REAL HYPERELLIPTIC SURFACES AND THE ORBIFOLD FUNDAMENTAL GROUP

FABRIZIO CATANESE PAOLA FREDIANI

Abstract. In this paper we finish the topological classification of real algebraic surfaces of Kodaira dimension zero and we make a step towards the Enriques classification of real algebraic surfaces, by describing in detail the structure of the moduli space of real hyperelliptic surfaces.

Moreover, we point out the relevance in real geometry of the notion of the orbifold fundamental group of a real variety, and we discuss related questions on real varieties (X, σ) whose underlying complex manifold X is a K(π, 1).

Our first result is that if (S, σ) is a real hyperelliptic surface, then the differentiable type of the pair (S, σ) is completely determined by the orbifold fundamental group exact sequence. This result allows us to determine all the possible topological types of (S, σ), and to prove that they are exactly 78.

It follows also as a corollary that there are exactly eleven cases for the topological type of the real part of S.

Finally, we show that once we fix the topological type of (S, σ) corresponding to a real hyperelliptic surface, the corresponding moduli space is irreducible (and connected).

We also give, through a series of tables, explicit analytic representations of the 78 components of the moduli space.

Keywords: real varieties, hyperelliptic surfaces, orbifold fundamental group.

AMS classification: 14P25; 14J15; 32Q55; 32J15 .

Introduction

The purpose of this paper is twofold: on the one side, we finish the topological classification of real algebraic surfaces of Kodaira dimension zero and we make a step towards the Enriques classification of real algebraic surfaces, by describing in detail the structure of the moduli space of real hyperelliptic surfaces; on the other hand, we point out the relevance in real geometry of the notion of the orbifold fundamental group.

Research performed in the realm of the EU Project ”E.A.G.E.R.” .

In order to illustrate the latter concept, let us begin by answering the reader’s question: what is a real variety?

A smooth real variety is a pair (X, σ), consisting of the data of a smooth complex manifold X of complex dimension n and of an antiholomorphic involution σ : X → X (an involution σ is a map whose square is the identity).

The quickest explanation of what antiholomorphic means goes as follows: the smooth complex manifold X is determined by the differentiable manifold M underlying X and by a complex structure J on the complexification of the real tangent bundle of M.

If instead we consider the same manifold M together with the complex structure −J, we obtain a complex manifold which is called the conjugate of X and denoted by X̄.

The involution σ is now said to be antiholomorphic if it provides an isomorphism between the complex manifolds X and X̄ (and then (X, σ) and (X̄, σ) are also isomorphic as pairs).

What are the main problems concerning real varieties? (we may restrict ourselves to the case where X is compact).

• Describe the isomorphism classes of such pairs (X, σ).

• Or, at least describe the possible topological or differentiable types of the pairs (X, σ).

• At least describe the possible topological types for the real parts X′ := X(R) = F ix(σ).

Remark 0.1. Recall that Hilbert’s 16-th problem is a special case of the last question but for the more general case of a pair of real varieties (Z ⊂ X, σ).

For a smooth real variety, we have the quotient double covering π : X → Y = X/ < σ >, and the quotient Y is called the Klein variety of (X, σ).

In dimension n = 1 the datum of the Klein variety is equivalent to the datum of the pair (X, σ), but this is no longer true in higher dimension, where we will need also to specify the covering π.

The covering π is ramified on the so-called real part of X, namely, X′ := X(ℝ) = F ix(σ), which is either empty, or a real submanifold of real dimension n.

If X′ := X(ℝ) = F ix(σ) is empty, the orbifold fundamental group of Y is just defined as the fundamental group of Y.

If instead X′ ≠ ∅, we may take a fixed point x_{0} ∈ F ix(σ) and observe that σ acts on the fundamental group π_{1}(X, x_{0}): we can therefore define the orbifold fundamental group as the semidirect product of the normal subgroup π_{1}(X, x_{0}) with the cyclic subgroup of order two generated by σ. It is easy to verify then that changing the base point does not alter the isomorphism class of the following exact sequence, yielding the orbifold fundamental group as an extension

1 → π_{1}(X) → π_{1}^{orb}(Y) → ℤ/2 → 1

(changing the base point only affects the choice of a splitting of the above sequence).

We claim that the orbifold fundamental group exact sequence is a powerful topological invariant of the pair (X, σ) in the case where X has large fundamental group.

To illustrate a concrete issue of such a statement, let us consider the case of a K(π, 1), i.e., the case where the universal cover of X is contractible: this case includes the case of complex tori, of hyperelliptic surfaces and their generalizations, as well as the case of quotients of the complex n-ball and of polydisks.

Then the homotopy type of X is determined by the fundamental group π and some interesting quite general open questions are the following ones (where, by abuse of language, we shall talk about orbifold fundamental group of a real variety, to refer to the above orbifold fundamental group exact sequence).

• If X is a compact complex manifold which is a K(π, 1), to what extent does π also determine the differentiable type?

• Same assumptions as above and fix the group π: when is then the

moduli space of those manifolds X irreducible or connected?

For instance, is it sufficient to put the further hypothesis that we only consider Kähler manifolds? (a vast literature already exists, cf. [Mos], [Yau], [Siu],[J-Y1], cf. also [Ca] for more examples and for further references)

• Assume that we consider real varieties, (X, σ) where X is as above a K(π, 1): does the orbifold fundamental group determine the differentiable type of the real variety?

• Fixing the orbifold fundamental group, when do we get a connected moduli space?

Our purpose here is thus to give an issue where the classification of real varieties can be given in terms of the orbifold fundamental group (concerning the previous questions, even the apparently easier case of real and complex tori appears to be unsettled as soon as the complex dimension becomes at least 3, at least if we drop the Kähler hypothesis; however, as it is going to be shown elsewhere, the complex manifolds isomorphic to the product of a curve C of genus at least 3 and of a complex torus of dimension 2 form a connected component of their moduli space, which admits however another component, corresponding to the examples of [Cal], providing on the underlying differentiable manifold a non Kähler complex structure with trivial canonical bundle).

Our first result, concerning the topological type of real hyperelliptic surfaces, can now be briefly stated as follows:

Theorem 0.2. Let (S, σ) be a real hyperelliptic surface. Then the differentiable type of the pair (S, σ) is completely determined by the orbifold fundamental group exact sequence.

In a sequel to this paper, we plan to show other issues (e.g., in the Kodaira classification of non algebraic real surfaces) where the topology of the pair (S, σ) is determined by the orbifold fundamental group exact sequence.

Returning to the case of real hyperelliptic surfaces, the previous theorem allows us to easily determine completely the possible topological types of (S, σ), and in particular we have the

Theorem 0.3. Real hyperelliptic surfaces fall into exactly 78 topological types.

and the

Corollary 0.4. Let (S, σ) be a real hyperelliptic surface. Then the real part S(ℝ)) is either

• a disjoint union of t tori, where 0 ≤ t ≤ 4

• a disjoint union of b Klein bottles, where 1 ≤ b ≤ 4.

• the disjoint union of one torus and one Klein bottle

• the disjoint union of one torus and two Klein bottles.

As the reader may guess, the above results are too complicated to be described in detail here in the introduction: therefore we will limit ourselves to illustrate the underlying philosophy by describing it in the much simpler case of the real elliptic curves.

Classically (cf. e.g. [A-G] for a modern account) real elliptic curves have been classified according to the number ν of connected components (these are circles) of their real part: ν can only attain the values 0, 1, 2 and completely determines the differentiable type of the involution.

The orbifold fundamental group explains easily this result: if there is a fixed point for σ, the orbifold fundamental group sequence splits and the action of σ on the elliptic curve C is completely determined by the action s of σ on H_{1}(C, ℤ).

