Proof This follows from [Reference Bădescu2, Theorems 8.6 and 8.10 and Lemma 8.7].

Proof We will prove this in three steps: first, when k is an algebraically closed field; second, when k is a separably closed field; and finally, in the case of a general field.

In the case when $k = \overline {k}$ , then it follows from [Reference Bădescu2, Section 10.24]. For future convenience, we will outline the proof. Let $g\colon X \rightarrow \mathbb {P}^1$ be an elliptic fibration given in Proposition 2.1. Let $S \subset \mathbb {P}^1$ be the image of the non-smooth locus of g, which is a closed subscheme of dimension $0$ . Then we can define the action

$$\begin{align*}A^{\prime} \times g^{-1}(\mathbb{P}^{1} \setminus S) \rightarrow g^{-1}(\mathbb{P}^{1} \setminus S) \end{align*}$$

as follows. For any k-algebra R, take $P \in g^{-1}(\mathbb {P}^{1}\setminus S) (R).$ Then we have the abelian scheme

$$\begin{align*}T:= g^{-1} (g(P)) = g^{-1} (\mathbb{P}^{1} \setminus S) \times_{\mathbb{P}^{1} \setminus S} g(P) \end{align*}$$

over $\mathop {\mathrm {Spec}} R$ with a section P, and we have a morphism $T \rightarrow A^{\prime}_{R}$ coming from the composition

$$\begin{align*}T \rightarrow g^{-1}(\mathbb{P}^{1}\setminus S) \hookrightarrow X \rightarrow A^{\prime}. \end{align*}$$

Then we have

$$\begin{align*}{A^{\prime}}_{R} \simeq {A^{\prime}}_{R}^{\vee} \rightarrow T^{\vee}, \end{align*}$$

and the right-hand side acts on T canonically. Therefore we get the desired action.

By the minimality of X (see [Reference Bădescu2, Theorem 10.21]), one can extend the above action to $\sigma \colon A^{\prime} \times X \rightarrow X$ (since $A^{\prime}$ is an elliptic curve, it is enough to extend the action of the generic point of $A^{\prime}$ ). Let n be the intersection number of a fiber of $\mathrm {alb}$ and a fiber of g.

Then we have a diagram

(2.1)

where $\phi (a, a'):= na + a'$ . This diagram is commutative since it is commutative over $\overline {k}$ as in [Reference Bădescu2, Section 10.24]. By the above diagram, $B := \mathrm {alb}^{-1}(0)$ is stable under the action of $G^{\prime} := A^{\prime}[n]$ given by $\sigma $ . Therefore we have the morphism $G^{\prime} \rightarrow \text {Aut}_{B/k}$ corresponding to $\sigma \colon G^{\prime} \times B \rightarrow B$ . Moreover, we get the desired isomorphism $(A^{\prime} \times B) /G^{\prime} \simeq X$ via the action $\sigma $ since it induces an isomorphism over $\overline {k}$ . We put

$$ \begin{align*} A &:= A^{\prime}/ \ker(G^{\prime} \rightarrow \text{Aut}_{B/k}), \\ G &:= G^{\prime}/ \ker(G^{\prime} \rightarrow \text{Aut}_{B/k}) \end{align*} $$

so that $G \rightarrow \text {Aut}_{B/k}$ is injective. These $A, B,$ and G satisfy the desired conditions, and it finishes the proof for the case where $k= \overline {k}$ .

Next, we treat the case where k is separably closed. Then by the argument in the case where $k=\overline {k}$ , we have a finite étale morphism $\pi \colon A^{\prime}_{\overline {k}} \times B_{\overline {k}}\rightarrow X_{\overline {k}}$ which is the quotient map by an action of $G^{\prime}_{\overline {k}}$ , where $G^{\prime}:= A^{\prime}[n]$ for some $n\in \{2,3,4,6,8,9\}$ . We shall prove that this morphism is defined over k. The morphism $\pi $ can be decomposed as

Here, $\pi _{\ast } \mathcal {O}_{A^{\prime}_{\overline {k}}\times B_{\overline {k}}}$ is decomposed as $\oplus _{\chi \in \widehat {G^{\prime}(\overline {k})}} \mathcal {F}_{\chi }$ , where $\mathcal {F}_{\chi }$ is a line bundle on $X_{\overline {k}}$ associated with a character $\chi \in \widehat {G^{\prime}(\overline {k})}$ . Since the n-multiplication map on the Picard scheme $\mathrm {Pic}_{X/k}$ is an étale morphism, each $\mathcal {F}_{\chi }$ descends to a line bundle on X, hence the morphism $\pi _{2}$ comes from the morphism

$$\begin{align*}\pi_{2,0} \colon Y:= \underline{\mathop{\mathrm{Spec}}}_{X}\oplus_{\chi \in \widehat{G^{\prime}(\overline{k})}}\mathcal{F}_{\chi} \rightarrow X \end{align*}$$

over k. Since $\pi _{2,0}$ is a finite étale morphism, the fiber over x consists of k-rational points. In particular, $\pi _{1}(0,0)$ descends to a k-rational point of Y. We can equip Y with the structure of abelian variety over k with the zero section $\pi _{1}(0,0)$ by the Albanese morphism. Therefore the morphism $\pi _{1}$ is a homomorphism over $\overline {k}$ between abelian varieties defined over k. Since the homomorphism scheme of abelian varieties is étale [Reference Edixhoven, Van der Geer and Moonen7, Proposition 7.14], the morphism $\pi _{1}$ is also defined over k. Hence our $\pi $ is defined over k, i.e., there exists a morphism $A^{\prime} \times B \rightarrow X$ . Through this morphism, we have an isomorphism $(A^{\prime}\times B)/G^{\prime}\rightarrow X$ . This isomorphism induces the desired isomorphism $ (A \times B)/G \rightarrow X, $ where

$$ \begin{align*} A &:= A^{\prime}/\ker (G^{\prime} \rightarrow \text{Aut}_{B/k}), \\ G &:= G^{\prime}/\ker (G^{\prime} \rightarrow \text{Aut}_{B/k}). \end{align*} $$

Finally, we show the general case. Note that, we have an elliptic fibration

$$\begin{align*}g' \colon X_{k^{\mathrm{sep}}} \rightarrow B_{k^{\mathrm{sep}}}/G_{k^{\mathrm{sep}}} \simeq \mathbb{P}^{1}_{k^{\mathrm{sep}}}. \end{align*}$$

Then by the argument in [Reference Creutz and Viray6, Proposition 5.6], we obtain an elliptic fibration

$$\begin{align*}g \colon X \rightarrow \mathbb{P}^{1} \end{align*}$$

over k such that $g_{k^{\mathrm {sep}}}$ is isomorphic to $g'$ . Using this g, we can prove the theorem in exactly the same way as in the case where $k = \overline {k}$ . It finishes the proof.

