {H}_{F}$ .

Since Hilbert modular forms are holomorphic functions on $\mathcal {H}_{F}$ invariant under the lattice $\mathcal {O}_{F}$ , they admit a q-expansion over $\mathcal {O}_{F}^{\vee }=\mathfrak {d}^{-1}$ , where $q=e^{2\pi i \sum \tau _{i}}$ ; see [Reference Goren24, Definition 3.1] for more details.

2.3.3 Hecke Operators and Hilbert Eigenforms

On the space of Hilbert modular forms of level

$$ \begin{align*} U_{0}(\mathfrak{N}) \cap G({\mathbb{Q}})^+, \end{align*} $$

one has Hecke operators $T(\mathfrak {n})$ for every integral ideal of $\mathcal {O}_{F}$ coprime with $\mathfrak {N}$ . The definition is analogous to the one for classical modular forms. For example, if $\mathfrak {p}$ does not divide $\mathfrak {N}$ and x is a totally positive generator of $\mathfrak {p}$ , one defines

$$ \begin{align*} (T(\mathfrak{p})f)(\underline{\tau}):=\text{Nm}(\mathfrak{p}) f(x\cdot\underline{\tau})+ \frac{1}{\text{Nm}(\mathfrak{p})}\sum\limits_{a\in\mathcal{O}_{F}/\mathfrak{p}}f(\gamma_{a} \cdot \underline{\tau}), \end{align*} $$

where $\gamma _{a}:= \bigg(\begin{matrix}1 & a \\ 0 & x\end{matrix}\bigg)$ . We recall the following definition.

Definition 2.2 A cuspidal Hilbert modular form (i.e., such that the $0$ -th Fourier coefficient $a_{0}(f)$ vanishes) is an eigenform if it is an eigenvector for every Hecke operator $T(\mathfrak {n})$ .

As in the case of classical modular forms, if f is an eigenform, normalised so that $a_{1}(f) = 1$ , then the eigenvalues of the Hecke operators are the Fourier coefficients, i.e., $T(\mathfrak {n})f = a_{\mathfrak {n}}(f)\cdot f$ ; moreover, they are algebraic integers lying in the number field $K(f):={\mathbb {Q}}((a_{\mathfrak {n}}(f))_{\mathfrak {n}})$ , as shown in [Reference Shimura36, $\S $ 2].

2.3.4 Eichler–Shimura for Hilbert Modular Forms

Blasius and Rogawski [Reference Blasius and Rogawski7], Carayol [Reference Carayol11], and Taylor [Reference Taylor40] proved that to any Hilbert eigenform, one can attach representations of $G_{F}$ , similarly to the classical case. More precisely, one has the following result.

Theorem 2.4 If f is a Hilbert eigenform for F of weight $(\underline {r},t)$ , level $\mathfrak {N,}$ and trivial nebentype character and $K(f)$ is the number field generated by its eigenvalues, then for every finite place $\lambda $ of $K(f),$ there is an irreducible $2$ -dimensional Galois representation

$$ \begin{align*} \rho_{f,\lambda}: G_{F} \longrightarrow \mathrm{GL}_{2}(K(f)_{\lambda}) \end{align*} $$

such that for every prime $w\nmid \mathfrak {N}\operatorname {Nm}_{K(f)/{\mathbb {Q}}}(\lambda )$ in F, $\rho _{f,\lambda }$ is unramified at w and

$$ \begin{align*} \det(1-X\rho_{f,\lambda}(\operatorname{Frob}_{w}))=X^{2}-a_{w}(f)X+\operatorname{Nm}_{F/{\mathbb{Q}}}^{t-1}(w). \end{align*} $$

Assume that f is of weight $(2,\ldots ,2 )$ . As in the classical case, we would like to have such Galois representations to be attached to opportune abelian varieties. The existence of abelian varieties associated with f was first considered by Oda in [Reference Oda31], and Blasius gave a conjectural solution to such a problem.