This situation gives rise to only two cases: s is diagonalizable, and C(ℝ) consists of two circles, or s is not diagonalizable, and C(ℝ) consists of only one circle.

If instead there are no fixed points, an easy linear algebra argument (cf. lemma 4.4) shows that s is diagonalizable, and the translation vector of the affine transformation inducing σ can be chosen to be 1/2 of the +1-eigenvector e_{1} of s.

In fact, σ is represented by an affine transformation (x, y) → s(x, y)+ (a, b), and s is not diagonalizable if and only if s(x, y) = (y, x) for a suitable choice of two basis vectors. The identity map, which equals the square of σ, is the transformation (x, y) → (x, y) + (a + b, a + b), thus a + b is an integer, and therefore the points (x, x − a) yield a fixed S^{1} on the elliptic curve.

The complete description of the moduli space of real hyperelliptic surfaces is too long to be reproduced in the introduction, we want here only to mention the following main result, which confirms a conjecture by Kharlamov, that more generally for all real Kähler surfaces of Kodaira dimension at most 1 the differentiable type of the pair (S, σ) should determine the deformation type (this is false already for complex surfaces if the Kodaira dimension equals 2, cf. [Ca1],[Ca2], [Ma]).

Theorem 0.5. Fix the topological type of (S, σ) corresponding to a real hyperelliptic surface. Then the moduli space of the real surfaces (S′, σ′) with the given topological type is irreducible (and connected).

Again, we wish to give the flavour of the argument by outlining it in the much simpler case of the elliptic curves. Assume for instance that our involution σ acts as follows: (x, y) → (y, x). We look then for a translation invariant complex structure J which makes σ antiholomorphic, i.e., we seek for the matrices J with J^{2} = −1 and with Js = −sJ.

The latter condition singles out the matrices ( \begin{matrix}a & b \\ −b & −a \end{matrix} )

while the first condition is equivalent to requiring that the characteristic polynomial be equal to λ^{2} + 1, whence, it is equivalent to the equation b^{2} − a^{2} = 1.

We get therefore a hyperbola with two branches which are exchanged under the involution J → −J, but, as we already remarked, J and −J yield isomorphic real elliptic curves, thus the moduli space consists of just one branch of the hyperbola.

This example serves also the scope of explaining the statement in the above theorem that the moduli space is irreducible (and connected): the hyperbola is an irreducible algebraic variety, but not an irreducible analytic space, since it is not connected (in general, moduli spaces of real varieties will be semianalytic spaces or semialgebraic real spaces).

We want to emphasize once more that an interesting question is to determine, in the realm of the real varieties whose topological type is determined by the orbifold fundamental group, those for which the corresponding moduli spaces are irreducible (respectively: connected).

It is now however time to recall what the hyperelliptic surfaces are, why they have this name and, last but not least, point out how crucial is the role of the hyperelliptic surfaces in the Enriques classification of algebraic surfaces.

As elliptic curves are exactly the curves such that the homogeneous coordinates of their points cannnot be uniformized by polynomials, yet they can be uniformized by entire holomorphic functions on ℂ, hyperelliptic varieties of dimension n were generally defined by Humbert and Picard through the entirely analogous property that the coordinates of their points, although not uniformizable by rational functions, could be uniformized by entire meromorphic functions on ℂ^{n}.

Among these varieties are clearly (nowadays) the Abelian varieties, and the classification of such varieties in dimension two was achieved by Bagnera and de Franchis who got the Bordin Prize in 1909 for their important result (the classification by Enriques and Severi, who got the same prize for it the year before, had serious gaps which were corrected only later on).

The missing surfaces, which are now called hyperelliptic, were described as quotients of the product of two elliptic curves by the action of a finite group G. For this reasons, some authors call these surfaces bielliptic surfaces (cf. [Be]).

The classification is in the end very simple and produces a list of 7 cases where the Bagnera de Franchis group G and its action is explicitly written down.

The reason to recall all this is that, as a matter of fact, an important ingredient in the proof of our theorems is to rerun the arguments of the proof of the classification theorem, which characterizes the hyperelliptic surfaces as the algebraic surfaces with nef canonical divisor K, with K^{2} = p_{g} = 0, q = 1, and moreover with Kodaira dimension equal to 0. This is done in section 1.

To keep close in spirit to the beautiful result of Bagnera and de Franchis we felt compelled to produce tables exhibiting simple and explicit actions for the 78 types of the real hyperelliptic surfaces: these are contained in the last section, and they summarize a lot of information that we could not give in a more expanded form.

Concerning now the Enriques classification of real algebraic surfaces, it has been focused up to now mostly on the classification of the topology of the real parts, the topological classification of real rational surfaces going back to Comessatti ([CO1][CO2][CO3]), as well as the classification of real abelian varieties ([CO3], see also [Si], [Se-Si]).

In the case of real K3 - surfaces we have the classification by Nikulin and Kharlamov ([Ni], [Kha]), for the real Enriques surfaces the one by Degtyarev and Kharlamov ([D-K1], [D-K3]).

Finally partial results on real ruled and elliptic surfaces have been obtained by Silhol ([Si]) and by Mangolte ([Man3]).

The paper is organized as follows:

in section 1 we moreover recall the description given by Bagnera and de Franchis of the hyperelliptic surfaces as quotients of a product of two elliptic curves E × F by the product action of a finite group G acting on E as a group of translations and on F via an action whose quotient is ℙ^{1}.

G is called the Bagnera de Franchis group (or symmetry group) and is a quotient of the fundamental group of the surfaces S.

In section 2 we observe that the orbifold fundamental group has a finite quotient Ĝ which contains G as a normal subgroup of index 2: Ĝ is called the extended Bagnera de Franchis group and its structure will be investigated in detail in section 4.

In the rest of section 2 we show that isomorphisms of real hyperelliptic surfaces lift to isomorphisms of the respective products of elliptic curves, compatibly with the identifications of the respective extended Bagnera de Franchis groups.

Section 3 is devoted first to showing that the representation of the orbifold fundamental group as a group of affine transformations of ℚ^{4} is uniquely determined, up to isomorphism, by the abstract structure of the group, what proves theorem 0.2 (by the way, we show in the course of the proof a fact hardly mentioned in the literature, namely, that the differentiable structure of a hyperelliptic surface is determined by the fundamental group).

Second, after recalling quite briefly the notion of moduli spaces for real varieties, we show that, once this affine representation is fixed, the moduli space for the compatible complex structures is irreducible and connected.

Section 4 recalls some known facts about antiholomorphic maps of elliptic curves and applies these results to the determination of the possible extended Bagnera de Franchis group which do in effect occur.

Section 5 determines the analytical actions of these groups on the two factors under the condition that the exact sequence 1 → G → Ĝ → ℤ/2 → 1 splits, while section 6 deals with the simpler case where there is no

splitting.

Section 7 gives the recipe to identify the real part S(ℝ) of our surfaces as a disjoint union of Klein bottles and tori.

Finally, section 8 applies the results developed insofar and achieves the classification of the 78 components of the moduli space, for which explicit analytical representations, describing the action of the extended Bagnera de Franchis group, are given through a series of tables.

Acknowledgements. We would like to thank V. Kharlamov for an interesting conversation concerning the status of the Enriques classification of algebraic surfaces, and for communicating his conjecture to us. Finally, S.T. Yau pointed out Calabi’s example ([Cal]) of diffeomorphic complex manifolds such that one is Kähler, while the other is not: with some work, this example shows that, without the Kähler hypothesis, the moduli space of complex manifolds which are K(π, 1)’s (for a fixed π) can be disconnected. The present research was performed in the realm of the Schwerpunkt ” Globale Methode in der Komplexe Geometrie”, and partially supported by the italian M.U.R.S.T. 40/100 program ”Geometria Algebrica”.

