ON THE BRAUER GROUP OF BIELLIPTIC SURFACES EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP WITH AN APPENDIX BY J ONAS B ERGSTRÖM AND S OFIA T IRABASSI A BSTRACT. We provide explicit generators of the torsion of the second cohomology of bielliptic surfaces, and we use these to study the pullback map between the Brauer group of a bielliptic surface and that of its canonical cover. We completely describe the loci where this map is injective, trivial, or neither one of them. Explicit examples are constructed for all possible different behaviors of this morphism. C ONTENTS 1. Introduction 2. Background and preliminary results 3. Generators for the torsion of the second cohomology for bielliptic surfaces 4. The Brauer map to another bielliptic surface 5. The Brauer map to the canonical cover Appendix A. The homomorphism lattice of two elliptic curves References 1 4 9 13 16 28 30 1. I NTRODUCTION Given a smooth complex projective variety Z its (cohomological) Brauer group is defined 2 ( Z, O ∗ ) . A morphism of projective varieties f : Z → Y induces, via as Br( Z ) := Hét Z tor pullbacks, a homomorphism f Br : Br(Y ) → Br( Z ), which we call the Brauer map induced by f . In [Bea09] Beauville studies this map in the case of a complex Enriques surface S and that of its K3 canonical cover π : X → S. More precisely the author of [Bea09] identifies the locus in the moduli space of Enriques surfaces in which πBr is not injective (and therefore trivial). In this paper we carry out a similar investigation in the case of bielliptic surfaces. A bielliptic surface is constructed by taking the quotient of a product of ellipic curves A × B by the action of a finite group G. They were classified in 7 different types by Bagnera– De Franchis (see [BDF10], or [Suw69] for a presentation in a more modern language), as illustrated in Table 1. Since the canonical bundle of a bielliptic surface S is a torsion element G Order of ωS in Pic(S) H 2 (S, Z)tor 1 Z/2Z 2 Z/2Z × Z/2Z 2 Z/2Z × Z/2Z 2 Z/2Z 3 Z/4Z 4 Z/2Z 4 Z/4Z × Z/2Z 4 0 5 Z/3Z 3 Z/3Z 6 Z/3Z × Z/3Z 3 0 7 Z/6Z 6 0 Type Table 1. Types of bielliptic surfaces and torsion of their second cohomology. 2 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP in Pic(S), it can be used to define an étale cyclic cover π : X → S, where X is an abelian variety isogenous to A × B. We then obtain a homomorphism between the respective Brauer groups: πBr : Br(S) → Br( X ). A very natural question is the following. Question. When is πBr injective? When is it trivial? As for Enriques surfaces, using the long exact exponential sequence and Poincaré duality, we have a non canonical isomorphism Br(S) ' H 2 ( X, Z)tor , so from the fourth column of Table 1, we easily see that this map is trivial when S is of type 4, 6 or 7. Thus we will limit ourselves to surfaces of type 1, 2, 3, and 5. We will find that the behavior of the Brauer map depends heavily on the geometry of the bielliptic surface S. Bielliptic surfaces of type 2 and 3 admit (by a construction which is written explicitly only in [Nue]) a degree 2 étale cover π̃ : S̃ → S, with S̃ a bielliptic surface of type 1 (see Paragraph 2.3 for more details). We first investigate the properties of the induced Brauer map for these types of covers finding how this behaves differently in the two cases: Theorem A. (a) If S is of type 2, then π̃Br : Br(S) → Br(S̃) is trivial. (b) If S is of type 3, then π̃Br : Br(S) → Br(S̃) is injective. The main tool behind our argument is a result of Beauville (see Section 2 for more details) which states that the kernel of the Brauer map of a cyclic étale cover X → X/σ is naturally isomorphic to the kernel of the norm map Nm : Pic( X ) → Pic( X/σ) quotiented by Im(1 − σ∗ ). We prove that a line bundle on S̃ is in the kernel of the norm map only if it is numerically trivial. Then we reach our conclusion by carefully computing the norm map of numerically trivial line bundles. The different behaviors of the two type of surfaces is motivated by the different "values" taken by the norm map on torsion elements of H 2 (S̃, Z): in the type 2 case they are sent to topologically trivial line bundles, while this is not true in the type 3 case. Theorem A is interesting in itself, and some parts of its proof will be useful in order to study the Brauer map to the canonical cover for bielliptic surfaces of type 2. We then turn our attention to the main focus of this paper: to give a complete description for the Brauer map to the canonical cover of any bielliptic surface. We first show, similarly to what happens for Enriques surfaces, that the Brauer map is injective for a general bielliptic surface. In particular we show the following general statement: Theorem B. Given a bielliptic surface S, let π : X → S be its canonical cover. If the two elliptic curves A and B are not isogenous, then the pullback map πBr : Br(S) → Br( X ) is injective. When A and B are isogenous one encounters the first examples of bielliptic surfaces with non injective Brauer map. It turns out that the behavior of the Brauer map depends on the explicit action of the group G on the product A × B. This is showed in Theorems 5.3, 5.8, 5.10, 5.14, 5.19, and 5.21, where we give necessary and sufficient conditions for the Brauer map to be injective, trivial, and, in the case of type 1 surfaces (whose Brauer group is not simple), neither trivial nor injective. Our very explicit results allow us to construct examples of bielliptic surfaces exhibiting each of the possible behaviors of the Brauer map. Unfortunately the statements are involved and it is not possible reproduce them here without a lengthy explanation of the notation used. The rough geometric picture is the following: Type 1: These surfaces are constructed by choosing two elliptic curves A and B and a 2torsion point on A. Thus the moduli space has dimension 2. In order to have a non injective Brauer map one can choose freely the elliptic curve B, but has only finitely many possibilities BRAUER GROUP OF BIELLIPTIC SURFACES 3 for A and the 2-torsion point. Thus we obtain a 1- dimensional family. In Example 5.9 we show that uncountably many such surfaces exist. On the other hand only countably many type 1 bielliptic surfaces can have a trivial Brauer map to their canonical cover. In fact, to obtain a trivial Brauer map one has to choose the ellipic curve B among those having complex multiplication. Type 2: These surfaces are constructed by choosing two elliptic curves A and B and two 2torsion points, one on A and one on B. Hences the moduli space has dimension 1. Similarly to what happens in the previous case, in order to have a non injective (and hence trivial) Brauer map only the choice of the curve B can be made freely, while A must be taken among finitely many possibilities. We show in this case (cf. Example 5.25(a)) that if the curves A and B are isomorphic, regardless of the choice of the torsion points, the Brauer map is trivial. Type 3: These surfaces are constructed by choosing one elliptic curve A and a 4-torsion point on it. Therefore the moduli space has dimension 1. In order to have a non injective (and hence trivial) Brauer map, A must be isogenous to the curve with j-invariant 1728. Thus there are only finitely many such surfaces. Type 5: These surfaces are constructed by choosing one elliptic curve A and a 3-torsion point on it. We deduce that the moduli space has dimension 1. In order to have a non injective (and hence trivial) Brauer map, A must be isogenous to the curve with j-invariant 0. Thus, as in the previous case, there are only finitely many such surfaces. The proof of Theorem B uses the same ideas as in the proof of Theorem A. In fact we can leverage on the fact that X and S have the same Picard number (as for the case of a bielliptic cover) to show that line bundles in the kernel of the norm map are topologically trivial. The result is then obtained by showing that line bundles in Pic0 ( X ) which are also in the kernel of the norm map are always in Im(1 − σ∗ ). As a corollary of both Theorem A and B we find an example of isogeny between two abelian varieties ϕ : X → Y such that the corresponding group homomorphism ϕBr is not injective, see Section 5.2. This paper is organized as follows. Section 2 contains all the background and preliminary results. More precisely we outline some classical facts on the geometry of bielliptic surfaces, and present the construction, due to Nuer, of the bielliptic covers of surfaces of type 2 and 3. We also expose the work of Beauville [Bea09] which allows us to study the kernel of the Brauer map in terms of the norm homomorphism of the cover. We conclude the section by describing the Neron–Severi group of a product of elliptic curves. In Section 3 we provide explicit generators for H 2 (S, Z)tor , when S is a bielliptic surface of type 1, 2, 3 or 5. We prove Theorem A in Section 4, while we completely describe the norm map to the canonical cover in Section 5. Here we also construct examples of bielliptic surfaces of every type in which the Brauer map behaves differently. In the Appendix, which is joint work of the second author of the main paper with J. Bergström, a structure theorem for the homomorphism ring of two elliptic curves is given in the case of j-invariant 0 or 1728. This will give, in turn, a really useful description of the Picard group of the product of such curves, which is fundamental to study the Brauer map of bielliptic surfaces of type 3 and 5. Notation. We are working over the field of complex numbers C. If X is a complex abelian variety over C, and n ∈ Z, then X [n] will denote the subscheme of n-torsion points of X, while n X : X → X will stand for the "multiplication by n isogeny". Given x ∈ X a point, then the translation by x will be denoted as t x . In addition, if dim X = 1 – that is, X is an elliptic curve – then Px will be the line bundle O X ( x − p0 ) ' t∗− x O X ( p0 ) ⊗ O X (− p0 ) in Pic0 ( X ), where p0 ∈ X is the identity element. For any smooth complex projective variety Y we will denote the identity homomorphism as 1Y (or simply 1 if there is no chance of confusion), while KY and ωY will stand for the canonical divisor class and the dualizing sheaf on Y, respectively. If D and E are two linearly 4 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP equivalent divisors on Y we will write D ∼ E; in addition OY ( D ) will denote the line bundle associated to the divisor D. Acknowledgement. All the authors were partially supported by the grant 261756 of the Research Councils of Norway. We are grateful to H. Nuer for sharing with us an early draft of [Nue] and the construction of a bielliptic cover of bielliptic surfaces of type 2 and 3. We would also like to thank T. Suwa for explaining many facts about the torsion of the H 2 ( X, Z) for bielliptic surfaces. 