Theorem 2.5 (Blasius)

Let f be a Hilbert eigenform for F of parallel weight $2$ . Denote by $K(f)$ the number field generated by the $a_{w}(f)$ for all w. Assume Conjecture 1.1. There exists a $[K(f):{\mathbb {Q}}]$ -dimensional abelian variety $A_{f}/F$ with $\mathcal {O}_{K(f)}$ -multiplication such that for all but finitely many of the finite places w of F at which $A_{f}$ has good reduction, we have

$$ \begin{align*} L\big(s,A_{f},K(f)\big)=L(f,s), \end{align*} $$

where the L-function $L(s,A_{f},K(f))$ is defined by the product of local factors

$$ \begin{align*} \det\big(1-\operatorname{Nm}(w)^{-s}\operatorname{Frob}_{w}|T_{v}(A)^{I_{w}}\big), \end{align*} $$

where v is a prime of K, w is a prime of $F,$ and w and v lay above distinct rational primes.

Proof If $K(f)={\mathbb {Q}}$ , this is precisely [Reference Blasius6, Theorem 1, p. 3]. As noticed by Blasius [Reference Blasius6, 1.10], the proof easily adapts to the general case (where the necessary changes are hinted at in the remarks in [Reference Blasius6, Sections 5.4., 5.7., and 7.6.]). ▪

Remark 2.6 The proof is completely different from the one of Shimura, since, as noticed in Theorem 2.2, we cannot obtain a non-trivial abelian variety as quotient of the Albanese variety of a Hilbert modular variety. Blasius instead considers the symmetric square of the automorphic representation of $\mathrm {GL}_{2,F}$ associated with the Hilbert eigenform; it is an automorphic representation of $\mathrm {GL}_{3,F,}$ and its base change to a quadratic imaginary field appears in the middle degree cohomology of a Picard modular variety. He then considers the associated motive and shows that its Betti realisation is the symmetric square of a polarised Hodge structure of type $(1,0), (0,1)$ . This gives a complex abelian variety A and Conjecture 1.1 (applied to the product of the Picard modular variety and A) allows us to conclude that A is defined over a number field containing F. He then finds the desired abelian variety inside the restriction of scalars of A over F.

Remark 2.7 Theorem 2.5 is known to hold unconditionally in many interesting cases (for example, when $n_{F}$ is odd, by the work of Hida). For more details, we refer the reader to [Reference Blasius6, Theorem 3] and references therein. The proof of such unconditional cases actually follows Shimura’s proof of Theorem 2.3, rather than the strategy described in the previous remark.

2.4 A Remark on Polarisations

To use Theorems 2.3 and 2.5 to produce F-points of a Hilbert modular variety, we need, of course, the abelian varieties produced to be principally polarised (up to isogeny would actually be enough for our applications, if the isogeny is defined over the base field F). An abelian variety over an algebraically closed base field always admits an isogeny to a principally polarised abelian variety. But since the same does not hold over number fields, some considerations are needed. The first observation is that every weight one Hodge structure of dimension $2$ with an action by $\mathcal {O}_{K(f)}$ is automatically $\mathcal {O}_{K(f)}$ -polarised, as explained for example in [Reference Edixhoven17, Appendix B].

As noticed in [Reference Blasius6, Remark 5.7.], Blasius first finds a principally polarised abelian variety A over a finite extension $L'/F$ . Actually, we can assume that A has a principal $\mathcal {O}_{K(f)}$ -polarisation $\lambda $ . As explained above, the proof then considers the Weil restriction of A to F, which is again principally $\mathcal {O}_{K(f)}$ -polarised. It is not hard to see that the construction of [Reference Blasius6, section 7] behaves well with respect to the $\mathcal {O}_{K(f)}$ -polarisation, and, therefore, the proof actually produces an $\mathcal {O}_{K(f)}$ -polarised abelian variety over F.

3.1 Weakly Compatible Systems

The definition of weakly compatible families presented is due to Serre, who called them strictly compatible in [Reference Serre35, p. I-11]. It follows from the Weil conjectures that the $\ell $ -adic Tate modules of abelian varieties form a weakly compatible system of Galois representations.