1. Basics on hyperelliptic surfaces

In this section we recall ([B-dF], [B-dF2] see also [Be], Chapitre VI, pp. 91-115, [B-P-VdV] pp. 147-149) the definition of hyperelliptic surfaces and their characterization in the realm of the Enriques classification of complex algebraic surfaces. We shall also briefly recall the main lines of the proof of the Bagnera - de Franchis classification theorem, since we shall repeatedly need modified or sharper versions of the arguments used therein.

Definition 1.1. A complex surface S is said to be hyperelliptic if S ≅ (E × F)/G, where E and F are elliptic curves and G is a finite group of translations of E with a faithful action on F such that F/G ≅ ℙ^{1}.

G ⊂ Aut(F), so G = T ⋊G′ (semidirect product), where T is a group of translations and G′ ⊂ Aut(F) consists of group automorphisms. Since F/G ≅ ℙ^{1}, then G′ ≠ 0, hence G′ ≅ ℤ/m, with m = 2, 3, 4, 6, by the following well known result.

Fact 1.2. Let F be an elliptic curve. Every automorphism of F is the composite of a translation and a group automorphism. The non trivial group automorphisms are the symmetry x ↦ −x and also: for the curve F_{i} = ℂ/(ℤ ⊕ ℤ · i), x ↦ ±ix. For the curve F_{ρ} = ℂ/(ℤ ⊕ ℤ · ρ), where ρ^{3} = 1 ≠ ρ, x ↦ ±ρx, and x ↦ ±ρ^{2}x.

Since G is abelian, as a group of translations of E, the product T ⋊G′ must be direct. We have the following result:

Theorem 1.3. (Bagnera - de Franchis) Every hyperelliptic surface is one of the following, where E, F are elliptic curves and G is a group of translations of E acting on F as specified:

1. (E × F)/G, G = ℤ/2 acting on F by symmetry.

2. (E × F)/G, G = ℤ/2 ⊕ ℤ/2 acting on F by x ↦ −x, x ↦ x + ϵ, where ϵ belongs to the group F_{2} of points of F of order 2.

3. (E × F_{i})/G, G = ℤ/4 acting on F_{i} by x ↦ ix.

4. (E × F_{i})/G, G = ℤ/4 ⊕ ℤ/2 acting on F_{i} by x ↦ ix, x ↦ x + (1 + i)/2.

5. (E × F_{ρ})/G, G = ℤ/3 acting on F_{ρ} by x ↦ ρx.

6. (E × F_{ρ})/G, G = ℤ/3 ⊕ ℤ/3 acting on F_{ρ} by x ↦ ρx, x ↦ x + (1 − ρ)/3.

7. (E × F_{ρ})/G, G = ℤ/6 acting on F_{ρ} by x ↦ −ρx.

Hyperelliptic surfaces are algebraic surfaces with p_{g} = 0, q = 1, K^{2} = 0, K nef. We have the following basic result of the Enriques classification of surfaces (Kodaira [KoIV] indeed proved that the same result holds more generally for compact complex surfaces provided one replaces the hypothesis q = 1 by b_{1} = 2).

Theorem 1.4. The complex surfaces S with K nef, K^{2} = 0, p_{g} = 0 and such that either S is algebraic with q = 1, or more generally b_{1} = 2, are hyperelliptic surfaces if and only if kod(S) = 0 (this is equivalent to requiring that all the fibers of the Albanese map be smooth of genus 1).

Proof. Let α : S ⟶ A be the Albanese map, q = 1, so A is a curve of genus 1. Let π : ℂ → A be the universal covering and let us consider the pull - back diagram:

[figure: S̃; α̃; S; α; ℂ; π; A]

All the fibers of α are smooth, therefore also the fibers of α̃ are smooth. So α and α̃ are C^{∞} bundles, by Ehresmann’s theorem. ℂ is contractible, so S̃ is diffeomorphic to the product ℂ × F. Thus we obtain a holomorphic map (since we have in fact a locally liftable holomorphic map f : ℂ → H/P SL(2, ℤ), and ℂ is simply connected)

f : ℂ → H = {τ ∈ ℂ | Im(τ) > 0},
t ↦ τ,

where

α̃^{−1}(t) ≅ ℂ/ℤ ⊕ τℤ.

By Liouville’s theorem we see that f is constant, therefore

S̃ ≅ ℂ × F.

If A = ℂ/Λ, then S ≅ (ℂ × F)/Λ, where Λ acts on ℂ by translations and on F by a map μ : Λ → Aut(F). Let Γ := Aut(F)/Aut^{0}(F), where Aut^{0}(F) are the automorphisms which are homotopic to the identity, and let ν : Λ → Γ be the induced map. We set Λ′ := kerν, Γ′ := ν(Λ). Then we get the following pull-back diagram:

[figure: S′ := S̃/Λ′; α′; ψ; S = S′/Γ′; α; A′ := ℂ/Λ′; A = A′/Γ′]

Λ′ acts by translations on F, so it acts as the identity on H^{0}(Ω_{F}^{1} ). Then there exist η, η′ ∈ H^{0}(Ω_{S′}^{1}) such that η ∧ η′ ≢ 0, thus K_{S′} ≡ 0 and q(S′) = 2, so S′ is a complex torus. The map ψ : S′ → S is an unramified covering of degree m, where m ∈ {2, 3, 4, 6}. Then mK_{S} = ψ_{∗}ψ^{∗}K_{S} = ψ_{∗}K_{S′} = 0. In particular 12K_{S} ≡ 0. The map α′ : S′ → A′ is a fibre bundle on an elliptic curve and S′ is a complex torus of dimension 2. Since b_{1} = 2, S is algebraic hence S′ is algebraic too and, by Poincaré’s reducibility theorem there exists a finite unramified covering A′′ → A′ yielding a product structure on the pull back S′′ of S′.

[figure: S′′ := A′′ ×_{A′} S′ ≅ A′′ × F; α′′; S′; α′; A′′ =: ℂ/Λ′′; A′ = ℂ/Λ′]

Then we have found that S ≅ (A′′ × F)/G, where G = Λ/Λ′′. Moreover, by choosing the covering A′′ → A′ minimal with the above property, one sees that E := A′′, F, G are as in the list by Bagnera de Franchis. Q.E.D.

2. Real conjugations on hyperelliptic surfaces

Let us now suppose that S is a real hyperelliptic surface, i.e. there is an antiholomorphic involution σ : S → S, and we consider the isomorphism class of the pair (S, σ). Since, by definition of the Albanese map α, fixed a point x_{0} ∈ S, α(x) = ∫_{x_{0}}^{x}, we obtain

α(σ(x)) = ∫_{x_{0}}^{σ(x)} = ∫_{x_{0}}^{σ(x_{0})} + ∫_{σ(x_{0})}^{σ(x)},

if we define

σ̄(γ) := ∫_{x_{0}}^{σ(x_{0})} +σ_{∗}(γ),

we get an induced antiholomorphic map on the Albanese variety σ̄ : A → A with the property that the following diagram commutes

[figure: S; α; σ; S; α; A = ℂ/Λ; σ̄; A = ℂ/Λ]

A direct calculation, or the remark that α(S) generates A and σ̄^{2} is the identity on α(S), assures that σ̄ is an antiholomorphic involution on A. Notice that, is S(ℝ) ≠ ∅, we may choose a point x_{0} with σ(x_{0}) = x_{0}, and then σ̄ will be a group homomorphism. We want to prove that σ lifts to a map σ̃ : A′′ × F → A′′ × F, where A′′ and F are as in the proof of the previous theorem. Observe that since we have the pull-back diagram

