Higher-dimensional Calabi-Yau varieties with dense sets of rational points
aa r X i v : . [ m a t h . AG ] F e b HIGHER-DIMENSIONAL CALABI-YAU VARIETIES WITHDENSE SETS OF RATIONAL POINTS
FUMIAKI SUZUKI
Abstract.
We construct higher-dimensional Calabi-Yau varieties defined overa given number field with Zariski dense sets of rational points. We give two ele-mentary constructions in arbitrary dimensions as well as another constructionin dimension three which involves certain Calabi-Yau threefolds containing anEnriques surface. The constructions also show that potential density holds for(sufficiently) general members of the families. Introduction
For a smooth projective variety X defined over a number field, one can askwhether the set of rational points is dense. It is expected that the set of rationalpoints reflects the positivity of the canonical bundle ω X (cf. the Bombieri-Langconjecture) and it is tempting to study the intermediate case ω X = O X . In thiscase, X belongs to the class of special varieties introduced by Campana [6] which isconjecturally the same as that of varieties on which rational points are potentiallydense, that is, Zariski dense after passing to some finite field extension. An interest-ing subcase is given by abelian varieties, for which potential density is well-known.It is challenging to consider the subcase given by Calabi-Yau varieties in a strictsense , i.e., smooth projective varieties X with ω X = O X and H ( X, Ω iX ) = 0 forall 0 < i < dim X , which are simply connected over C . For elliptic K3 surfacesand K3 surfaces with infinite automorphism groups, potential density holds due toBogomolov-Tschinkel [4]. Moreover, there are several works on K3 surfaces overthe rational numbers with Zariski dense sets of rational points; for instance, quarticK3 surfaces have long been studied [9, 15, 21, 22].Very little is known in higher dimensions. It is stated by Tschinkel [23, afterProblem 3.5] that it would be worthwhile to find non-trivial examples of Calabi-Yau threefolds over number fields with Zariski dense sets of rational points. It isonly recent that the first such examples were actually obtained: Bogomolov-Halle-Pazuki-Tanimoto [1] studied Calabi-Yau threefolds with abelian surface fibrationsand showed potential density for threefolds (including simply connected ones) con-structed by Gross-Popescu [10]. However, it is not immediately clear whether theirmethod can be used to determine the minimal field extensions over which rationalpoints become Zariski dense.In this short note, we construct higher-dimensional Calabi-Yau varieties in astrict sense defined over a given number field with Zariski dense sets of rationalpoints. We give two elementary constructions in arbitrary dimensions (Section 2,3) as well as another construction in dimension three which involves certain Calabi-Yau threefolds containing an Enriques surface (Section 5). The constructions alsoshow that potential density holds for (sufficiently) general members of the families.The third construction is a by-product of the author’s attempt to analyze indetail the recent construction due to Ottem-Suzuki [16] of a pencil of Enriquessurfaces with non-algebraic integral Hodge classes. For the third construction, our Date : February 26, 2021. example contains no abelian surface and is a unique minimal model in its birationalequivalence class, thus a theorem of Bogomolov-Halle-Pazuki-Tanimoto [1, Theorem1.2] cannot be applied. For all the constructions, elliptic fibration structures arecrucial.We work over a number field unless otherwise specified.
Acknowledgements.
The author wishes to thank Lawrence Ein, Yohsuke Mat-suzawa, John Christian Ottem, Ramin Takloo-Bighash, and Burt Totaro for inter-esting discussions. 2.
Construction I
The first idea is to construct elliptic Calabi-Yau varieties in a strict sense (i.e., H ( X, Ω iX ) = 0 for all 0 < i < dim X and simply connected over C ) whose basespaces are rational and which admits infinitely many sections. For that purpose, weintroduce variants of Schoen’s Calabi-Yau fiber products of rational elliptic surfaces[18].We let S ⊂ P × P be a smooth hypersurface of bi-degree (1 , f : S → P . Moreover, via the secondprojection, S is the blow-up of P along the nine points given by the intersectionof two cubic curves, hence rational. For n ≥
3, we let Y ⊂ P × P n − be asmooth hypersurface of bi-degree (1 , n ). The first projection restricts to a fibrationinto Calabi-Yau hypersurfaces g : Y → P and Y is again rational via the secondprojection. We assume that over any point in P either f or g is smooth, whichis satisfied if Y is general with respect to S . We define X = S × P Y and let π : X → P be the natural projection. Lemma 2.1.
