## 1. Introduction

Let $X$ be a ramified cover of an abelian variety over a number field $K$, i.e. there is a finite surjective non-étale morphism $X\to A$ with $X$ normal and irreducible. The aim of this paper is to prove novel results on the distribution of rational points on $X$.

In this work we are guided by Lang's conjectures on varieties of general type, and by a question of Serre on covers of abelian varieties. By Kawamata's structure result for ramified covers of abelian varieties [Reference KawamataKaw81], for every *ramified* cover $X\to A$ of an abelian variety $A$ over a number field $K$, there is a finite field extension $L/K$ and finite étale cover $X'\to X_L$ such that $X'$ dominates a positive-dimensional variety of general type over $L$. Assuming Lang's conjecture [Reference JavanpeykarJav20, Reference LangLan86], it follows that the $K$-rational points $X(K)$ are *not* Zariski-dense in $X$. Despite Faltings’ deep finiteness theorems for integral (and rational) points on subvarieties of abelian varieties [Reference FaltingsFal94], we are far from proving such desired non-density results for rational points on ramified covers of abelian varieties (even in dimension two) over number fields.

Lang's aforementioned conjecture predicts that, for any abelian variety $A$ over a number field $K$ with $A(K)$ dense and any ramified cover $\pi :X\to A$, the set $A(K)\setminus \pi (X(K))$ is again dense. Motivated by the inverse Galois problem, Serre also raised a related question [Reference SerreSer08, § 5.4, Problem], a positive answer to which would lead to the same conclusion. Our main results stated in the following (see Theorems 1.3 and 1.4) confirm the density of $A(K)\setminus \pi (X(K))$ unconditionally for all abelian varieties, and thereby provide novel evidence for Lang's conjectures on rational points and answer, in part, Serre's question.

We stress that there are no prior results concerning the distribution of rational points on ramified covers of abelian varieties, except for some very special cases, and the same goes for integral points on ramified covers of $\mathbb {G}_m^n$. Our current work provides the *first* step towards understanding the Diophantine properties of ramified covers of abelian varieties.

The conclusion of our main theorem that $A(K)\setminus \pi (X(K))$ is dense bears a strong resemblance to the geometric formulation of Hilbert's irreducibility theorem. In fact, this point of view is quite fruitful and key to many of the results of this paper.

In its simplest form, Hilbert's irreducibility theorem states that, given a number field $K$ and an irreducible polynomial $p(t,x_1,\ldots,x_s)$ with coefficients in $K$, there exist infinitely many $a \in K$ such that the specialization $p(a,x_1,\ldots,x_s)$ is an irreducible polynomial in the variables $x_1,\ldots,x_s$. A similar statement applies to the simultaneous specialization of finitely many irreducible polynomials.

Following the work of several authors, Hilbert's irreducibility theorem has been recast in geometric language as follows. An irreducible polynomial $p(t,x)$ defines in a natural way an irreducible cover (that is, a finite surjective morphism) of an open subvariety of $\mathbb {P}_1$ over $K$: this is obtained simply by projecting the zero locus of $p$ on the $t$-line. Hilbert's irreducibility theorem can then be interpreted as stating that ‘many’ fibres of this cover are irreducible over $K$ and hence, in particular, that they have no $K$-rational points; in turn, this is also equivalent to ‘many’ rational points of $\mathbb {P}_1$ not being the image of rational points in the zero locus of $p$. Similar constructions can be made in the case of several variables, and this point of view is encapsulated in the following definition of Serre (for this and much more about the Hilbert property we refer the reader to the books of Serre [Reference SerreSer08, Chapter 3] and Fried and Jarden [Reference Fried and JardenFJ08]).

Definition 1.1 An integral, quasi-projective variety $X$ over a field $k$ satisfies the *Hilbert property over* $k$ if, for every finite collection of finite surjective morphisms $(\pi _i:Y_i\to X)_{i=1}^n$ with $Y_i$ a normal (integral) variety over $k$ and $\deg \pi _i \geq 2$, the set $X(k)\setminus \bigcup _{i=1}^n \pi _i(Y_i(k))$ is dense in $X$.

In this language, Hilbert's irreducibility theorem states that every projective space over a number field $K$ has the Hilbert property. As the Hilbert property is a birational invariant, it follows that every rational variety over $K$ has the Hilbert property.

The arithmetic nature of the ground field is of course essential for the truth of Hilbert's irreducibility theorem, and a field $k$ is called *Hilbertian* if $\mathbb {P}_1$ has the Hilbert property over $k$ (or equivalently, if $\mathbb {P}_n$ has the Hilbert property over $k$ for every $n$). In this paper we focus on the case of finitely generated fields of characteristic zero, which are known to be Hilbertian.

Clearly not every variety has the Hilbert property: for example, in [Reference Corvaja and ZannierCZ17] it is shown that, as a consequence of the Chevalley–Weil theorem, a smooth proper variety over a number field with the Hilbert property is geometrically simply connected. Nonetheless, the literature is by now very rich with examples of varieties with the Hilbert property. Several proofs exist for the case of projective space (see, for example, [Reference Bombieri and GublerBG06, § 9.6], [Reference Fried and JardenFJ08, Chapter 11], [Reference SchinzelSch00, § 4.4], [Reference SerreSer97, Chapter 9], [Reference SerreSer08, Chapter 3] and [Reference VölkleinVöl96]). Note that connected reductive algebraic groups have been shown in [Reference Colliot-Thélène and SansucCTS87] to have the Hilbert property over Hilbertian fields; note that these varieties are rational over $\overline {k}$, but not necessarily over $k$). Some sporadic examples of varieties with the Hilbert property are also known (such as the K3 surface $x^4+y^4 = z^4+w^4 \subset \mathbb {P}_{3,\mathbb {Q}}$, which is not even geometrically unirational [Reference Corvaja and ZannierCZ17, Theorem 1.4]; see also [Reference DemeioDem19] and [Reference DemeioDem20]).

Hilbert's irreducibility theorem and its extensions have led to a number of applications (see, for example, [Reference Dvornicich and ZannierDZ07, Reference ZannierZan00]), typically motivated by the following observation: if a certain desired object can be realized *geometrically*, and if the base space of the geometric construction has the Hilbert property, then the situation can be specialized to an *arithmetic* one. The point is of course that various tools are available on the geometric side that might make certain constructions more natural than they would be if they were carried out directly on the arithmetic side. Concretely, this has been implemented for example by Néron [Reference NéronNér52], who used this idea to construct elliptic curves defined over $\mathbb {Q}$ of large rank. By far the most important such application is to the inverse Galois problem (this was also Hilbert's original motivation): if a group $G$ can be realized as the Galois group of a cover $Y \to X$ defined over $\mathbb {Q}$, with $X$ having the Hilbert property, then $G$ is a Galois group over $\mathbb {Q}$. Noether famously used this idea to prove that $S_n$ is a Galois group over $\mathbb {Q}$ for all $n$.

Given the significance of the Hilbert property for number-theoretical questions, one would wish to extend Hilbert's theorem to other classes of varieties in addition to the rational varieties. However, there are serious restrictions to possible generalizations of the classical results, starting with the aforementioned fact that the Hilbert property can only hold for geometrically simply connected varieties [Reference Corvaja and ZannierCZ17].

Nevertheless, it is conjectured that, for a variety $X$ over a finitely generated field $k$ of characteristic zero with $X(k)$ Zariski-dense, the only obstruction to the Hilbert property should indeed come from unramified covers (see Conjecture 1.7 for a precise statement). Further taking into account that the original version of Hilbert's irreducibility theorem does not need to contend with any unramified covers at all (since $\mathbb {P}_n$ is geometrically simply connected), it seems natural to rule out such covers at the outset. This leads to the following notion.

### Definition 1.2 (Corvaja–Zannier)

A smooth proper variety $X$ over a field $k$ has the *weak-Hilbert property over* $k$ if, for all finite collections of finite surjective *ramified* morphisms $(\pi _i:Y_i\to X)_{i=1}^n$ with each $Y_i$ an integral normal variety over $k$, the set $X(k) \setminus \bigcup _{i=1}^n \pi _i(Y_i(k))$ is Zariski-dense in $X$.

Note that we restrict ourselves to finite surjective *morphisms*, as opposed to Corvaja and Zannier who considered dominant *rational maps* of finite degree in their definition of the weak-Hilbert property (see [Reference Corvaja and ZannierCZ17, § 2.2]). By a standard argument using Stein factorization, one can easily show that these two definitions are equivalent (see, for example, the proof of Theorem 7.11). In addition, we stress that the weak-Hilbert property is a birational invariant amongst smooth proper varieties (see Proposition 3.1). As a consequence, all our results concerning this property automatically propagate between smooth proper varieties within the same birational equivalence class.

The weak-Hilbert property can be applied to the inverse Galois problem similarly as the classical Hilbert property; see Remark 1.5. It is studied (using some of the results of this paper) in [Reference Bary-Soroker, Fehm and PetersenBSFP22, Reference JavanpeykarJav22, Reference Gvirtz-Chen and MezzedimiGCM21].