Recall that $\rho _{v}$ is said to be unramified at a place w of L if the image of the inertia at w is trivial. If $\rho _{v}$ is attached to the v-adic cohomology of a smooth proper variety defined over a number field, the smooth and proper base change theorems (see, for example, [Reference Deligne14, I, Theorems 5.3.2 and 4.1.1]) imply that $\rho _{v}$ is unramified at every place $w\notin S_{v}$ such that X has good reduction at w.

3.2 Key Proposition

Fix a finite set of places S of L, containing all the archimedean places. To prove Theorem 1.A and Theorem 1.B, we need the following proposition.

Remark 3.2 If we start with an abelian variety $A/L$ with $\mathcal {O}_{K}$ -multiplication, we can produce a system

$$ \begin{align*} \rho_{v}: G_{L} \longrightarrow \mathrm{GL}(T_{v}(A)), \end{align*} $$

which satisfies the four conditions of (𝒮) for S the union of infinite places and the set of places of bad reduction; see [Reference Ribet33, $\S $ 3]. Proposition 3.1 hence implies that A is modular; i.e., there exists a Hilbert modular form for the totally real field L such that

$$ \begin{align*} L(A/L,s)=L(f,s) \end{align*} $$

up to a finite number of Euler factors. Unconditionally, it has been proven that elliptic curves over real quadratic fields are modular (see [Reference Freitas, Le Hung and Siksek21]), and, more generally, the work of Taylor and Kisin implies that elliptic curves over L become modular (in this sense) after a totally real extension $L'/L$ . See [Reference Buzzard9, Theorem 1.16] and reference therein.

In the proof of Proposition 3.1, we use Conjecture 1.2 and the following result due to Serre (for the proof, see [Reference Serre34, 4.9.4]).

Proposition 3.3 (Serre)

Let q be a power of $\ell $ . Let $r: G_{E} \to \mathrm {GL}_{2}(\mathbb {F}_{q})$ be a continuous homomorphism, where $q=\ell ^{t}$ and E is a local field of residue characteristic $p\neq \ell $ and discrete valuation $v_{E}$ . Let $e_{E}:=v_{E}(p)$ and $c\geq 0$ be an integer such that the image via r of the wild inertia of E has cardinality $p^{c}$ . We denote by $n(r,E)$ the exponent of the conductor of r. We have

$$ \begin{align*} n(r,E)\leq 2\big(1+e_{E}\cdot c + \tfrac{e_{E}}{p-1}\big). \end{align*} $$

We also need to compute the weight and the conductor of the modular forms produced by Conjecture 1.2. As anticipated in Remark 1.1, this is a known result under some assumptions. The weight part stated in the following theorem is a special case of the work [Reference Gee, Liu and Savitt23]; the conductor part follows from automorphy lifting methods or can be seen as a consequence of the main theorem of [Reference Fujiwara22].

Theorem 3.4 [Reference Gee, Liu and Savitt23, Reference Fujiwara22]

Let $\ell>5$ and $\bar {\rho }: G_{L}\to \mathrm {GL}_{2}(\overline {\mathbb {F}}_{\ell })$ be an irreducible totally odd representation such that its determinant is the cyclotomic character and it is finite flat at all places $w\mid \ell $ . Assume, furthermore, that $\bar {\rho }$ satisfies the Taylor–Wiles assumption; namely,

(TW) $$ \begin{align} \bar{\rho}_{\mid G_{L(\zeta_{\ell})}} \text{ is irreducible}, \end{align} $$

where $\zeta _{\ell }$ is a primitive $\ell $ -th root of unity. Then if $\bar {\rho }$ is modular, there exists a Hilbert modular form of parallel weight $2$ and conductor equal to the Artin conductor of $\bar {\rho }$ giving rise to $\bar {\rho }$ .

We are now ready to prove Proposition 3.1.

Proof Our goal is to apply Serre’s conjecture to $\bar {\rho }_{v}$ , the reduction modulo v of the representation $\rho _{v}$ , for infinitely many $v\notin S_{K}$ , where $S_{K}$ is the following finite set of primes of $K:$

$$ \begin{align*} S_{K}=\{ v: \ v\mid \ell \text{ and } w\mid \ell \text{ for some }w\in S \text{ or } \ell \text{is ramified in } L\}. \end{align*} $$

Let v be such a prime, and let $\ell $ be its residue characteristic.