2. B ACKGROUND AND PRELIMINARY RESULTS 2.1. Bielliptic surfaces. A complex bielliptic (or hyperelliptic) surface S is a minimal smooth projective surface over the field of complex numbers with Kodaira dimension κ (S) = 0, irregularity q(S) = 1, and geometric genus p g (S) = 0. By the work of Bagenera– De Franchis (see for example [Bad01, 10.24-10.27]), the canonical bundle ωS has order either 2, 3, 4 or 6 in Pic(S), and S occurs as a finite étale quotient of a product A × B of elliptic curves by a finite group G acting on A by translations and on B such that B/G ' P1 . More precisely we have the following classification result. Theorem 2.1 (Bagnera–De Franchis [BDF10], [Suw69, Theorem at p. 473], [BM77, p. 37]). A bielliptic surface is of the form S = A × B/G, where A and B are elliptic curves and G a finite group of translations of A acting on B by automorphisms. They are divided into seven types according to G as shown in Table 1. There are natural maps aS : S → A/G and g : S → B/G ' P1 which are both elliptic fibrations. The morphism as is smooth, and coincides with the Albanese morphism of S. On the other hand, g admits multiple fibers, corresponding to the branch points of the quotient B → B/G, with multiplicity equal to that of the associated branch point. The smooth fibers of aS and g are isomorphic to B and A, respectively. We will denote by a and b the classes of these fibers in Num(S), H 2 (S, Z) and H 2 (S, Q). It is well known (see for example [Ser90a, p. 529]) that a and b span H 2 (S, Q) and satisfy a2 = b2 = 0, ab = | G |. Furthermore, we have the following description of the second cohomology of S: Proposition 2.2. The decomposition of H 2 (S, Z) is described according to the type of S and the multiplicities (m1 , . . . , ms ) of the singular fibers of g : S → P1 as follows: Type (m1 , . . . , ms ) H 2 (S, Z) H 2 (S, Z)tor Z[ 21 a] ⊕ Z[b] ⊕ Z/2Z ⊕ Z/2Z Z/2Z × Z/2Z 1 (2, 2, 2, 2) 2 (2, 2, 2, 2) Z[ 12 a] ⊕ Z[ 21 b] ⊕ Z/2Z Z/2Z 3 (2, 4, 4) Z[ 14 a] ⊕ Z[b] ⊕ Z/2Z Z/2Z 4 (2, 4, 4) Z[ 41 a] ⊕ Z[ 12 b] 0 5 (3, 3, 3) Z[ 13 a] ⊕ Z[b] ⊕ Z/3Z Z/3Z 6 (3, 3, 3) 0 7 (2, 3, 6) Z[ 31 a] ⊕ Z[ 13 b] Z[ 16 a] ⊕ Z[b] 0 Proof. See [Ser90a, Tables 2 and 3]. The computation of the torsion of H 2 (S, Z) can be also found in [Iit70, Ser91, Suw69, Ume75]. BRAUER GROUP OF BIELLIPTIC SURFACES 5 Since H 2 (S, OS ) = 0, the first Chern class map c1 : Pic(S) → H 2 (S, Z) is surjective, so the Néron-Severi group NS(S) ' H 2 (S, Z). Modulo torsion we then get Num(S) = Z[ a0 ] ⊕ Z[b0 ] where a0 = 1 a ord (ωS ) and b0 = ord (ωS ) b. |G| 2.2. Canonical covers. Let S be a bielliptic surface and denote by n the order of its canonical bundle. Then, by a classical construction (see for example [BM98, Section 2]), ωS induces an étale cyclic cover πS : X → S, called the canonical cover of S. From now on, when there is no confusion, we will omit the subscript S and write simply π : X → S. If we let λS := | G |/ ord (ωS ), we have that G ' Z/nZ ⊕ Z/λS Z, and X is the abelian surface sitting as an intermediate quotient / S ' A × B/G 5 A×B ' π X ' A × B/H where H ' Z/λS Z. The abelian surface X thus comes with homomorphisms of abelian varieties p A : X → A/H and p B : X → B/H with kernels isomorphic to B and A, respectively. Denoting by a X and bX the classes of the fibers A and B in Num( X ), we have a X · bX = λS and the embedding π ∗ : Num(S) ,→ Num( X ) satisfies n bX . (2.1) π ∗ a0 = a X , π ∗ b0 = λS There is a fixed-point-free action of the group Z/nZ on the abelian variety X such that the quotient is exactly S. We will denote by σ ∈ Aut( X ) a generator of Z/nZ. In what follows it will be useful to have an explicit description of σ when S is of type 1, 2, 3, or 5. Suppose first that S is of type 1, 3, or 5, so G is cyclic, H is trivial, and X ' A × B. If S is of type 3, then the j-invariant of B is 1728, and B admits an automorphism ω : B → B of order 4. If S is of type 5, B has j-invariant 0 and admits an automorphism ρ of order 3 (see for example [BM77, p. 37], [Bad01, List 10.27] or [BHPvdV15, p. 199]). With this notation we have that the automorphism σ of A × B inducing the covering π is given by if S is of type 1, ( x + τ, −y), (2.2) σ( x, y) = ( x + e, ω (y)), if S is of type 3, ( x + η, ρ(y)), if S is of type 5, where τ, e, and η are points of A of order 2, 4, and 3 respectively. We remark that different choices for the automorphism ρ and ω - there are two possible choices in each case- will lead to isomorphic bielliptic surfaces. If S is otherwise of type 2, then there are points θ1 ∈ A and θ2 ∈ B, both of order two, such that X is the quotient of A × B by the involution ( x, y) 7→ ( x + θ1 , y + θ2 ). If we denote by [ x, y] the image of ( x, y) through the quotient map, we have that (2.3) σ[ x, y] = [ x + τ, −y], where τ ∈ A is a point of order 2, τ 6= θ1 . 2.3. Covers of bielliptic surfaces by other bielliptic surfaces. When G is not a cyclic group, or when G is cyclic, but the order of G is not a prime number, then the bielliptic surface S admits a cyclic cover π̃ : S̃ → S, where S̃ is another bielliptic surface. This construction, together with the statement of Lemma 2.3, appears explicitly in the unpublished 6 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP work of Nuer [Nue],and is implicit in the work of Suwa [Suw69, p. 475]. The main point that we will need in Section 4 is the description of the pull-back map Num(S) → Num(S̃). Lemma 2.3. (i) Let S be a bielliptic surface such that ord(ωS ) is not a prime number and take d a proper divisor of n. Then there is a bielliptic surface S̃ sitting as an intermediate étale cover between S and X, πS̃ X / S̃ 4/ S π̃ πS such that ord (ωS̃ ) = ord (ωS ) d and π̃ ∗ a0 = a˜0 , π̃ ∗ b0 = db˜0 , where a˜0 , b˜0 are the natural generators of Num (S̃). (ii) Let S be a bielliptic surface with λS > 1, i.e., with G not cyclic. Then there is a bielliptic surface S̃ sitting as an intermediate étale cover between S and A × B A×B πS̃ / S̃ π̃ /4 S πS such that λS̃ = 1, ord(ωS̃ ) = ord(ωS ) and π̃ ∗ a0 = λS a˜0 , π̃ ∗ b0 = b˜0 , where a˜0 , b˜0 are the natural generators of Num (S̃). In what follows we will need a more explicit construction of S̃, when S is either of type 2 or 3. Example 2.4. (a) Suppose that S is a bielliptic surface of type 3. Then the canonical bundle has order 4. In addition the canonical cover X of S is a product of elliptic curves, that is X ' A × B. Using the notation of (2.2), we obtain S̃ from A × B by taking the quotient with respect to the involution ( x, y) 7→ ( x + 2e, −b). Thus we have that S̃ is a bielliptic surface of type 1. The map π̃ : S̃ → S is an étale double cover with associated involution σ̃. Hence, given s ∈ S̃, we can see it as an equivalence class [ x, y] of a point ( x, y) ∈ A × B. Then we have an explicit expression for σ̃: (2.4) σ̃(s) = [ x + e, ω (y)]. (b) Suppose that S is a bielliptic surface of type 2, so the group G is isomorphic to the product Z/2Z × Z/2Z. Then we obtain S̃ from A × B by taking the quotient with respect to ( x, y) 7→ ( x + τ, −y), where we are using the notation of (2.3). Thus, as in 2.4, S̃ is a bielliptic surface of type 1 and each s ∈ S̃ can be written as an equivalence class [ x, y] of a point ( x, y) ∈ A × B. If we denote again by σ̃ the involution induced by the cover π̃ : S̃ → S, we have the following: (2.5) σ̃(s) = [ x + θ1 , y + θ2 ]. 2.4. Norm homomorphisms. Let π : X → Y be a finite locally free morphism of projective varieties of degree n. To it we can associate a group homomorphism Nmπ : Pic( X ) → Pic(Y ) called the norm homomorphism associated to π. This is constructed in the following manner. First, one lets B := π∗ O X , and defines a morphism of sheaves of multiplicative monoids N : B → OY : given s a section of B on an open set U, let ms be the endomorphism of B (U ) induced by the multiplication by s; we set N (s) := det(ms ) ∈ OY (U ) (see [Gro61, § 6.4, and §6.5] or [Sta19, Lemma 0BD2] ). The restriction of N to invertible sections induces a morphism of sheaves of groups N : B ∗ → OY∗ . Now, given L an invertible sheaf on X, π∗ L is an invertible B-module and, as such is represented by a cocycle {uij , Ui } for an open BRAUER GROUP OF BIELLIPTIC SURFACES 7 cover {Ui } of Y. Observe that uij ∈ B ∗ (Uij ). The fact that N is multiplicative ensures that also the vij := N (uij ) satisfies the cocycle condition and so uniquely identifies a line bundle Nmπ ( L) on Y. The map L 7→ Nmπ ( L) is a group homomorphism by [Gro61, (6.5.2.1)]. In addition [Gro61, (6.5.2.4)] ensures that (2.6) Nmπ (π ∗ M ) ' M⊗n , and we also have the following important property: Proposition 2.5. Given two finite locally free morphism π1 : X → Y and π2 : Y → Z, then Nmπ2 ◦π1 = Nmπ2 ◦ Nmπ1 Proof. See [Gro67, Lemma 21.5.7.2]. Suppose now that π : X → Y is an étale cyclic cover of degree n. Then there is a fixedpoint-free automorphism σ : X → X of order n such that Y ' X/σ. In addition we can L −1 ⊗ h write B ' nh= with M a line bundle of order n in Pic(Y ). In this particular setting 0 M the norm homomorphism satisfies some additional useful properties. First, as Nmπ behaves well with base change ( [Gro61, Proposition 6.5.8]), it is not difficult to see that (2.7) Nmπ ◦(1X − σ∗ ) = 0. In additon, as discussed by Beauville in [Bea09], we have that (2.8) π ∗ Nmπ ( L) ' n O (σ h )∗ L h =0 In fact, by the definiton of pushforward of divisors ( [Gro67, Definition 21.5.5]), if L ' O X (∑ ai · Di ) with prime divisors on X, then Nmπ ( L) ' OY (∑ ai · π∗ Di ). Therefore (2.8) −1 h ∗ follows form the fact that for a prime divisor D we have that π ∗ π∗ D ∼ ∑nh= 0 ( σ ) D. Remark 2.6 (Pic0 trick). In what follows it will be important to provide elements in the kernel of the Norm homomorphism. We will often use the following trick. Let π : X → Y be an étale morphism of degree n and suppose that there is a line bundle L on X such that Nmπ ( L) ∈ Pic0 (Y ). Then there is an element α ∈ Pic0 ( X ) such that Nmπ ( L ⊗ α) is trivial. In fact, as abelian varieties are divisible groups, it is possible to find β ∈ Pic0 (Y ) such that β⊗n ' Nmπ ( L)−1 . Then, by (2.6) we get Nmπ ( L ⊗ π ∗ β) ' Nmπ ( L) ⊗ β⊗n ' OY . We conclude this paragraph by saying that, from now on, if there is no possibility of confusion, we will omit the subscript when denoting the norm. That is we will write Nm instead of Nmπ 2.5. Brauer groups and Brauer maps. For a scheme X, the cohomological Brauer group 2 ( X, O ∗ ). For complex Br0 ( X ) is defined as the torsion part of the étale cohomology group Het X ∗ ) in the analytic topology. In addition, varieties, this is isomorphic to the torsion of H 2 ( X, O X when X is quasi-compact and separated, by a theorem of Gabber (see, for example, [dJ] for more details) the cohomological Brauer group of X is canonically isomorphic to the Brauer group Br( X ) of Morita-equivalence classes of Azumaya algebras on X. For what it concerns the present document, we will only be concerned with smooth complex projective varieties, therefore all these three groups will be isomorphic and will be denoted simply by Br( X ). Furthermore we will only speak of the Brauer group of X, without any additional connotation. If S is a bielliptic surface, the exponential sequence yields that H 3 (S, Z) ' H 2 (S, OS∗ ), so that the Brauer group of S is isomorphic to the torsion of H 3 (S, Z). By Poincaré duality and the universal coefficients theorem, the torsion of H 3 (S, Z) is (non canonically) isomorphic 8 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP to the torsion of H 2 (S, Z), so the isomorphism type of the Brauer group of S can be deduced in terms of Proposition 2.2. Crucial to our purposes will be the following result of Beauville which describes the kernel of the Brauer map πBr when π is a cyclic étale cover. Proposition 2.7 ( [Bea09, Prop. 4.1]). Let π : X → S be an étale cyclic covering of smooth projective varieties. Let σ be a generator of the Galois group of π, Nm : Pic( X ) → Pic(S) be the norm map and πBr : Br(S) → Br( X ) be the pullback. Then we have a canonical isomorphism Ker(πBr ) ' Ker Nm/(1 − σ∗ ) Pic( X ). 