Our main (though not only) focus in this paper is the weak-Hilbert property for abelian varieties over finitely generated fields of characteristic zero.

Theorem 1.3 Let $k$ be a finitely generated field of characteristic zero, let $A$ be an abelian variety over $k$, let $\Omega \subset A(k)$ be a Zariski-dense subgroup, and let $(\pi _i:Y_i \to A)_{i=1}^n$ be a finite collection of ramified covers with each $Y_i$ a normal integral variety. Then, there is a finite index coset $C\subset \Omega$ such that, for every $c$ in $C$ and every $i=1,\ldots, n$, the scheme $Y_{i,c}$ has no $k$-points. In particular, if $A(k)$ is dense, then $A$ has the weak-Hilbert property over $k$.

We prove a stronger conclusion on the scheme-theoretic fibres $Y_{i,c}$, assuming however in addition that the ramified covers $\pi _i:Y_i\to A$ do not have any non-trivial étale subcovers. Here, we say that $Y_i\to A$ has no non-trivial étale subcovers if, for every cover $X_i\to A$ of degree greater than one such that $Y_i\to A$ factors over $X_i\to A$, we have that $X_i\to A$ is ramified. Note that such a covering $Y_i\to A$ is ramified if it is of degree greater than one. Indeed, otherwise $Y_i\to A$ would be an étale subcover of itself.

Theorem 1.4 Let $k$ be a finitely generated field of characteristic 0, let $A$ be an abelian variety over $k$, let $\Omega \subset A(k)$ be a Zariski-dense subgroup and, for $i=1,\ldots, n$, let $Y_i$ be a normal integral variety, and let $\pi _i:Y_i \to A$ be a finite surjective morphism with no non-trivial étale subcovers. Then, there is a finite index coset $C\subset \Omega$ such that, for every $c$ in $C$ and every $i=1,\ldots, n$, the $k$-scheme $Y_{i,c}$ is integral.

### Remark 1.5 (Inverse Galois problem)

Let $G$ be a finite group, let $X$ be a normal geometrically integral variety over $\mathbb {Q}$, and let $X\to A$ be a $G$-Galois cover of an abelian variety $A$ over $\mathbb {Q}$ with no non-trivial étale subcovers. Then, if $A(\mathbb {Q})$ is dense, by our version of Hilbert's irreducibility theorem for abelian varieties (Theorem 1.4), there is a Galois number field $K$ with Galois group $G$.

Note that Theorems 1.3 and 1.4 represent the usual parallelism between irreducible fibres and fibres with no rational points. In addition, as alluded to previously, one can easily extend these results to obtain similar conclusions for finite collections of dominant *rational maps* of finite degree (see Theorem 7.11). Furthermore, we stress that the assumption that the subgroup $\Omega$ is Zariski-dense in Theorems 1.3 and 1.4 can not be weakened to the assumption that $\Omega$ is merely infinite (see Remark 7.9).

The two results directly generalize [Reference ZannierZan10, Theorem 2] in which the case of an abelian variety isomorphic to a power of a non-CM elliptic curve was handled. In addition, we stress that in Theorem 1.3 one cannot expect the integrality of the fibres $Y_{i,c}$. Indeed, the presence of non-trivial étale subcovers obstructs the desired integrality; see Remark 7.8 for a precise statement.

### Remark 1.6 (About our proofs)

Working with pairs $(A,\Omega )$ in Theorems 1.3 and 1.4 has several technical advantages. For example, an induction argument allows one to reduce from a finite collection of covers to just a single cover (Lemma 4.14). Moreover, by a specialization argument, we can easily pass from number fields to arbitrary finitely generated fields of characteristic zero (see § 7.1). Furthermore, using local–global arguments, it suffices to construct a *single* point in $\Omega$ over which the fibre is integral (Corollary 5.4). Finally, using structure results for vertically ramified covers (Lemma 2.17), we establish a product theorem for pairs $(A,\Omega )$ which allows us to consider only pairs $(A,\Omega )$ with $\Omega$ cyclic; see § 5.2 for precise statements.

As already hinted at, the weak-Hilbert property is conjectured to hold as soon as the ‘obvious’ necessary conditions are met; more precisely, one has the following (see [Reference CampanaCam11, § 4] for the definition of a ‘special’ smooth proper connected variety).

### Conjecture 1.7 (Campana, Corvaja–Zannier)

Let $X$ be a smooth proper geometrically connected variety over a finitely generated field $k$ of characteristic zero. Then the following are equivalent.

1. There is a finite field extension $L/k$ such that $X_L$ has the weak-Hilbert property over $L$.

2. There is a finite field extension $M/k$ such that $X(M)$ is dense in $X$.

3. The smooth proper connected variety $X_{\overline {k}}$ is special (in the sense of Campana [Reference CampanaCam11, § 4]).

Frey and Jarden [Reference Frey and JardenFJ74] proved that an abelian variety $A$ over a finitely generated field $k$ of characteristic zero admits a finite extension $L/k$ such that $A(L)$ is Zariski-dense in $A$ (see also [Reference JavanpeykarJav21a, § 3] and [Reference Hassett and TschinkelHT00]). As abelian varieties are special [Reference CampanaCam11, § 4], by combining Frey and Jarden's result with Theorem 1.3 we obtain a proof of Conjecture 1.7 for any variety which is birational to a smooth projective connected variety with trivial tangent bundle.

Corollary 1.8 Let $k$ be a finitely generated field of characteristic zero. Let $X$ be a smooth proper geometrically connected variety over $k$ which is birational to a smooth proper variety with trivial tangent bundle. Then $X_{\overline {k}}$ is special [Reference CampanaCam11, § 4], and there is a finite field extension $L/k$ such that $X_L$ has the weak-Hilbert property over $L$.

Note that in view of Campana's perspective on special varieties, it is natural to study the influence of some positivity condition on the tangent bundle; the varieties under consideration in this work being precisely those with trivial tangent bundle. According to a classical theorem of Mori, a smooth projective variety with ample tangent bundle is geometrically isomorphic to a projective space, so that the Hilbert property for such varieties follows from Hilbert's original irreducibility theorem (after a possible field extension).

Motivated by Conjecture 1.7 and inspired by fundamental properties of special varieties [Reference CampanaCam11] and of varieties with a dense set of rational points, we establish several basic facts about the class of varieties with the weak-Hilbert property in § 3, first and foremost among them the following product theorem.

Theorem 1.9 Let $X$ and $Y$ be smooth proper varieties over a finitely generated field $k$ of characteristic zero with the weak-Hilbert property over $k$. Then $X\times Y$ has the weak-Hilbert property over $k$.

Theorem 1.9 was obtained jointly by the third-named author and Olivier Wittenberg, and we are grateful to Olivier Wittenberg for allowing us to include this result here.

It seems worthwhile pointing out that Theorem 1.9 is the ‘weak-Hilbert’ analogue of Bary-Soroker, Fehm and Petersen's product theorem for the (usual) Hilbert property [Reference Bary-Soroker, Fehm and PetersenBSFP14]. This product theorem was stated by Serre as a problem in [Reference SerreSer08, § 3.1].

Note that Theorem 1.9 actually reproves the product theorem of [Reference Bary-Soroker, Fehm and PetersenBSFP14] when the base field is finitely generated and of characteristic zero. Indeed, if $X$ and $Y$ satisfy the Hilbert property (hence, weak-Hilbert property) over a finitely generated field $k$ of characteristic zero, then $X\times Y$ has the weak-Hilbert property over $k$ by Theorem 1.9. However, because $X$ and $Y$ are geometrically simply connected (by Corvaja and Zannier's theorem [Reference Corvaja and ZannierCZ17, Theorem 1.6]), it follows that $X\times Y$ is geometrically simply connected (by the product property for étale fundamental groups). Thus, the smooth proper variety $X\times Y$ is geometrically simply connected and has the weak-Hilbert property over $k$. Therefore, by definition, it has the Hilbert property over $k$.

In addition to proving Theorem 1.9, in § 3 we also show that the weak-Hilbert property of a variety $X$ is inherited by both its étale covers and its surjective images under some natural assumptions (in particular, including smooth surjective images), thus building a toolkit that simplifies proving new instances of the weak-Hilbert property.

To conclude this introduction we describe the ingredients that go into the proof of Theorem 1.3, which is the main result of this paper. Our argument follows the same broad lines as the proof of [Reference ZannierZan10, Theorem 2], which handled the special case of $A=E^n$ being the power of a non-CM elliptic curve with a *non-degenerate* point $P$, that is, a rational point $P$ generating a Zariski-dense subgroup. The method is based on the following idea, which we simplify slightly for ease of exposition. Suppose that $A$ is defined over a *number field* $K$. Given a cover $\pi : Y \to A$, one proves that there is a prime $\mathfrak {p}$ of $\mathcal {O}_K$ and a torsion point $\zeta \in A(\mathcal {O}_K/\mathfrak {p})$ that does not lift to $Y(\mathcal {O}_K/\mathfrak {p})$. Any rational point $Q \in A(K)$ reducing to $\zeta$ modulo $\mathfrak {p}$ then also does not lift to $Y(K)$, and, because this condition is adelically open and using the group structure on $A$, we get the desired Zariski-dense subset of points that do not lift to $Y$. Naturally this requires that there is at least one $Q \in A(K)$ reducing to $\zeta$: the last task remaining is then to prove the existence of such a point $Q$.