[figure: S′′ ≅ A′′ × F; α′′; ϕ; S; α; A′′ = ℂ/Λ′′; A = ℂ/Λ]

it suffices to prove that the involution σ̄ on A lifts to A′′. In fact then σ lifts to S′′ as a fibre product and so we have an induced action on A′′ × F, preserving α′′. We need the following Lemma 2.1. Let π : Y ⟶ X be a connected covering space, and let g be a homeomorphism of X. Choose x_{0} ∈ X, y_{0} ∈ Y with π(y_{0}) = x_{0}, and let z_{0} = g(x_{0}), w_{0} ∈ Y with π(w_{0}) = z_{0}. Then there exists a lift g̃ of g with g̃(z_{0}) = w_{0} if and only if, given a path δ̃ from y_{0} to w_{0} and setting δ = π ∘ δ̃, and considering the isomorphism Δ : π_{1}(X, z_{0}) →^{≅} π_{1}(X, x_{0}) such that Δ(γ) = δγδ^{−1}, and similarly Δ̃, we have

Δg_{∗}(H) = H, where H = H_{y_{0}} = π_{∗}(π_{1}(Y, y_{0})).

Proof. Consider the diagram of pointed spaces

[figure: (Y, y_{0}); π; (Y, w_{0}); π; (X, x_{0}); g; (X, z_{0})]

Then g̃ exists and it is unique if and only if g_{∗}(H_{y_{0}}) = H_{w_{0}}; applying Δ, iff

Δg_{∗}(H_{y_{0}}) = π_{∗}(Δ̃π_{1}(Y, w_{0})).

But Δ̃π_{1}(Y, w_{0}) = π_{1}(Y, y_{0}), whence g̃ exists and it is unique iff Δg_{∗}(H) = H. Q.E.D.

Corollary 2.2. Under the above notation there exists a lift g̃ of g if and only if Δg_{∗}(H) is a conjugate of H.

Proof. There exists g̃ if and only if there exists w′_{0} ∈ π^{−1}(z_{0}) such that Δ′g_{∗}(H) = H. This is equivalent to say that Δg_{∗}(H) is a conjugate of H (we conjugate by δδ′^{−1}, where δ′ = π ∘ δ̃′ and δ̃′ is a path from y_{0} to w′_{0}). Q.E.D.

Corollary 2.3. There exists a lift g̃ with a fixed point if and only if there exists x_{0} ∈ F ix(g) and a conjugate subgroup H′ of H such that g_{∗}(H′) = H′. Furthermore then, if g has order n then also g̃ has order n (since g̃^{n} is a lift of the identity and has a fixed point).

Now we go back to our situation. Since the fundamental groups of S′, resp. S′′ give rise to subgroups of π_{1}(S) which are the pull backs of π_{1}(A′), resp. π_{1}(A′′) under the bundle homotopy exact sequence

1 → π_{1}(F) → π_{1}(S) → π_{1}(A) → 1,

we obtain that they are preserved under σ_{∗} if and only if the corresponding subgroups of π_{1}(A) are preserved under σ̄_{∗}. In the latter case all the fundamental groups are abelian and we need to prove that σ̄ lifts to A′′, or equivalently that Λ′′ is σ̄ - invariant. We observe that Λ′ is invariant by σ̄, since Λ′ is the kernel of the topological monodromy ν : Λ → Γ = Aut(F)/Aut^{0}(F), which is induced by σ_{∗}, hence it is σ̄ - equivariant. We observe that while Λ′ is canonically defined as the kernel of the topological monodromy of α, it is a priori not clear that Λ′′ be canonically defined (we shall indeed prove later that Λ′′ is the centre of the fundamental group of S). However, using the list of Bagnera - de Franchis, case by case, we see that Λ′′ is a characteristic subgroup of Λ and therefore that it is σ̄ - invariant.

In the cases 1, 3, 5, 7 of the list of Bagnera - de Franchis (1.3) there is nothing to prove, since we have Λ′′ = Λ′. In case 2 we find Λ′′ = 2Λ. In fact we have G = ℤ/2 × ℤ/2, acting on A′′ × F as:

(x_{1}, x_{2}) ↦ (x_{1} + η, −x_{2}),
(x_{1}, x_{2}) ↦ (x_{1} + η′, x_{2} + ϵ).

Analogously in case 6, Λ′′ = 3Λ, in case 4, Λ′′ = 2Λ ∩ Λ′. Therefore we always find that σ(Λ′′) = Λ′′ and thus we get a lift

[figure: A′′ × F; σ̃; A′′ × F; S; σ; S]

Definition 2.4. The extended symmetry group Ĝ is the group generated by G and σ̃.

Ĝ is the group of homeomorphisms of S′′ which lift the group {1, σ}: hence we have the following extension that will be studied in the next section

(∗) 0 → G →^{i} Ĝ →^{π} ℤ/2 → 1

With the same arguments as above we obtain the following

Theorem 2.5. Let (S, σ), (Ŝ, σ̂) be isomorphic real hyperelliptic surfaces (i.e. there exists ψ : S →^{≅} Ŝ such that ψ^{−1}σ̂ψ = σ). Then the respective extended symmetry groups Ĝ are the same for S and Ŝ. Moreover let S = (E×F)/G, Ŝ = (Ê×F̂)/G be two Bagnera - De Franchis realizations. Then there exists an isomorphism Ψ : E×F → Ê×F̂, of product type (i.e. Ψ = Ψ_{1} × Ψ_{2}) commuting with the action of Ĝ, and inducing the given isomorphism ψ : S →^{≅} Ŝ.

Proof. ψ induces an isomorphism ψ_{∗} of the Albanese varieties, which is compatible with the antiholomorphic involutions σ̄ and ˆ σ̄ and such that we have a commutative diagram

[figure: S; α; ψ; Ŝ; α̂; A; ψ_{∗}; Â]

Whence, by taking the coverings associated to the subgroups and points corresponding under ψ and ψ_{∗}, we obtain isomorphisms ψ̃, ψ_{1} and a commutative diagram

[figure: S′′; α′′; ψ̃; Ŝ′′; α̂′′; A′′; ψ_{1}; Â′′]

We observe that if (E × F)/G is a Bagnera - de Franchis realization of S, then E ≅ A′′ and there is an isomorphism of (S′′ → A′′) with (E × F → E), commuting with the action of G. We obtain therefore a commutative diagram

[figure: E × F; ψ̃; Ê × F̂; E; ψ_{1}; Ê]

where, moreover, ψ̃ and ψ_{1} are real. Since ψ̃ preserves the fibres of the two projections, we have

ψ̃(e, f) = (ψ_{1}(e), ψ_{2}(e, f)).

Let us fix an origin 0 ∈ E, then, since ψ_{2} is an affine map, we can write

ψ_{2}(e, f) = ψ_{2}(0, f) + r(e),

where r is a holomorphic homomorphism r : E → P ic^{0}(F̂) (P ic^{0}(F̂) is the group of translations of F̂). By G - equivariance, for all g ∈ G ⊂ P ic^{0}(E), we have

ψ_{2}(e + g, g(f)) = g(ψ_{2}(0, f) + r(e)).  (1)

But the left hand side of 1 equals

ψ_{2}(0, g(f)) + r(e + g) = ψ_{2}(0, g(f)) + r(e) + r(g),

thus if we let g_{∗} be the linear part of g : F̂ → F̂, and we look at the linear part of 1 with respect to e, we obtain

g_{∗}(r(e)) ≡ r(e).