2.6. The Neron–Severi of a product of elliptic curves. In this paragraph we want to describe Num( A × B) when A and B are two elliptic curves. We will do so by using the identification of Num( X ) ' NS( X ) which holds for abelian surfaces. We believe that many of these topics might be well known by experts, but we were not able to find a rigorous literature, thus we wrote this for the reader convenience. In the first part of this paragraph we will follow closely the narrative of [HLT19]. Let A be an elliptic curve over C with identity element p0 , then there is a lattice Λ such that A ' C/Λ. Identify A with its dual and consider P A the normalized Poincaré bundle on A × A: P A ' O A× A (∆ A ) ⊗ pr1∗ O A (− p0 ) ⊗ pr2∗ O A (− p0 ) where ∆ A ⊂ A × A is the diagonal divisor and pr1 , pr2 are the projections of A × A onto the first and second factor respectively. Observe that if x is a point in A, then the topologically trivial line bundle Px is simply P A | A×{ x} ' P A |{ x}× A . Given another elliptic curve B, line bundles L A and L B on A and B respectively, and a morphism ϕ : B → A, we define a line bundle on the product A × B (2.9) L( L A , L B , ϕ) := (1 A × ϕ)∗ P A ⊗ pr∗A L A ⊗ pr∗B L B where pr A and prB are the projections onto A and B respectively. As a direct consequence of the see-saw principle it is possible to see that, if M A and MB are two other line bundles on A and B, and ψ : B → A is another homomorphism, then L ( L A ⊗ M A , L B ⊗ M B , ϕ + ψ ) ' L ( L A , L B , ϕ ) ⊗ L ( M A , M B , ψ ). In addition, the universal property of the dual abelian variety ensures that every line bundle L ∈ Pic( A × B) is of the form L( L A , L B , ϕ) for some invertible sheaves L A and L B and a morphism ϕ. Therefore we have an isomorphism Pic( A × B) ' Pic( A) × Pic( B) × Hom( B, A). If we quotient by numerically trivial line bundles, we find that (2.10) H 2 ( A × B, Z) ' Num( A × B) ' Z · [ B] × Z · [ A] × Hom( B, A), where [ A] and [ B] are the classes of the fibers of the two projections. Let us denote by l (deg( L A ), deg( L B ), ϕ) the first Chern class of L( L A , L B , ϕ). Then every class in Num( A × B) can be written as l (m, n, ϕ) for some integers n and m and an isogeny ϕ. In what follows we will often refer to line bundles (or numerical classes) in Hom( B, A) as elements of the Hompart of Pic( A × B) (or of Num( A × B)). For our purposes it will be really important to pick explicit generators for Num( A × B) to see how the automorphism σ acts on H 2 ( A × B, Z). In order to do that, we need to investigate the Z-module structure on Hom( B, A). So suppose that there is a nontrivial isogeny ϕ : B → A. Then we know that Hom( B, A) has rank 1 if A does not have complex multiplication, and 2 otherwise (more details about elliptic curves with complex multiplication can be found in the Appendix). Suppose the first, so that there exists an isogeny ψ : B → A such that l (0, 0, ψ) generates the Hom-part of H 2 ( A × B, Z). We will call such isogeny a generating isogeny for Num( A × BRAUER GROUP OF BIELLIPTIC SURFACES 9 B). Observe that, since l (0, 0, ψ) is necessarily a primitive class, ψ cannot factor through any "multiplication by n" map. That is, we cannot write ψ = n · ψ0 for any n. In particular, for any integer n we have that Ker ψ does not contain B[n] as a subscheme. Suppose now that A has complex multiplication, and again fix a non trivial isogeny ϕ : B → A. Then also B has complex multiplication, and Hom( B, A) is a rank 2 free Z-module. We pick generators ψ1 and ψ2 , and we have that for any line bundle L on A × B there are two integers h and k such that (2.11) L ' L( M A , MB , h · ψ1 + k · ψ2 ), where M A and MB are element of Pic( A) and Pic( B) respectively. In addition we can write (2.12) H 2 ( A × B, Z) = hl (1, 0, 0), l (0, 1, 0), l (0, 0, ψ1 ), l (0, 0, ψ2 )i . In the particular cases in which the j-invariant of B is either 0 or 1728, then Theorem A.1 in the Appendix yields a more accurate description. In fact, if we denote by λ B : B → B the automorphism ρ or ω (see again the Appendix or Paragraph 2.2), we have that there exist an isogeny ψ : B → A such that, in (2.11) and (2.12) we can take ψ1 = ψ and ψ2 = ψ ◦ λ B . So we have that (2.13) H 2 ( A × B, Z) = hl (1, 0, 0), l (0, 1, 0), l (0, 0, ψ), l (0, 0, ψ ◦ λ B )i . In this case we say that ψ is again a generating isogeny for H 2 ( A × B, Z). Observe again the isogenies ψi , as well as ψ, cannot factor through the multiplication by an integer or they could not generate the whole Hom( B, A). 3. G ENERATORS FOR THE TORSION OF THE SECOND COHOMOLOGY FOR BIELLIPTIC SURFACES In this section we give explicit generators for the torsion of H 2 (S, Z) in terms of the reduced multiple fibers of the elliptic fibration g : S → P1 . More precisely we will prove the following statement: Proposition 3.1. Let S = A × B/G be a bielliptic surface. Denote by Di the reduced multiple fibers of g : S → P1 with the same multiplicity. Then the torsion of H 2 (S, Z) is generated by the classes of differences Di − D j for i 6= j. The reader who is familiar with the work of Serrano might find similarities between the above statement and Serrano’s description of the torsion of H 2 ( X, Z) when there is an elliptic fibration ϕ : X → C with multiple fibers (cfr. [Ser90b, Corollary 1.5 and Proposition 1.6]). However in [Ser90b] it is used the additional assumption that h1 ( X, O X ) = h1 (C, OC ). This clearly does not hold in our context. Before proving Proposition 3.1 we need two preliminary Lemmas. Lemma 3.2. Let g : S → P1 be an elliptic pencil with connected fibers. Let D1 and D2 be two reduced multiple fibers. Let m1 and m2 be the corresponding multiplicities. Then, for all non negative integers n, (3.1) D1 nD2 . Proof. The statement is obvious for n = 0, so one has to check for n > 0. By contradiction, assume D1 ∼ nD2 , and let F be the generic fiber of g. Then h0 (S, OS ( F )) = h0 (P1 , g∗ OS ( F )) = h 0 ( P 1 , O P1 ( 1 ) ⊗ g ∗ O S ) = h0 (P1 , OP1 (1)) = 2. 10 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP Since h0 (S, OS ( D1 )) ≤ h0 (S, OS (m1 D1 )) = h0 (S, OS ( F )), it follows that h0 (S, OS ( D1 )) ≤ 2. The absurd hypothesis is used here: if D1 ∼ nD2 , then, since the supports of D1 and D2 are disjoint, H 0 (S, OS ( D1 )) has at least two independent sections, and therefore the dimension of H 0 (S, OS ( D1 )) is 2. Thus, since D12 = 0 implies that there are no basepoints (see for example [Bea96, II.5]), the map is actually a morphism ϕ| D1 | : S −→ P1 . Note that both D1 and nD2 are fibers of this morphism. Let now C be the generic fiber of ϕ (which is irreducible by semicontinuity). Since C · D1 = 0, one gets C · F = 0 for any fiber F of g. This implies that g and ϕ| D1 | have the same generic fiber. So one can write C = F for a fiber F of g. But then D1 ∼ F ∼ m1 D1 , which in turn implies that OS ( D1 )⊗(m1 −1) ' OS , which is a contradiction. Lemma 3.3. Let S = A × B/G be a bielliptic surface with its fibrations f : S → A/G and g : S → P1 . Let D1 and D2 be two reduced multiple fibers of g. Then the restriction of OS ( D1 − D2 ) to the generic fiber of aS is trivial. Proof. Let F = g−1 ( p) be a smooth fiber of g. Here p is the orbit G · y of a point y ∈ B not fixed under any element of G. We will choose an embedding of A into S via an isomorphism ϕ : A → F such that we get a commutative diagram j A A×B ϕ π i F S aS ψ A/G where i is just the natural inclusion of the fiber F into S and π is the quotient map. To this end we let ϕ : A → F be the isomorphism a 7→ G · ( x, y) and j be the embedding a 7→ ( x, y). Since the multiple fibers Di are images of A × {yi }, i = 1, 2, where the yi ∈ B are points fixed under a subgroup of G of order equal to the multiplicity of Di , we have that π ∗ OS ( D1 − D2 ) = p∗ O B (y1 − y2 ) where p B is the projection A × B → B and y1 , y2 ∈ B are the points corresponding to D1 , D2 , respectively. Then ϕ∗ i∗ OS ( D1 − D2 ) ' j∗ π ∗ OS ( D1 − D2 ) ' j∗ p∗B O B (y1 − y2 ) As p ◦ j is the constant map we have that this is clearly trivial. Hence ϕ∗ i∗ OS ( D1 − D2 ) is trivial, and since ϕ is an isomorphism we deduce the statement. For the remainder, we identify F and A via the isomorphism ϕ defined in the proof above. So we get the following commutative triangle. i A (3.2) S aS ψ A/G Note that ψ is an isogeny of degree | G |. In particular we have also that the dual isogeny ψ∗ : Pic0 (S) → Pic0 ( A) has degree | G | (see, for example [BL13, Proposition 2.4.3]). BRAUER GROUP OF BIELLIPTIC SURFACES 11 With these observations we are now ready to start proving Proposition 3.1. We first remark that by the canonical bundle formula for elliptic fibrations (see e.g. [Bad01, Thm. 7.15]) applied to g : S → P1 we can write ωS ' g∗ OP1 (−2) ⊗ OS ∑(mk − 1) Dk k where the Dk are the multiple fibers of g of multiplicity mk . Choosing points p, q on P1 giving rise to the fibers mi Di and m j D j we get that (3.3) K S ∼ − Di − D j + ∑ ( m k − 1 ) Dk . k6=i,j Since ωS is a nontrivial element in Pic0 (S) we conclude that the classes of − Di − D j and ∑k6=i,j (mk − 1) Dk coincide in H 2 (S, Z). Moreover, we observe that KS restricts trivially to A, so ωS yields a nontrivial element in Ker ψ∗ . Note that if Di and D j have the same multiplicity m, the difference Di − D j induces a (possibly trivial) torsion element in H 2 (S, Z) of order m. We prove Proposition 3.1 by showing that a sufficient number of these is nontrivial so to generate the torsion of H 2 (S, Z). We proceed by a case by case analysis, studying separately bielliptic surfaces of type 1, 2, 3, and 5. The key point in the argument is the observation that, if [ Di − D j ] is trivial, then the line bundle OS ( Di − D j ) belongs to Pic0 (S). In addition, using Lemma 3.2 and the diagram (3.2), we would have that ψ∗ OS ( Di − D j ) ' OS , in particular OS ( Di − D j ) ∈ Ker ψ∗ , while Lemma 3.3 ensures that OS ( Di − D j ) cannot be OS . A closer study of the structure of Ker ψ∗ ' Ĝ will bring us to the desired conclusion. 3.1. Type 1 bielliptic surfaces. In this case we have that Ker ψ∗ is the reduced group scheme Z/2Z and the fibration g : S → P1 has four multiple fibers all of multiplicity 2. Hence, up to reordering the indices (3.3) yields (3.4) K S ∼ Di − D j + D k − D l . In particular, as the canonical divisor is algebraically equivalent to 0, for distinct indices i, j, k, and l we have that D j − Di is algebraically equivalent to Dk − Dl . Thus we get three classes in H 2 (S, Z) [ D1 − D2 ] = { D1 − D2 , D3 − D4 }, (3.5) [ D1 − D3 ] = { D1 − D3 , D2 − D4 }, [ D1 − D4 ] = { D1 − D4 , D2 − D3 }, which a priori are neither distinct nor nontrivial. Since H 2 (S, Z)tors is isomorphic to the Klein 4-group, we need to show that they are indeed different classes and are not zero. Note that, if two classes are equal, since they both are 2-torsion and the third classes is clearly equal to the sum of the first two, then the remaining class would be trivial. Thus it will be enough to show that for any two distinct indices the divisor Di − D j is not algebraically equivalent to 0. Suppose otherwise that for some indices we have that OS ( Di − D j ) ∈ Pic0 (S), then (3.4) would imply that also OS ( Dk − Dl ) would be in Pic0 (S). The above discussion yields that both OS ( Di − D j ) and OS ( Di − D j ) are nontrivial elements of Ker ψ∗ , which has only one nontrivial element, ωS . Then we can write ωS ' OS ( Di − D j ) ⊗ OS ( Dk − Dl ) ' ωS⊗2 ' OS , which brings a contradiction, and thus we may conclude. 12 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP 3.2. Type 2 bielliptic surfaces. Here H 2 (S, Z)tors ' Z/2Z, Ker(ψ∗ ) ' Z/2Z × Z/2Z and like in the previous case there are four multiple fibers, each of multiplicity 2. As above we get the three classes induced by D1 − D2 , D1 − D3 and D1 − D4 , and we want to show that they cannot be all trivial. Suppose that two of these classes, say [ D1 − D2 ] and [ D1 − D3 ], are trivial in H 2 (S, Z). For i = 2, 3 set Li := OS ( D1 − Di ) and Mi := OS ( Di − D4 ), then the Li ’s and the Mi ’s determine nontrivial elements of Ker ψ∗ , which has only three nonzero elements. We deduce that some of these must be the same line bundle. The only option which will not contradict Lemma 3.3 would be that Li ' M j for som i 6= j. But then we would have ωS ' Li ⊗ M j ' Li⊗2 ' OS , which would be a contradiction. Hence at most one of the three classes can be trivial, and indeed one is actually trivial because the two nontrivial classes must coincide, implying the third is trivial. 