We first compute the conductor $N(\bar {\rho }_{v})$ . Since $\bar {\rho }_{v}$ is unramified outside $S_{v}$ , the conductor is divisible only by primes in S. We then apply Proposition 3.3 to $E=L_{w}$ and $r=\bar {\rho }_{v}$ . The image of $\bar {\rho }_{v}$ s contained in $\mathrm {GL}_{2}(\mathbb {F}_{\ell ^{t}})$ , where $t\leq [K:{\mathbb {Q}}]=n_{K}$ . The cardinality of this group is $\ell ^{t}(\ell ^{2t}-1)(\ell ^{t}-1)$ . Let $W_{w}$ denote the wild inertia subgroup of $G_{L_{w}}$ . If $\ell $ satisfies the following congruences:

(⋆) $$ \begin{align} \ell^{n_{K}} \not\equiv \begin{cases} \pm 1 \mod p &\text{ if } p\neq 2,3\\ \pm 1 \mod 8 &\text{ if } p=2,\\ \pm 1, 4,7 \mod 9 &\text{ if } p=3, \end{cases} \end{align} $$

then the same congruences hold for $\ell ^{t},$ and hence $\bar {\rho }_{v}(W_{w})$ is trivial if $p\neq 2,3$ and is at most $p^{5}$ if $p=2$ and at most p if $p=3$ . Hence for $v\notin S$ laying above $\ell $ satisfying the above conditions, using that $e_{E}\leq [L:{\mathbb {Q}}]=n_{L}$ , the inequality of Proposition 3.3 implies that

$$ \begin{align*} n_{K}(\bar{\rho}_{v},L_{w})\leq \begin{cases} 2(1+n_{L}) &\text{ if } p\neq 2,3,\\ 2(1+6n_{L}) &\text{ if } p=2,\\ 2(1+2n_{L}) &\text{ if } p=3. \end{cases} \end{align*} $$

Writing $\mathfrak {p}_{w}$ for the prime ideal of L corresponding to w, we hence find that the conductor of $\bar {\rho }_{v}$ divides

$$ \begin{align*} \mathfrak{C}:= \prod\limits_{\substack{w\in S,\\ w\nmid 2,3}} \mathfrak{p}_{w}^{2+2n_{L}}\cdot \prod\limits_{\substack{w\in S,\\ w\mid 2}}\mathfrak{p}_{w}^{2+12n_{L}}\cdot \prod\limits_{\substack{w\in S,\\ w\mid 3}}\mathfrak{p}_{w}^{2+4n_{L}}. \end{align*} $$

Finally, notice that $\rho _{v}$ is odd thanks to the condition on the determinant, and, moreover, [Reference Barnet-Lamb, Gee, Geraghty and Taylor5, Proposition 5.3.2] implies that there exists a density one set of primes such that $(\bar {\rho }_{v})_{\mid G_{L(\zeta _{\ell })}}$ is irreducible, i.e., (TW) is satisfied.

We can apply Conjecture 1.2 to $\bar {\rho }_{v}$ for $v\in \Sigma $ , where $\Sigma $ is the infinite set of primes $v\mid \ell $ such that $v\not \in S_{K}$ , $\ell $ satisfies (⋆); $\bar {\rho }_{v}$ is absolutely irreducible and satisfies (TW). We have produced infinitely many $f_{v}$ Hilbert modular eigenform, which by Theorem 3.4 are of parallel weight 2 and level dividing $\mathfrak {C}$ . Their Fourier coefficients are defined over a ring $\mathcal {O}(v)\subset \mathcal {O}_{K},$ and at a prime $\lambda \mid v,$ the associated Galois representation $\rho _{f_{v},\lambda }$ is isomorphic to $\bar {\rho }_{v}$ modulo $\lambda $ . Since the space of Hilbert modular form of fixed weight and with conductor dividing $\mathfrak {C}$ is finite dimensional (see [Reference Freitag20, Theorem 6.1]), we can find at least one Hilbert modular eigenform f of parallel weight 2 and level dividing $\mathfrak {C}$ defined over some $\mathcal {O}\subset \mathcal {O}_{K}$ such that for infinitely many of the v above, the same property holds for $\rho _{f,\lambda }$ for $\lambda \mid v$ . This implies that for all $w\not \in S,$ the congruence

$$ \begin{align*} a_{w}(f)\equiv a_{w} \mod \lambda\mid v \end{align*} $$

holds for infinitely many primes $\lambda ,$ and hence, $a_{w}(f)= a_{w}$ , as required. ▪