The fiber product X is a Calabi-Yau n -fold in a strict sense.Remark . If S C is very general, it is classical that the elliptic fibration f : S C → P C admits infinitely many sections, which implies that the same holds for the naturalprojection X C → Y C . This provides examples of Calabi-Yau varieties in a strictsense defined over C containing infinitely many rational divisors in all dimensions ≥
3. We will construct below such an example over a given number field.
Proof of Lemma 2.1.
It is immediate to see that X is smooth. The fiber product X is a complete intersection in P × P × P n − of hypersurfaces of tri-degree (1 , , , , n ). We have ω X = O X by the adjunction formula andan easy computation shows that H i ( X, O X ) = 0 for all 0 < i < n .For simple connectedness over C , Schoen proved this result for n = 3 (see [17,Lemma 1.1] for the strategy). In fact, his method works for n ≥ U ⊂ P C be the open subset over which π : X C → P C is smoothand let V = π − ( U ). The natural map π | V : V → U is topologically locally trivialand we let F be a fiber. We note that π : X C → P C admits a section since f and g do, so does π | V : V → U . Then we have a commutative diagram π ( F ) / / π ( V ) / / / / (cid:15) (cid:15) (cid:15) (cid:15) π ( U ) (cid:15) (cid:15) (cid:15) (cid:15) v v π ( X C ) / / / / π ( P C ) v v where the upper row is exact by the homotopy long exact sequence. Chasing thediagram and using the fact that π ( P C ) is trivial, we are reduced to showing that π ( F ) has the trivial image in π ( X C ). Writing F = F × F , where F is a fiber of f and F is a fiber of g , we see that the Van-Kampen theorem implies π ( F ) = π ( F ) × π ( F ) . IGHER-DIMENSIONAL CALABI-YAU VARIETIES WITH DENSE SETS OF RATIONAL POINTS Now it is enough to verify that the image of π ( F ) and π ( F ) in π ( X C ) are trivial.This is immediate since π ( F ) → π ( X C ) (resp. π ( F ) → π ( X C )) factors throughthe fundamental group of a section of X C → S C (resp. X C → Y C ) and since S C (resp. Y C ) is simply connected (it is rational). The proof is complete. (cid:3) We give a construction of X defined over Q such that X ( Q ) is Zariski dense. Westart from constructing S defined over Q such that the elliptic fibration f : S → P admits infinitely many sections over Q , or equivalently, the generic fiber E/ Q ( t )admits a Q ( t )-rational point and the Mordel-Weil group E ( Q ( t )) has a positiverank. The construction is as follows.Let C ⊂ P be an elliptic curve defined over Q with a Zariski dense set of Q -rational points. Let O ∈ C ( Q ) (resp. P ∈ C ( Q )) be the origin (resp. a non-torsionpoint). Let D ⊂ P be another elliptic curve defined over Q which goes throughboth O and P and which intersects transversally with C . Let S ⊂ P × P bethe hypersurface of bi-degree (1 ,
3) defined over Q corresponding to the pencil ofelliptic curves generated by C and D . A zero-section of f : S → P is given by the( − O and the Mordel-Weil group E ( Q ( t )) has a positive rank since theimage of the specialization homomorphism E ( Q ( t )) → C ( Q ) has a positive rank.We conclude that S has the desired property.Let Y be smooth, defined over Q , and general so that over any point in P either f or g is smooth. Then X = S × P Y is a Calabi-Yau n -fold in a strict sense definedover Q . Moreover the elliptic fibration X = S × P Y → Y admits infinitely manysections over Q by construction. We note that the set Y ( Q ) is Zariski dense since Y is rational over Q . The following theorem is now immediate: Theorem 2.3.