3.3. Type 3 bielliptic surfaces. Here H 2 (S, Z)tors ' Z/2Z and Ker(ψ∗ ) ' Z/4Z, but now we have two fibers of multiplicity 4 and one of multiplicity 2. Denote by E the reduced multiple fiber of multiplicity 2 and by D1 , D2 the reduced multiple fibers of multiplicity 4. By the canonical bundle formula, we get KS ∼ E − D1 − D2 . Then in H 2 (S, Z) we have the following equalities [ E − 2D1 ] = [ D2 − D1 ], and [ E − 2D2 ] = [ D1 − D2 ]. We need to show that they are not both trivial. Suppose by contradiction they are both zero in H 2 (S, Z), then, as before we have that OS ( E − 2D1 ) and OS ( E − 2D2 ) are non trivial elements of Ker ψ∗ . Since both these line bundles have order two in Pic(S), and Ker ψ∗ has only one element of order 2, we deduce that OS ( E − 2D1 ) ' OS ( E − 2D2 ). But then ωS⊗2 ' OS ( E − D1 − D2 )⊗2 ' OS ( E − 2D1 ) ⊗ OS ( E − 2D2 ) ' OS ( E − 2D1 )⊗2 ' OS which is impossible because ωS is of order 4. Therefore E − 2D1 and E − 2D2 induce the same nontrivial torsion element of H 2 (S, Z). 3.4. Type 5 bielliptic surfaces. Here H 2 (S, Z)tors ' Z/3Z, Ker(ψ∗ ) ' Z/3Z and there are three multiple fibers, each of multiplicity 3. By the canonical bundle formula, we get KS ∼ − Di − D j + 2Dk = ( Dk − Di ) + ( Dk − D j ). Again, KS is algebraically equivalent to zero, so we get that [ Dk − Di ] = [ D j − Dk ] in H 2 (S, Z). Running through the indices we get the two classes [ D1 − D2 ] = { D1 − D2 , D3 − D1 , D2 − D3 }, [ D1 − D3 ] = { D1 − D3 , D3 − D2 , D2 − D1 }. We need to show that they are distinct and both nontrivial. Observe that if they were the same class then both classes would be trivial, so it is enough to show that they are not the zero class. Again suppose by contradiction that [ Dk − Di ] = 0 in H 2 (S, Z), then we can write ωS ' OS ( D1 − D2 ) ⊗ OS ( D1 − D3 ), with OS ( D1 − D2 ) and OS ( D1 − D3 ) for nontrivial elements in Ker(ψ∗ ). Neither OS ( D1 − D2 ) nor OS ( D1 − D3 ) can be isomorphic to the canonical bundle ωS , or we would have BRAUER GROUP OF BIELLIPTIC SURFACES 13 OS ( Dk − Di ) ' OS , contradicting Lemma 3.3. As Ker ψ∗ has only two nontrivial elements, we necessarily have OS ( D1 − D2 ) ' OS ( D1 − D3 ) and so OS ( D2 − D3 ) ' OS , which contradicts again Lemma 3.3, thus we can conclude. 4. T HE B RAUER MAP TO ANOTHER BIELLIPTIC SURFACE Let S be a bielliptic surface of type 2 or 3. Then by Examples 2.4 and 2.4 there is a 2:1 cyclic cover π̃ : S̃ → S, where S̃ is a bielliptic surface of type 1. As in paragraph 2.3 , we will denote by σ̃ the involution induced by π̃. In this section we are concerned with studying the Brauer map π̃Br : Br(S) → Br(S̃). Surpisingly we reach two antipodal conclusions, depending on the type of the bielliptic surface in object. Recall that, as S̃ is a bielliptic surface of type 1, the elliptic fibration q B : S̃ → P1 has four multiple fibers D1 , . . . , D4 of multiplicity 2, corresponding to the four 2-torsion points of B. We will denote by τij the line bundle OS̃ ( Di − D j ). 4.1. Bielliptic surfaces of type 2. Suppose that S is of type 2, and note that the involution σ̃ acts on the set of the Di ’s by exchanging them pairwise. Up to relabeling we can assume that σ̃∗ D1 ∼ D2 and σ̃∗ D3 ∼ D4 . By (2.8), we therefore have that (4.1) π̃ ∗ (Nm(τ13 )) ' τ13 ⊗ σ∗ τ13 ' τ13 ⊗ τ24 ' ωS̃ , where the last equality is a consequence of (3.4).Thus, if we denote by γ the generator of Ker π̃ ∗ , we get that Nm(τ13 ) ∈ {ωS , ωS ⊗ γ} ⊂ Pic0 (S). Then we can use the Pic0 trick (Remark 2.6) and find a β ∈ Pic0 (S) such that Nm(π̃ ∗ β ⊗ τ13 ) is trivial. Lemma 4.1. In the above notation, the line bundle π˜∗ β ⊗ τ13 does not belong to the image of 1 − σ̃∗ Before going forward with the proof, let us notice how, as an easy corollary, we get Corollary 4.2. If S is of type 2, then the induced map πBr : Br(S) → Br(S̃) is trivial. Proof of Lemma 4.1. We will show that the class of τ13 in H 2 (S̃, Z) is not in the image of 1 − σ̃∗ . Denote by [τij ] the algebraic equivalence class of the line bundle τij . Then, by Proposition 2.2 and (3.5), for every L in Pic(S̃) there are integers n, m, and h, and k such that n c1 ( L) = · a + m · b + h · [τ13 ] + k · [τ14 ] 2 Then it is easy to see that (1 − σ̃∗ )c1 ( L) = 2h · [τ13 ] + 2k · [τ14 ] = 0. But on the other side we have that c1 (π˜∗ β ⊗ τ13 ) = [τ13 ] is not trivial, thus π˜∗ β ⊗ τ13 cannot possibly lie in the image of (1 − σ̃∗ ), and the lemma is proved. 4.2. Bielliptic surface of type 3. In this paragraph we aim to show the following statement Theorem 4.3. If S is a bielliptic surface of type 3, then the Brauer map π̃Br : Br(S) → Br(S̃) induced by the cover π̃ : S̃ → S, where S̃ is bielliptic of type 1, is injective. We will use 2.7 and show that Ker(Nm)/ Im(1 − σ∗ ) is trivial. There are two main key steps: (1) We first study the norm map when applied to numerically trivial line bundles; (2) then we prove that all the line bundles L in Ker(Nm) are numerically trivial. 14 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP 4.2.1. Norm of numerically trivial line bundles. We will use the notation of Example 2.4. Observe that we have the following diagram (4.2) S̃ aS̃ π̃ A/G /S ϕ , aS / A/H Where G ' Z/2Z, and H is Z/4Z. Remark 4.4. Note that the bottom arrow, ϕ, is an isogeny of degree 2. As the vertical arrows are the Albanese maps of S̃ and S respectively, we have that π̃ ∗ : Pic0 (S) → Pic0 (S̃) coincides with the isogeny dual to ϕ. In particular it is surjective. Our first step in the study of the norm homomorphism for numerically trivial line bundles is to see how it behaves when applied to the generator of the torsion of H 2 (S̃, Z). In order to do that, we remark that the automorphism ω acts on B[2] with at least one fixed point, the one corresponding to the identity element of B. Since ω has order 4, it cannot act transitively on the remaining three points on B[2]. Thus the action has at least two fixed points. We deduce that σ̃ acts on the set of the reduced multiple fibers by leaving fixed at least two of them, let us say D1 and D2 . If the action were trivial, then we would have that all the line bundles τij are invariant under the action of σ̃ and as a consequence they would be pullbacks of line bundles coming from S. We would deduce that all the torsion classes of H 2 (S̃, Z) are pullbacks of classes from H 2 (S, Z), which is impossible. Thus we know that σ̃ exchanges D3 and D4 . Then we can prove the following Lemma. ⊗n ⊗m Lemma 4.5. Let n and m be two integers. Then the norm of the line bundle τ13 ⊗ τ14 is zero ⊗n ⊗m if and only if n and m have the same parity. In addition we have that Nm(τ13 ⊗ τ14 ) is not in Pic0 (S) if n and m are not congruent modulo 2. Proof. Observe first of all that, thanks to the above discussion, the line bundle τ34 ' τ13 ⊗ τ14 is invariant with respect to the action of σ̃. In particular we can write τ34 ' π̃ ∗ τ where τ is a line bundle on S whose algebraic equivalence class is the only nontrivial class in H 2 (S, Z). ⊗n ⊗m Now, if n and m are both even, then τ13 ⊗ τ14 is the trivial line bundle, and there is nothing to prove. Otherwise, if n and m are odd, then ⊗n ⊗m Nm(τ13 ⊗ τ14 ) ' Nm(τ34 ) ' τ ⊗2 ' OS . Conversely suppose that n and m are not congruent modulo 2. Up to exchanging n and m ⊗n ⊗m we can assume that m is even, while n is odd. Then τ13 ⊗ τ14 ' τ13 . Again by (2.8) we get π̃ ∗ Nm(τ13 ) ' τ13 ⊗ σ̃∗ τ13 ' τ34 ' π̃ ∗ τ. We deduce that Nm(τ13 ) is either equal to τ or to τ ⊗ ωS⊗2 . In any case it is not algebraically equivalent to zero and so the statement is proven. Remark 4.6. (a) Observe that τ34 is in the image of 1 − σ̃∗ , as we have that τ34 ' OS̃ ( D3 ) ⊗ σ̃∗ OS̃ (− D3 ). (b) We will see in what follows that the different behavior of the norm map applied to torsion classes is what determines the contrast between the type 2 and type 3 bielliptic surfaces. In particular, the fact that the norm map of a torsion class is not necessarily algebraically trivial is what does not allow us to use Remark 2.6 in order to provide a non trivial class in Ker(Nm)/ Im(1 − σ̃∗ ) Now we turn our attention to the elements of Pic0 (S̃) whose norm is trivial. We will show that they never determine nonzero classes in Ker(Nm)/ Im(1 − σ̃∗ ). BRAUER GROUP OF BIELLIPTIC SURFACES 15 Lemma 4.7. Denote by Nm : Pic(S̃) → Pic(S) the norm homomorphism. Let L ∈ Pic0 (S̃), such that Nm( L) = OS . Then the class of L in H 1 (Z/2Z, Pic(S̃)) is trivial. Proof. We have to show that such L is in the image of the morphism 1 − σ̃∗ . By Remark 4.4, we can write L ' π̃ ∗ M with M ∈ Pic0 (S). Then our assumption warrants that OS ' Nm( L) ' M⊗2 . We deduce that M is a 2-torsion point in Pic0 (S). Now we know that Pic0 (S)[2] is a group scheme isomorphic to Z/2Z × Z/2Z. Let γ be the element ωS⊗2 ∈ Pic0 (S)[2] then we can find β ∈ Pic0 (S)[2], β nontrivial, such that Pic0 (S)[2] = {OS , γ, β, γ ⊗ β}. In particular, as π̃ ∗ γ ' OS̃ , (4.3) Ker(Nm) ∩ Pic0 (S̃) = {OS̃ , π̃ ∗ β}. Now we aim at producing a line bundle α ∈ Pic0 (S̃) ∩ Im(1 − σ̃∗ ), α 6' OS̃ . Thus we will have that Pic0 (S̃) ∩ Im(1 − σ̃∗ ) is a nontrivial subgroup of Ker(Nm) ∩ Pic0 (S̃). From (4.3) we deduce that Ker(Nm) ∩ Pic0 (S̃) = Pic0 (S̃) ∩ Im(1 − σ̃∗ ) and so the statement. To this aim let e ∈ A0 := A/G the image of the point e ∈ A defining the involution σ̃ (see (2.4)). Denote also by p0 the identity element of A0 ; observe that by the construction of bielliptic surfaces e 6= p0 . Consider the following line bundle on S̃: α := a∗S̃ (O A0 ( p0 ) ⊗ te∗ O A0 (− p0 )). Clearly α is a nontrivial element in Pic0 (S̃). In addition by (2.4) we see that α ' a∗S̃ (O A0 ( p0 )) ⊗ σ̃∗ a∗S̃ (O A0 (− p0 )), and therefore it is in the image of 1 − σ̃∗ . Thus we can conclude. 4.2.2. Injectivity of the Brauer map. We are now ready to prove Theorem 4.3. We will do so by showing the following statement. Proposition 4.8. If L ∈ Ker(Nm) then L is numerically trivial. Before proceeding with the proof, let us show how this implies Theorem 4.3. Let L be a line bundle in the kernel of the norm map. Then Proposition 4.8 yields that ⊗n ⊗m L ' α ⊗ τ13 ⊗ τ14 for some positive integers n and m, and for some α ∈ Pic0 (S̃). Write again α ' π̃ ∗ β, and observe that if n and m are not congruent modulo 2, then by Lemma 4.5,we get Nm( L) ' β⊗2 ⊗ τ, which is not algebraically trivial. We deduce that n and m must have the same parity. Now we apply the first part of Lemma 4.5 and see that α ∈ Ker(Nm). In particular Lemma 4.7 implies that α ∈ Im(1 − σ̃∗ ), and so the class of L in Ker(Nm)/ Im(1 − σ̃∗ ) is the same as the class of τ34 . But Remark 4.6(a) tells us that the latter is trivial and so Theorem 4.3 is proved. Proof of Proposition 4.8. Let L in the kernel of the norm map. Lemmas 2.3 and 2.3 imply that π̃ ∗ Num(S) is a sublattice of index 2 of Num(S̃). In particular L⊗2 is numerically equivalent to the pullback of a line bundle from S. Thus we can write ⊗n ⊗m L⊗2 ' π̃ ∗ M ⊗ α ⊗ τ13 ⊗ τ14 16 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP for some positive integers n and m, and for some α ∈ Pic0 (S̃). Again, by Remark 4.4 we can write α ' π̃ ∗ β for some β ∈ Pic0 (S), and so, up to substituting M with M ⊗ β we have that ⊗n ⊗m L⊗2 ' π̃ ∗ M ⊗ τ13 ⊗ τ14 . If we show that M is numerically trivial we can conclude. Observe that ⊗n ⊗m OS ' Nm( L) ⊗ Nm( L) ' Nm( L⊗2 ) ' M⊗2 ⊗ Nm(τ13 ⊗ τ14 ) ' M⊗2 ⊗ τ ⊗(n+m) , where the last equality is a consequence of Lemma 4.5. As τ is numerically trivial we conclude that the same is true for M. 5. T HE B RAUER MAP TO THE CANONICAL COVER In this section we study the Brauer map πBr : Br(S) → Br( X ) when S is a bielliptic surface and X is its canonical cover. Then there is an n to 1 étale cyclic cover π : X → S, where n denotes the order of the canonical bundle ωS . Thus, as in the previous section, we can use Beauville’s work [Bea09] to study the kernel of the πBr via the norm homomorphism Nm : Pic( X ) → Pic(S). As in the other cases the Brauer group is trivial, we can assume that S is of type 1, 2, 3, or 5. Recall that, independently from the case at hand, there are two elliptic curves A and B such that X is isogenous to A × B. In what follows we will see that the geometry of the Brauer maps depends much on the geometry of A × B, and in particular on whether there are isogenies between A and B or not. Throughout this section we will use the notation established in paragraph 2.2. 