3.3 Proof of Theorem 1

Recall that, as in Section 1.2, the system $(\rho _{v})_{v}$ is required to satisfy the following additional property:

(𝒮.5) $$ \begin{align} \text{the field generated by } a_{w} \text{ for every } w \text{ is exactly } K. \end{align} $$

In other words, we have $K(f)=K$ , where f is the Hilbert modular form for L produced in Proposition 3.1.

Proof Starting with our initial datum of Galois representations, we have produced a Hilbert modular form f for L. We can then apply Theorem 2.5, which gives an abelian variety $A_{f}$ over L of dimension $[K:{\mathbb {Q}}]$ and an embedding of $\mathcal {O}_{K}$ into $\mathrm {End}(A)$ . For all but finitely many $w\mid p$ at which $A_{f}$ has good reduction

$$ \begin{align*} \det(1-X\rho_{A_{f},v}(\text{Frob}_{w}))=1-a_{w}(f)X+N_wX^{2}, \end{align*} $$

where v is a finite prime of K not dividing p and $\rho _{A_{f},v}$ is the $G_{L}$ -representation on $T_{v}(A_{f})$ , the v-adic Tate module of $A_{f}$ . We have therefore produced an abelian variety $A_{f}$ as stated in Theorems 1.A and 1.B.

We just need to stress that we do not require any conjectural statement in the case $n_{L}=1$ . Indeed, we can use Theorem 2.3 in place of Blasius’ conjectural version, and Serre’s conjecture is fully known thanks to the work of Khare–Wintenberger [Reference Khare and Wintenberger27, Theorem 1.2]. ▪

3.4 A Corollary

We rephrase the main results of the section as needed to prove Theorem 2.

Corollary 3.5 Assume that, in the setting of Theorems 1.A and 1.B, we also have a representation

$$ \begin{align*} \rho : G_{L}\longrightarrow \mathrm{GL}_{2}(\mathcal{O}_{K}/\mathfrak{N}), \end{align*} $$

for some integral ideal $\mathfrak {N}\subset \mathcal {O}_{K}$ , such that for all pairs $(v, a)$ , where v is a place of K and $a\in \mathfrak {N}-\{0\}$ , such that $v^{a}$ divides $\mathfrak {N}$ , the reductions of $\rho $ and $\rho _{v}$ modulo $v^{a}$ agree. Then there exists an $n_{K}$ -dimensional abelian variety $A/L$ with good reduction at all w outside $S,$ and the action of $G_{L}$ on $A[\mathfrak {N}]$ is given by $\rho $ .

4.1 Recap on the Integral Finite Descent Obstruction

Let $Y/F$ be a smooth, geometrically connected variety (not necessarily proper) over a number field F. Let S be a finite set of places of $F,$ and, as before, assume that S contains the archimedean places and all places of bad reduction of Y. Choose and fix a smooth model $\mathcal {Y}$ of Y over $\mathcal {O}_{F,S}$ . In this section, we recall the definition of the set $\mathcal {Y}^{f-cov}$ , which will correspond to the adelic points of $\mathcal {Y}$ that are unobstructed by all Galois covers. To make the paper self contained, we recall the discussion from [Reference Helm and Voloch26, Section 2] (where the authors work with affine curves). We then recall that, for Hilbert modular varieties, a point unobstructed by finite covers admits an infinite tower of twists of covers with a compatible system of lifts of adelic points along the tower (following [Reference Helm and Voloch26, Proposition 1]).