Since there is a g ∈ G such that g_{∗} ≠ IdentityF̂, we infer that r(e) is constant, or equivalently that ψ_{2}(e, f) = ψ_{2}(f). Q.E.D.

Proposition 2.6. Let (S = (E × F)/G, σ) be a real hyperelliptic surface, and let σ̃ : E×F → E×F be a lift of σ. Then the antiholomorphic map σ̃ is of product type.

Proof. As in the proof of the previous theorem, we have

σ̃(e, f) = (σ_{1}(e), σ_{2}(e, f))

since σ̃ preserves the fibration onto E. Then, after choosing an origin 0 ∈ E, we have

σ_{2}(e, f) = σ_{2}(0, f) + r(e),

where r : E → P ic^{0}(F) is an antiholomorphic homomorphism. We know that σ̃ normalizes the group G, in particular if we take some

element g ∈ G, such that g_{∗} ≠ Id_{F}, σ̃ has a matrix ( \begin{matrix}a & 0 \\ b & c \end{matrix} ) while

g_{∗} = ( \begin{matrix}1 & 0 \\ 0 & ξ \end{matrix} ) with ξ ≠ 1. We must have

( \begin{matrix}a & 0 \\ b & c \end{matrix} ) ( \begin{matrix}1 & 0 \\ 0 & ξ̄ \end{matrix} ) ( \begin{matrix}a & 0 \\ b & c \end{matrix} )^{−1} = ( \begin{matrix}1 & 0 \\ 0 & θ \end{matrix} )

whence b = 0. Q.E.D.

Now we would like to see when the lift σ̃ is an involution. Let us denote by ϕ : A′′ × F → S the map in the diagram.

Remark 2.7. Let σ̃ be a lift of σ, we have two different cases:

1. ∃z ∈ S such that σ(z) = z. Then, since the covering is Galois, for all z′ ∈ ϕ^{−1}(z), there exists a lift σ̃ of σ such that σ̃(z′) = z′. But then σ̃^{2} lifts the identity and has a fixed point, therefore it is the identity.

2. F ix(S) = ∅.

Let z, w ∈ S, ∀z′ ∈ ϕ^{−1}(z), ∀w′ ∈ ϕ^{−1}(w), ∃! lift σ̃, with σ̃(z′) = w′.

3. Orbifold fundamental groups and the topological type of a real hyperelliptic surface

Let (X, σ) be a smooth real variety (i.e. a smooth complex variety together with an antiholomorphic involution σ) of complex dimension n. Then we have a double covering π : X → Y = X/ < σ > ramified on X′ = X(ℝ) = F ix(σ). X′ is a real submanifold of codimension n, hence the map

π_{1}(X − X′) → π_{1}(X)

is surjective if n ≥ 2 and it is an isomorphism for n ≥ 3. The singularities of Y are contained in Y′ = π(X′) and there we have a local model ℝ^{n} × (ℝ^{n}/(−1)). Therefore Y is smooth for n = 2 and topologically singular for n ≥ 3. The local punctured fundamental group π_{1}(Y − Y′)_{loc} is isomorphic to ℤ for n = 2, while it is isomorphic to ℤ/2 for n ≥ 3. This means that the kernel of the surjection

π_{1}(Y − Y′) → π_{1}(Y)

is normally generated by loops γ around the components of Y′. If n ≥ 3, then we have automatically γ^{2} = 1.

Definition 3.1. Assume dim_{ℂ}X = n ≥ 2. Then the orbifold fundamental group of Y (or of (X, σ)) is defined to be π_{1}(Y − Y′)/ < γ^{2} >.

Remark 3.2. 1. If n ≥ 3, the orbifold fundamental group of Y co-

incides, as observed, with the fundamental group of Y − Y′.

2. If X′ = ∅, π_{1}^{orb}(Y) = π_{1}(Y) and we have an exact sequence

1 → π_{1}(X) → π_{1}^{orb}(Y) → ℤ/2 → 1  (2)

3. For n ≥ 2 we have the exact sequence 2, since we have

\xymatrix{1 \ar[r] & π_{1}(X − X′) \ar[d] \ar[r] & π_{1}(Y − Y′) \ar[d] \ar[r] & ℤ/2 \ar[d] \ar[r] & 1 \\ 1 \ar[r] & π_{1}(X) \ar[r] & π_{1}^{orb}(Y) \ar[r] & ℤ/2 \ar[r] & 1}

and the kernel of the map π_{1}(X − X′) → π_{1}(X) is normally generated by the above γ^{2}’s.

Let X̃ be the universal covering of X, so that X = X̃/π_{1}(X). The exact sequence 2 defines a group which is the group of liftings of σ to X̃, so that Y = X̃/π_{1}^{orb}(Y). If X′ = X(ℝ) ≠ ∅, let us choose a base point x_{0} ∈ X′. Then σ acts on π_{1}(X, x_{0}) by an automorphism of order 2, which is obtained by conjugation, since we have the following

Lemma 3.3. If X′ ≠ ∅, then the exact sequence 2 always splits.

Proof. The proof follows immediately from corollary 2.3. Q.E.D.

Remark 3.4. The topological invariants of (X, σ) are: the topological invariants of X(ℝ), the topological invariants of Y and π_{1}^{orb}(Y).

We have the following

Theorem 3.5. Let (S, σ) be a real hyperelliptic surface. Then the topological and also the differentiable type of the pair (S, σ) is completely determined by the orbifold fundamental group exact sequence.

Proof. We want first of all to show how Π := π_{1}(S) determines the topological (actually differentiable) type of the hyperelliptic surface S. Consider the exact homotopy sequence associated with the covering ψ : S′ → S described in the proof of theorem 1.4.

1 → Ω′ = π_{1}(S′) ≅ ℤ^{4} → Π → G′ → 1  (3)

where G′ = G/_{translations}. The exact sequence

1 → Γ = [Π, Π] → Π → Λ = Π/[Π, Π] → 1  (4)

is given by the induced map of fundamental groups associated to the Albanese map and Γ ≅ ℤ^{2} is the fundamental group of the fibre of α. Ω′ is then the kernel of the action of Π on Γ by conjugation (Γ ⊂ Ω′ and Ω′/Γ = Λ′ in our previous notation). Now Ω′ is a representation of G′ and Γ is a subrepresentation. An easy calculation shows that (Ω′)^{G′} = Λ′′, which yields a direct sum Ω = Λ′′ ⊕ Γ, such that Π/Ω = G. Notice that since Ω′ is abelian and by 3, we know that (Ω′)^{G′} is the centre of Π, so Λ′′ is the centre of Π. Since Ω ⊂ Ω′, the universal cover X̃ ≅ ℝ^{4} of S is homeomorphic to Ω′⊗ℝ on which Ω′ acts freely by translations. We have S ≅ (Ω′⊗ℝ)/Π, thus it suffices to show that the exact sequence 3 determines the action of Π on the universal covering Ω′ ⊗ ℝ. The action is given by a group of affine transformations of ℝ^{4} and since Ω′ is the sugroup of translations, the action of G′ on the torus Ω′⊗ℝ/Ω′ has a linear part which is determined by the conjugation action of G′ on Ω′. Since G′ is cyclic, we may find a lift λ(g) to Π of its generator g. Now, for all g′ ∈ G′ we have a lift λ(g′) in Π and if m = |G′|, we have λ(g)^{m} ∈ Ω′. Every element γ ∈ Π can be uniquely written as ω′λ(g′), with ω′ ∈ Ω′, g′ ∈ G′. Since λ(g)^{m} ∈ Ω′ and it is invariant by conjugation by λ(g), we have λ(g)^{m} ∈ (Ω′)^{G′} = Λ′′, therefore we let g act on (Λ′′ ⊗ ℝ) ⊕ (Γ ⊗ ℝ) by