5.1. The norm of numerically trivial line bundles. Our first step will be proving the following proposition, which will allows us to study the norm map from a strictly numerical point of view. Proposition 5.1. Let L ∈ Pic0 ( X ) ∩ Ker(Nm). Then L is in Im(1 − σ∗ ). Before going any further we need to describe more precisely our setting and introduce some notation. Observe first that, if we let as in 2.2 p A : X → A/H and p B : X → B/H the two elliptic fibrations of the abelian variety X, then Pic0 ( X ) is generated by p∗A Pic0 ( A/H ) and p∗B Pic0 ( B/H ), thus we can write any L ∈ Pic0 ( X ) as p∗A α ⊗ p∗B β, where α ∈ Pic0 ( A/H ), β ∈ Pic0 ( B/H ). In this notation we have the following. Lemma 5.2. For every β ∈ Pic0 ( B/H ) we have that p∗B β is in the image of 1 − σ∗ . In particular these line bundles are in the kernel of the norm homomorphism. Proof. We suppose first that G is cyclic and so the group H is trivial, and X ' A × B. We proceed with a case by case analysis. Type 1 case. Since abelian varieties are divisible groups, there exist γ ∈ Pic0 ( B) such that ⊗ γ 2 ' β. Then by (2.2) we have that (1 − σ∗ ) p∗B γ ' p∗B γ ⊗ (σ∗ p∗B γ)−1 ' p∗B 2∗B γ ' p∗B β, and the statement is proven in this case. Type 3 case. In this case the j-invariant of B is 1728 and there is an automorphism ω of B of order 4. Consider the map 1 − ω : B → B. Since this is not trivial it is an isogeny, and in particular (1 − ω )∗ : Pic0 ( B) → Pic0 ( B) is surjective. Let γ ∈ Pic0 ( B) such that (1 − ω )∗ γ ' β, then by (2.2) we have (1 − σ∗ ) p∗B γ ' p∗B (1 − ω )∗ γ ' p∗B β, and the statement is proven in this case. BRAUER GROUP OF BIELLIPTIC SURFACES 17 Type 5 case. This case is similar to the previous one in which instead of ω we use the automorphism ρ. We note that (1 − ρ) : B → B is non trivial, and so an isogeny. In particular the dual map (1 − ρ)∗ : Pic0 ( B) → Pic0 ( B) is surjective and we can find γ such that (1 − ρ)∗ γ ' β. Again (2.2) yields: (1 − σ∗ ) p∗B γ ' p∗B (1 − ρ)∗ γ ' p∗B β, and the statement is proven. In order to conclude we need to analize the case of bielliptic surfaces of type 2. Under this assumption the group H is not trivial but it is cyclic of order 2. Let B0 := B/H and observe that we have the following diagram X pB B0 σ −1 B 0 /X pB / B0 So let, as in the type 1 case, γ ∈ Pic0 ( B0 ) such that 2∗B0 γ ' β, then we will have again that (1 − σ∗ ) p∗B γ ' p∗B β and the proof is concluded. Now let L = p∗A α ⊗ p∗B β ∈ Pic0 ( X ) such that Nm( L) ' OS . Lemma 5.2 implies that also p∗A α is in the kernel of the norm homomorphism. In addition we have that the class of L in H 1 ( G, Pic( A × B)) is just the class of p∗A α. We have a commutative diagram X pA π A/H /S ϕ , aS / A/G where the bottom arrow is an isogeny of degree n. In particular we can write p∗A α ' π ∗ M with M ∈ Pic0 (S). In addition we have that OS ' Nm( p∗A α) ' M⊗n , thus we have that p∗A Pic0 ( A/H ) ∩ Ker(Nm) = π ∗ Pic0 (S)[n] . It easy to see that the right-hand-side above is a group isomorphic to the cyclic group of order n. Since Im(1 − σ∗ ) is a subgroup of∗ the kernel of the norm, if we provide an element 0 ∗ of order n in p A Pic ( A/H ) ∩ Im(1 − σ ) we would conclude that p∗A Pic0 ( A/H ) ∩ Im(1 − σ∗ ) = p∗A Pic0 ( A/H ) ∩ Ker(Nm) and consequently the statement of Proposition 5.1. Let p0 be the identity element of A/H, using the notation of 2.2 and (2.3) we set O A ( p0 ) ⊗ t∗τ (O A (− p0 )), if S is of type 1, O ∗ if S is of type 2, A/H ( p0 ) ⊗ tτ 0 (O A/H (− p0 )), γ := ∗ O A ( p0 ) ⊗ te (O A (− p0 )), if S is of type 3, ∗ O A ( p0 ) ⊗ tη (O A (− p0 )), if S is of type 5; where τ 0 is the image of τ under the isogeny A → A/H. Then γ is a nontrivial element of Pic0 ( A/H ) with the desired property. In addition, by (2.2) (2.3), we have that p∗A γ ' (1 − σ∗ ) p∗A O A ( p0 ), and so we can conclude. 18 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP Now we are ready to start our investigation of the Brauer map πBr : Br(S) → Br( X ). We first put ourselves in the special situation in which there are no nontrivial morphisms between A and B. 5.2. The Brauer map when the two elliptic curves are not isogenous. If there are no isogenies between A and B, the the lattice Num( X ) has rank 2 and it is generated by the classes of the two fibers, a X and bX . In addition, π ∗ Num(S) is a sublattice of Num( X ) of index n. So, let L be in the kernel of the norm map. We have that L⊗n is numerically equivalent to the pullback of a line bundle from S. More precisely we can write L⊗n ' π ∗ L0 ⊗ p∗A α ⊗ p∗B β ' π ∗ M ⊗ p∗B β, with β ∈ Pic0 ( B/H ). Lemma 5.2 ensures that π ∗ M is in the kernel of the norm map. In particular, M is an n-torsion element in Pic(S). We deduce that it is numerically trivial, and so L was numerically trivial to start with. Now we apply Proposition 5.1 and deduce the following statement. Theorem 5.3. If S := A × B/G is a bielliptic surface such that the elliptic curves A and B are not isogenous, then the Brauer map to the canonical cover πBr : Br(S) → Br( X ) is injective. Before going to the next case, observe that if S is a bielliptic surface of type 2, then we have the following diagram A×B ϕ πS̃ X / S̃ πS π̃ / S. If A and B are not isogenous Theorem 5.3) above implies that the Brauer map induced by πS is injective. On the other side, the results of this paragraph imply that the Brauer map induced by πS ◦ ϕ is trivial. Then the Brauer map induced by ϕ cannot be injective and we have Corollary 5.4. If ϕ : X → Y is an isogeny of abelian varieties, the map ϕBr : Br(Y ) → Br( X ) is not necessarily injective. 5.3. The Brauer map when the two elliptic curves are isogenous. Suppose now that A and B are isogenous. Our first step will be to use the description the Picard group and of the Neron–Severi of A × B that we outlined in 2.6 in order to find the image of 1 − σ∗ and the numerical type of line bundles in the kernel of the Norm homomorphism when S is a cyclic bielliptic surface. We begin with the following Lemma. Lemma 5.5. Suppose that G is a cyclic group, so that X ' A × B. If L ∈ Pic( A × B) is in the kernel of the Norm map, then c1 ( L) = l (0, 0, ϕ) for some isogeny ϕ : B → A. Proof. By the result of 2.6 we have that c1 ( L) = l (m, n, ϕ) for two integers n and m and an isogeny ϕ. Suppose that S is of type 1 and L is in the kernel of the norm map. We have that L ⊗ σ∗ L is trivial. In particular c1 ( L ⊗ σ∗ L) is zero. But then we get the following 0 = c1 ( L ⊗ σ ∗ L ) = c1 ( L ) + σ ∗ c1 ( L ) = l (m, n, ϕ) + l (m, n, − ϕ) = l (2m, 2n, 0). We conclude that n = m = 0. At the same time, if S is of type 5, we have that 0 = c1 ( L) + σ∗ c1 ( L) + (σ2 )∗ c1 ( L) = l (3m, 3n, 0). BRAUER GROUP OF BIELLIPTIC SURFACES 19 Finally, if S is of type 3 we get 0 = c1 ( L) + σ∗ c1 ( L) + (σ2 )∗ c1 ( L) + (σ3 )∗ c1 ( L) = l (4m, 4n, 0), so the statement is proven. We turn now our attention to the Brauer map in general and we study it by performing a case by case analysis on the different type of bielliptic surfaces. 5.3.1. Bielliptic surfaces of type 1. In this paragraph we study the Brauer map to the canonical cover of bielliptic surfaces of type 1. If B does not have complex multiplication, we fix, once and for all, ψ : B → A be a generating isogeny. Otherwise we fix ψi : B → A, for i = 1, 2 two generators of Hom( B, A). Our first step is to describe (1 − σ∗ ) Pic( A × B). Lemma 5.6. Let S be a bielliptic surface of type 1, and consider L ∈ (1 − σ∗ ) Pic( A × B), then there exist three integers m, h and k, and a line bundle β ∈ Pic0 ( B) such that ( if B does not have complex multiplication; L( Pτ⊗m , β, 2h · ψ) L' ⊗ m L( Pτ , β, 2h · ψ1 + 2k · ψ2 ) if B has complex multiplication. Proof. We do the complex multiplication case, the other is similar. Let M ∈ Pic( A × B), then by the results of 2.6 we have that M ' L( M A , MB , h · ψ1 + k · ψ2 ). We can write M A ' O A (n · p0 ) ⊗ α and MB ' O B (m · q0 ) ⊗ γ for q0 the identity element of B, some integers n and m and some topologically trivial line bundles α and γ. With this notation we find that σ∗ M ' L t∗τ O A (n · p0 ) ⊗ α, O B (m · q0 ) ⊗ γ−1 ⊗ (−h · ψ1 − k · ψ2 )∗ Pτ , −h · ψ1 − k · ψ2 . Observe that as γ ranges in all Pic0 ( B) also β := γ⊗2 ⊗ (h · ψ1 + k · ψ2 )∗ Pτ ranges in the whole Pic0 ( B). In addition we have that (1 − σ∗ ) M ' L Pτ⊗n , β, 2h · ψ1 + 2k · ψ2 Remark 5.7. It is not difficult to check that, for any two integers h and k L(0, 0, 2h · ψ1 + 2k · ψ2 ) = L(0, 0, h · ψ1 + k · ψ2 ) ⊗ σ∗ L(0, 0, h · ψ1 + k · ψ2 )−1 , and so it is in Im(1 − σ∗ ). We are now ready to prove one of the main statements of this section: Theorem 5.8. Suppose that S is a bielliptic surface of type 1 whose canonical cover is A × B with A and B isogenous elliptic curves. Then the Brauer map to the canonical cover of S is not injective if, and only if, one of the following mutually exclusive conditions is satisfied: (1) the elliptic curve B (and so A) does not have complex multiplication and ψ∗ Pτ is trivial; (2) the elliptic curve B (and so A) has complex multiplication and we have that at least one of the following line bundles is trivial (5.1) L1 := ψ1∗ Pτ , L2 := ψ2∗ Pτ , L3 := (ψ1 + ψ2 )∗ Pτ Proof. We deal with the complex multiplication case that is slightly more involved. The argument for the other case is very similar. Before explaining the details of our reasoning we would like to give, for the reader convenience, a quick outline of the proof. The key observation is that the assumption on the line bundles (5.1) are equivalent to the triviality of the norm of one of the following invertible sheaves: (5.2) M1 := (1 × ψ1 )∗ P A , M2 := (1 × ψ2 )∗ P A , M3 := (1 × (ψ1 + ψ2 ))∗ P A 20 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP being topologically trivial. Therefore, if the assumptions are verified, we can use the Pic0 trick (Remark 2.6) to provide an element in the kernel of the norm map. Such an element will give by construction a nontrivial class in Ker Nm / Im(1 − σ∗ ). Conversely, if neither of the line bundles is trivial, then an element in the kernel of the norm map will be forced to be numerically equivalent to (1 × 2 · ϕ)∗ P A for some isogeny ϕ ∈ Hom( B, A). Then we will apply Lemma 5.6 and see that such a line bundle lies in Im(1 − σ∗ ), so no element of Pic( A × B) yields a nontrivial class in Ker Nm / Im(1 − σ∗ ). Now, for the complete argument, observe first that by (2.8) and the see-saw principle, it is easy to check that, for every α in Pic0 ( A) and every isogeny ϕ : B → A, π ∗ Nm((1 × ϕ)∗ P A ⊗ p∗A α) ' (1 × ϕ)∗ P A ⊗ (1 × ϕ)∗ P A−1 ⊗ ⊗ p∗A α⊗2 ⊗ p∗B ϕ∗ Pτ (5.3) ' p∗A α⊗2 ⊗ p∗B ϕ∗ Pτ . Suppose first that one of the three line bundles in (5.1) is trivial. To fix the ideas we can assume that ψ1∗ Pτ is trivial, the argument is identical in the other cases. Then by (5.3) we have that Nm((1 × ψ1 )∗ P A ) is in the kernel of π ∗ , so in particular it is in Pic0 (S). We can therefore apply Pic0 trick (Remark 2.6) and find γ ∈ Pic0 (S) such that the norm of (1 × ψ1 )∗ P A ⊗ π ∗ γ is trivial. But by Lemma 5.6 we have that (1 × ψ1 )∗ P A ⊗ π ∗ γ is not in the image of 1 − σ∗ and so it defines a non trivial class in Ker Nm / Im(1 − σ∗ ), and one direction of the statement is proven. Conversely suppose that there is a line bundle L on X which identifies a non trivial class in Ker Nm / Im(1 − σ∗ ). By Lemmas 5.5 and 5.6 we can write L ' (1 × h · ψ1 + k · ψ2 )∗ P A ⊗ p∗A α ⊗ p∗B β, for two integers h and k, and two topologically trivial line bundles α and β. Not that h and k cannot be both even, for otherwise Lemma 5.6 and Remark 5.7 yield that [ L] = [ p∗A α ⊗ p∗B β] ∈ Ker Nm / Im(1 − σ∗ ) which, by Proposition 5.1, implies that [ L] = 0 . Thus we can assume that one between h and k is odd. Then by Lemma 5.2 and Lemma 5.6 we have that L ' (1 × ψi )∗ P A ⊗ p∗A α ⊗ M, or L ' (1 × ψ1 + ψ2 )∗ P A ⊗ p∗A α ⊗ M, with M in Im(1 − σ∗ ). In particular one of the following line bundles in 5.2 has trivial norm. We deduce by (5.3) that one of the line bundles in (5.1) is trivial and the statement is proved. Example 5.9. (a) Suppose that A ' B. If A does not have complex multiplication, then we can take ψ = ±1 A . In particular we have that ψ∗ Pτ is never trivial and the Brauer map is injective. (b) Suppose again that A ' B and that the j-invariant of A is 1728. Then End( A) ' Z[i ] and the multiplication by i induces an automorphism ω of A of order 4, and we can take 1 A and ω as generators of End( A). Suppose that Pτ is a fixed point† of the dual automorphism ω ∗ . Then (1 A + ω )∗ Pτ is zero and the Brauer map is not injective. (c) We can also use a similar argument to construct uncountably many Type 1 byelliptic surfaces with non injective Brauer map. Let B any elliptic curve without complex multiplication and chose θ a point of over 2 on B. Let A := B/θ and ψ : B → A the quotient map. This is a degree 2 isogeny, so it is primitive and hence generating. If τ is the only point of order 2 in Ker ψ∗ , then we have that the data A, τ, B uniquely identify a Type 1 bielliptic surface which has a non injective Brauer map. †For example we can identify A with its dual and ω ∗ with ω and take τ = ( 1 , 1 ) + Λ, where Λ =< 1, i > 2 2 A ' C/Λ. BRAUER GROUP OF BIELLIPTIC SURFACES 21 In order to complete our description of the Brauer map for type 1 bielliptic surfaces we need to give necessary and sufficient conditions for it to be trivial. To this aim we want to provide two distinct non-zero classes in Ker Nm / Im(1 − σ∗ ). We can assume that the Brauer map is already non-injective, and so the condition of Theorem 5.8 are satisfied. Suppose first that B does not have complex multiplication. And consider L in the kernel of the norm map, yielding a non trivial class in Ker Nm / Im(1 − σ∗ ). Then, as before, we have that L ' (1 × h · ψ)∗ P A ⊗ p∗A α ⊗ p∗B β. Again by Lemma 5.6 we can assume that h is odd, and the same result also yields that in Ker Nm / Im(1 − σ∗ ) the class of L and that of (1 × ψ)∗ P A ⊗ p∗A α are the same. Since ψ∗ Pτ is trivial, (5.3) implies that (1 × ψ)∗ P A ⊗ p∗A γ is in the kernel of the norm map for some γ ∈ Pic0 ( A). So Nm( p∗A (α ⊗ γ−1 )) ' OS and as before p∗A (α ⊗ γ−1 ) lies in the image of (1 − σ∗ ). We deduce that, in Ker Nm / Im(1 − σ∗ ), [ L] = [(1 × ψ)∗ P A ⊗ p∗A γ] = [(1 × ψ)∗ P A ⊗ p∗A δ], for every δ ∈ Pic0 ( A) such that (1 × ψ)∗ P A ⊗ p∗A δ is in the kernel of the norm homomorphism. In particular there is only one non-trivial element in Ker πBr . Thus we can assume that B has complex multiplication and that, as before, we have fixed ψ1 and ψ2 a system of generators for Hom( A, B). Suppose that only one among the line bundles (5.1) is trivial, for example L1 , and as usual take L in the kernel of the norm map. As before we can write L ' Mi ⊗ p∗A α ⊗ M with M in the image of (1 − σ∗ ) and Mi one of the line bundles appearing in (5.2). We deduce that i = 1 and that the class of L in Ker Nm / Im(1 − σ∗ ) is equal to the class of M1 ⊗ p∗A γ for every γ ∈ Pic0 ( A) such that Nm( M1 ⊗ p∗A γ) is trivial. Thus, there is just one non-zero class and the Brauer map is again non trivial. Finally suppose that two (and so all) line bundles in (5.1) are trivial. We have that both M1 and M2 are in the kernel of the norm map. In addition M1 ⊗ M2−1 ' (1 × (ψ1 − ψ2 ))∗ P A , which by Lemma 5.6 is not in the image of (1 − σ∗ ). Therefore we deduce that they determine two different classes in Ker Nm / Im(1 − σ∗ ), and hence the Brauer map is trivial. We have thus proven the following statement Theorem 5.10. The Brauer map to the canonical cover of a type 1 bielliptic surface is trivial if, and only if, the elliptic curves A and B are isogenous, B has complex multiplication, and all the line bundles in (5.1) are trivial. Example 5.11. (a) If A ' B then the Brauer map is never trivial. Suppose otherwise that there are ψ1 and ψ2 generators of End( A) such that both ψ1∗ Pτ and ψ2∗ Pτ are zero. Then we can write 1 A = h · ψ1 + k · ψ2 and we would get that Pτ ' 1∗A Pτ is trivial, reaching an obvious contradiction. (b) Let now A ' C/Z[2i ] and let τ the point (0, i ) + Z[2i ]. The elliptic curve B := A/ < τ > has j-invariant 1728 and Hom( B, A) is generated by the isogenies ψ1 := ϕ2 and ψ2 := ϕ2 ◦ λ B , where ϕ2 : B → A denotes the isogeny induced by multiplication by 2 (see Example A.8 in the Appendix). Observe that ϕ2∗ ( Pτ ) ' ϕ2∗ (O A (τ − p0 ) ' O A ( ϕ2 (τ ) − ϕ( p0 )) ' O B Thus we have that ψ1∗ Pτ ' ψ2∗ Pτ ' O B and the Brauer map is trivial. 5.3.2. Bielliptic surfaces of type 3. Let now S be a bielliptic surface of type 3. Then the canonical cover of S is isomorphic to A × B with j( B) = 1728 and multiplication by i induces and automorphism ω of B of order 4, ω. By the discussion in 2.6, it is possible to find a generating isogeny ψ such that Num( X ) = hl (1, 0, 0), l (0, 1, 0) l (0, 0, ψ), l (0, 0, ψ ◦ ω )i , 22 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP We fix, once and for all, such a ψ and prove the following Lemma, which yields a precise description of (1 − σ∗ ) Pic( X ). Lemma 5.12. Let ϕ : B → A and isogeny. Then there are two integers h and k such that ϕ = h · ψ + k · ψ ◦ ω. Then the line bundle (1 × ϕ)∗ P A ∈ Im(1 − σ∗ ) if and only if h+k is even. Proof. Let T : Hom( B, A) → Hom( B, A) the linear operator obtained by pre composing by (1B − ω ). Then, using that ω 2 = −1B , it is not difficult to show that an isogeny ϕ as in the statement is in the image of T if, and only if, h + k is even. Hence if h + k is not even, (1 × ϕ ) ∗ P A ∈ / Im(1 − σ∗ ). Suppose now that ϕ = h · ψ + k · ψ ◦ ω with h + k an even number. Then, by the above argument, we can find an isogeny γ : B → A such that ϕ = γ ◦ (1B − ω ). Then we have (1 − σ∗ )(1 × γ)∗ P A ' (1 × γ ◦ (1B − ω ))∗ P A ⊗ p∗B ω ∗ γ∗ Pe−1 ' (1 × ϕ)∗ P A ⊗ p∗B ω ∗ γ∗ Pe−1 . By Lemma 5.2, elements of the form p∗B β with β ∈ Pic0 ( B) are in the image of (1 − σ∗ ), so we conclude that (1 × ϕ)∗ P A ∈ Im(1 − σ∗ ). Remark 5.13. Observe that this Lemma implies easily that the quotient Hom( B, A)/ Im(1 − σ∗ ), where we are identifying Hom( B, A) with the corresponding subgroup of Num( A × B), is cyclic generated by the coset (1 A × ψ)∗ P A + Im(1 − σ∗ ). Now we are ready to start studying the kernel for the Brauer map πBr : Br(S) → Br( X ). Our main result is the following Theorem 5.14. Let S is a bielliptic surface of type 3 with canonical cover A × B such that A and B are isogenous. Then the Brauer map to the canonical cover is identically zero if, and only if, (1B + ω )∗ ψ∗ P2e is trivial Proof. For any isogeny ϕ : B → A, α ∈ Pic0 ( A) and β ∈ Pic0 ( B), using that the norm of p∗B β is trivial by Lemma 5.2, we have that π ∗ Nm((1 × ϕ)∗ P A ⊗ p∗A α ⊗ p∗B β) ' (1 × ϕ)∗ P A ⊗ p∗A α⊗ (1 × ϕ ◦ ω )∗ P A ⊗ p∗B ω ∗ ϕ∗ Pe ⊗ p∗A α⊗ (5.4) (1 × − ϕ)∗ P A ⊗ p∗B (−1B )∗ ϕ∗ P2e ⊗ p∗A α⊗ (1 × − ϕ ◦ ω )∗ P A ⊗ p∗B (−ω )∗ ϕ∗ P3e ⊗ p∗A α⊗ ' p∗A α⊗4 ⊗ p∗B (1 + ω )∗ ϕ∗ P2e . Suppose that (1B + ω )∗ ψ∗ P2e ' O B . Since P2e is a two torsion point, this is equivalent to asking that (1B − ω )∗ ψ∗ P2e is also trivial. Then (5.4) implies that the norms of (1 × ψ)∗ P A and of (1 × ψ ◦ ω )∗ P A lie in Pic0 (S). Then using the Pic0 -trick (Remark 2.6) and Lemma 5.12 we can find a non zero class in Ker Nm / Im(1 − σ∗ ), and the Brauer map is trivial. Conversely, let L be a line bundle defining a nontrivial class in Ker Nm / Im(1 − σ∗ ). Then as we did in the case of type 1 surfaces, we can write L ' (1 × h · ψ + k · ψ ◦ ω )∗ P A ⊗ p∗A α ⊗ p∗B β with α and β in Pic0 ( A) and Pic0 ( B). Lemma 2.3 implies that the integer h + k is odd or we would have that p∗A α is in the kernel of the norm map, and consequently, by Proposition 5.1, L ∈ Im(1 − σ∗ ) . Thus we can write L ' M ⊗ M0 BRAUER GROUP OF BIELLIPTIC SURFACES 23 where M0 is in the image of 1 − σ∗ , and M is numerically equivalent to (1 × ψ)∗ P A (this is a consequence of Lemma 5.12 and Remark 5.13). We deduce that M is in the kernel of the norm map. But then (5.4) implies that (1 + ω )∗ ψ∗ P2e is trivial, proving the statement. Example 5.15. Suppose that A ' B, so we can take ψ = 1 A . If P2e is a fixed point of ω, then we have that P A yields a nontrivial element in Ker Nm / Im(1 − σ∗ ). Conversely, if P2e is not a fixed point of ω we will have that the Brauer map is injective. 5.3.3. Bielliptic surfaces of type 5. Let S be a bielliptic surface of type 5. We will solve this case in a similar fashion as for bielliptic surfaces of type 3. In the type-5 case, the canonical cover is isomorphic to an abelian surface A × B with j( B) = 0. As already seen, B admits an automorphism ρ of order 3 such that ρ2 + ρ + 1 = 0. Again, thanks to Theorem 5.3 we need to study only the case in which A and B are isogenous. Also in this case, by the results of 2.6, there is generating isogeny ψ : B → A such that Num( X ) = hl (1, 0, 0), l (0, 1, 0) l (0, 0, ψ), l (0, 0, ψ ◦ ρ)i . With this notation, we prove a statement analogous to Lemma 5.12: Lemma 5.16. Let ϕ : B → A and isogeny. Then there are two integers h and k such that ϕ = h · ψ + k · ψ ◦ ρ. If h + k is not divisible by 3, then (1 × ϕ)∗ P A ∈ / Im(1 − σ∗ ). Conversely if 3 divides h + k, then (1 × ϕ)∗ P A ⊗ p∗B β ∈ Im(1 − σ∗ ), for every β ∈ Pic0 ( B). Proof. The argument is completely analogous to the proof of Lemma 5.12, after observing that, if T : Hom( B, A) → Hom( B, A) is the operator defined by pre composing with 1B − ρ, then the image of T are exactly the homomorphism h · ψ + k · ψ ◦ ρ such that 3 divides k + h. Remark 5.17. This Lemma implies easily that the quotient of the Hom-part of Num( A × B) by the action of 1 − σ∗ is isomorphic to Z/3Z with elements (1 A × ψ)∗ P A + Im(1 − σ∗ ) and (1 A × ψ + ψ ◦ ρ)∗ P A + Im(1 − σ∗ ) = (1 A × 2 · ψ)∗ P A + Im(1 − σ∗ ). We will also need the following statement: Lemma 5.18. Let B an elliptic curve with j-invariant 0 and β an element Pic0 ( B). Consider the following line bundles P1 := (2 · ρ + 1B )∗ β, , Pρ := (2 · ρ + 1B )∗ ρ∗ β, and P1+ρ := (2 · ρ + 1B )∗ (1B + ρ)∗ β. If one of them is trivial then they are all trivial. Proof. Observe first that (2 · ρ + 1B )∗ ρ∗ β ' ρ∗ (2 · ρ + 1B )∗ β. Since ρ is an automorphism the triviality of Pρ is equivalent to the triviality of P1 . In addition as P1+ρ ' P1 ⊗ Pρ we have that if P1 and Pρ are both trivial, then also P1+ρ is trivial. It remains to show that if P1+ρ ' O B , then also P1 and Pρ are trivial. We not that P1+ρ ' O B if, and only if, ρ∗ P1+ρ ' O B . On the other side we have ρ∗ P1+ρ ' ρ∗ (2 · ρ + 1B )∗ (1B + ρ)∗ β ' ρ∗ (ρ − 1B )∗ β ' (−2 · ρ − 1B )∗ β ' P1−1 . We conclude that the triviality of P1+ρ is equivalent to the triviality of P1 as required by the statement. Now we are ready to prove the main result of this paragraph: Theorem 5.19. Let S be an bielliptic surface of type 5 such that the two elliptic curves A and B are isogenous. Let ψ be a generating isogeny, then we have that the Brauer map πBr : Br(S) → Br( A × B) is trivial if, and only if, the line bundle (2 · ρ + 1B )∗ ψ∗ Pη ' O B . 