Let $\pi : \mathcal {X} \to \mathcal {Y}$ be a map of $\mathcal {O}_{F,S}$ -schemes, such that it becomes a Galois covering over $\overline {F}$ . Such a map is called a geometrical Galois cover of $\mathcal {Y}$ . Denote by $\operatorname {Tw}(\pi )$ the set of isomorphism classes of twists of $\pi $ , i.e., of maps $\pi ':\mathcal {X}'\to \mathcal {Y}$ that become isomorphic to $\pi $ over $\overline {F}$ . We have

$$ \begin{align*} \mathcal{Y}(\mathcal{O}_{F,S})= \bigcup\limits_{\pi ' \in \operatorname{Tw}_{0}(\pi)} \pi' \big(\mathcal{X}'(\mathcal{O}_{F,S})\big), \end{align*} $$

where $\operatorname {Tw}_{0}(\pi )$ is a suitable finite subset of $\operatorname {Tw}(\pi )$ (for a more detailed discussion, we refer the reader to [Reference Skorobogatov38, pp. 105, 106]), and $\pi ':\mathcal {X}'\to \mathcal {Y}$ is a twist of $\pi $ . In what follows, w denotes a place of F.

Definition 4.1 We define $\mathcal {Y}^{f-cov}(\mathcal {O}_{F,S})=\mathcal {Y}^{f-cov}$ as the set of $ (P_{w})_{w} \in \prod _{w \notin S} \mathcal {Y}(\mathcal {O}_{F_{w}})$ such that, for each geometrical Galois cover $\pi $ , we can write

$$ \begin{align*} P_{w}= \pi' (Q_{w}), \ \ \ \forall w \notin S \end{align*} $$

for some $\pi ' \in \operatorname {Tw}_{0}(\pi )$ and $(Q_{w})_{w}\in \prod _{w \notin S} \mathcal {X}'(\mathcal {O}_{F_{w}})$ .

A few words to justify the equivalence between the two definitions are needed. This is explained in [Reference Helm and Voloch26, Proposition 1] for curves, and it relies on results from [Reference Stoll39] (notably [Reference Stoll39, Lemma 5.7]Footnote ³). In [Reference Stoll39], Stoll works with projective varieties and their rational points, but what he says still holds true for the integral points of non-projective varieties. Once such differences are taken into account, the proof works in the same way in our setting.

Clearly, we have $\mathcal {Y}(\mathcal {O}_{F,S})\subset \mathcal {Y}^{f-cov},$ and so if $ \mathcal {Y}^{f-cov}$ is empty, then $\mathcal {Y}(\mathcal {O}_{F,S})$ has to be empty as well. What can be said when $ \mathcal {Y}^{f-cov}$ contains a point?

From now on, we specialise to the case of Hilbert modular varieties (and their twists).

4.2 Integral Points on Hilbert Modular Varieties

Recall the notation from Section 2.2.1. Let $Y_{K}(\mathfrak {N})$ be the $n_{K}$ -dimensional ${\mathbb {Q}}$ -scheme described in Section 2.2.1 and let N be the integer such that $\mathfrak {N}\cap \mathbb {Z}=(N)$ . The set of twists of $\pi : Y_{K}(\mathfrak {N}) \to Y_{K}(1)$ over a number field F corresponds to the set of Galois representations $\rho : G_{F} \to \mathrm {GL}_{2}( {\mathcal {O}_{K}}/\mathfrak {N})$ whose determinant is the cyclotomic character $\chi : G_{F} \to (\mathbb {Z}/N\mathbb {Z})^{\times }$ . Moreover, a point $x \in Y_{K}(1)(F)$ lifts to a F-rational point of the twist of $Y_{K}(\mathfrak {N})$ corresponding to a representation $\rho $ , if and only if $\rho $ describes the action of $G_{F}$ on the $\mathfrak {N}$ -torsion of the underlying abelian variety $A_{x}$ (as an $\mathcal {O}_{K}$ -module).