Proof $(3) \Rightarrow (2) \Rightarrow (1)$ is clear. We shall show $(1)\Rightarrow (3)$ . Let $\mathcal {X}$ be a smooth proper algebraic space $\mathcal {O}_{K}$ -model of X. Let $\widehat {\mathcal {X}}$ be a formal completion of $\mathcal {X}_{\widehat {\mathcal {O}_{K}}}$ along its special fiber $\mathcal {X}_{k}$ , where $\widehat {\mathcal {O}_{K}}$ is a completion of $\mathcal {O}_{K}$ . We note that $\mathcal {X}_{k}$ is a bielliptic surface over k. Indeed, since $\mathcal {X}_{k}$ is a principal Cartier divisor on $\mathcal {X}$ , we have $\omega ^{\otimes m}_{\mathcal {X}/\mathcal {O}_{K}} \simeq \mathcal {O}_{\mathcal {X}}$ , where m is the order of $\omega _{X/K}$ . Therefore $\omega _{\mathcal {X}_{k}/k}^{\otimes m}$ is trivial, and $\mathcal {X}_{k}$ is a minimal surface of Kodaira dimension $0$ with the same Betti numbers as X. Hence $\mathcal {X}_{k}$ is a bielliptic surface, and we have $H^{2}(\mathcal {X}_{k}, \mathcal {O}_{\mathcal {X}_{k}})=0$ . By [Reference Fantechi, Göttsche, Illusie, Kleiman, Nitsure and Vistoli9, Corollary 8.5.5], one can lift an ample line bundle $L_{k}$ on $\mathcal {X}_{k}$ to $\widehat {L}$ on $\widehat {\mathcal {X}}$ . By Grothendieck’s existence theorem [Reference Knutson18, Theorem 6.3], the line bundle $\widehat {L}$ also lifts to L on $\mathcal {X}_{\widehat {\mathcal {O}_{K}}}$ as an invertible sheaf. The categorical equivalence of Grothendieck’s existence theorem implies that for any coherent sheaf F on $\mathcal {X}_{\widehat {\mathcal {O}_{K}}}$ , $F\otimes L^{\otimes n}$ is globally generated for sufficiently large n. This also holds over a finite scheme cover of $\mathcal {X}_{\widehat {\mathcal {O}_{K}}}$ , hence L is an ample line bundle. Therefore, we have a smooth projective scheme model $\mathcal {X}_{\widehat {\mathcal {O}_{K}}}$ over $\widehat {\mathcal {O}_{K}}$ . Let $\phi \colon \omega _{\mathcal {X}_{\widehat {\mathcal {O}_{K}}}/\widehat {\mathcal {O}_{K}}}^{\otimes m}\simeq \mathcal {O}_{\mathcal {X}_{\widehat {\mathcal {O}_{K}}}}$ be the base change of the isomorphism $\omega ^{\otimes m}_{\mathcal {X}/\mathcal {O}_{K}} \simeq \mathcal {O}_{\mathcal {X}}$ . We take a cyclic covering $\mathcal {Y}$ associated with $\phi $ , i.e.,

$$\begin{align*}Y := \underline{\mathop{\mathrm{Spec}}}_{\mathcal{X}_{\widehat{\mathcal{O}_{K}}}} \bigoplus_{r=0}^{m-1} \omega_{\mathcal{X}_{\widehat{\mathcal{O}_{K}}/\mathcal{O}_{K}}}^{\otimes r}, \end{align*}$$

where the module on the right-hand side is equipped with the algebra structure given by $\phi $ . Since m is $2, 3, 4,$ or $6$ (see [Reference Bombieri, Mumford, Baily and Shioda3, p. 37]), $\mathcal {Y}$ is finite étale over $\mathcal {X}_{\widehat {\mathcal {O}_{K}}}$ . The base change $Y_{\overline {\widehat {K}}}$ of the generic fiber $Y := \mathcal {Y}_{\widehat {K}}$ is an abelian surface since its canonical sheaf is trivial and its first Betti number is not equal to zero. Moreover, by Hensel’s lemma, there exists a finite unramified extension $L/ \widehat {K}$ such that $Y_{L}$ admits an L-rational point, i.e., $Y_{L}$ has a structure of an abelian variety. Therefore, by [Reference Bosch, Lütkebohmert and Raynaud4, Proposition 1.4.2], the $\mathcal {O}_L$ -model $\mathcal {Y}_{\mathcal {O}_{L}}$ satisfies the Néron mapping property, and $\mathcal {Y}$ also satisfies the Néron mapping property by a descent argument. Using [Reference Bosch, Lütkebohmert and Raynaud4, Theorem 7.2.1], the smooth proper Néron model $\mathcal {Y}$ descends to the smooth proper Néron model $\mathcal {Y}_{0}$ over $\mathcal {O}_{K}$ whose generic fiber is $Y_{0}$ which is the cyclic covering of X associated with $\phi _{0}$ . By the Néron mapping property, the cyclic action on $Y_{0}$ can be extended to that on $\mathcal {Y}_{0}$ , and this action is free since its base change to $\widehat {\mathcal {O}_{K}}$ is the cyclic action on $\mathcal {Y}$ . Therefore, we can take a finite étale quotient $\mathcal {Y}_{0}/\mu _{m}$ which is separated of finite type. Hence it gives a smooth proper scheme model over $\mathcal {O}_{K}$ of X. Note that $\mathcal {Y}_0$ is projective over $\mathcal {O}_{K}$ by [Reference Bosch, Lütkebohmert and Raynaud4, Theorem 6.4.1], hence $\mathcal {Y}_{0}/\mu _{m}$ is also a smooth projective.