(e, f) ↦ (e + \frac{1}{m}λ(g)^{m}, g(f))

where g acts on Γ by conjugation. It is immediate that the above action is precisely the one yielding S as (E × F)/G = (Λ′′ ⊗ ℝ ⊕ Γ ⊗ ℝ)/Π and that the way we described it is completely dictated by Π as an abstract group. Thus we have proven that Π determines the topological type of S. Let us now consider the ramified covering E × F → E × F/Ĝ, where Ĝ is the extended symmetry group defined in 2.4. Then we have the following exact sequences

1 → Ω → Π → G → 1
∥ ∩ ∩
1 → Ω → Π̂ → Ĝ → 1

where Π̂ is the orbifold fundamental group,

1 → Ω′ → Π → G′ → 1
∥ ∩ ∩
1 → Ω′ → Π̂ → Ĝ′ → 1

We want to describe how Π̂ acts on Ω′ ⊗ ℝ. In order to understand the action of Π̂ it suffices to describe the action of a suitable element σ in Π̂ − Π. σ acts on Ω′ ⊗ ℝ by an affine transformation: v ↦ Av + b, of which we know the linear part A, that is determined by the conjugation action of σ on Ω′. On the other hand, we have σ^{2} ∈ Π, therefore we know the affine map σ^{2}. Since we have

σ^{2}(v) = A^{2}v + Ab + b =: A^{2}v + b′,

we are able to determine b uniquely from A and b′, in the case in which (A + I) is invertible. We also know that σ is of product type, and that A^{2} = A_{1}^{2} ⊕A_{2}^{2} has the property that A_{1}^{2} = I. Thus W = ℝ_{1}^{2} = Λ′′ ⊗ ℝ splits as a direct sum of eigenspaces W^{+} ⊕ W^{−} and we can recover the translation vector b_{1}^{+} by what we have remarked above. Whereas on W^{−} = Im(A_{1} − I) we can change coordinates by a translation in such a way that the action of σ is linear on W^{−}. Therefore we can restrict our attention to the second component A_{2}. If (−1) is not an eigenvalue of A_{2}, we are done. Since σ is antiholomorphic, (−1) is an eigenvalue of A_{2} if and only if A_{2}^{2} = I. Therefore it remains to treat the case in which A_{2}^{2} = I and (A_{2}g)^{2} = I, ∀g ∈ G. In this case, the second component of σ is given by σ_{2}(x) = A_{2}x + c. Let us consider another element in Π̂ − Π, s = g ∘ σ, where g ∈ G is not a translation. We know the action of σ ∘ s and of s ∘ σ, since they are in Π.

σ_{2} ∘ s_{2}(x) = A_{2}(gA_{2}x + c′) + c,
s_{2} ∘ σ_{2}(x) = gA_{2}(A_{2}x + c) + c′ = gx + gA_{2}c + c′.

Looking at the translation terms, we get knowledge of gA_{2}c + c′ and A_{2}c′ + c.

We argue by looking at the rank of the matrix ( \begin{matrix}gA_{2} & I \\ I & A_{2} \end{matrix} ) which is

applied to the vector ( \begin{matrix}c \\ c′ \end{matrix} ) .

It has the same rank as

rank ( \begin{matrix}g & I \\ A_{2} & A_{2} \end{matrix} ) = rank ( \begin{matrix}g − I & I \\ 0 & A_{2} \end{matrix} ) = 2 + rk(g − I).

Then we are done, since (g − I) is invertible, because g is holomorphic and it is not a translation, thus it does not have 1 as eigenvalue. So the proof of the theorem is concluded, since we have shown that the exact sequence of the orbifold fundamental group completely determines the action of Π̂ on Ω ⊗ ℝ. Q.E.D.

Corollary 3.6. The topological type of a real hyperelliptic surface (S, σ) determines:

1. the Bagnera - de Franchis group G,

2. the extension 1 → G → Ĝ → ℤ/2 → 1

3. the topological type of the action of (Ĝ ⊃ G) on E and F, or equivalently the affine equivalence class of the representation of Π̂/Γ ⊃ Π/Γ on Λ′′ ⊗ ℚ and of Π̂/Λ′′ ⊃ Π/Λ′′ on Γ ⊗ ℚ. In particular it determines the topological type of the real elliptic curve (E/G, Ĝ/G).

Remark 3.7. Notice that the representation ρ of Π̂ in the group of the affine tranformations of Ω ⊗ ℚ takes values in A(2, ℤ, ℚ) × A(2, ℤ, ℚ), where A(2, ℤ, ℚ) = {v ↦ Bv + β, |B ∈ GL(2, ℤ), β ∈ ℚ^{2}}.

We want now to discuss in some greater generality than needed for our present purposes the notion of moduli space of real varieties. Since we will only assume that X is a complex manifold, we have to adopt the point of view of Kodaira-Spencer-Kuranishi (cf. [Ca3], [K-M], [Ko]).

For a general complex manifold X we have the Kuranishi family of deformations of X, ϕ : X → B. Here, the base B of the Kuranishi family is a complex analytic subset of the vector space H^{1}(X, Θ_{X}) corresponding to the complex structures J which satisfy Kuranishi’s integrability equation.

If now (X, σ) is real, we want to see when a neighbouring complex structure J is such that the differentiable map σ remains antiholomorphic (whence we get a deformation X_{t} of the complex manifold X such that the new pair (X_{t}, σ) is still real).

The corresponding equation writes down simply as

σ_{∗}J = −Jσ_{∗}, or equivalently,

−σ_{∗}Jσ_{∗} = J.

We see immediately that this condition means that J lies in the fixed locus of the involution induced by σ.

Interpreting now J as a harmonic representative of a Dolbeault cohomology class θ ∈ H^{1}(X, Θ_{X}), we are going to show more precisely that σ induces a complex antilinear involution σ^{∗} on the vector space H^{1}(X, Θ_{X}), such that the real part B(ℝ) consists of the deformations of X for which σ remains real.

Recall that the complex structure J_{0} of X induces a splitting of the complexified real tangent bundle of X, T_{X} ⊗ ℂ = T^{0,1} ⊕ T^{1,0} where T^{0,1} = T̄^{1,0}.

The form θ = Σ_{ij̄}θ_{ij̄}(∂/∂z_{i} ⊗ dȳ_{j}) can be interpreted as yielding a linear map θ′ : T^{0,1} → T^{1,0} such that the new subbundle T_{θ}^{1,0} = {(u, v) ∈ T^{0,1} ⊕ T^{1,0} | v = θ′(u)}.

θ′ is completely determined by the matrix ϕ = θ_{ij̄}, and then the subspace T_{θ}^{0,1} = {(u, v) ∈ T^{0,1}⊕T^{1,0} | u = θ̄′(v)}, where θ̄′ is determined by the matrix ϕ̄ = θ̄_{īj} in the chosen coordinate basis.

Saying that σ is antiholomorphic amounts to saying that its differential induces complex linear isomorphisms σ_{∗} : T^{0,1} → T^{1,0} and σ_{∗} : T^{1,0} → T^{0,1} which are inverses and conjugate to each other.

Whence, if A is the matrix of σ_{∗} : T^{0,1} → T^{1,0}, A^{−1} = Ā is the matrix of σ_{∗} : T^{1,0} → T^{0,1}.

We want now to write down the condition that σ be antiholomorphic for the new complex structure induced by the form θ.