Despite the basic strategy being the same, however, we need to introduce several new ingredients with respect to [Reference ZannierZan10], and to reinterpret various parts of Zannier's approach in a more general context. We now briefly describe where the main novelties lie.

Some technical reductions, carried out in § 4, show that multiple variants of the weak-Hilbert property for abelian varieties are essentially all equivalent. These variants describe how the weak-Hilbert property interacts with the group structure of $A$, and allow us to pass from statements about fibres of covers having no $k$-rational points to statements about fibres being irreducible over $k$. The core argument of our proof is contained in § 6, where we study the case of abelian varieties possessing a non-degenerate point. Unlike in the case of $A=E^n$, which can essentially be reduced to the analysis of a single elliptic curve, we do not have at our disposal the full strength of Serre's open image theorem [Reference SerreSer72]. In addition, some explicit computations with torsion points that are accessible in dimension one would become extremely cumbersome in general. We bypass these problems by giving a more streamlined construction of the torsion point $\zeta$ (see § 6.1) and by replacing the open image theorem by an appeal to several deep results in the Kummer theory of abelian varieties (see § 6.2).

All that is left to do is then to extend the result to all abelian varieties. Up to $k$-isogeny, any abelian variety $A$ over $k$ is a direct product of $k$-simple abelian varieties $A_i$, and if $A(k)$ is Zariski-dense, then $A_i(k)$ is Zariski-dense for all $i$. As any point of infinite order on a *simple* abelian variety is non-degenerate, defining $\Omega _i = \langle P_i\rangle$ for $P_i$ a point of infinite order in $A_i(k)$, we may conclude at this point that, for every $i$, the pair $(A_i,\Omega _i)$ satisfies the conclusion of Theorem 1.3. The results of § 3 (in particular, Theorem 1.9) can then be used to show that $A$ has the weak-Hilbert property over $k$. However, the more precise version given by Theorem 1.3 (or Theorem 1.4) does not follow as easily. This is why in § 5.2 we extend the techniques developed in § 3 to prove a more specific version of Theorem 1.9, in which the factors are abelian varieties and we also take into account a (Zariski-dense) subgroup $\Omega$. A key observation in this section is that, given a cover $Z\to A$, the existence of a *single* point $P\in \Omega$ for which the fibre $Z_P$ is integral implies that there is a finite index coset $C$ of $\Omega$ such that, for every $c$ in $C$, the fibre over $c$ has no $K$-points; see Corollary 5.4. Using the invariance of the weak-Hilbert property under isogeny, it is then a fairly straightforward matter to deduce from this the general case of our result for $k$ a number field. Finally, we use a specialization argument to reduce from finitely generated fields of characteristic zero to number fields.

### Outline of paper

In § 2 we gather some preliminaries. Notably, we provide a structure result for vertically ramified covers of products (Lemma 2.17). In the following section we prove that the class of varieties over a finitely generated field of characteristic zero with the weak-Hilbert property is closed under products, finite étale covers, and smooth images. In § 4 we introduce the class of (PB)-covers, i.e. ramified covers of abelian varieties with no non-trivial étale subcovers. We prove several basic properties of (PB)-covers, and provide links between variants of the Hilbert property for abelian varieties. In § 5 we prove a product theorem, analogous to that obtained in § 3, that applies to a *variant* of the weak-Hilbert property specific to abelian varieties. Then, in § 6, we prove that this property holds for abelian varieties over number fields endowed with a non-degenerate point. Finally, in § 7 we prove the theorems stated in the introduction: the results of the previous sections suffice to handle the case of the ground field being a finite extension of $\mathbb {Q}$, and the general case is then proven by reduction to the number field case.

## 2. Preliminaries

### 2.1 Notation

Throughout the paper we let $K$ denote a number field, whereas we write $k$ for a general field (unless otherwise specified). For a number field $K$, we denote by $M_K$ the set of its places, and by $M_K^{\operatorname {fin}}$ the subset of finite places.

A *variety* over a field $k$ is an integral separated scheme of finite type over $k$. If $X$ is a variety over $k$ and $A\subset k$ is a subring, we define a *model* for $X$ over $A$ to be a pair $(\mathcal {X},\phi )$, where $\mathcal {X}$ is a separated scheme of finite type over $A$ and $\phi :\mathcal {X}\times _A k\to X$ is an isomorphism; we often omit $\phi$ from the notation.

A morphism $\pi :Y\to X$ of normal varieties over $k$ is a *cover of* $X$ *(over* $k$*)* if $\pi$ is finite and surjective.

For a morphism $f:Y \rightarrow X$ of schemes, and a point $c \in X$, we denote the scheme-theoretic fibre of $f$ over $c$ by $Y_c$, or by $f^{-1}(c)$ when we need to specify the morphism to avoid ambiguity.

Let $A$ be an abelian variety over a field $k$. For a prime $\ell$ different from the characteristic of $k$, we let $T_{\ell } A := \varprojlim _{n \to \infty } A[\ell ^n]$ denote the $\ell$-adic Tate module of $A$, where $A[\ell ^n]$ is, by convention, the full geometric torsion subgroup $A[\ell ^n](\overline {k})$. We similarly denote by $A[\ell ^\infty ]$ the union of all $A[\ell ^n](\overline {k})$ for $n \geq 1$. We denote by $\operatorname {Gal}(k'/k)$ the Galois group of a (possibly infinite) Galois field extension $k'/k$, and simply by $\Gamma _k$ the absolute Galois group of $k$, namely $\Gamma _{k} = \operatorname {Gal}(\overline {k}/k)$. For a rational (respectively, $\ell$-adic) number $a\neq 0$, we define $v_{\ell }(a)$ to be the unique integer such that $a={a_1}/{a_2}\cdot \ell ^{v_{\ell }(a)}$, with $a_1, a_2 \in \mathbb {Z}$ (respectively, $a_1, a_2 \in \mathbb {Z}_\ell$) and $\ell \nmid a_i, \ i=1,2$. If $a=0$, we let $v_{\ell }(a):= \infty$ by convention.

### 2.2 Unramified morphisms

As regards unramified morphisms, we follow the conventions of the Stacks project [Sta20, Tag 02G3]: in particular, a morphism of schemes $Y\to X$ is unramified if and only if it is locally of finite type and its diagonal is an open immersion [Sta20, Tag 02GE]. We say that a morphism locally of finite type (e.g. a cover $Y \to X$) is *ramified* if it is not unramified. We need the following lemma.

Lemma 2.1 Let $f:X\to S$ be a morphism of normal proper varieties over a field $k$ and let $\pi :Z\to X$ be a finite surjective ramified morphism. Assume that the branch locus $D$ of $\pi :Z\to X$ dominates $S$ (i.e. $f(D) = S$). Then, for every point $s$ in $S$, the morphism $Z_s\to X_s$ is finite surjective ramified.

Proof. A morphism of varieties $V\to W$ over $k$ is unramified if and only if, for every $w$ in $W$, the morphism $V_w\to \operatorname {Spec} k(w)$ is unramified (i.e. étale); see [Sta20, Tag 00UV]. Now, let $s$ be a point of $S$. To show that the finite surjective morphism $Z_s\to X_s$ is ramified, let $d\in D$ be a point lying over $s$. Then, by the definition of the branch locus, $Z_d\to \operatorname {Spec} k(d)$ is ramified. Note that $Z_d = Z_s\times _{X_s} d$ as schemes over $d=\operatorname {Spec} k(d)$. As the fibre of $Z_s\to X_s$ over $d$ is ramified, it follows that $Z_s\to X_s$ is ramified.

An unramified cover $X\to Y$ of varieties might not be étale [Reference CutkoskyCut18, Exercise 21.89], but this holds whenever the target is geometrically unibranch [Sta20, Tag 0BQ2], as we now show. (Recall that a scheme $Y$ is *equidimensional* if every irreducible component has the same dimension.)

Lemma 2.2 Let $X$ be a geometrically unibranch integral scheme and let $\pi :Y\to X$ be a finite surjective unramified morphism of schemes with $Y$ equidimensional. Then $\pi$ is étale.