Proof Let $\mathcal {A}'$ be the Néron model of $A/G$ . Then the natural morphism $\pi : \mathcal {A} \rightarrow \mathcal {A}'$ is an étale isogeny. Indeed, by [Reference Bosch, Lütkebohmert and Raynaud4, Proposition 7.3.6], we have a morphism $\pi ' \colon \mathcal {A}' \rightarrow \mathcal {A}$ with $\pi ' \circ \pi = n$ , where n is the order of $G(\overline {K})$ . Since n is coprime to p, $\pi ' \circ \pi $ is étale [Reference Bosch, Lütkebohmert and Raynaud4, Lemma 7.3.2]. Hence, $\pi $ is quasi-finite and flat [Reference Bosch, Lütkebohmert and Raynaud4, Proposition 2.4.2, Lemma 7.3.1]. Moreover, the kernel of $\pi $ is étale, since $\pi ' \circ \pi $ is étale. So $\pi $ is étale. Let $\mathcal {K}$ be the kernel of $\pi $ , which is étale separated of finite type over $\mathcal {O}_{K}$ . Then $\mathcal {G}$ is canonically isomorphic to $\mathcal {K}$ , since for any smooth $\mathcal {O}_{K}$ -scheme Z,

$$ \begin{align*} \mathcal{K}(Z) &= \ker (\mathcal{A}(Z)\rightarrow \mathcal{A}'(Z))\\ &= \ker (A(Z_{K})\rightarrow (A/G)(Z_{K})) \\ &= G(Z_{K}). \end{align*} $$

Proof First we shall show $(1)\Rightarrow (2)$ . By Lemma 2.5, we have $\mathcal {X} \rightarrow S:= \mathop {\mathrm {Spec}} \mathcal {O}_{K}$ which is a smooth projective scheme $\mathcal {O}_{K}$ -model of X. Note that $\mathrm {Pic}_{\mathcal {X}_{\kappa (s)}/\kappa (s)}$ is smooth of dimension $1$ for any geometric point s on S since $\mathcal {X}_{\kappa (s)}/\kappa (s)$ is a bielliptic surface over a field of characteristic different from $2$ and $3$ . Let $\mathrm {Pic}_{\mathcal {X}/S}$ be the Picard functor, and $\mathcal {Y} := \mathrm {Pic}^{0}_{\mathcal {X}/S} \hookrightarrow \mathrm {Pic}_{\mathcal {X}/S}$ is the open closed subgroup scheme whose fiber over any geometric point s of S is equal to the identity component of $\mathrm {Pic}_{\mathcal {X}_{s}/\kappa (s)}$ (see [Reference Fantechi, Göttsche, Illusie, Kleiman, Nitsure and Vistoli9, Theorem 9.4.8, Proposition 9.5.20]). Therefore, $\mathcal {Y}$ is an abelian scheme over S. For the fixed rational point $x\in \mathcal {X}(\mathcal {O}_{K})$ , we have the corresponding morphism $\mathrm {alb} \colon \mathcal {X} \rightarrow \mathrm {Pic}^{0}_{\mathcal {Y}/S}$ which sends x to $0$ . Note that this morphism is flat since one can check it fiberwise (see [Reference Bosch, Lütkebohmert and Raynaud4, Proposition 2.4.2]). Since $\mathrm {Pic}^{0}_{\mathcal {Y}/S}$ is a smooth proper scheme $\mathcal {O}_{K}$ -model of the Albanese variety of X, the elliptic curve A admits good reduction. Moreover, one can show that $\mathcal {Z}:= \mathcal {X}\times _{\mathrm {Pic}^{0}_{\mathcal {Y}/S}}S$ is a smooth proper scheme $\mathcal {O}_{K}$ -model of B, where $S \rightarrow \mathrm {Pic}^{0}_{\mathcal {Y}/S}$ is the $0$ -section. Indeed, by the definition $\mathcal {Z}$ is proper and flat over S, and whose geometric fibers are smooth. Therefore B admits good reduction too.

Next, we shall show $(2)\Rightarrow (1)$ . Let $\mathcal {A}, \mathcal {B}$ be the Néron models of A, B (which is the unique smooth proper scheme $\mathcal {O}_{K}$ -model of the elliptic curves $A, B$ ). Moreover, let $\mathcal {G}$ be the Néron model of G. Since A admits good reduction, the scheme G consists of spectra of fields that are unramified over K (recall that G is a quotient of the n-torsion points of $A^{\prime}$ , where $A^{\prime}$ is an elliptic curve which is isogenous to A and $n \in \{ 2,3,4,6,8,9 \}$ ). Therefore a Néron model of G is a finite étale group scheme over $\mathcal {O}_{K}$ . By the Néron mapping property, one can extend the action of G on $A \times B$ to the action of $\mathcal {G}$ on $\mathcal {A} \times \mathcal {B}$ . Be Lemma 2.7, $\mathcal {G} \hookrightarrow \mathcal {A}$ is a closed immersion of group schemes. Therefore the above action $\mathcal {G} \times \mathcal {A} \times \mathcal {B} \rightarrow \mathcal {A} \times \mathcal {B}$ is a free action by the locally free group scheme. Hence one can take a finite étale quotient $\mathcal {A} \times \mathcal {B} \rightarrow \mathcal {X}$ , where $\mathcal {X}$ is separated of finite type over $\mathcal {O}_{K}$ . Therefore one can show that $\mathcal {X}$ is a smooth proper scheme $\mathcal {O}_{K}$ -model of X, and it finishes the proof.