Again, this means that σ_{∗} : T_{θ}^{1,0} → T_{θ}^{0,1}, i.e., that for each vector u, the image σ_{∗}(u, θ′(u)) = (Āϕ(u), A(u)) satisfies the equation of T_{θ}^{0,1}, namely, we have

Āϕ(u) = ϕ̄A(u).

Again we can equivalently define

σ^{∗}(ϕ) := Aϕ̄A

write the above (since it must hold for each vector u) as

ϕ = σ^{∗}(ϕ).

It is now obvious that σ^{∗} is complex antilinear, and it is an involution since

σ^{∗}^{2}(ϕ) = AĀϕĀA = ϕ.

By what we have shown above, it follows right away that B(ℝ), being the intersection of a complex analytic space with the fixed part of a complex antilinear involution, is a real analytic space.

In general, by a result of Wavrik (cf. [Wav]), if the function h : B → ℤ given by

h(t) = h^{0}(X_{t}, Θ_{X_{t}})

is constant on the germ B, then the quotient

B/Aut(X) is a local moduli space.

The group Aut(X) does not however act on the set B(ℝ), therefore we take the smaller subgroup

Aut^{σ}(X) : = {ϕ ∈ Aut(X)|ϕ^{∗}(σ) = σ}, where ϕ^{∗}(σ) := ϕσϕ^{−1}.

We finally can define the real local moduli space as

Definition 3.8. Under the assumption that the function h^{0}(X_{t}, Θ_{X_{t}}) is constant on the base of the Kuramishi family, the real local moduli space of (X, σ)is defined as the quotient B(ℝ)/Aut^{σ}(X).

Remark 3.9. Therefore, the real local moduli space is just a real semi algebraic space, and it maps neither surjectively nor one to one to the real part of the complex local moduli space.

Remark 3.10. Once we have local moduli spaces for the varieties or manifolds under consideration, the standard procedure is to consider the global moduli space as the set of isomorphism classes of such varieties and to use the local moduli spaces as giving local charts. For instance, if we have local moduli spaces for a certain class of complex manifolds, these charts provided by the local moduli spaces yield for our global moduli space the structure of a complex analytic space, possibly non Hausdorff.

Likewise, for real algebraic varieties such that Wavrik’s condition holds, the global moduli space is a semianalytic space.

Now, one can give different definitions for polarized algebraic varieties, and again one has to distinguish between the real part of the quotient of a Hilbert scheme, and the quotient of its real part. However, the approach via the Kuranishi family is particularly suitable for the case of hyperelliptic surfaces, as we are going now to see.

Theorem 3.11. The moduli space of real hyperelliptic surfaces of a given topological type is irreducible.

Proof. Let us fix an orbifold fundamental group Π̂, then theorem 3.5 and corollary 3.6 tell us that Π̂ determine the representation ρ : Π̂ → A(2, ℤ, ℚ)^{2}. Such a representation induces a representation of Ĝ′ into GL(2, ℤ)^{2}:

Ĝ′ ⟶^{(λ_{1},λ_{2})} GL(2, ℤ)^{2},

where Ĝ′ is the group fitting into the exact sequence

(#) 1 → G′ → Ĝ′ → ℤ/2 → 1

We observe that giving a hyperelliptic surface with the given topological type amounts to giving two complex structures J_{1}, J_{2} on ℤ^{2} ⊗ℚ such that λ_{i}(G′) consists of ℂ - linear maps, whereas λ_{i}(Ĝ′−G′) consists of ℂ - antilinear maps (i = 1, 2).

Therefore we have to solve the following problem:

given a representation λ : Ĝ′ → GL(2, ℤ), find all the complex structures J such that for a generator g′ of G′ we have

λ(g′)J = Jλ(g′),

and for a σ ∉ G′ we have

λ(σ)J = −Jλ(σ).

We have of course to keep in mind that, if M is a fixed differentiable manifold with an involution σ : M → M, and we look for the complex structures J which make σ an antiholomorphic involution, for each solution J we shall also find the solution −J.

But, as we already remarked in the introduction, σ provides an isomorphism between the complex manifolds (M, J) and (M, −J), and clearly it conjugates the involution σ to itself.

Therefore, if we shall see that the parameter space for our complex structures J will consist of exactly two irreducible components, exchanged by the involution J → −J, it will follow that the moduli space is irreducible.

We shall prove in the next section that the extension (#) always splits (see corollary 4.3), therefore we may assume that σ^{2} = 1. In particular we can find a basis in such a way that either ζ := λ(σ) =

( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) or ζ = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ).

If we now set B := λ(g′), since G′ ≅ ℤ/q for q = 2, 3, 4, 6, we have either ζBζ = B, or ζBζ = B^{−1}. We have two different cases:

Case I: ζ := λ(σ) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ).

Then, if we set B = λ(g′) = ( \begin{matrix}b_{11} & b_{12} \\ b_{21} & b_{22} \end{matrix} ), we have again two cases:

1. ζBζ = B,

2. ζBζ = B^{−1}.

In case 1 we find ( \begin{matrix}b_{11} & b_{12} \\ b_{21} & b_{22} \end{matrix} ) = ( \begin{matrix}b_{11} & −b_{12} \\ −b_{21} & b_{22} \end{matrix} ), whence, since det(B) =

1, we have B = ±Id. But we shall see in 4.7 that this case occurs if and only if q = 2 and B = −Id.

In case 2 we find ( \begin{matrix}b_{11} & −b_{12} \\ −b_{21} & b_{22} \end{matrix} ) = ( \begin{matrix}b_{22} & −b_{12} \\ −b_{21} & b_{11} \end{matrix} ), whence b_{11} =

b_{22} =: b, b^{2} − b_{12}b_{21} = 1, the characteristic polynomial is λ^{2} − 2bλ + 1. But since either B = −Id, or B has no real eigenvalues, b^{2}−1 < 0, thus

b = 0, whence B = ± ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and we may assume B = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )

by changing generator of G′.

Case II: ζ = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). We have again the two cases 1 and 2.

Case II 1: We find ( \begin{matrix}b_{11} & b_{12} \\ b_{21} & b_{22} \end{matrix} ) = ( \begin{matrix}b_{22} & b_{21} \\ b_{12} & b_{11} \end{matrix} ), thus B = ( \begin{matrix}b & c \\ c & b \end{matrix} )

and since det(B) = b^{2} −c^{2} = 1, we have b+c = ±1, b−c = ±1, whence c = 0 and B = −Id.

Case II 2: We have ( \begin{matrix}b_{22} & b_{21} \\ b_{12} & b_{11} \end{matrix} ) = ( \begin{matrix}b_{22} & −b_{12} \\ −b_{21} & b_{11} \end{matrix} ) which yields

B = ( \begin{matrix}b_{11} & c \\ −c & b_{22} \end{matrix} ). But b_{11}b_{22} +c^{2} = 1, thus either c = 0 and B = −Id,

or b_{11}b_{22} ≤ 0.

If c = ±1, b_{11}b_{22} = 0. By exchanging the two basis vectors e_{1} and e_{2} we may assume c = −1 and by replacing B with B^{−1} we can assume b_{22} = 0.

Thus B = ( \begin{matrix}b & −1 \\ 1 & 0 \end{matrix} ) and either b = ±1 if G′ = ℤ/6 or G′ = ℤ/3,

or b = 0, B = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) if G′ = ℤ/4.

Assume then |c| ≥ 2. Since |b_{11} + b_{22}| < 2, we have two possible cases: b_{11} + b_{22} = 0, b_{11} + b_{22} = ±1.

In the first case we find B = ( \begin{matrix}b & c \\ −c & −b \end{matrix} ) and −b^{2} + c^{2} = 1, thus

b = 0, c = ±1, a contradiction.