24 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP Proof. The argument is really similar to what happens for type 3 bielliptic surfaces. We first note that, for any isogeny ϕ : B → A, and every α and β in Pic0 ( A) and Pic0 ( B) respectively, we have that (5.5) π ∗ Nm((1 × ϕ)∗ P A ⊗ p∗A α ⊗ p∗B β) ' p∗A α⊗3 ⊗ p∗B (2 · ρ + 1B )∗ ϕ∗ Pη . Suppose first that (2 · ρ + 1B )∗ ψ∗ Pη is trivial. Then (5.5) ensures that the norm of M1 := (1 × ψ)∗ P A is topologically trivial. By Lemma 5.16 we know that no line bundle numerically equivalent to M1 is in the image of 1 − σ∗ . Thus we use the Remark 2.6 to provide an element in Ker Nm inducing a non trivial class in Ker Nm / Im(1 − σ∗ ). Conversely, assume that L is a line bundle in Ker Nm whose class in Ker Nm / Im(1 − σ∗ ) is not trivial. As before we can write L ' (1 × h · ψ + k · ψ ◦ ρ)∗ P A ⊗ p∗A α ⊗ p∗B β. We apply Lemma 5.16 and write L ' M ⊗ M0 with M0 ∈ Im(1 − σ∗ ) and M a line bundle numerically equivalent to one of the following (5.6) M1 := (2 · ρ + 1B )∗ ψ∗ Pη , and M1+ρ := (2 · ρ + 1B )∗ (1 + ρ)∗ ψ∗ Pη . Clearly M is in the kernel of the norm map, which, by (5.5) implies that one among the following is trivial: P1 := (2 · ρ + 1B )∗ ψ∗ Pη , and P1+ρ := (2 · ρ + 1B )∗ (1B + ρ)∗ ψ∗ Pη . We conclude by applying Lemma 5.18 and deducing that P1 ' O B . Example 5.20. Suppose that A ' B. Note that the isogeny ϕ := (2 · ρ + 1B ) : B → has degree 3, and its kernel is contained in B[3] which has order 9. If η is in the kernel of ϕ then the bielliptic surface obtained by the action of σ( x, y) = ( x + η, ρ(y)) has trivial Brauer map. Otherwise the Brauer map is injective. 5.3.4. Bielliptic surfaces of type 2. We kept last the bielliptic surfaces of type two since for them we need an ad hoc argument. Let therefore S be a bielliptic surface of type 2 and denote by X its canonical cover. Then X ' A × B/ < t(θ1 ,θ2 ) > for two elliptic curves A and B and θ1 and θ2 points of order 2 in A and B respectively. Let us fix generators for Hom( B, A): if B does not have complex multiplication then Hom( B, A) =< ψ > with ψ : B → A an isogeny; otherwise there are two isogenies ψ1 , ψ2 : B → A such that Hom( B, A) =< ψ1 , ψ2 > . Our goal is to prove the following statement. Theorem 5.21. In the above notation the Brauer map πBr : Br(S) → Br( X ) is not injective if, and only if, one of the following conditions is satisfied: (1) the elliptic curve B does not have complex multiplication and either ψ(θ2 ) is not the identity element of A or ψ∗ Pθ1 is not trivial. (2) the elliptic curve B has complex multiplication and not all of the following elements are the identity element in the elliptic curve they belong to {ψ1 (θ2 ), ψ2 (θ2 ), ψ1∗ Pθ1 , ψ2∗ Pθ1 , (ψ1 + ψ2 )(θ2 ), (ψ1 + ψ2 )∗ ( Pθ1 )} Before proceeding with the proof we need to set up some notation. Recall that we have the following diagram A×B φ πS̃ X / S̃ πS π̃ /S where S̃ is a bielliptic surface of type 1. We have that S ' X/σ, S̃ ' A × B/σ̃ and X ' A × B/Σ, where Σ denotes the translation t(θ1 ,θ2 ) . We are going to deal just with the case in BRAUER GROUP OF BIELLIPTIC SURFACES 25 which B hax complex multiplication. The proof in the other case will be identical, provided that one drops one of the two generators. We first observe the following fact: Lemma 5.22. In the notation above suppose that B has complex multiplication and let Li be the line bundle (1 × ψi )∗ P A , for i = 1, 2. Then the conditions of Theorem 5.21 are satisfied if, and only if, for every γ ∈ Pic0 ( A × B) one of the following line bundles is not Σ-invariant: L1 ⊗ γ, L2 ⊗ γ, L1 ⊗ L2 ⊗ γ. (5.7) Proof. By see-saw, it is easy to see that Σ∗ [(1 × ψi )∗ P A ⊗ γ] ' (1 × ψi )∗ P A ⊗ γ ⊗ p∗A Pψi (θ2 ) ⊗ p∗B ψi∗ Pθ1 , Σ∗ [(1 × (ψ1 + ψ2 ))∗ P A ⊗ γ] ' (1 × (ψ1 + ψ2 ))∗ P A ⊗ γ ⊗ p∗A Pψ1 +ψ2 (θ2 ) ⊗ p∗B (ψ1 + ψ2 )∗ Pθ1 ; the statement follows directly. Proof of the sufficiency of the conditions of the Theorem 5.21. Suppose that the conditions of the statement are satisfied. Then, by Lemma 5.22, one of the line bundles (5.7) is not Σinvariant. Suppose first that L1 ⊗ γ is not Σ-invariant for every topologically trivial γ. Thus we have that l (0, 0, ψ1 ) is not in φ∗ Num( X ). We deduce that (5.8) 2 · ψ1 ∈ / (1 − σ̃)∗ φ∗ Num( X ). Otherwise we would have 2 · ψ1 = (1 − σ̃)∗ φ∗ ϕ = (1 − σ̃)∗ (h · ψ1 + k · ψ2 ) = 2h · ψ1 + 2k · ψ2 . Therefore h = 1, k = 0 and φ∗ ϕ = ψ1 , contradicting our previous conclusion. Now consider the line bundle L := Nmφ ((1 × ψ1 )∗ P A ). We want to show that there is β ∈ Pic0 ( X ) such that NmπS ( L ⊗ β) is trivial. We use the functoriality of the norm map (Proposition 2.5) and we obtain that NmπS ( L) ' Nmπ̃ ◦ NmπS̃ ((1 × ψ1 )∗ P A ). Observe that by (5.3) we have that πS̃∗ NmπS̃ ((1 × ψ1 )∗ P A ) is numerically trivial. Therefore we have that NmπS̃ ((1 × ψ1 )∗ P A ) is itself numerically trivial. This implies that Nmπ̃ ◦ NmπS̃ ((1 × ψ)∗ P A ) ∈ Pic0 (S). In fact if we have that NmπS̃ ((1 × ψ1 )∗ P A ) := α ∈ Pic0 (S̃) then we write α ' π̃ ∗ γ and we have that NmπS ( L) ' Nmπ̃ ◦ NmπS̃ ((1 × ψ1 )∗ P A ) ' γ⊗2 . On the other side if NmπS̃ ((1 × ψ1 )∗ P A ) := T a numerically trivial but not algebraically trivial line bundle, then as in (4.1) we have that Nmπ̃ ( T ) is topologically trivial. Thus, as before, we obtain β such that NmπS ( L ⊗ β) ' OS via the Pic0 trick (Remark 2.6). In order to determine the non injectivity of the Brauer map we have to ensure that L ⊗ β is not in Im(1 − σ∗ ). Suppose that this were not the case, and consider the following commutative diagram A×B φ σ̃ X / A×B σ φ / X. 26 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP Then c1 (φ∗ L) ∈ (1 − σ̃)∗ φ∗ Num( X ). However the properties of the norm (see (2.8)) ensures that c1 (φ∗ L) = l (0, 0, 2 · ψ1 ), thus we would have that l (0, 0, 2 · ψ1 ) ∈ φ∗ Num( X ), contradicting (5.8). If L2 ⊗ γ is not Σ-invariant for every γ ∈ Pic0 ( A × B), then we proceed as before by exchanging the role of ψ1 and ψ2 . Thus, it remain only to see what happen if L1 ⊗ L2 ⊗ γ is not Σ-invariant for every γ. In this case we will have that l (0, 0, ψ1 + ψ2 ) ∈ / φ∗ Num( A × B), ∗ and so either l (0, 0, ψ1 ) or l (0, 0, ψ2 ) are not in the image of φ . Without loss of generality we can assume the first. Then we will still have (5.8) and we can repeat the above argument. In order to complete the proof of Theorem 5.21 we need to show that if all (1 × ψ1 )∗ P A , (1 × ψ2 )∗ P A , and (1 × (ψ1 + ψ2 ))∗ P A are Σ-invariant then the Brauer map to X is injective. Observe that, under this assumptions, we can write (1 × ψ1 )∗ P A ' φ∗ L1 , (1 × ψ2 )∗ P A ' φ∗ L2 , and (1 × (ψ1 + ψ2 ))∗ P A ' φ∗ L3 . for some line bundles L1 , L2 , and L3 in Pic( X ). Then for α ∈ Pic0 ( X ) write φ∗ α ' p∗A α1 ⊗ p∗B α2 we have φ∗ (πS∗ NmπS ( Li ⊗ α)) ' φ∗ ( Li ⊗ α ⊗ σ∗ ( Li ⊗ α)) ' p∗A α1⊗2 ⊗ p∗B (ψ1∗ Pτ ), φ∗ (πS∗ NmπS ( L1 ⊗ L2 ⊗ α)) ' φ∗ ( L1 ⊗ L2 ⊗ α ⊗ σ∗ ( L1 ⊗ L2 ⊗ α)) ' p∗A α1⊗2 ⊗ p∗B (ψ1∗ Pτ ⊗ ψ2∗ Pτ ); where, in both cases, the last equality is again given by (5.3). Observe that neither the ψi ’s nor ψ1 + ψ2 can factor through the multiplication by 2 isogeny, or we would have that ψ1 and ψ2 cannot generate Hom( B, A). In particular, we cannot have that neither ψi∗ Pτ nor (ψ1 + ψ2 )∗ Pτ can be trivial. We deduce that φ∗ (πS∗ NmπS ( Li ⊗ α)) 6' O A× B , φ∗ (πS∗ NmπS ( L1 ⊗ L2 ⊗ α)) 6' O A× B . In particular we obtained the following lemma: Lemma 5.23. In the above notation, if the conditions of Theorem 5.21 are not satisfied, then line bundles numerically equivalent to Li or L1 ⊗ L2 are not in the kernel of the norm map NmπS . Before going further we need an intermediate step: Lemma 5.24. For any integer n, Li⊗2n and ( L1 ⊗ L2 )⊗2n are in Im(1 − σ∗ ) Proof. Obviously it is enough to show that Li⊗2 is in the image of (1 − σ∗ ). To this aim, we pull Li ⊗ σ∗ Li back to A × B and apply (5.3). We see that φ∗ ( Li ⊗ σ∗ ( Li )) ∈ p∗B Pic0 ( B) ⊆ A × B, and we deduce that γ := Li ⊗ σ∗ ( Li ) is a line bundle in p∗B Pic0 ( B/H ). By 5.2 we know that γ ∈ Im(1 − σ∗ ). Thus we can write Li⊗2 ' γ ⊗ σ∗ Li ⊗ Li−1 . Conclusion of the Proof of Theorem 5.21. Let M is a line bundle such that NmπS ( M ) ' OS , we will show that M is in the image of (1 − σ∗ ). Using (2.8), we know that M ⊗ σ∗ M ' O X . By pulling back via φ we get that φ∗ M ⊗ σ̃∗ φ∗ M is again trivial and by the proof of 5.5 we see that c1 (φ∗ M ) = l (0, 0, h · ψ1 + k · ψ2 ) for two integers h and k. Then we can write φ∗ M ' (1 × h · ψ1 )∗ P A ⊗ (1 × k · ψ2 )∗ P A ⊗ γ ' φ∗ ( L1⊗h ⊗ L2⊗k ) ⊗ γ, BRAUER GROUP OF BIELLIPTIC SURFACES 27 for some γ in Pic0 ( A × B). Therefore φ∗ ( M ⊗ L1⊗−h ⊗ L2⊗−k ) ' γ, and we deduce that M ' L1⊗h ⊗ L2⊗k ⊗ α for some α ∈ Pic0 ( X ). If h and k are both even, then by Lemma 5.24 we know that α ∈ Ker NmπS , and the class of M in Ker NmπS / Im(1 − σ∗ ) is exactly [α]. We apply Proposition 5.1 and deduce that [ M ] = 0. We will now show that neither one between h and k can be odd. Suppose otherwise that h and k are not both even. For example, assume that h is odd and k is even, the proof in the other cases is very similar. Under this hypothesis, Lemma 5.24 ensures that L1 ⊗ α is in the kernel of the norm map. But this contradicts Lemma 5.23, and our proof is complete Example 5.25. (a) Suppose that A ' B, then the isogenies ψ1 and ψ2 are indeed isomorphisms and thus the Brauer map can never be injective. (b) Let B be an elliptic curve without complex multiplication and consider θ2 a point of order 2 in B. Let A be the elliptic curve B/ < θ2 > and ψ : B → A the quotient map. The dual map ψ∗ has degree 2. Let θ1 ∈ A the point such that ψ∗ Pθ1 is trivial and let τ be another order 2 element of A. All this data identify a bielliptic surface of type 2 whose Brauer map to the canonical cover is injective. 