Proof Shrinking $\mathop {\mathrm {Spec}} R$ , we may assume that R is smooth over $\mathbb {Z}$ . For $X \in \mathrm {Shaf}$ , one can associate a rational point $x\in X (F)$ and we have a description $X \simeq (A \times B)/G$ as in Lemma 2.2. By Proposition 2.8, the elliptic curves A and B admit good reduction at any height $1$ prime $\mathfrak {p} \in \mathop {\mathrm {Spec}} R$ . Therefore there exists a map $\mathrm {Shaf} \rightarrow \mathrm {Shaf}_{\text {ab}}$ , where $\mathrm {Shaf}_{\text {ab}}$ is the set of pairs of F-isomorphism classes of elliptic curves both of which admit good reduction at any height $1$ prime of $\mathop {\mathrm {Spec}} R$ . By [Reference Faltings, Wüstholz, Grunewald, Schappacher and Stuhler8, VI, Section 1, Theorem 2], $\mathrm {Shaf}_{\text {ab}}$ is a finite set. Therefore, by Lemma 2.11 below, we obtain the desired finiteness.

Proof For fixed $A \times B \in \mathrm {Shaf}_{\text {ab}}$ , let $X\in \mathrm {Shaf}$ be a bielliptic surface lying in its fiber. Then, X is isomorphic to $(A \times B) /G_{X}$ for some finite étale subgroup scheme $G_{X} \hookrightarrow A$ and some embedding $\alpha _{X} \colon G_{X} \hookrightarrow \text {Aut}_{B/F}$ . Since the order of $G_{X}$ is bounded, the candidates of $G_{X}\hookrightarrow A$ are finitely many. Hence we may fix an embedding $\iota \colon G := G_{X} \hookrightarrow A$ . Let $S_{A, B, \iota }$ be the set of embeddings $\alpha \colon G \hookrightarrow \text {Aut}_{B/F}$ such that $(A\times B) / G$ is a bielliptic surface, where the quotient is taken with respect to the action $\iota \times \alpha $ . Let $\sim $ be the equivalence relation on $S_{A,B, \iota }$ given by $B(F)$ -conjugate. More precisely, for $\alpha , \alpha ' \in \text {Aut}_{B/F}(G)$ , $\alpha \sim \alpha '$ if and only if there exists $b \in B(F)$ such that $t_{b} \alpha t_{-b} = \alpha '$ , where $t_{b} \colon B \rightarrow B$ is the translation which sends $0$ to b. In this case, we have the F-isomorphism

$$\begin{align*}((A \times B)/G)^{(\iota \times \alpha)} \simeq ((A \times B)/G)^{(\iota \times \alpha')} \end{align*}$$

induced by $\mathrm {id} \times t_{b}$ , where the left-hand (resp. right-hand) side is the quotient with respect to the action $\iota \times \alpha $ (resp. $\iota \times \alpha '$ ). Therefore, it suffices to show the finiteness of $S_{A,B,\iota }/\sim $ . Since $(B/G)_{\overline {F}}^{(\alpha _{\overline {F}})}$ (the quotient with respect to the action $\alpha _{\overline {F}}$ ) is $\mathbb {P}^{1}_{\overline {F}}$ , the image H of $\alpha _{\overline {F}} (G(\overline {F}))$ in $\text {Aut}(B_{\overline {F}})/B(\overline {F}) \simeq \text {Aut}(B_{\overline {F}},0)$ is non-trivial. Indeed, if this is trivial, then the quotient is given by an isogenous elliptic curve, and we have a contradiction. Take an element $g = g_{\alpha } \in G(\overline {F})$ such that the image of $\alpha _{\overline {F}} (g)$ generates H. Let $b_{\alpha } \in B(\overline {F})$ be a fixed point of $\alpha _{\overline {F}} (g)$ . Since $\alpha _{\overline {F}}(G(\overline {F}))$ is commutative, we have a decomposition $G(\overline {F}) \simeq H_{1}^{\alpha } \times H_{2}^{\alpha }$ , with $\alpha (H_{1}^{\alpha }) \subset \text {Aut} (B_{\overline {F}},b_{\alpha })$ and $\alpha (H_{2}^{\alpha }) \subset B(\overline {F}) \subset \text {Aut} (B_{\overline {F}})$ (see the arguments in [Reference Bădescu2, Section 10.26, List 10.27]). We denote the projection $G(\overline {F}) \rightarrow H_{i}^{\alpha }$ by $p_{i}^{\alpha }$ . Since $\text {Aut} (B_{\overline {F}}, b_{\alpha })$ is $\mathbb {Z}/2\mathbb {Z}$ , $\mathbb {Z}/4\mathbb {Z}$ , or $\mathbb {Z}/6\mathbb {Z}$ , there are classification of $H_{1}^{\alpha } \hookrightarrow \text {Aut} (B_{\overline {F}}, b_{\alpha })$ and $H_{2}^{\alpha } \hookrightarrow B(\overline {F})$ (see [Reference Bădescu2, List 10.27]). In particular, we remark that if $H_{2}^{\alpha }$ is non-trivial then $H_{2}^\alpha $ is $\mathbb {Z}/2\mathbb {Z}$ (the case of $(a_{2})$ or $(c_{2})$ in [Reference Bădescu2, List 10.27]) or $\mathbb {Z}/3\mathbb {Z}$ (the case of $(c_{1})$ in [Reference Bădescu2, List 10.27]). In both cases, we note that $\# H_{2}^{\alpha }$ divides $\# H_{1}^{\alpha }$ .