Proof. By [Sta20, Tag 04HJ], there is a surjective étale morphism $f:U\to X$ such that $Y_U :=Y\times _{\pi, X, f} U$ has a finite disjoint union decomposition

such that each $V_j\to U$ is a closed immersion. Refining this decomposition if necessary, we may assume that each $V_j$ is connected. As $X$ is a geometrically unibranch integral scheme and $U\to X$ is a surjective étale morphism, it follows that $U$ is a disjoint union of integral schemes. Indeed, each connected component of $U$ is reduced because $U \to X$ is étale and $X$ is integral, hence reduced. Furthermore, each connected component of $U$ is irreducible, for two irreducible components would meet at some point $u$, and the local ring at $u$ would have two minimal primes, contradicting the fact that $X$ is geometrically unibranch [Sta20, Tag 06DM]. Let $U_i$ be the connected component containing the image of $V_j \to U$. Then the restriction of the closed immersion $V_j\to U$ to the integral scheme $U_i$ is a dominant closed immersion, as $V_j \hookrightarrow Y_U$ is an open and closed immersion, $Y_U \to U$ is finite surjective and $Y_U$ is equidimensional (because $Y$ is equidimensional). It follows that each non-trivial $V_j\to U_i$ is an isomorphism. This implies that $Y_U\to U$ is étale, so that $Y\to X$ is étale by étale descent.

The following consequence is well known and used repeatedly throughout the paper.

Lemma 2.3 Let $X$ be an integral normal noetherian scheme, and let $\pi : Y \to X$ be a finite surjective morphism of integral schemes. Then $\pi$ is either ramified or étale.

### 2.3 Galois closures

Let $\pi : Y \to X$ be a cover of normal varieties over a field $k$ of characteristic zero, and let $G(Y/X)$ be the automorphism group of $Y$ over $X$. The arguments in the proof of [Reference CutkoskyCut18, Proposition 21.67] show the following.

Proposition 2.4 The canonical homomorphism $G(Y/X) \to \operatorname {Aut}(k(Y) / k(X))^{\operatorname {opp}}$ is an isomorphism.

The cover $\pi : Y \to X$ is called *Galois* if $\#G(Y/X) = \deg \pi$. In this case we also write $\operatorname {Gal}(Y/X)$ for $G(Y/X)$. We let $\widehat {Y}\to X$ be the normalization of $X$ in the Galois closure of $k(Y)$ over $k(X)$; we note that the composed cover $\widehat {Y}\to X$ (which we commonly refer to as the *Galois closure of* $\pi : Y\to X$) is Galois. By Proposition 2.4, the morphism $\pi : Y \to X$ is Galois if and only if the field extension $k(Y) / k(X)$ is. Moreover, if $\pi : Y \to X$ is étale, then $\pi$ is Galois if and only if it is Galois in the sense of [Sta20, Tag 03SF].

Definition 2.5 If $\pi : Y \to X$ is Galois with Galois group $G=\mathrm {Gal}(Y/X)$ and $H$ is a subgroup of $G$, we let $Y/H$ be the normalization of $X$ in $k(Y)^H$, where $k(Y)^H$ is the subfield of $k(Y)$ fixed by $H$. Note that $Y/H$ is a normal (integral) variety over $k$, and that one could equivalently describe $Y/H$ as the (geometric) quotient of $Y$ by $H$.

By Galois theory for $k(X)\subset k(Y)$ and Zariski's main theorem that a birational cover of a normal variety is an isomorphism, we have the following geometric version of Galois correspondence.

Proposition 2.6 Assume that $\pi : Y \to X$ is a Galois cover with group $G$. There is a bijection between subgroups $H$ of $G$ and intermediate covers $Y \to Z \to X$ with $Z$ normal and integral. The correspondence is given by $H \mapsto [Y \to Y/H \to X]$.

Remark 2.7 (i) Let $k \subset F \subset F'$ be field extensions such that $\overline {k} \cap F'=k$ (i.e. $\overline {k}$ and $F'$ are linearly disjoint over $k$) and $F'/F$ is finite. Let $\widehat {F'}/F$ be the Galois closure of $F'/F$, let $M/F{\overline {k}}$ be the Galois closure of $F'{\overline {k}}/F{\overline {k}}$ and write $L = \widehat {F'} \cap \overline {k}$. Note that $L$ is a finite extension of $k$. Then $\widehat {F'}$ is also the Galois closure of $F'L$ over $FL$; hence, in particular, we have $\widehat {F'} \otimes _{L} \overline {k} \cong \widehat {F'} \overline {k} = M$.

(ii) Proposition 2.4 and remark (i) have the following immediate consequence. If $W \xrightarrow {\phi } V$ is a cover of varieties over $k$, $\widehat {W} \rightarrow V$ is the Galois closure of $\phi$, and $\operatorname {Spec} L /\operatorname {Spec} k$ is the normalization of $\operatorname {Spec} k$ in $\widehat {W}$, then $\widehat {W} \times _{\operatorname {Spec} L} \operatorname {Spec} \overline {k}$ is the Galois closure of $W \times _k \overline {k} \xrightarrow {\phi } V\times _k \overline {k}$.

Remark 2.8 Let $E'/E$ be a finite separable extension, and $\widehat {E'}/E$ be its Galois closure, of Galois group $G$, and let $H$ be such that $\widehat {E'}^H = E'$. Then $\widehat {E'}\otimes _E E' \cong \oplus _{r \in G/H} \widehat {E'}$ as $\widehat {E'}$-algebras.

The following proposition and its corollary tell us how Galois closure of covers behaves under smooth base change.

Proposition 2.9 Let $F/E$ be a finite separable field extension, let $E'/E$ be a field extension and let $\widehat {F}/F$ be the Galois closure of $F$ over $E$. Assume that $F_{E'}:= E' \otimes _E F$ is a field, and let $\widehat {F_{E'}}$ be the Galois closure of $E' \otimes _E F$ over $E'$. There exists a surjective morphism $E' \otimes _E \widehat {F} \rightarrow \widehat {F_{E'}}$ that restricts to the identity on $E'\otimes _E F$.

Proof. There is a canonical embedding $\iota :F \hookrightarrow E' \otimes _E F \hookrightarrow \widehat {F_{E'}}$ that is the identity on $E$. The field $\widehat {F_{E'}}$ is normal over $E'$, and contains $F$ (through the embedding $\iota$). Hence, there exists an embedding $\widehat {F}\hookrightarrow \widehat {F_{E'}}$ which restricts to the identity on $E$. Consider the morphism $\varphi : E' \otimes _E \widehat {F} \rightarrow \widehat {F_{E'}}$ which, by the universal property of the tensor product, is induced by the embedding $E'\hookrightarrow E' \otimes _E F \hookrightarrow \widehat {F_{E'}}$ and the natural embedding $\widehat {F} \hookrightarrow \widehat {F_{E'}}$. The field $\widehat {F}$ is generated over $E$ by the roots $\alpha _1,\ldots,\alpha _n$ of a separable polynomial with coefficients in $E$, and these same roots also generate $\widehat {F_{E'}}$ over $E'$. As both $E'$ and $\alpha _1,\ldots,\alpha _n$ are contained in the image of $\varphi$, this shows that $\varphi$ is surjective as desired.

Corollary 2.10 Let $Z \rightarrow X$ be a cover of normal varieties over $k$, let $X'\to X$ be a smooth morphism of varieties over $k$, and let $\widehat {Z} \rightarrow Z \rightarrow X$ be the Galois closure of $Z\to X$. Then $Z' := Z \times _X X'$ and $\widehat {Z} \times _X X'$ are normal. Assume that $Z' := Z \times _X X'$ is connected. Then $Z'$ is an integral normal scheme and, if $Z''\rightarrow Z' \rightarrow X'$ is the Galois closure of $Z' \rightarrow X'$, then there is an open and closed embedding $Z'' \rightarrow \widehat {Z} \times _X X'$ that commutes with projection to $Z'$.

Proof. By [Sta20, Tag 034F], the schemes $Z'$ and $\widehat {Z} \times _X X'$ are normal. In particular, if $Z'$ is connected, then it is integral (as it is connected and normal). Define

Note that

The result now follows from Proposition 2.9 and the universal property of normalization.

### 2.4 Action on the fibres

Let $X$ be a normal variety over a field $k$. Let $G$ be a finite group and let $\phi :Y \rightarrow X$ be an étale (right) $G$-torsor over $X$, so that $Y$ is also normal. In this section we discuss various properties of the action of $G$ on the fibres of $\phi$. Let $\overline {y_0}: \operatorname {Spec} \overline {k}\rightarrow Y$ be a geometric point of $Y$ and let $\overline {x_0} = \phi \circ \overline {y_0}$ be the corresponding geometric point of $X$.

##### The left $G$-action

There is a left $G$-action on $Y_{\overline {x_0}}$, defined as follows:

This induces a morphism $\iota _{\overline {y_0}}:G \rightarrow \operatorname {Aut}(Y_{\overline {x_0}})$. It is straightforward to check that $\iota _{\overline {y_0}}(G)$ consists of the group $\operatorname {Aut}_G(Y_{\overline {x_0}})$ of all automorphisms of $Y_{\overline {x_0}}$ that commute with the right $G$-action.