The set X ( Q ) is Zariski dense. Construction II
We let X ⊂ ( P ) n +1 be a smooth hypersurface of multi-degree (2 n +1 ). Thefollowing is an immediate consequence of the Lefschetz hyperplane section theorem: Lemma 3.1. If n ≥ , then X is a Calabi-Yau n -fold in a strict sense. We give a construction of X defined over Q such that X ( Q ) is Zariski dense. Werecall that a multi-section of an elliptic fibration is saliently ramified if it is ramifiedin a point which lies in a smooth elliptic fiber. The following theorem is due toBogomolov-Tschinkel: Theorem 3.2 ([2, 3]) . Let φ : E → B be an elliptic fibration over a number field K . If there exists a saliently ramified multi-section M such that M ( K ) is Zariskidense, then E ( K ) is Zariski dense.Proof. We sketch the proof for the convenience of the reader. It is obvious thatthe subset φ ( M ( K )) ⊂ B is Zariski dense. Let φ J : J → B be the correspondingJacobian fibration. We consider a rational map τ : E J , p dp − M φ ( p ) , where d = deg( M /B ). Then τ ( B ) cannot be contained in the m -torsion part J [ m ] for any positive integer m . Now Merel’s theorem (or Mazur’s theorem when K = Q ) implies that there exists a non-empty Zariski open subset U ⊂ B such thata rational point p b ∈ M ( K ) is non-torsion on the fiber J b for any b ∈ φ ( M ( K )) ∩ U .Finally, the fiberwise action of the Jacobian fibraion on E translates rational pointson M , which concludes the proof. (cid:3) FUMIAKI SUZUKI
We start from constructing a smooth hypersurface X ⊂ P × P of bi-degree(2 ,
2) defined over Q such that X ( Q ) is Zariski dense. We set P × P = Proj Q [ S , T ] × Proj Q [ S , T ] . For instance, we can take X to be the hypersurface defined by the equation S S T + S T (2 S + 2 S T + 3 T ) + T T ( S + T ) = 0 . Then X is an elliptic curve with a non-torsion point defined over Q , hence X ( Q )is Zariski dense.For n >
1, we set( P ) n +1 = Proj Q [ S , T ] × · · · × Proj Q [ S n +1 , T n +1 ]and inductively define X n ⊂ ( P ) n +1 to be a general hypersurface of multi-degree(2 n +1 ) defined over Q containing X n − as the fiber of the projection pr n +1 : X n → P over T n +1 = 0. Lemma 3.3.
The hypersurface X n is smooth and X n − is a saliently ramifiedmulti-section of the elliptic fibration pr , ··· ,n − : X n → ( P ) n − .Proof. For smoothness, it is enough to show that X n is smooth around X n − . Thisis obvious because X n − is a fiber of the flat proper morphism pr n +1 : X n → P and X n − is smooth by induction. We are reduced to showing the second assertion.Let B ⊂ ( P ) n − be the branch locus, that is, the set of critical values of pr , ··· ,n − : X n − → ( P ) n − . We note that this morphism is generically finite, butnot finite when n >
3. We only need to prove that the fiber of pr , ··· ,n − : X n → ( P ) n − over a general point in B is smooth. Let Σ ⊂ X n be the set of criticalpoints of pr , ··· ,n − : X n → ( P ) n − . By generality, it is sufficient to show thatdim Σ ∩ X n − = n −
3. This can be checked by a direct computation as follows.The equation of X n ⊂ ( P ) n +1 can be written as S n +1 F + S n +1 T n +1 G + T n +1 H = 0 , where F = 0 defines X n − ⊂ ( P ) n . If we write F as F = S n F + S n T n F + T n F , then the set Σ ∩ X n is defined by S n +1 = 2 S n F + T n F = S n F + 2 T n F = G = 0 . The first three equations define the ramification locus, that is, the set of criticalpoints of pr , ··· ,n − : X n − → ( P ) n − , which is of dimension n −
2, thus the fourequations together define a closed subset of dimension n − (cid:3) Now Theorem 3.2 implies:
Theorem 3.4.