We can define the morphism of sets

$$\begin{align*}\phi \colon S_{A,B,\iota} \rightarrow B(F); \alpha \mapsto \sum_{g \in G(\overline{F})}\alpha_{\overline{F}}(g) (0). \end{align*}$$

Since $\# H_{2}^{\alpha } | \# H_{1}^{\alpha }$ , we have

$$\begin{align*}\sum_{g\in G(\overline{F})} \alpha_{\overline{F}}(p_{2}^{\alpha}(g))(0) = \sum_{h \in H_{2}^{\alpha}} \# H_{1}^{\alpha} \cdot \alpha_{\overline{F}}(h)(0) =0. \end{align*}$$

Let n be the order of $G(\overline {F})$ , and $h^{\alpha ,g} \in \text {Aut}(B_{\overline {F}},0)$ the automorphism such that

$$\begin{align*}\alpha_{\overline{F}}(p_{1}^{\alpha}(g)) = t_{b_{\alpha}} h^{\alpha,g} t_{b_{\alpha}}^{-1}. \end{align*}$$

Since

$$\begin{align*}h^{\alpha,g_{0}}\left(\sum_{g\in G(\overline{F})} h^{\alpha,g}\right) =\sum_{g\in G(\overline{F})} h^{\alpha,g} \end{align*}$$

for non-trivial $h^{\alpha ,g_{0}}$ (one can take $g_{0} = g_{\alpha }$ ), the image $(\sum _{g\in G(\overline {F})} h^{\alpha ,g}) (B_{\overline {F}})$ is contained in the fixed locus of $h^{\alpha , g_{0}}$ . Since $(\sum _{g\in G(\overline {F})} h^{\alpha ,g}) (B_{\overline {F}})$ is connected and containing $0$ , we have $\sum _{g\in G(\overline {F})} h^{\alpha ,g} = 0 \in \text {End} (B_{\overline {F}})$ . Combining with

$$\begin{align*}\alpha_{\overline{F}}(g)(0) = \alpha_{\overline{F}}(p_{1}^{\alpha}(g))(0) + \alpha_{\overline{F}}(p_{2}^{\alpha}(g))(0), \end{align*}$$

we have

$$ \begin{align*} \phi (\alpha) &= \sum_{g\in G(\overline{F})} \alpha_{\overline{F}}(p_{1}^{\alpha}(g))(0) \\ &= \sum_{g\in G(\overline{F})} ( h^{\alpha,g} (-b_{\alpha}) + b_{\alpha})\\ &= \sum_{g\in G(\overline{F})} (1 - h^{\alpha,g}) (b_{\alpha})\\ &= n b_{\alpha}. \end{align*} $$

Since we fix the isomorphism class of G, possible isomorphism classes $H_{1}^{\alpha }, H_{2}^{\alpha }$ are at most finitely many (in fact unique). Therefore, possible $\alpha |_{H_{2}^{\alpha }}$ are at most finitely many. Moreover, if $n b_{\alpha }$ is fixed, then possible $b_{\alpha }$ are finitely many, and $\alpha |_{H_{1}^{\alpha }}$ are at most finitely many since possible embeddings $H_{1}^{\alpha } \rightarrow \text {Aut}(B_{\overline {F}}, b_{\alpha })$ are finitely many. Hence each fiber of $\phi $ is finite. We note that $\phi (t_{b} \alpha t_{b}^{-1}) = n(b_{\alpha }+b)$ . Therefore $\phi $ induces the morphism with finite fibers

$$\begin{align*}S_{A,B,\iota}/ \sim \rightarrow B(F)/ n B(F). \end{align*}$$

By [Reference Néron29] (see also [Reference Lang and Néron22, Section 1]), the abelian group $B(F)$ is finitely generated, so the right-hand side is finite. It finishes the proof.

Proof We take a finite set of finite prime numbers S and an elliptic curve E over $\mathbb {Q}$ satisfying the following:

1. (1) $2 \notin S,$

2. (2) E has good reduction outside S,

3. (3) The Mordell-Weil rank of E is 0, and the analytic rank of E is 0,

4. (4) Any point of $E[2]$ is a $\mathbb {Q}$ -rational point.

Note that the assumption $(2)$ ensures that S is non-empty by [Reference Ogg31]. Moreover, by Kolyvagin’s result [Reference Kolyvagin19] (see also [Reference Kolyvagin and Satake20, Theorem 1]) and the assumption (3), the Tate-Shafarevich group of E is known to be finite. We can take, for example, S as $\{3,5 \}$ and E as the elliptic curve over $\mathbb {Q}$ with Cremona label $15a2$ . We shall show that $\mathrm {Shaf}'$ is an infinite set for this S.

By Tate’s argument [Reference Mazur27, footnote 27] and the assumptions (2) and (3), we can take infinitely many isomorphism classes of E-torsors $C_{i}$ over $\mathbb {Q}$ ( $i=1,2, \ldots $ ) such that $C_{i, \mathbb {Q}_{p}}$ is a trivial torsor for any $p \notin S$ . Since is finite, we may assume that there exists a prime number $p \in S$ such that the isomorphism classes of $E_{\mathbb {Q}_{p}}$ -torsor $C_{i, \mathbb {Q}_{p}}$ are all distinct. By restricting the action of the torsor structure, we have a free action $\sigma _{1}$ of $E[2]$ on $C_{i}$ . We fix a $\mathbb {Z}/2\mathbb {Z}$ -basis $P, Q$ of $E[2] \simeq (\mathbb {Z}/2\mathbb {Z})^{\oplus 2}$ , that are $\mathbb {Q}$ -rational points by the assumption (4). We consider the morphism

$$\begin{align*}\sigma_{2} \colon E[2] \times E \rightarrow E; (aP+bQ, R) \mapsto (-1)^a R + bQ, \end{align*}$$