##### Decomposition group

There is a natural left $\Gamma _{k(x_0)}$-action on $Y_{\overline {x_0}}$ given by

where ${{^\gamma }{\overline {y'}}{}}$ denotes the composition $\operatorname {Spec} \overline {k(x_0)} \xrightarrow {\gamma } \operatorname {Spec} \overline {k(x_0)} \xrightarrow {\overline {y'}} Y$. As the $G$-action is defined over $k$, the action (2) commutes with the right $G$-action, hence yields a morphism $\Gamma _{k(x_0)} \rightarrow \operatorname {Aut}_G(Y_{\overline {x_0}})$. Composing with the inverse of the isomorphism $\iota _{\overline {y_0}} : G \to \operatorname {Aut}_G(Y_{\overline {x_0}})$ we get a morphism

called the *decomposition morphism* of $\overline {y_0}$ under $\phi$. The image of $\mathfrak {D}_{\overline {y_0}}$ is called the *decomposition group* of $\overline {y_0}$ and denoted by $D_{\overline {y_0}}$. Finally, if $P$ is an $L$-rational point of $X$ for some field $k \subseteq L \subseteq \overline {k}$, a decomposition group of $P$ under $\phi$ is any subgroup of the form $D_{\overline {y_0}}$ for some geometric point $\overline {y_0}$ of $Y$ whose image in $X(\overline {k})$ is the geometric point corresponding to $P$.

Remark 2.11 Note that the morphism $\mathfrak {D}_{\overline {y_0}}$ is the unique morphism that sends $\gamma \in \Gamma _{k(x_0)}$ to the unique element $g \in G$ such that ${{^\gamma }{\overline {y_0}}{}} = \overline {y_0} \cdot g$.

##### Compatibility with subcovers

Let $Y_1 \xrightarrow {\phi _1} Y_2 \xrightarrow {\phi _2} X$ be finite étale morphisms of $k$-schemes of finite type such that the composition $\phi := \phi _2 \circ \phi _1$ is an étale $G$-torsor. Suppose that $Y_2 = Y_1/H$ for some subgroup $H\subset G$. (By Proposition 2.6, the latter is automatically satisfied if $Y_1$ and $Y_2$ are integral normal schemes.) Again let $\overline {y_0}$ be a geometric point of $Y_1$ and $\overline {x_0}$ be its image in $X$. By our discussion of the left $G$-action, we have a commutative diagram:

where the map in the upper row is the left $G$-action (1).

Observe that $\phi _2^{-1}(\overline {x_0})$ is isomorphic to $G/H$, with the isomorphism preserving the left $G$-action. The commutativity of (3) implies the following lemma.

Lemma 2.12 The fibre $\phi _2^{-1}(\overline {x_0})$ contains no $k$-rational points if and only if the decomposition group $D_{\overline {y_0}}$ acts with no fixed points on $\phi _2^{-1}(\overline {x_0})$. Moreover, $D_{\overline {y_0}}$ acts with no fixed points on $\phi _2^{-1}(\overline {x_0})$ if and only if $D_{\overline {y_0}} \subset G$ acts with no fixed points on $G/H$.

The irreducibility of a fibre of a Galois cover is equivalent to the absence of rational points on the fibres of certain subcovers, as is shown implicitly in [Reference SerreSer08, Proposition 3.3.1].

Proposition 2.13 Let $\pi : Y \to X$ be a Galois cover with group $G$, and let $x \in X(k)$ be a rational point. Suppose that $\pi$ is étale at $x$. The scheme $Y_x$ is reducible over $k$ if and only if there is a subgroup $H \subsetneq G$ such that the fibre of $Y/H \to X$ over $x$ has a $k$-rational point.

##### Specialization

From now on, we assume that $k$ is a number field $K$. Given an étale $G$-torsor $\phi :Y\to X$ as before and a place $v \in M_{K}^{\operatorname {fin}}$, we say that a point $x_0 \in X(K_v)$ is of *good reduction* for $\phi$ if there exists an étale $G$-torsor $\psi :\mathcal {Y}\rightarrow \mathcal {X}$ over $\operatorname {Spec} \mathcal {O}_{K_v}$ extending $\phi$ such that $x_0$ extends to a morphism $\operatorname {Spec} \mathcal {O}_{K_v} \rightarrow \mathcal {X}$, where $\mathcal {O}_{K_v}$ is the ring of integers of $K_v$. We fix a geometric point $\overline {y_0} \in Y$ lying over $x_0$, and have a commutative diagram:

where $\mathcal {O}_{K_v^{\operatorname {ur}}}$ denotes the ring of integers of the maximal unramified extension $K_v^{\operatorname {ur}} \subset \overline {K_v}$ of $K_v$. Diagram (4) induces morphisms:

where the isomorphism $\widehat {\mathbb {Z}} \cong \Gamma _{\mathbb {F}_v}$ sends the topological generator $1 \in \widehat {\mathbb {Z}}$ to the Frobenius $x \mapsto x^{\# \mathbb {F}_v}$.

Proposition 2.14 There exists a morphism $\operatorname {Gal}(K_v^{\operatorname {ur}}/K_v) \to G$ such that the following diagram commutes.

Proof. The fibres of $\psi : \mathcal {Y} \to \mathcal {X}$ over $\psi (\overline {y_0})$ and over $\psi (\overline {y_0}_v)$ are both identified with $G$, hence with each other, in a $G$-equivariant way.

The image of $1 \in \widehat {\mathbb {Z}}$ under $\mathfrak {D}_{{\overline {y_0}_v}}$ is known as the Frobenius element of $\overline {y_0}_v$, and is denoted by $\operatorname {Fr}_{\phi,\overline {y_0}_v}$. We also use the notation $\operatorname {Fr}_{\phi,\overline {y_0}}:= \operatorname {Fr}_{\phi,\overline {y_0}_v}$; note that this is well defined. If $\overline {y}^1_v$ and $\overline {y}^2_v$ lie above the same point $x_v \in \mathcal {X}(\mathbb {F}_v)$, and $\overline {y}^2_v=\overline {y}^1_v \cdot g$, then we have that $\operatorname {Fr}_{\phi,\overline {y}^2_v}=g^{-1}\cdot \operatorname {Fr}_{\phi,\overline {y}^1_v}\cdot g$. In particular, the conjugacy class of $\operatorname {Fr}_{\phi,\overline {y_0}_v}$ depends only on the base point $x_v$. Thus, when there is no risk of confusion, we also use the notation $\operatorname {Fr}_{x_v}$ to indicate the Frobenius element $\operatorname {Fr}_{\phi,\overline {y_0}_v}$ of any geometric point $\overline {y_0}_v$ above $x_v$. If $x_0 \in X(K_v)$ reduces to $x_v$ in $\mathcal {X}({\mathbb {F}_v})$, we also use the notation $\operatorname {Fr}_{x_0}:= \operatorname {Fr}_{x_v}$.

Proposition 2.15 Let $\phi :Y \rightarrow X$ be an étale cover of (normal) varieties over $K$, whose Galois closure $\widehat {Y}\rightarrow X$ has Galois group $G$, and let $H \subset G$ be such that $Y \cong \widehat {Y}/H$ as $X$-covers. Let $v$ be a finite place of $\mathcal {O}_{K}$ and $x_0 \in X(K_v)$ be a point with good reduction for $\phi$. Then, $\phi ^{-1}(x_0)(K_v) \neq \emptyset$ if and only if $\operatorname {Fr}_{{x_0}}$ acts on $G/H$ with at least one fixed point (note that, for $g \in G$, the condition that $g$ acts with at least one fixed point on $G/H$ depends only on the conjugacy class of $g$).

Proof. Let $\overline {x_0}:\operatorname {Spec} \overline {K_v} \to {x_0} \rightarrow X$ be a geometric point lying over $x_0$. We have that $\phi ^{-1}(x_0)(K_v)\neq \emptyset$ if and only if there exists a point in $\phi ^{-1}(\overline {x_0})$ fixed by $\operatorname {Fr}_{{x_0}}$. By Lemma 2.12 and Proposition 2.14, such a point exists if and only if $\operatorname {Fr}_{{x_0}}$ acts with at least one fixed point on $G/H$.

### 2.5 Vertically ramified covers

The following structure result shows that, roughly speaking, a cover of $X\times Y$ which is ‘vertically ramified’ splits as a product, up to a finite étale cover. The fact that such a structure result might be true was first observed after many fruitful discussions between the third-named author and Olivier Wittenberg.

The following structure result is used twice in this paper. First, we use it to prove the product property for varieties with the weak-Hilbert property (Theorem 1.9). We then use it to prove a similar product property of a variant of the Hilbert property for abelian varieties (see Proposition 5.6).

Definition 2.16 Let $X, Y$ be proper smooth varieties over $k$ and $\pi : Z \to X \times Y$ be a ramified cover. We say that $\pi$ is *vertically ramified* over $X$ if there exists a dense open subscheme $U \subset X$ such that $\pi$ is unramified (hence, étale; see Lemma 2.3) over $U \times Y$.