The set X n ( Q ) is Zariski dense for any n ≥ .Remark . For a smooth hypersurface X in ( P ) n +1 of multi-degree (2 n +1 ), thebirational automorphism group Bir( X C ) is infinite by Cantat-Oguiso [7]. It wouldbe possible to give a proof of the density of rational points by using the action ofBir( X C ). IGHER-DIMENSIONAL CALABI-YAU VARIETIES WITH DENSE SETS OF RATIONAL POINTS Calabi-Yau threefolds containing an Enriques surface
In this section, we work over the complex numbers. We construct a Calabi-Yauthreefold containing an Enriques surface and prove basic properties of the threefold,which will be used in Section 5.Let P = P P ( O ⊕ ⊕ O (1)). On the projective bundle P , we consider a map ofvector bundles u : O ⊕ → O (2 H ) ⊕ O (2 H ) , where H (resp. H ) is the pull-back of the hyperplane section class on P (resp.the tautological class on P ). Let X be the rank one degeneracy locus of u . Lemma 4.1. If u is general, X is a Calabi-Yau threefold. We have the topologicalEuler characteristic χ top ( X ) = c ( T X ) = − and the Hodge numbers h , ( X ) = 2 and h , ( X ) = 44 .Proof. Since the vector bundle O (2 H ) ⊕O (2 H ) is globally generated, X is a smooththreefold by the Bertini theorem for degeneracy loci. Another projective modelof X is defined as the zero set of a naturally defined section of O (1) ⊕ on theprojective bundle e P = P P ( O (2 H ) ⊕ O (2 H )). Let e H be the tautological class on e P . The adjunction formula gives ω X = O X ( e H − H ). On the other hand, e H − H is the class of the intersecction X ∩ P P ( O (2 H )), which is empty, thus we have O X ( e H − H ) = O X . It follows that ω X = O X . The rest of the statement is aconsequence of a direct computation using the conormal exact sequence and theKoszul resolution of the ideal sheaf of X in e P . The proof is complete. (cid:3) We assume that u is general in what follows. Lemma 4.2.
The threefold X admits an elliptic fibration φ : X → P . Moreover, X contains an Enriques surface S and the linear system | S | defines a K3 fibration ψ : X → P .Remark . There are several examples of Calabi-Yau threefolds containing anEnriques surface. For instance, see Borisov-Nuer [5].
Proof of Lemma 4.2.
A natural projection P → P restricts to a surjection φ : X → P with the geometric generic fiber a complete intersection of two quadrics in P ,which is an elliptic curve. The morphism φ has equidimensional fibers, hence it isflat.Moreover, the map u restricts to a map of vector bundles v : O ⊕ → O (2 , ⊕ O (0 , P P ( O ⊕ ) = P × P . The intersection X ∩ P P ( O ⊕ ) is the rank one degeneracylocus of v , which is an Enriques surface S by generality (see [16, Lemma 2.1]). Thenthe linear system | S | defines a K3 fibration ψ : X → P by [5, Proposition 8.1].The proof is complete. (cid:3) Corollary 4.4.
The threefold X contains no abelian surface.Proof. If X contains an abelian surface A , then it is easy to see that the linearsystem | A | defines an abelian surface fibration η : X → P . Then pulling backNS( P × P ) R by ( ψ, η ) : X → P × P defines a two-dimensional linear subspaceof NS( X ) R which does not contain an ample divisor. Therefore we should have ρ ( X ) ≥
3. This is impossible since we have ρ ( X ) = h , ( X ) = 2 by Lemma 4.1. (cid:3) Lemma 4.5.
Nef( X ) = Eff( X ) = R ≥ [ H ] ⊕ R ≥ [ S ] . FUMIAKI SUZUKI
Proof.