which is well-defined and gives the action of $E[2]$ on E. We put

$$\begin{align*}X_{i} := (C_{i} \times E)/ E[2], \end{align*}$$

where the quotient is taken with respect to the action $\sigma := \sigma _{1} \times \sigma _{2}$ . Clearly, $X_{i}$ is a bielliptic surface over $\mathbb {Q}$ . Moreover, $X_{i}$ has good reduction outside S by the same argument as in the proof of Proposition 2.8. We shall show that $X_{i} (i=1,2, \ldots )$ represent infinitely many $\mathbb {Q}$ -isomorphism classes. Let $A_{i}$ be the Albanese torsor $\mathrm {Alb} (X_{i})$ of $X_{i}$ . It suffices to show that $A_{i} (i=1,2, \ldots )$ represent infinitely many $\mathbb {Q}$ -isomorphism classes. Let $f_{i} \colon X_{i} \rightarrow C_{i}/ E[2]$ be the natural morphism. Fix a $\overline {\mathbb {Q}}$ -rational point of $C_{i, \overline {\mathbb {Q}}}$ so that $C_{i, \overline {\mathbb {Q}}}$ is equipped with the structure of an elliptic curve, which is isomorphic to $E_{\overline {\mathbb {Q}}}$ . Since

$$\begin{align*}(E_{\overline{\mathbb{Q}}} \times E_{\overline{\mathbb{Q}}})/E[2] \simeq X_{i, \overline{\mathbb{Q}}} = (C_{i, \overline{\mathbb{Q}}} \times E_{\overline{\mathbb{Q}}}) /E[2] \rightarrow C_{i, \overline{\mathbb{Q}}} / E[2] \simeq E_{\overline{\mathbb{Q}}} / E[2] \rightarrow E_{\overline{\mathbb{Q}}} \end{align*}$$

is an Albanese morphism by the proof of Lemma 2.2, we have $A_{i} \simeq C_{i}/E[2]$ . Therefore, $A_{i}$ admits $E/E[2] \simeq E$ -torsor structure, whose class in Weil–Châtelet group $H^1(\mathrm {Gal}(\overline {\mathbb {Q}}/\mathbb {Q}), E)$ is $2[C_{i}]$ . Here, $C_{i}$ is a class represented by the E-torsor $C_{i}$ . We recall that $[C_{i, \mathbb {Q}_{p}}] \in H^1 (\mathrm {Gal} (\overline {\mathbb {Q}_{p}},\mathbb {Q}_{p}), E_{\mathbb {Q}_{p}})$ is a non-trivial class. Moreover, since $p \neq 2$ by the assumption (1), the group $H^1 (\mathrm {Gal} (\overline {\mathbb {Q}_{p}},\mathbb {Q}_{p}), E_{\mathbb {Q}_{p}}) [2] \simeq E(\mathbb {Q}_{p})/ 2 E(\mathbb {Q}_{p})$ is a finite group. Since the classes $[C_{i,\mathbb {Q}_{p}}] \in H^1 (\mathrm {Gal} (\overline {\mathbb {Q}_{p}},\mathbb {Q}_{p}), E_{\mathbb {Q}_{p}})$ are all distinct, $\{2[C_{i,\mathbb {Q}_{p}}] \} \subset H^1 (\mathrm {Gal} (\overline {\mathbb {Q}_{p}},\mathbb {Q}_{p}), E_{\mathbb {Q}_{p}})$ is an infinite set, and it finishes the proof.

Proof See [Reference Bosch, Lütkebohmert and Raynaud4, Proposition 3.5.3].

Proof Let Z be a smooth $\mathcal {O}_{K}$ -scheme, and $u_{K} \colon Z_{K} \rightarrow \mathcal {Y}_{K}$ a K-morphism. We shall extend the domain of $u_{K}$ to Z. By Proposition 3.1, there exists an open subscheme $U \subset Z$ and $u \colon U \rightarrow \mathcal {Y}$ , such that U contains the generic fiber $Z_{K}$ and all the generic points of the special fiber, and u is the extension of $u_{K}$ . Using u, one has a finite étale covering $\widetilde {U}:= U \times _{\mathcal {Y}}\mathcal {X} \rightarrow U$ . Since Z is a regular scheme, by taking the normalization of Z in $\widetilde {U}$ and using the Zariski–Nagata purity theorem, one can extend this finite étale covering to $\widetilde {Z} \rightarrow Z$ . By the Néron mapping property of $\mathcal {X}$ , we have $\mathcal {X}(\widetilde {U})=\mathcal {X}(\widetilde {U}_{K})= \mathcal {X}(\widetilde {Z})$ . Therefore, the base change morphism $\widetilde {u} \colon \widetilde {U} \rightarrow \mathcal {X}$ can be extended to $\widetilde {u}\colon \widetilde {Z} \rightarrow \mathcal {X}$ . Then we have a morphism $u_{\widetilde {Z}} \colon \widetilde {Z} \rightarrow \mathcal {Y}$ . Consider the equalizer diagram

(3.1)

where the vertical arrows are injective since Z, $\widetilde {Z}$ , and $\widetilde {Z} \times _{Z}\widetilde {Z}$ are reduced schemes which are flat over $\mathcal {O}_{K}$ and $\mathcal {Y}$ is separated over $\mathcal {O}_{K}$ . It suffices to show that $u_{\widetilde {Z}} \in \mathcal {Y}(\widetilde {Z})$ is contained in the equalizer of the upper right arrows. Since the middle vertical map sends $u_{\widetilde {Z}}$ to $u_{K}$ , which is contained in the equalizer of the lower right arrows, it finishes the proof.

Proof Let $\mathcal {X}$ be a proper smooth scheme in Lemma 2.5. Then the cyclic covering over $\mathcal {O}_{K}$ associated with a fixed isomorphism $\omega _{\mathcal {X}/\mathcal {O}_{K}}^{\otimes m} \simeq \mathcal {O}_{\mathcal {X}}$ is the Néron model of its generic fiber as in the proof in Lemma 2.5. By the valuative criterion of properness, the model $\mathcal {X}$ is a weak Néron model of X, so by Lemma 3.2, it finishes the proof.

Proof By the assumption, the scheme $(\overline {\mathcal {A}}\times \overline {\mathcal {B}})/\mathcal {G}$ is a finite étale quotient of $\overline {\mathcal {A}} \times \overline {\mathcal {B}}$ . Therefore $(\overline {\mathcal {A}} \times \overline {\mathcal {B}})/\mathcal {G}$ is a regular scheme which is proper over $\mathcal {O}_{K}$ . Since its smooth locus is $(\mathcal {A} \times \mathcal {B})/\mathcal {G}$ , by the valuative criterion of the properness and [Reference Liu and Tong26, Lemma 3.1], one can show that $(\mathcal {A}\times \mathcal {B})/\mathcal {G}$ is a weak Néron model of X. By Lemma 3.2, it finishes the proof.