Lemma 2.17 Let $X,Y$ be proper smooth varieties over $k$ and $\pi : Z \to X \times Y$ be a ramified cover. Let $U \subset X$ be a dense open subscheme such that $\pi$ is unramified over $U \times Y$ (so that $\pi$ is vertically ramified over $X$). Assume, furthermore, that the geometric fibres of the composition $Z \to X \times Y \xrightarrow {p_1} X$ are connected and $U(k) \neq \emptyset$. Then there exists a commutative diagram

where:

(1) $X', Y', Z'$ are normal varieties over $k$;

(2) $X' \to X$ is a ramified cover;

(3) $Z' \to Z$ and $Y' \to Y$ are finite étale;

(4) $Z'$ is a connected component of the fibred product $Z \times _Y Y'$; in particular, if $Z \times _Y Y'$ is connected, the upper square is Cartesian;

(5) $Z' \rightarrow X' \rightarrow X$ is the Stein factorization of $Z' \rightarrow X$.

Proof. Let $x\in U(k)$ be a $k$-rational point. Then $Z_x\to \{x\} \times Y$ is a finite étale morphism. Let $Y'\to Z_x\to Y$ be the Galois closure of $Z_x\to Y$, and observe, in particular, that $Y' \to Y$ is finite étale. Let $Z' \subset Z\times _{Y} Y'$ be a connected component of the pull-back of $Z\to Y$ along $Y'\to Y$.

There are natural maps $Z' \to Z \to X \times Y \to X$ and $Z' \to Y'$ which induce a morphism $Z' \to X \times Y'$. Let $Z'\to X'\to X$ be the Stein factorization of the composed morphism $Z'\to X\times Y'\to X$, which (together with the obvious map $Z' \to Y'$) gives a natural morphism $Z'\to X'\times Y'$. We claim that this map is an isomorphism.

To prove this, note that $Z'$ is normal, as $Z'\subset Z \times _Y Y'\to Z$ is finite étale and $Z$ is normal. Moreover, the morphism $Z'\to X'\times Y'$ is finite and surjective. By [Reference GrothendieckGro71, Proposition X.1.2], the Stein factorization of $Z'\to X$ is étale over $U\subset X$. By [Reference GrothendieckGro63, Corollaire 7.8.7], over the étale locus the Stein factorization commutes with taking fibres, so the Stein factorization of $Z'_x\to \operatorname {Spec} k(x)$ is given by $\operatorname {Spec} \Gamma (Z'_x, \mathcal {O}_{Z'_x})$. As $Z'_{x} = Z_{x} \times _Y Y'$ is a disjoint union of copies of $Y'$ (as it follows from Remark 2.8 and a standard normalization argument, noting that $Y'\to Z_{x} \to Y$ is the Galois closure of $Z_x \to Y$), we see that $Z'_x \to X'_x \times Y'$ is an isomorphism (as it is a finite surjective morphism between the same number of copies of $Y'$). It follows that $Z'\to X'\times Y'$ is an isomorphism over a dense open subset, hence it is a birational morphism. As $Z'$ and $X'\times Y'$ are integral normal varieties over $k$, it follows as claimed that $Z'\to X'\times Y'$ is an isomorphism by Zariski's main theorem (see [Reference GrothendieckGro61, Corollaire 4.4.9]).

We have thus constructed the desired diagram and shown parts (1) and (3). Parts (4) and (5) are true by construction. As for part (2), we already know that $X' \to X$ is finite (it arises as the finite part of the Stein factorization of $Z' \to X$) and surjective because $Z' \to X$ is. It remains to show that $X' \to X$ is ramified; if it were not, $Z' \to X \times Y' \to X \times Y$ would be étale, hence also $Z' \to Z \to X \times Y$ would be étale. As we already know that $Z' \to Z$ is surjective and étale, by the cancellation property for étale morphisms we would get that $\pi : Z \to X \times Y$ is also étale, a contradiction.

## 3. The weak-Hilbert property

Throughout this section, we let $k$ be a field of characteristic zero, unless otherwise specified. The goal of this section is to prove that the class of varieties with the (potential) weak-Hilbert property (Definition 1.2) has several features in common with Campana's class of special varieties [Reference CampanaCam11].

We begin by showing that the weak-Hilbert property is a birational invariant among smooth proper geometrically connected varieties.

### Proposition 3.1 (Birational invariance)

Let $X$ and $X'$ be smooth proper geometrically connected varieties over $k$. Suppose that $X$ and $X'$ are birational over $k$. Then $X$ has the weak-Hilbert property over $k$ if and only if $X'$ has the weak-Hilbert property over $k$.

Proof. We denote by $\operatorname {Cov}(X)$ (respectively, $\operatorname {Cov}(X')$) the category of covers of $X$ (respectively, $X'$).

As $X$ and $X'$ are birational over $k$, we may choose:

(i) a dense open subscheme $U$ of $X$ with $\mathrm {codim}_X(X\setminus U)\geq 2$;

(ii) a dense open subscheme $U'$ of $X'$ with $\mathrm {codim}_{X'}(X'\setminus U')\geq 2$; and

(iii) an isomorphism $\sigma : U' \to U$ with inverse $\sigma ' : U \to U'$.

We let $\eta _X, \eta _{X'}$ be the generic points of $X, X'$, and we denote the isomorphism $\sigma '|_{\eta _X}:\eta _X \rightarrow \eta _{X'}$ as $\iota$.

We define the functor $N\sigma ^*:\operatorname {Cov}(X) \rightarrow \operatorname {Cov}(X')$ (respectively, $N(\sigma ')^*:\operatorname {Cov}(X') \rightarrow \operatorname {Cov}(X)$) as sending a cover $Y \to X$ to the relative normalization $Y' \to X'$ of $X'$ in the cover $\iota _*( Y|_{\eta _X}) \rightarrow \eta _{X'}$ (respectively, in the cover $\iota ^*( Y|_{\eta _X}) \rightarrow \eta _{X'}$).

Clearly, $N(\sigma ')^*$ and $N\sigma ^*$ are inverse natural equivalences. We claim that these functors send étale covers to étale covers.

Let $Y \to X$ be an étale cover. We then have that $\sigma ^*Y\rightarrow U'$ is étale as well. As $U'$ is normal, it follows that $\sigma ^*Y$ is normal as well. Hence, $((N\sigma ^*)Y)|_{U'}\cong \sigma ^*Y$ as $U'$-schemes. In particular, $(N\sigma ^*)Y \rightarrow X'$ is finite and étale over the complement of a codimension-two closed subscheme of the base. As $X'$ is smooth and $(N\sigma ^*)Y$ is normal, by Zariski–Nagata purity [Reference GrothendieckGro71, Théorème X.3.1], $(N\sigma ^*)Y \rightarrow X'$ is étale, which concludes the proof of the claim (the proof for $N(\sigma ')^*$ being analogous).

Therefore, because $N(\sigma ')^*$ and $N\sigma ^*$ are natural inverses, Lemma 2.3 implies that ramified covers $\pi : Y \to X$ give rise to *ramified* covers of $X'$ (through $N\sigma ^*$), and conversely (through $N(\sigma ')^*$), which immediately implies that $X$ has the weak-Hilbert property if and only if $X'$ does.

Remark 3.2 Let $X$ be a smooth proper variety over $k$ with the weak-Hilbert property over $k$, let $U$ be a dense open of $X$. Then, for every finite collection of ramified covers $(Z_i \xrightarrow {\pi _i} U)_i$, we have that $U(k) \setminus \bigcup _i \pi _i(Z_i(k))$ is Zariski-dense in $U$. Indeed, letting $Z'_i$ be the normalization of $X$ in $Z_i$, this becomes an immediate consequence of applying the definition of the weak-Hilbert property to $X$ and the family of ramified covers $(Z'_i \xrightarrow {\pi '_i} X)_i$.

Remark 3.3 If $Y \xrightarrow {\pi } X$ is a cover of varieties over $k$ and $Y$ is not geometrically connected, then $Y(k)= \emptyset$, so that $\pi (Y(k))=\emptyset$. In particular, when studying the weak-Hilbert property for $X$, one may always restrict to covers $Y\to X$ with $Y$ geometrically connected (hence, geometrically integral).

### 3.1 Images of varieties with the weak-Hilbert property

In this subsection we show that, under suitable assumptions on a map of varieties $X \to Y$, if the variety $X$ has the weak-Hilbert property then so does $Y$.

#### Proposition 3.4 (Finite étale images)

Let $X\xrightarrow {\phi } Y$ be a finite étale morphism of smooth proper varieties over $k$. If $X$ has the weak-Hilbert property over $k$, then $Y$ has the weak-Hilbert property over $k$.

Proof. As $X$ and $Y$ are integral, it follows that $\phi$ is surjective. Let $( \pi _i : Z_i \to Y)_{i=1,\ldots,n}$ be a finite collection of ramified covers over $k$. By base change, we obtain finite surjective morphisms $\varphi _i : Z_i \times _Y X \to X$.