Using the moduli interpretation of Section 2.2.1, we can construct a model of $Y_{K}(\mathfrak {N})$ over $\mathbb {Z}$ , which is smooth over $\mathbb {Z}[1/b]$ , for some natural number b, divisible by N. To be more precise ,b depends on the level structure and the discriminant of K. Fixing such a model, which we denote by $\mathcal {Y}_{K}(\mathfrak {N})$ , we can talk about $\mathcal {O}_{F,S}$ -points of $Y_{K}(\mathfrak {N})$ , for any number field F and set of places S containing the archimedean places and the ones dividing b. Such $\mathcal {O}_{F,S}$ -points then correspond to abelian varieties (with some extra structure), having good reduction outside S. Recall that N is assumed to be bigger than $3$ , since it is important to have a fine moduli space. For example, the affine line is the moduli space of elliptic curves and has plenty of $\mathbb {Z}$ -points, even though there are no elliptic curves defined over $\mathbb {Z}$ .

We are ready to study the finite descent obstruction for the $\mathcal {O}_{L,S}$ -points of $\mathcal {Y}_{K}(\mathfrak {N})$ and its twists, where L is a totally real field (the fact that L is totally real will be used only in the next section). Let

$$ \begin{align*} \rho : G_{L}\longrightarrow \mathrm{GL}_{2}(\mathcal{O}_{K}/\mathfrak{N}) \end{align*} $$

be a representation whose determinant is the cyclotomic character. Assume that S contains the places w of ramification of $\rho $ . Under this assumption, arguing as above, we can consider $\mathcal {Y}_\rho $ to be the S-integral model of the twist of $\mathcal {Y}_{K}(\mathfrak {N})$ corresponding to $\rho $ . From now on, we assume that $\mathcal {Y}_\rho (\mathcal {O}_{L,S})$ is non-empty. The next lemma relates a point $(P_{w})_{w} \in \mathcal {Y}_\rho ^{f-cov}$ to a system of Galois representations as considered in the previous section.

Proof We first check, using Lemma 4.1, that an unobstructed point corresponds to a system of Galois representations as described above, and then we show that every such system enjoys the desired properties.

Thanks to Proposition 4.1, we can fix a compatible system of lifts of $(P_{w})_{w}$ on $\mathcal {Y}_\rho ^{f-cov}(\mathcal {O}_{L,S})$ . In particular, for each $\mathfrak {M}$ divisible by $\mathfrak {N}$ , we obtain a twist $\mathcal {Y}_{K}(\mathfrak {M})'$ of $\mathcal {Y}_{K}(\mathfrak {M})$ and a compatible family of points $(P_{w})_{\mathfrak {M}}$ of $\mathcal {Y}_{K}(\mathfrak {M})'$ lifting $(P_{w})_{w}$ . We remark here that the latter compatible family of points depends on $(P_{w})_{w}$ . Indeed, a priori, we cannot simply lift $\rho $ to mod $\mathfrak {M}$ coefficients.

By the interpretation of $\mathcal {Y}_{K} (\mathfrak {M})'$ as moduli space of abelian varieties discussed above, the point $(P_{w})_{\mathfrak {M}}$ corresponds to an abelian variety $A_{w}/ L_{w}$ of dimension $n_{K}$ , with good reduction and $\mathcal {O}_{K}$ -multiplication and prescribed $\mathfrak {M}$ -torsion. The other conditions are easily checked as at the end of proof of [Reference Helm and Voloch26, Theorem 2].

The fact that the action of $G_{L_{w}}$ on $T_{v}(A_{w})$ is given by the restriction of $\rho _{v}$ to the decomposition group at w ensures that ( $\mathcal {S}.1$ ) and ( $\mathcal {S}.2$ ) are satisfied. Moreover, since $A_{w}$ has good reduction, $A_{w}[v]\simeq \bar {\rho }_{v}$ is a finite flat group scheme over $O_{K_{w}}$ for all $w\mid \ell $ if $v\mid \ell $ ; this implies that ( $\mathcal {S}.3$ ) also holds.