Proof It suffices to show the first statement. Let $X \simeq (A \times B)/G$ be a bielliptic surface with a rational point $x \in X(K)$ , as in the assumption. Let n be the integer as in Remark 2.3. Since we are working with a strictly Henselian discrete valuation field, the following hold.

• • The group scheme G consists of K-rational points.

• • Every connected component of the special fiber of a Néron model $\mathcal {A}$ has a k-rational point. Especially, every component is geometrically connected.

• • Let

  $$\begin{align*}\Phi\colon X(K) \rightarrow A/G(K) \rightarrow A(K) \simeq \mathcal{A}(\mathcal{O}_{K}) \rightarrow \pi_{0}(\mathcal{A}_{k})/n\pi_{0}(\mathcal{A}_{k}) \end{align*}$$
  be the composition of the Albanese morphism with the reduction maps. Here, the second arrow is an isomorphism given by the n-multiplication map as in Remark 2.3. Then, there exist K-valued points $x_{0}=x, x_{1}, \ldots , x_{s} \in X(K)$ which give a complete set of representatives of $\mathrm {Im} \Phi $ .

For each rational points $x_{i}$ , we have a description

$$\begin{align*}X \simeq (A_{i} \times B_{i}) /G_{i} \end{align*}$$

given in Remark 2.3. Here, by the definition every $A_{i}$ is the Albanese variety of X, so every $A_{i}$ is naturally isomorphic to each other. The only difference is that the zero section of $A_{i}$ is given by $\mathrm {alb}(x_{i})$ . As in Remark 2.3, each $G_{i}$ consists of K-rational points.

Let $\mathcal {A}_{i}, \mathcal {B}_{i}, \mathcal {G}_{i}$ be the Néron models of $A_{i}, B_{i}, G_{i}$ . By the Néron mapping property, one can extend the group action to Néron models. By Lemma 2.7, a Néron model $\mathcal {G}_{i}$ is a finite étale subgroup scheme of $\mathcal {A}_{i}$ . Let $\mathcal {X}_{i}$ be the finite étale quotient $(\mathcal {A}_{i} \times \mathcal {B}_{i})/\mathcal {G}_{i}$ . As before, the quotient $\mathcal {X}_{i}$ is smooth separated of finite type over $\mathcal {O}_{K}$ . Let $\mathcal {X}$ be the scheme obtained by gluing $\mathcal {X}_{i}$ together on the generic fibers. By definition, $\mathcal {X}$ is a smooth finite type scheme over $\mathcal {O}_{K}$ satisfying $\mathcal {X}_{K}\simeq X$ . We shall prove that this $\mathcal {X}$ is the desired Néron model. First, we shall prove that $\mathcal {X}$ is separated over $\mathcal {O}_K$ . Let R be a discrete valuation ring over $\mathcal {O}_{K}$ , and F the fraction field of R. By the valuative criterion of separatedness, it is enough to show that $\mathcal {X}(R) \rightarrow \mathcal {X}(F)$ is injective. Assume that there exist $t_{1}, t_{2} \colon \mathop {\mathrm {Spec}} R \rightarrow \mathcal {X}$ which are different R-valued points going to the same F-valued point. Clearly, each $t_{l}$ factors through some $\mathcal {X}_{i_{l}} \hookrightarrow \mathcal {X}$ . If $t_{l}$ factors through the same component $\mathcal {X}_{i}$ , by the separatedness of $\mathcal {X}_{i}$ , we have $t_{1}=t_{2}$ . Therefore each $t_{l}$ factors through $\mathcal {X}_{i_{l}}$ with $i_{1}\neq i_{2}$ and each $t_{l}$ does not factors through the generic fiber X. Let $s_{l} := \mathrm {alb}(t_{l}) \in \mathcal {A}_{i_{l}} (R),$ where $\mathrm {alb}$ is the extension of the Albanese morphism to $\mathcal {X} \rightarrow \mathcal {A}$ . The morphism $\mathrm {alb}|_{\mathcal {X}_{i_{l}}}$ is written as the following composition,

$$\begin{align*}\mathcal{X}_{i_{l}} := (\mathcal{A}_{i_{l}} \times \mathcal{B}_{i_{l}}) /\mathcal{G}_{i_{l}} \rightarrow \mathcal{A}_{i_{l}}/\mathcal{G}_{i_{l}} \rightarrow \mathcal{A}_{i_{l}}, \end{align*}$$

where the last map is given by n-multiplication morphism. As remarked above, for any $0\leq i,j \leq s$ , the varieties $\mathcal {A}_{i}$ and $\mathcal {A}_{j}$ are naturally isomorphic (we denote it by $\mathcal {A}$ ), and there exists the following commutative diagram.

(3.2)

Here $t_{i,j}$ is a translation on $\mathcal {A}$ which sends $\mathrm {alb}(x_{i})$ to $\mathrm {alb}(x_{j})$ , and $n_{\mathcal {A}_{i}}$ is n-multiplication map as the group scheme $\mathcal {A}_{i}$ . Note that we identify $\mathrm {alb}(x_{i})$ with its extension to the $\mathcal {O}_{K}$ -valued point of $\mathcal {A}$ . Let $\mathfrak {m}$ be the maximal ideal of $\mathop {\mathrm {Spec}} R$ . Since $t_{l} (\mathfrak {m})$ is contained in the special fiber of $\mathcal {X}_{i_{l}}$ , the point $s_{l} (\mathfrak {m})$ is contained in the special fiber of $\mathcal {A}$ . By the above description of $\mathrm {alb}|_{\mathcal {X}_{i_{l}}}$ , the point $s_{l}(\mathfrak {m})$ lies in the component which is contained in $\Pi _{i_{l}}$ , where

$$\begin{align*}\Pi_{i} := \mathrm{Im} (n_{\mathcal{A}_{i}} \colon \pi_{0}(\mathcal{A}_{k}) \rightarrow \pi_{0}(\mathcal{A}_{k})). \end{align*}$$