28 EUGENIA FERRARI, SOFIA TIRABASSI, AND MAGNUS VODRUP A PPENDIX A. T HE HOMOMORPHISM LATTICE OF TWO ELLIPTIC CURVES J ONAS B ERGSTRÖM AND S OFIA T IRABASSI The main goal of this appendix is to give a structure theorem for the Z-module Hom( B, A) where A and B are two complex elliptic curves with j( B) = 0, 1728. This result has been used in 2.6 above in order to make a clever choice of generators for Num( A × B) which in turn has allowed an accurate description of the action of the automorphism σ∗ on the Neron–Severi group of the product A × B when S is a bielliptic surface of type 3 or 5. If B is an elliptic curve with j-invariant 0 or 1728, then B admits an automorphism λ B of order 3 or 4 respectively. The main result of this Appendix is that the group Hom( B, A) can be completely described in terms of λ B and an isogeny ψ : B → A. More precisely we have the following statement: Theorem A.1. Let A and B two isogenous complex elliptic curves with j( B) is either 0 or 1728. Then there exist an isogeny ψ : B → A such that Hom( B, A) =< ψ, ψ ◦ λ B > . This appendix is organized in three main parts. In the first we outline some classical results about imaginary quadratic fields and their orders. The second is concerned with complex elliptic curves with complex multiplication. Theorem A.1 is proven in A.3. The key idea of our argument is to describe Hom( B, A) as a fractional ideal of End( B) homothetic to End( B). This is done by observing that the class number of End( B) is 1. A.1. Preliminaries on orders in imaginary quadratic fields. An imaginary quadratic field √ is a subfield K ⊆ C of the form Q( −d), with d a positive, square-free integer. The discriminant of K is the integer dk defined as ( −d, if d ≡ 3 mod 4, dK = −4d, otherwise. The ring of integers of K, OK is the largest subring of K which is a finitely generated abelian group. Then we have that OK = Z[δ], where ( √ 1+ − d , if d ≡ 3 mod 4, (A.1) δ= √2 −d, otherwise. An order in an imaginary quadratic field K is a subring O of OK which properly contains Z. It turns out that O ' Z + Z · (nδ) for some positive integer n. Given an order O in an imaginary quadratic field K, a fractional ideal of O is a non-zero finitely generated sub O -module of K. For every fractional ideal M of O there is an α ∈ K ∗ and an ideal a of O such that M = α · a. We will need the following notions. Definition A.2. (i) Two fractional O -ideals M and M0 are homothetic if there is α ∈ K ∗ such that M = αM0 . (ii) A fractional O -ideal is invertible if there is a fractional ideal M0 such that M · M0 = O . The set of invertible O -ideals is denoted by I (O). (iii) A fractional O -ideal M is principal if it is of the form α · O for some α ∈ K ∗ . So principal ideals are precisely the fractional ideals homothetic to O . The set of principal O -ideals is denoted by P(O). Principal ideals are clearly invertible. In general not all fractional ideals are invertible, but they are so if O = OK (see also [Cox11, Proposition 5.7]). The quotient Cl(O) := I (O)/P(O) STRUCTURE OF Hom( A, B) 29 describes the homothety classes of invertible O -ideals. It is a group with the product and it is called the ideal class group of O . Its order is finite and is called the class number of O . When O = OK , then the class number of O is exactly the class number of the field K, which is a function of the discriminant of K (see [Cox11, Theorem 5.30(ii)]). More generally the class number of O is a general function of dK and [O : OK ]. √ Example A.3. If K is either Q(i ) or Q( −3), then all the fractional ideals of OK are homothetic to OK . In fact the class number of the field K in this cases is 1, as it was computed by Gauss in his book Disquisitiones arithmeticae . A.2. Elliptic curves with complex multiplication. The importance of orders in the study of the geometry of elliptic curves is that they describe the endomorphism rings of a complex elliptic curves: Theorem A.4. Let A be an elliptic curve over C, then End( A) is either isomorphic to Z or to an order in an imaginary quadratic field. Proof. See [Sil09, Theorem VI.5.5]. We say that a (complex) elliptic curve has complex multiplication if its endomorphisms ring is larger than Z. Observe that in this case End( A) ⊗ Q is a quadratic field K and End( A) is an order in K. Given a complex elliptic curve A there is a canonical way to identify its endomorphisms ring with a subring of C. More generally let A and B two elliptic curves, then there are two lattices Λ A and Λ B in C such that A ' C/Λ A and B ' C/Λ B . Given a complex number ζ such that ζ · Λ B ⊆ Λ A , the map Φζ : C → C defined by z 7→ ζ · z descends to an (algebraic) homomorphism ϕζ : B → A. It is possible to show (see [Sil09, VI.5.3(d)]) that any morphism of elliptic curves preserving the origin is obtained in this way, and in particular we get an isomorphism of abelian groups (A.2) Hom( B, A) ' {ζ ∈ C | ζ · Λ B ⊆ Λ A } ⊆ C. By setting B = A we get a ring isomorphism End( A) ' O := {ζ ∈ C | ζ · Λ A ⊆ Λ A } ⊆ C. The isomorphism ζ 7→ ϕζ is characterized as the unique isomorphism f : O → End( A) such that, for any ζ ∈ O and for every invariant form ω on A we have that f (ζ )∗ ω = ζ · ω ( [Sil13, II.1.1]). Notation A.5. For an elliptic curve with complex multiplication A such that End( A) ' Z + Z · nδ, we will denote by λ A the isogeny ϕnδ : A → A and we will say that A has complex multiplication by λ A . It is clear that, with this identification, End( A) =< 1 A , λ A > as a Z-module. Example A.6. (a) Suppose that B is an elliptic curve with j-invariant 0. Then we can write √ 2πi ) and End( B) ' OK = B '√ C/Λ B , with Λ B =< 1, e 3 >. Then End( B) ⊗ Q ' Q( −3√ 1+ −3 1+ −3 Z[ 2 ]. We have that λ B is induced by the multiplication by and is an automor2 phism of B satisfying λ2B + λ B + 1B = 0. This is exactly the automorphism which in 2.2 was denoted by ρ and which was used to construct bielliptic surfaces of type 5. (b) Suppose now that the j-invariant of B is 1728. Then we can take Λ B =< 1, i > and we have that End( B) ⊗ Q ' Q(i ). The endomorphisms ring of B is isomorphic to Z[i ] and the multiplication by i induces an automorphism λ B such that λ2B = −1B . This is the automorphism ω of B used to construct bielliptic surfaces of type 3 in 2.2. 30 JONAS BERGSTRÖM AND SOFIA TIRABASSI A.3. Proof of the main result. We are now ready to provide a proof for Theorem A.1. Our key point will be the following: Claim: the Z-module Hom( B, A) is isomorphic to a fractional ideal of OK . Before proceeding with showing that this Claim is true, let us see how it implies the statement. We do this applying Example A.3 and deducing that all fractional OK -ideals are homothetic to OK . Therefore there exist α ∈ K ∗ such that M ' α · OK = α· < 1, δ >=< α, α · δ >, where δ is like in (A.1). But then we have that Hom( B, A) =< ϕα , ϕα ◦ λ B >, and the statement is true. Proof of the Claim. Let Λ A =< 1, τ > a lattice in C such that A ' C/Λ A , and denote by K ⊆ C the quadratic field End( B) ⊗ Q. Then the ring End( B) is exactly the ring of integers OK . Observe that this is isomorphic to a lattice in C, and that B ' C/OK (See Example A.6). By (A.2) we can identify M := Hom( B, A) as a finitely generated subgroup of C. Composition on the right with endomorphism of B gives to M a structure of OK -module. Let α 6= 0 denote an element of a := Hom( A, B), identifyied with a subgroup of C. Then clearly α · M ⊆ OK . We deduce that M ⊆ K is a fractional ideal of OK , and the Claim is proven. Remark A.7. (a) For any order O in a quadratic extension of Q a representative of each homothety class of fractional ideals can be given as I · O 0 , where O 0 ⊇ O is an over order and I is an invertible fractional ideal (see [Mar18]). The over order S can be given a Z-basis of the form {1, δ · f } where f is a positive integer. For any pair of isogenous complex elliptic curves with complex multiplication we have that Hom( B, A) is a fractional End( B) ideal. In addition, if we assume that End( B) has class number 1, we have that B ' C/ End( B). In fact, under this assumption [Cox11, Corollary 10.20] yields that, there is just one elliptic curve up to isomorphism with endomorphism ring End(B). In conclusion, demanding that End( B) has class number 1 (instead of j( B) being either 0 or 1728) is sufficient for Theorem A.3 to hold. So Theorem A.3 will hold for the 13 isomorphism classes of complex elliptic curves B for which End( B) has discriminant -3, -4,-7, -8, -11, -12, -16, -19, -27, -28, -43, -67, -164 (see [Sil13, Example 11.3.2]). (b) It is clear from the proof that the role of A and B can be exchanged, so we have proven a structure theorem for Hom( A, B) when the endomorphism ring of one of the two curves has class number 1. Theorem A.3 is not constructive, in the sense that it does not provide a way to determine the isogeny ψ such that ψ and ψ ◦ λ B generate Hom( B, A). We conclude this appendix by constructing ψ in an easy example. Example A.8. Let Λ be the lattice < 1, 2i >⊆ C, and consider A := C/Λ. Consider the 2-torsion point τ := (0, i ) + Λ of A and let B be the quotient A/ < τ >. It is clear that B has j-invariant 1728. We claim that Hom( B, A) =< ϕ2 , ϕ2 ◦ λ B >. We use first (A.2) and identify Hom( B, A) with a lattice in C. Given α = ( a + bi ) ∈ Hom( B, A), we have that both α and α · i must be elements of Λ. We deduce that both a and b must be even integers and so Hom( B, A) =< 2, 2 · i >. We conclude by observing that λ B is the automorphism of B induced by multiplication by i. R EFERENCES [Bad01] L. Badescu. Algebraic surfaces. Springer Science & Business Media, 2001. STRUCTURE OF Hom( A, B) 31 [BDF10] G Bagnera and M. De Franchis. Le nombre ρ de Picard pour les surfaces hyperelliptiques. Rend. Circ. Mat. Palermo, 10, 1910. [Bea96] A. Beauville. Complex Algebraic Surfaces. Cambridge University Press, 1996. [Bea09] A. Beauville. On the Brauer group of Enriques surfaces. Math. Res. Lett., 16(6):927–934, 2009. [BHPvdV15] W. Barth, K. Hulek, C. Peters, and A. van de Ven. Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Springer Berlin Heidelberg, 2015. [BL13] C. Birkenhake and H. Lange. Complex abelian varieties, volume 302. Springer Science & Business Media, 2013. [BM77] E. Bombieri and D. Mumford. Enriques’ classification of surfaces in char. p. II. In Complex analysis and algebraic geometry, pages 23–42. Springer Science & Business Media, 1977. [BM98] T. Bridgeland and A. Maciocia. Fourier-Mukai transforms for quotient varieties. arXiv preprint math/9811101, 1998. [Cox11] D. A. Cox. Primes of the form x2 + ny2 : Fermat, class field theory, and complex multiplication, volume 34. John Wiley & Sons, 2011. [dJ] J. de Jong. A result of Gabber. https://www.math.columbia.edu/ dejong/papers/2-gabber.pdf. [Gau66] C. F. Gauss. Disquisitiones arithmeticae, volume 157. Yale University Press, 1966. [Gro61] A. Grothendieck. éléments de géométrie algébrique : Ii. étude globale élémentaire de quelques classes de morphismes. Publications Mathématiques de l’IHÉS, 8:5–222, 1961. [Gro67] A. Grothendieck. éléments de géométrie algébrique : Iv. étude locale des schémas et des morphismes de schémas, quatrième partie. Publications Mathématiques de l’IHÉS, 32:5–361, 1967. [HLT19] K. Honigs, L. Lombardi, and S. Tirabassi. Derived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic. Math. Z., 2019. Published online first. DOI:10.1007/s00209-019-02362-1. [Iit70] S. Iitaka. Deformations of compact complex surfaces ii. J. Math. Soc. Japan, 22(2):247–261, 04 1970. [Mar18] S. Marseglia. Computing the ideal class monoid of an order. To appear on the Journal of the London Mathematical Society. ArXiv preprint: 1805.09671, 2018. [Nue] H. Nuer. Moduli spaces of bridgeland stable objects on bielliptic surfaces,. In preparation. [Ser90a] F. Serrano. Divisors of bielliptic surfaces and embeddings in P4 . Math. Z., 203(3):527–533, 1990. [Ser90b] F. Serrano. Multiple fibres of a morphism. Comment. Math. Helv., 65(2):287–298, 1990. [Ser91] F. Serrano. The Picard group of a quasi-bundle. manuscripta mathematica, 73(1):63–82, Dec 1991. [Sil09] J. H. Silverman. The arithmetic of elliptic curves, volume 106. Springer Science & Business Media, 2009. [Sil13] J. H. Silverman. Advanced topics in the arithmetic of elliptic curves, volume 151. Springer Science & Business Media, 2013. [Sta19] The Stacks project authors. The stacks project. https://stacks.math.columbia.edu, 2019. [Suw69] T. Suwa. On hyperelliptic surfaces. J. Fac. Sci. Univ. Tokyo Sect. I, 16:469–476 (1970), 1969. [Ume75] H. Umemura. Stable vector bundles with numerically trivial chern classes over a hyperelliptic surface. Nagoya Math. J., 59:107–134, 1975. D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF B ERGEN , A LLÉGATEN 41, B ERGEN , N ORWAY Email address: [email protected], [email protected], [email protected] D EPARTMENT OF M ATHEMATICS , S TOCKHOLM U NIVERSITY, K RAFTRIKET Email address: [email protected], [email protected] HUS 6, S TOCKHOLM , S WEDEN