Finally, we need to show that ( $\mathcal {S}.4$ ) is satisfied. With the three conditions above, one can show, as in the proof of Proposition 3.1, that the conductor of $\bar {\rho }_{v}$ divides a fixed ideal $\mathfrak {C}$ of L. If w is such that $A_{w}/L_{w}$ is supersingular at v, then $A_{w}[v]\simeq \bar {\rho }_{v}$ is absolutely irreducible. If there existed infinitely many v such that $\bar {\rho }_{v}$ is absolutely reducible, we could then write the semisemplification of $\bar {\rho }_{v}$ as direct sum of $\phi $ and $\chi _{\ell }\phi ^{-1}$ , for some character $\phi $ . Since $\bar {\rho }_{v}\simeq A_{w}[v]$ for $w\mid p$ , we then have that $A_{w}$ is ordinary, and hence $\phi $ is unramified at w. We also know that the conductor of $\phi $ divides $\mathfrak {C}$ ; hence, if $w\equiv 1 \mod \mathfrak {C,}$ we have

$$ \begin{align*} a_{w}(A_{w}):=\mathrm{Tr}\big(\operatorname{Frob}_{w}, T_{v}(A_{w})\big) \equiv \chi_{\ell}(\operatorname{Frob}_{w})+1 \mod v. \end{align*} $$

Since $\chi _{\ell }(\operatorname {Frob}_{w})=p^{[L_{w}:{\mathbb {Q}}_{p}]}$ , we showed that if we had infinitely many v such that $\bar {\rho }_{v}$ is absolutely reducible, we would find $a_{w}(A_{w})=p^{[L_{w}:{\mathbb {Q}}_{p}]}+1$ . Since the Weil bound says that $|a_{w}|\leq 2\sqrt {p^{[L_{w}:{\mathbb {Q}}_{p}]}}$ , we reached a contradiction. ▪

We are now ready to prove the main theorem about descent obstruction for Hilbert modular varieties

4.3 Proof of Theorem 2

We do not treat the cases $n_{L}=1$ and $n_{L}>1$ separately, but, as in the proof of Theorem 1, we emphasize that we do not need any conjectural statement in the case $n_{L}=1$ , since we have Shimura’s unconditional result, Theorem 2.3.

Proof Thanks to Lemma 4.2, a point in $\mathcal {Y}_\rho ^{f-cov} (\mathcal {O}_{L,S})$ , which is assumed to be non-empty, gives rise to a compatible system of representations of $G_{L}$ , denoted by $\{\rho _{v}\}_{v}$ . Let E be the the subfield of K generated by $\mathrm {tr} (\rho _{v}(\operatorname {Frob}_{w}))$ for all w. If $E=K$ , Corollary 3.5 produces an $\mathcal {O}_{L,S}$ -abelian variety A with $\mathcal {O}_{K}$ -multiplication, such that $G_{L}$ acts on $A[\mathfrak {N}]$ via $\rho $ . To conclude the proof, we just need to see how A corresponds to a point $P\in \mathcal {Y}_{\rho }(\mathcal {O}_{L,S})$ . The only issue that is not clear from the quoted corollary is whether A is principally $\mathcal {O}_{K}$ -polarised, but this follows from the discussion in Section 2.4.

If E is strictly contained in K, i.e., condition (𝒮.5) is not satisfied, we consider $S_{E}$ , the Hilbert modular variety associated with E and of level $\mathfrak {N} \cap \mathcal {O}_{E}$ . For the same reason as above, the system $\{\rho _{v}\}$ corresponds to a $\mathcal {O}_{L,S}$ -point P of the twist by $\rho $ of $S_{E}$ . The embedding $\mathrm {Res}_{E/{\mathbb {Q}}}\mathrm {GL}_{2}\hookrightarrow \mathrm {Res}_{K/{\mathbb {Q}}}\mathrm {GL}_{2}$ induces a map of Shimura varieties

$$ \begin{align*} r :S_{E} \longrightarrow Y_{K}(\mathfrak{N}), \end{align*} $$

and therefore on their twists by $\rho $ . Via r, we can regard P as an $\mathcal {O}_{L,S}$ -point of $\mathcal {Y}_\rho $ , which completes the proof of the theorem. The only difference is that the abelian variety constructed in this case is not primitive. This concludes the proof of Theorem 2. ▪