Let $Z'$ be a connected component of $Z_i \times _Y X$. As $Z'\subset Z_i\times _Y X$ is open and closed, the morphism $Z'\to Z_i$ is finite étale, hence surjective (by connectivity of $Z'$ and $Z_i$). It follows that $\dim Z'=\dim Z_i = \dim Y = \dim X$. In particular, because $\varphi _i|_{Z'}:Z'\to X$ is finite, it is also surjective. Furthermore, the morphism $\varphi _i|_{Z'}:Z'\to X$ is ramified. Indeed, because $X$ is smooth, if $Z' \to X$ is unramified, then it is étale (Lemma 2.3). It follows that, as $Z'\to X$ is étale and $X\to Y$ is étale, the composition $Z' \to X \to Y$ is étale, which by the surjectivity of $Z' \to Z_i$ implies that $Z_i \to Y$ is étale, contradicting our assumption that it is ramified.

Let $Z'_{ij}$ be the connected components of $Z_i \times _Y X$. We may apply the weak-Hilbert property of $X$ to the ramified covers $\varphi _{i}|_{Z'_{ij}} : Z'_{ij} \to X$ to see that $X(k) \setminus \bigcup _{i,j} \varphi _i(Z'_{ij}(k))$ is dense in $X$. Applying the surjective morphism $\phi$ to this dense set of rational points we get a dense set of rational points on $Y$. We claim that this dense set is contained in $Y(k)\setminus \bigcup _i \pi _i(Z_i(k))$. Indeed, suppose by contradiction that there is a point $Q \in X(k)\setminus \bigcup _{i,j} \varphi _i(Z'_{ij}(k))$ such that $\phi (Q)\in \pi _i(Z_i(k))$ for some $i$. Then, by the universal property of the product $Z_i \times _Y X$, the point $Q$ belongs to $\varphi _i((Z_i \times _Y X) (k)) \subset \bigcup _{i,j} \varphi _i(Z'_{ij}(k))$, contradicting our assumption.

Remark 3.5 The converse to Proposition 3.4 is false. Indeed, let $E$ be an elliptic curve over $\mathbb {Q}$ with $E(\mathbb {Q})$ dense, and let $E'\to E$ be a finite étale morphism with $E'(\mathbb {Q})=\emptyset$; such data are easily seen to exist. Then $E$ has the weak-Hilbert property over $\mathbb {Q}$ by Faltings’ theorem, whereas $E'$ has no $\mathbb {Q}$-points and, thus, does not have the weak-Hilbert property over $\mathbb {Q}$. However, there is a number field $K$ such that $E_K'$ has the weak-Hilbert property over $K$. More generally, assuming $k$ is a finitely generated field of characteristic zero, we show a converse to Proposition 3.4 in which we allow an extension of the base field; see Theorem 3.16 for a precise statement.

A surjective morphism $f:X\to Y$ of varieties over $k$ is said to have *no multiple fibres in codimension one* if, for every point $y$ in $Y$ of codimension one, the scheme-theoretic fibre $X_y$ has an irreducible component which is reduced.

#### Proposition 3.6 (Fibrations with no multiple fibres)

Let $X\xrightarrow {\phi } Y$ be a surjective morphism of smooth proper varieties over $k$ with geometrically connected generic fibre and no multiple fibres in codimension one. If $X$ has the weak-Hilbert property over $k$, then $Y$ has the weak-Hilbert property over $k$.

Proof. Let $( \pi _i : Z_i \to Y)_{i=1,\ldots,n}$ be a finite collection of ramified covers of $Y$ over $k$. To prove the statement, we have to show that $Y(k) \setminus \bigcup _i \pi _i(Z_i(k))$ is Zariski-dense in $Y$. Let $V \subset X$ be the smooth locus of $\phi$, and let $U$ be the image $\phi (V)$. Note that $V$ is a dense open subscheme of $X$, so that $U \subset Y$ is also an open subscheme. Let $\psi :V\to U$ denote the induced (smooth) morphism. As $\phi$ has no multiple fibres in codimension one, the complement of $U$ in $Y$ is of codimension at least two. As $V\xrightarrow {\phi |_{V}}Y$ is smooth, the morphism $Z_i\times _Y V\rightarrow Z_i$ is smooth. In particular, by [Sta20, Tag 034F] and the normality of $Z_i$, the scheme $W_i:=Z_i\times _Y V$ is normal.

We claim that $W_i$ is integral. Since $k$ is of characteristic zero and $X$ is smooth, the generic fibre $X_{ {k(Y)}}$ of $\phi :X\to Y$ is smooth. As $X_{ {k(Y)}}$ is geometrically connected (by assumption), $X_{ {k(Y)}}$ is geometrically integral. It follows that the same is true for the generic fibre of $V \to Y$. In particular, the generic fibre of $W_i=Z_i\times _Y V\to Z_i$, being the base change of the generic fibre of $V \to Y$, is geometrically integral as well. Applying [Reference LiuLiu06, Proposition 3.8] to $(W_i)_{\overline {k}}\to (Z_i)_{\overline {k}}$, we deduce that $W_i$ is integral.

Define $\pi '_i:W_i \rightarrow V$ to be the natural projection. We claim that $\pi '_i$ is ramified. In fact, assume by contradiction that it is not. We have the following commutative diagram, which is Cartesian in the upper left corner.

As $\pi '_i$ and $\psi$ are both smooth, so is the morphism $W_i \to U$. Moreover, $W_i \to Z_i|_{U}$ is smooth and surjective (being a base change of the smooth surjective morphism $\psi$). Hence, applying [Sta20, Tag 02K5] to $(f,q,p)=(W_i \to Z_i|_{U}, \pi _i|_{U}, W_i \to U)$, we obtain that $\pi _i|_{U}$ is smooth, hence étale, which by the Zariski–Nagata theorem implies that $\pi _i$ is étale: contradiction.

As $X$ has the weak-Hilbert property, by Remark 3.2, the set $S:=V(k)\setminus \bigcup _{i=1}^n \pi '_i(W_i(k))$ is dense in $X$. By the universal property of the fibred product $W_i=Z_i\times _Y V$, following the same argument used at the end of the proof of Proposition 3.4, we conclude that $\psi (S) \subset Y(k) \setminus \bigcup _i \pi _i(Z_i(k))$ is Zariski-dense in $Y$, as required.

#### Theorem 3.7 (Smooth images)

Let $X\xrightarrow {\phi } Y$ be a smooth proper morphism of smooth proper varieties over $k$. If $X$ has the weak-Hilbert property over $k$, then $Y$ has the weak-Hilbert property over $k$.

Proof. Let $X\to X'\to Y$ be the Stein factorization of $\phi :X\to Y$. Note that $X\to X'$ is a smooth proper morphism and that $X'\to Y$ is finite étale. In particular, because $X\to X'$ has no multiple fibres in codimension one, it follows that $X'$ has the weak-Hilbert property over $k$ by Proposition 3.6. Then, as $X'$ has the weak-Hilbert property over $k$ and $X'\to Y$ is a finite étale morphism of smooth proper varieties over $k$, we conclude that $Y$ has the weak-Hilbert property over $k$ (Proposition 3.4), as required.

### 3.2 Arithmetic refinements

An *arithmetic refinement of a variety* $Z$ *over* $k$ is the data of a finite index set $J$ and, for every $j$ in $J$, a cover $\psi _j:W_j\to Z$ with $W_j$ normal and geometrically integral, with the property that $Z(k)\subset \bigcup _{j\in J}\psi _j(W_j(k))$. When ‘testing’ the weak-Hilbert property for a variety $X$, one can replace a given cover by arithmetic refinements. Let us be more precise.

Let $n\geq 1$ be an integer and, for $i=1,\ldots, n$, let $\pi _i:Z_i\to X$ be a ramified cover of normal projective geometrically integral varieties over $k$. For every $i$, let $J_i$ be a finite set. For $i=1,\ldots, n$ and $j$ in $J_i$, let $\psi _{i,j}:W_{ij}\to Z_j$ be a cover. Assume that, for $i=1,\ldots, n$, the collection $\{\psi _{i,j}:W_{ij}\to Z_i\}_{j\in J_i}$ is an arithmetic refinement of $Z_i$. Then

Thus, when ‘testing’ the weak-Hilbert property for a variety (Definition 1.2), one may replace each cover $Z_i\to X$ in a given finite collection of covers $(Z_i\to X)_i$ by an arithmetic refinement of $Z_i$.

### 3.3 Chevalley–Weil for finitely generated fields of characteristic zero

Let $f:X\to Y$ be a finite étale surjective morphism of proper schemes over a field $k$. When $k$ is a number field, a classical result (which goes back to Chevalley and Weil [Reference Chevalley and WeilCW32]) shows that there exists a finite extension $L$ of $k$ such that every $k$-rational point of $Y$ lifts to an $L$-rational point of $X$. The same statement holds when $k$ is a finitely generated field of characteristic zero; because we were unable to find this precise statement in the literature, we include a short proof.