Since the linear systems | H | and | S | define the fibrations φ : X → P and ψ : X → P respectively, the divisors H and S are semi-ample but not big, so theirclasses give extremal rays in Eff( X ). This finishes the proof. (cid:3) Corollary 4.6.
The threefold X is a unique minimal model in its birational equiv-alence class. Moreover, Bir( X ) = Aut( X ) and these groups are finite.Proof. Let f : X X ′ be a birational map with X ′ a minimal model. Then f can be decomposed into a sequence of flops by a result of Kawamata [12, Theorem1]. Note that any flopping contraction of a Calabi-Yau variety is given by a codi-mension one face of the nef cone (see [11, Theorem 5.7]). Since the codimensionone faces R ≥ [ H ] and R ≥ [ S ] of Nef( X ) give fibrations, it follows that f is infact an isomorphism. For the last statement, we note that Oguiso proved that theautomorphism group of any odd-dimensional Calabi-Yau variety in a wider sensewith ρ = 2 is finite. The proof is complete. (cid:3) Proposition 4.7.
The threefold X is simply connected.Proof. Applying [13, Lemma 5.2.2] to the K3 fibration ψ : X → P , whose smoothfibers are simply connected, we are reduced to showing that 2 S is the only onemultiple fiber of ψ . The K3 fibration ψ : X → P and the morphism X → P givenby the linear system | H | induce a morphism X → P × P , which is the blow-up of anon-normal complete intersection X in P × P of three hypersurfaces of bi-degree(1 ,
2) along the non-normal locus with the exceptional divisor S . Now we only needto prove that the first projection pr : X → P admits no multiple fibers. Thisfollows from the Lefschetz hyperplane section theorem. The proof is complete. (cid:3) Construction III
We recall a system of affine equations for an Enriques surface introduced byColliot-Th´el`ene–Skorobogatov–Swinnerton-Dyer [8].
Proposition 5.1 ([8], Proposition 4.1, Example 4.1.2; [14], Proposition 1.1) . Let k be a field of characteristic zero. Let c, d, f ∈ k [ t ] be polynomials of degree such that c · d · ( c − d ) · f is separable. Let S be the affine variety in A = Spec k [ t, u , u , u ] defined by u − c ( t ) = f ( t ) u , u − d ( t ) = f ( t ) u . Then a minimal smooth projective model S of S is an Enriques surface. Theprojection S → A = Spec k [ t ] induces an elliptic fibration S → P with reduceddiscriminant c · d · ( c − d ) · f which admits double fibers over f = 0 whose reductionsare smooth elliptic curves. Now we apply Proposition 5.1 to k = Q and c = − t − t + 8 , d = t − t + 162 , f = t + 12 . Let S be the corresponding Enriques surface. We prove: Lemma 5.2.
The surface S has a Zariski dense set of Q -rational points.Proof. We follow the strategy of the proof of [20, Proposition 5]. It will be easierto work on the K3 double cover e S . By setting w = t + 12 , v = wu , v = wu , we obtain the defining equations of its affine model e S in A = Spec Q [ t, u , v , v , w ]: u − ( − t − t + 8) = v , u − t − t + 162 = v , t + 12 = w . IGHER-DIMENSIONAL CALABI-YAU VARIETIES WITH DENSE SETS OF RATIONAL POINTS The projection e S → C = Spec Q [ t, w ] / ( t +12 − w ) defines an elliptic fibration e S → C with reduced discriminant c · d · ( c − d ).We consider the curve E ⊂ e S cut out by u = t − , v = 2 t − , which is isomorphic to the affine curve in A = Spec Q [ t, v , w ] defined by12 ( t + 1)( t + 2) = v , t + 12 = w . Then E gives an elliptic curve E . We prove that E ( Q ) is dense. Let O ∈ E be thepoint given by t = − , v = 0 , w = − P ∈ E be the point given by t = 7 , v = − , w = 5 . We only need to prove that P − O ∈ Pic ( E ) is of infinite order. It is a simplematter to check that the map( t, v , w ) (cid:18) t − w − t − w − , v t − w − (cid:19) gives a transformation into the Weierstrass model y = x − x + 144and sends O (resp. P ) to the point at infinity (resp. ( x, y ) = (0 , x, y ) = (0 ,