By the above commutative diagram, as subsets of the component group $\pi _{0}(\mathcal {A}_{k})$ , we have

$$\begin{align*}\Pi_{i_{l}} = t_{0,i_{l}}\Pi_{0}= (\mathrm{alb}(x_{i_{l}})-\mathrm{alb}{(x_{0}}))\Pi_{0}. \end{align*}$$

Since we have $i_{1}\neq i_{2}$ , by the definition of $x_{i}$ , we have that $\Pi _{i_{1}}$ and $\Pi _{i_{2}}$ are disjoint in $\pi _{0}(\mathcal {A}_{k}).$ So we have $s_{1} \neq s_{2}$ . However, since $s_{1}$ and $s_{2}$ give the same F-valued point of $\mathcal {A}$ , we have $s_{1}=s_{2}$ by the separatedness of $\mathcal {A}$ . This is a contradiction, thus $\mathcal {X}$ is separated over $\mathcal {O}_{K}$ .

Next, we shall prove that the family $\{\mathcal {X}_{i}\}_{i=0, \ldots , s}$ is a weak Néron model of X. Let $y \in X(K)$ be any K-valued point of X. By definition, there exists a unique $x_{i} \in X(K)$ which represents $\Phi (y)$ . Let $\mathrm {alb}(y)' \in \mathcal {A}(\mathcal {O}_{K})$ be the unique extension of $\mathrm {alb}(y)$ . Then the special fiber of $\mathrm {alb} (y)'$ lies in the union of components corresponding to $t_{0,i} \Pi _{0}=\Pi _{i}$ . We note that this union of components is the image of the n-multiplication map on the special fiber induced by $n_{\mathcal {A}_{i}}$ (see [Reference Bosch, Lütkebohmert and Raynaud4, Lemma 7.3.1 and 7.3.2]). Since $\mathcal {O}_{K}$ is strictly Henselian, the fiber of $\mathrm {alb}(y)'$ along $n_{\mathcal {A}_{i}}$ has a component $\mathop {\mathrm {Spec}} \mathcal {O}_{K}$ , i.e., there exists a lift $y^{\prime}_{1} \in \mathcal {A}(\mathcal {O}_{K})$ of $\mathrm {alb}(y)'$ along $n_{\mathcal {A}_{i}}$ . Let $y_{1}\in A(K)$ be the K-rational point of A induced by $y^{\prime}_{1}$ (so $y_{1}$ is a lift of $\mathrm {alb}(y)$ along the n-multiplication map on $A_{i}$ ). Let $y_{2} := (-y_{1}) \cdot y \in X(K)$ , where $\cdot $ denotes the action $A_{i}\times X \rightarrow X$ given in the proof of Lemma 2.2. Then by the definition (see the commutative diagram (2.1) in the proof of Lemma 2.2), we have $\mathrm {alb}(y_{2}) =\mathrm {alb} (x_{i})$ , and therefore $y_{2}$ lies in $B_{i}(K)$ and we have $y=\overline {(y_{1}, y_{2})}$ . We can extend $(y_{1}, y_{2}) \in A_{i} \times B_{i}$ to $(y^{\prime}_{1}, y^{\prime}_{2}) \in \mathcal {A}_{i}\times \mathcal {B}_{i}(\mathcal {O}_{K})$ . Therefore $y^{\prime} := \overline {(y^{\prime}_{1}, y^{\prime}_{2})} \in \mathcal {X}_{i}(\mathcal {O}_{K})$ gives a desired extension of y.

Finally, we shall prove that $\mathcal {X}$ satisfies the Néron mapping property. This part is essentially the same as in the proof of Lemma 3.2, but we include it for the sake of completeness. Let Z be any smooth $\mathcal {O}_{K}$ -scheme, and $u_{K} \colon Z_{K}\rightarrow X$ be a K-morphism. By Remark 2.4, it suffices to show the existence of an extension of u. Moreover, we may assume that Z has an irreducible special fiber (in the general case, one can glue them). By Proposition 3.1 (2), there exists an extension of $u_{K}$ to an $\mathcal {O}_{K}$ -rational map $u\colon Z \dashrightarrow \mathcal {X}_{i}$ for some $0 \leq i \leq s$ . Therefore, there exists an open subscheme $U \hookrightarrow Z$ where u is defined, such that U contains $Z_{K}$ and the generic point of $Z_{k}$ . Then the finite étale covering

$$\begin{align*}\widetilde{U}:= U\times_{\mathcal{X}_{i}} (\mathcal{A}_{i} \times \mathcal{B}_{i}) \rightarrow U \end{align*}$$

can be extended to the finite étale covering $\widetilde {Z} \rightarrow Z$ by taking the normalization of Z in $\widetilde {U}$ and using the Zariski–Nagata purity theorem. By the Néron mapping property, we have

$$\begin{align*}\mathcal{A}_{i} \times \mathcal{B}_{i}(\widetilde{U}) = \mathcal{A}_{i} \times \mathcal{B}_{i} (\widetilde{U}_{K}) = \mathcal{A}_{i} \times \mathcal{B}_{i}(\widetilde{Z}), \end{align*}$$

so the base change morphism $\widetilde {u} \colon \widetilde {U} \rightarrow \mathcal {A}_{i} \times \mathcal {B}_{i}$ can be extended to $\widetilde {u} \colon \widetilde {Z}\rightarrow \mathcal {A}_{i} \times \mathcal {B}_{i}$ . Hence we have a morphism $u_{\widetilde {Z}} \colon \widetilde {Z} \rightarrow \mathcal {X}_{i}$ , and by the uniqueness of extension, one can descends $u_{\widetilde {Z}}$ to $u_{Z}$ which is a desired extension of $u_{K}$ . It finishes the proof.