Theorem 3.8 If $f : X \to Y$ is a finite étale surjective morphism of proper schemes over a finitely generated field $k$ of characteristic zero, then there is a finite field extension $L/k$ such that $Y(k)\subset f(X(L))$.

Proof. By standard spreading out arguments, we may choose a regular $\mathbb {Z}$-finitely generated integral domain $A\subset k$, a proper model $\mathcal {X}$ for $X$ over $A$, a proper model $\mathcal {Y}$ for $Y$ over $A$, and a finite étale surjective morphism $F:\mathcal {X}\to \mathcal {Y}$ extending $f$. By properness of $\mathcal {Y}$ over $A$, for every $k$-point $y:\operatorname {Spec} k\to Y$, there exist a dense open subscheme $U_y\subset \operatorname {Spec} A$ whose complement in $\operatorname {Spec} A$ is of codimension at least two and a morphism $U_y\to \mathcal {Y}$ extending the morphism $y:\operatorname {Spec} k\to Y$. Pulling back $U_y\to \mathcal {Y}$ along $F:\mathcal {X}\to \mathcal {Y}$, we obtain a finite étale surjective morphism $V_y:= U_y\times _{\mathcal {Y}} \mathcal {X}\to U_y$ of degree $\deg (f)$. By purity of the branch locus [Reference GrothendieckGro71, Théorème X.3.1], the finite étale morphism $V_y\to U_y$ extends to a finite étale morphism $\overline {V_y}\to \operatorname {Spec} A$. Since the set of isomorphism classes of finite étale covers of $\operatorname {Spec} A$ with bounded degree is finite by the Hermite–Minkowski theorem for arithmetic schemes [Reference Harada and HiranouchiHH09], the set of isomorphism classes of the $\overline {V_y}$ (and, thus, $V_y$) appearing above (with $y\in Y(k)$) is finite. In particular, we may choose a finite field extension $L/k$ such that, for all $k$-points $y:\operatorname {Spec} k\to Y$ and every connected (hence, irreducible) component $V'$ of $V_y$, the function field $K(V')$ of $V'$ is contained in $L$. This readily implies that $Y(k)\subset f(X(L))$, as required.

### 3.4 Applying the Chevalley–Weil theorem

Let $X$ and $Y$ be smooth projective geometrically connected varieties over $k$. We stress that projectivity is only assumed for technical reasons, as it is used to ensure the existence of certain Weil restrictions in the proof of Theorem 3.9.

Let $\pi :Z\to X\times Y$ be a cover which is vertically ramified (see Definition 2.16) over $X$. We apply the Chevalley–Weil theorem (Theorem 3.8) and use Weil's restriction of scalars to construct suitable arithmetic refinements of the vertically ramified morphism $Z\to X\times Y$. The precise statement we prove reads as follows.

#### Theorem 3.9 (Arithmetic refinement)

Let $k$ be a finitely generated field of characteristic zero. Let $X$ and $Y$ be smooth projective geometrically connected varieties over $k$. Let $\pi :Z\to X\times Y$ be a cover which is vertically ramified over $X$. Assume that $Z(k)$ is dense in $Z$, so that, in particular, $X(k)$ and $Y(k)$ are dense in $X$ and $Y$, respectively. There exists an arithmetic refinement $\{f_j:W_j\to Z\}_{j\in J}$ of $Z$ such that, for every $j$ in $J$, the Stein factorization of $W_j\to X\times Y\to X$ is ramified over $X$.

Proof. Let $Z\to S\to X$ be the Stein factorization of the composed map $Z\to X\times Y\to X$ and note that $S$ is a normal geometrically integral variety over $k$. If $S\to X$ is ramified, there is no need to replace $Z\to X\times Y$ by an arithmetic refinement, and we are done. Thus, we may and do assume that $S\to X$ is unramified, hence étale (Lemma 2.3). Note that the finite surjective morphism $Z\to X\times Y$ factors over a finite surjective morphism $Z\to S\times Y$ which is vertically ramified with respect to $S\times Y\to S$ (note that $S \rightarrow X$ is étale). As the geometric fibres of the composed morphism $Z\to S\times Y\to S$ are connected and $S(k)$ is dense, it follows from Lemma 2.17 that there is the following commutative diagram.

Here $S' \to S$ is a ramified cover, $Y' \to Y$ is finite étale and $\psi :Z'\cong S'\times Y'\to Z$ is finite étale. In particular, because $k$ is a finitely generated field of characteristic zero, it follows from the Chevalley–Weil theorem (Theorem 3.8) that there is a finite field extension $L/k$ such that $Z(k)\subset \psi (Z'(L))$.

Consider the induced morphism of Weil restriction of scalars $R_{L/k}(\psi _L):R_{L/k} Z'_L \to R_{L/k} Z_L$; see [Reference Bosch, Lütkebohmert and RaynaudBLR90, Chapter 7.6]. Let $\Delta :Z\to R_{L/k} Z_L$ be the diagonal morphism. Let $(W_j)_{j\in J}$ be the connected components of $Z\times _{\Delta, R_{L/k} Z_L} R_{L/k} Z'_L$ such that $W_j(k)\neq \emptyset$ and let $f_j:W_j\to Z$ be the natural morphism. As $W_j$ is normal connected and with a $k$-rational point, it is geometrically integral. Moreover, note that, by the defining properties of the Weil restriction of scalars, the finite set of covers $\{f_j:W_j\to Z\}_{j\in J}$ is an arithmetic refinement of $Z$.

After base change to $\overline {k}$, the fibre product $Z\times _{\Delta, R_{L/k} Z_L} R_{L/k} Z'_L$ is given by $Z'_{\overline {k}}\times _{Z_{\overline {k}}} \ldots \times _{Z_{\overline {k}}} Z'_{\overline {k}}$. Hence, the morphism $(W_j)_{\overline {k}} \rightarrow X_{\overline {k}}$ factors as $(W_j)_{\overline {k}} \rightarrow Z'_{\overline {k}} \rightarrow Z_{\overline {k}} \rightarrow X_{\overline {k}}$, with the morphism $(W_j)_{\overline {k}} \rightarrow Z'_{\overline {k}}$ being finite étale. From this it follows that the Stein factorization of $(W_j)_{\overline {k}}\to X_{\overline {k}}$ factors over the Stein factorization of $Z'_{\overline {k}} = S'_{\overline {k}}\times Y'_{\overline {k}}\to X_{\overline {k}}$ which implies that the Stein factorization of $(W_j)_{\overline {k}}\to X_{\overline {k}}$ is ramified over $X_{\overline {k}}$. As Stein factorization commutes with flat base change (in particular, base change to $\overline {k}$ in our case), we deduce that the Stein factorization of $W_j\to X$ is ramified over $X$, as required.

#### Example 3.10 (Wittenberg)

It can happen that the geometric fibres of $Z\to X$ are connected. Indeed, let $E'\to E$ be a degree-two isogeny between elliptic curves over $k$, and let $G$ be its kernel. Let $G$ act on $E'$ by translation and on $\mathbb {P}_{1,k}$ by the involution $x\mapsto -x$. Define

The natural morphism $Z\to X\times Y$ is a finite surjective ramified morphism of degree two which is vertically ramified with respect to $X\times Y\to X$. In fact, the branch locus of $Z\to X\times Y$ lies over precisely two points of $X$. Note that the geometric fibres of $Z\to X\times Y\to X$ are connected, so that the Stein factorization of $Z\to X$ is étale (even an isomorphism) over $X$. Define $Z' = E'\times \mathbb {P}_{1,k}$. Then $Z'\to Z$ is a finite étale morphism (as the action of $G$ on $E'\times \mathbb {P}_{1,k}$ is free), and the Stein factorization of the composed morphism $Z'\to Z\to X\times Y\to X$ is given by $Z\to X'\to X$. Note that $X'\to X$ is ramified (as the action of $G$ on $X'$ is not free). This example shows that it is necessary to pass to a finite étale cover of $Z$ to guarantee that the Stein factorization is ramified over the base.

### 3.5 Products

We are now ready to show that a product of varieties satisfying the weak-Hilbert property over a finitely generated field $k$ of characteristic zero has the weak-Hilbert property over $k$.

Proof of Theorem 1.9 Let $X$ and $Y$ be smooth proper varieties over $k$ with the weak-Hilbert property over $k$. We may assume, without loss of generality, by Chow's lemma and Proposition 3.1, that $X$ and $Y$ are projective (so that we may appeal to Theorem 3.9). We aim to show that $X\times Y$ has the weak-Hilbert property over $k$. For $i=1,\ldots, n$, let $\pi _i:Z_i\to X\times Y$ be a ramified cover with $Z_i$ a normal proper geometrically integral variety over $k$ (we remind the reader of Remark 3.3). Let $\psi _i:S_i\to X$ denote the Stein factorization of $Z_i\to X\times Y\to X$, so that $Z_i\to S_i$ has geometrically connected fibres. By Theorem 3.9 we may replace each vertically ramified cover $Z_i \rightarrow X \times Y$ with an arithmetic refinement