12) defines a point of infinite order. This followsfrom a theorem of Lutz and Nagell [19, VIII. Corollary 7.2] since the y -value 12 isnon-zero and 12 does not divide 4 · ( − + 27 · .Moreover, E is a saliently ramified multi-section of the elliptic fibration e S → C .Indeed, it is easy to verify that E → C is branched over t = − , w = ± , while t = − c · d · ( c − d ) = ( − t − t + 8) (cid:18) t − t + 162 (cid:19) (cid:18) − t + 11 t (cid:19) . Now Theorem 3.2 shows that e S ( Q ) is dense. This in turn implies that S ( Q ) isdense. The proof is complete. (cid:3) We define a compactification of S in P × P as follows. We set P × P = Proj Q [ X , X , X ] × Proj Q [ Y , Y , Y ] . On P × P , we consider the map of vector bundles v : O ⊕ → O (2 , ⊕ O (0 , × (cid:18) X X X Y + 6 Y + 4 Y Y − Y Y − Y + 15 Y Y − Y Y + Y (cid:19) . Let S ′ be the rank one degeneracy locus of v . It is straightforward to see that S ′ isindeed a compactification of S . The surface S ′ is a local complete intersection, soin particular, Gorenstein. The surface S ′ has isolated singular points and blowingup along the points gives a crepant resolution S → S ′ .Finally, we give the construction of the Calabi-Yau theefold. On the projectivebundle P = P P ( O ⊕ ⊕ O (1)), we let u : O ⊕ → O (2 H ) ⊕ O (2 H ) FUMIAKI SUZUKI be general among all maps defined over Q which restrict to v on P P ( O ⊕ ) = P × P .We define X to be the rank one degeneracy locus of u . Theorem 5.3.
The threefold X is a Calabi-Yau threefold in a strict sense definedover Q with a Zariski dense set of Q -rational points. Moreover, X satisfies thefollowing geometric properties:(1) X C admits K3 and elliptic fibrations;(2) X C contains no abelian surface;(3) X C is a unique minimal model in its birational equivalence class;(4) Bir( X C ) = Aut( X C ) and these groups are finite.Proof. One can check that X is smooth and the proofs in Section 4 still go through.It remains to show that X ( Q ) is Zariski dense. By a similar argument to that inLemma 3.3, S ′ is a saliently ramified multi-section of the elliptic fibration φ : X → P . Since S ′ ( Q ) is dense by Lemma 5.2, the result follows from Theorem 3.2. Theproof is complete. (cid:3) One can be more explicit. We fix a non-zero element Z ∈ H ( P , O P ( H − H )).Let u be given by the matrix (cid:18) P Q R P Q R (cid:19) , where P = X , Q = X , R = X and P = ( X + X ) Z + ( X Y + X Y ) Z + 2 Y + 6 Y + 4 Y Y − Y ,Q = ( X + X ) Z + ( X Y + X Y ) Z + 2 Y − Y + 15 Y Y − Y ,R = ( X + X ) Z + ( X Y + X Y ) Z + Y + Y . Then
Macaulay 2 shows that the corresponding X is smooth and that the discrim-inant locus ∆ ⊂ P of φ : X → P and the branch locus B ⊂ P of φ | S : S → P meet properly. Consequently, the set X ( Q ) is Zariski dense. References [1] Bogomolov, F. A., Halle, L. H., Pazuki, F., Tanimoto, S.:
Abelian Calabi-Yau threefolds:N´eron models and rational points , Math. Res. Lett. (2018), no. 2, 367–392.[2] Bogomolov, F. A., Tschinkel, Y.: Density of rational points on Enriques surfaces , Math. Res.Lett. (1998), no. 5, 623–628.[3] Bogomolov, F. A., Tschinkel, Y.; On the density of rational points on elliptic fibrations , J.Reine Angew. Math. (1999), 87–93.[4] Bogomolov, F. A., Tschinkel, Y.:
Density of rational points on elliptic K3 surfaces , Asian J.Math. 4 (2000), no. 2, 351–368.[5] Borisov, L. A., Nuer, H. J.: On (2 , complete intersection threefolds that contain an En-riques surface , Math. Z. 284 (2016), no. 3-4, 853–876.[6] Campana, F.: Orbifolds, special varieties and classification theory , Ann. Inst. Fourier (Greno-ble) (2004), no. 3, 499–630.[7] Cantat, S., Oguiso, K.: Birational automorphism groups and the movable cone theorem forCalabi-Yau manifolds of Wehler type via universal Coxeter groups , Amer. J. Math. (2015), no. 4, 1013–1044.[8] Colliot-Th´el`ene, J. -L., Skorobogatov, A. N., Swinnerton-Dyer, P.:
Double fibres and doublecovers: paucity of rational points , Acta Arith. (1997), no. 2, 113–135.[9] Elkies, N. D. : On A + B + C = D , Math. Comp. 51 (1988), no. 184, 825–835.[10] Gross, M., Popescu, S.: Calabi-Yau threefolds and moduli of abelian surfaces. I , CompositioMath. (2001), no. 2, 169–228.[11] Kawamata, Y:
Crepant blowing-up of -dimensional canonical singularities and its applica-tion to degenerations of surfaces , Ann. of Math. (2) (1988), no. 1, 93–163. IGHER-DIMENSIONAL CALABI-YAU VARIETIES WITH DENSE SETS OF RATIONAL POINTS [12] Kawamata, Y.: Flops connect minimal models , Publ. Res. Inst. Math. Sci. (2008), no. 2,419–423.[13] Koll´ar, J.: Shafarevich maps and plurigenera of algebraic varieties , Inventiones mathematicae113.1 (1993): 177-215.[14] Lafon, G.:
Une surface d’Enriques sans point sur C (( t )), C. R. Math. Acad. Sci. Paris 338(2004), no. 1, 51–54.[15] Logan, A., McKinnon, D., van Luijk, R.: Density of rational points on diagonal quarticsurfaces , Algebra Number Theory (2010), no. 1, 1–20.[16] Ottem, J. C., Suzuki, F.: A pencil of Enriques surfaces with non-algebraic integral Hodgeclasses , Math. Ann. (2020), no. 1-2, 183–197.[17] Schoen, C.:
Complex multiplication cycles on elliptic modular threefolds , Duke Math. J. (1986), no. 3, 771–794.[18] Schoen, C.: On fiber products of rational elliptic surfaces with section , Math. Z. 197 (1988),no. 2, 177–199.[19] Silverman, J. H.:
The arithmetic of elliptic curves , second edition, Graduate Texts in Math-ematics, 106, Springer, Dordrecht, 2009.[20] Skorobogatov, A.:
Enriques surfaces with a dense set of rational points , Appendix to: “Hasseprinciple for pencils of curves of genus one whose Jacobians have a rational 2-division point,close variation on a paper of Bender and Swinnerton-Dyer” [in
Rational points on algebraicvarieties , 117–161, Progr. Math., 199, Birkh¨auser, Basel, 2001; 1875172] by J.-L. Colliot-Th´el`ene, in
Rational points on algebraic varieties , 163–168, Progr. Math., 199, Birkh¨auser,Basel.[21] Swinnerton-Dyer, H. P. F. : A + B = C + D revisited , J. London Math. Soc. (1968),149–151.[22] Swinnerton-Dyer, P.: Density of rational points on certain surfaces , Algebra Number Theory (2013), no. 4, 835–851.[23] Tschinkel, Y.: Geometry over nonclosed fields , in
International Congress of Mathematicians.Vol. II , 637–651, Eur. Math. Soc., Z¨urich.
UCLA Mathematics Department, Box 951555, Los Angeles, CA, 90095-1555
Email address ::