In the second case the matrix B has the form ( \begin{matrix}b & c \\ −c & ±1 − b \end{matrix} ), we

may assume b > 0, and we have ±b − b^{2} + c^{2} = 1.

The equation ±b − b^{2} + c^{2} = 1 is equivalent to c^{2} = 1 + b^{2} ∓ b = (b ∓ 1)^{2} ± b, whence ±b = (c − b ± 1)(c + b ∓ 1). Taking absolute values in the last equation, we see that we cannot have |c| ≥ 2.

We have therefore obtained the following possibilities for ζ and B:

• Case I 1: ζ = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), B = −Id.

• Case I 2: ζ = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), B = −Id, or B = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ).

• Case II 1: ζ = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), B = −Id.

• Case II 2: ζ = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), B = −Id, or B = ( \begin{matrix}1 & −1 \\ 1 & 0 \end{matrix} ), or

B = ( \begin{matrix}−1 & −1 \\ 1 & 0 \end{matrix} ), or B = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ).

We now look for a complex structure J = ( \begin{matrix}d_{11} & d_{12} \\ d_{21} & d_{22} \end{matrix} ) such that

ζJζ = −J,
BJ = JB.

Case I: the first equation reads ( \begin{matrix}d_{11} & −d_{12} \\ −d_{21} & d_{22} \end{matrix} ) = −J, thus d_{11} =

d_{22} = 0, and since the characteristic polynomial for J is λ^{2} + 1, we have det(J) = 1, equivalently d_{12}d_{21} = −1, whence J has the form

( \begin{matrix}0 & −d \\ 1/d & 0 \end{matrix} ).

Whence, as promised, the parameter space consists of the two branches of an hyperbola, which are exchanged upon multiplication by −1.

Case II: the equation ζJζ = −J reads ( \begin{matrix}d_{22} & d_{21} \\ d_{12} & d_{11} \end{matrix} ) = −J, thus we

obtain J = ( \begin{matrix}a & b \\ −b & −a \end{matrix} ), with a^{2} − b^{2} = −1, whence we can write

a = (c−1/c)/2, b = (c+1/c)/2 and the conclusion is exactly as before. Let us now consider the commutation relation BJ = JB.

If B = −Id, it is obviously verified.

If B = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ), we have cases I and II:

In case I the commutation relation yields ( \begin{matrix}−d & 0 \\ 0 & −1/d \end{matrix} ) = ( \begin{matrix}−1/d & 0 \\ 0 & −d \end{matrix} ),

whence d^{2} = 1, i.e. d = ±1, J = ± ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ).

In case II the commutation relation yields ( \begin{matrix}b & −a \\ −a & b \end{matrix} ) = ( \begin{matrix}b & a \\ a & b \end{matrix} ),

whence a = 0 and b^{2} = 1, i.e. b = ±1, J = ± ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ).

Assume now B = ( \begin{matrix}1 & −1 \\ 1 & 0 \end{matrix} ), then we are in case II and the com-

mutation relation reads ( \begin{matrix}a + b & a + b \\ a & b \end{matrix} ) = ( \begin{matrix}a + b & −a \\ −a − b & b \end{matrix} ), whence

b = −2a and since b^{2} − a^{2} = 1, we have a = ±1/\sqrt{3}.

We observe that if J commutes with B, it commutes also with B^{2}, thus, up to a base change, we have solved also the case G′ = ℤ/3.

Q.E.D.

4. The extended symmetry group

In this section we want to describe the extended Bagnera - de Franchis group, fitting into the exact sequence:

(∗) 0 → G →^{i} Ĝ →^{π} ℤ/2 → 1

.

Since G is a normal Abelian subgroup of Ĝ, conjugation induces an action of Ĝ/G = ℤ/2 on G.

Recall that, once such action is specified, the equivalence classes of such extensions are in bijective correspondence with the elements of

H^{2}(ℤ/2, G) = \frac{ker(σ − 1)}{(1 + σ)G} = \frac{G^{ℤ/2}}{(1 + σ)G},

where ℤ/2 =< σ >.

Let us consider the antiholomorphic elements of the extended Bagnerade Franchis group, which will be denoted by σ̃ (they are the lifts of σ in Ĝ). We showed in proposition 2.6 that their action is also of product type, whence we can restrict our preliminary investigation to the question: which extended Bagnera- de Franchis groups act as group of diholomorphic automorphisms of an elliptic curve?

Definition 4.1. A diholomorphic action of an extended Bagnera- de Franchis group Ĝ on an elliptic curve is an action such that G is precisely the subgroup of holomorphic automorphisms in Ĝ, while the elements in Ĝ − G act as antiholomorphic automorphisms.

Lemma 4.2. Consider a diholomorphic action of an extended Bagnerade Franchis group Ĝ on an elliptic curve. Then the square of an antiholomorphic map σ̃ in Ĝ is a translation.

Proof. Let σ̃^{2} = g ∈ G. Passing to the universal cover we can write σ̃(z) = az̄ + b. We have then σ̃^{2}(z) = aāz + ab̄ + b = g(z). We know that g(z) = ξz + c with ξ^{n} = 1, n = |G|. Since ξ = |a|^{2} ∈ ℝ_{>0}, we obtain ξ = 1.

Q.E.D.

Corollary 4.3. If the group G is cyclic, then the extension (∗) always splits.

Proof. Since G = G′, for any lift σ̃ of σ, we must have σ̃^{2} = Id ∈ G, so (∗) splits. Q.E.D.

We want now to recall the description of the antiholomorphic maps σ acting on an elliptic curve C, whose square σ^{2} is a translation.

Write C = ℂ/Γ, σ(z) = az̄ + b, σ^{2}(z) = aāz + ab̄ + b.

Whence

|a|^{2} = 1,  (5)

The following condition must be verified:

aΓ̄ = Γ

which is clearly equivalent to the existence of integers m, n, m′, n′ such that:

a = m + nτ ∈ Γ,  (6)
aτ̄ = m′ + n′τ, m′, n′ ∈ ℤ, with mn′ − nm′ = −1.  (7)

We may rewrite 5 as

(m + nτ)(m + nτ̄) = m^{2} + n^{2}|τ|^{2} + mn(τ + τ̄) = 1  (8)

We may assume that τ lies in the modular triangle, i.e., that |τ| ≥ 1, |Reτ| ≤ 1/2. Then Im(τ) ≥ \sqrt{3}/2, and since a = m + nτ, Im(a) = n Im(τ) ≤ 1,

and we conclude that |n| ≤ 1.

We have then the following cases:

• n = 0: then, since |a| = 1, m = a = −n′ = ±1.

Moreover, 7 tells us that 2(Re) τ = −n′m′, whence we either have Re(τ) = 0 or Re(τ) = −1/2.

• observing that if |n| = 1, by 8 we infer m^{2} + 1 ≤ m^{2} + |τ|^{2} = 1 ± (τ + τ̄) ≤ 1 + |m|, thus |m| ≤ 1, |τ| = 1, giving rise to the following two cases:

• |n| = 1, m = 0, |τ| = 1 (thus a = ±τ)

• |n| = 1, |m| = 1, |τ| = 1 and since 1 + |τ|^{2} = 1 − mn(τ + τ̄), we may also assume Re(τ) = −1/2, thus m = n, a = m(1 + τ).

Lemma 4.4. Assume now that σ is an antiholomorphism of an elliptic curve whose square is a translation of finite order d. Then, we may choose the origin in the universal cover ℂ in such a way that σ(z) = az̄ + b, with a as above, b ∈ (1/d)ℤ for a ≠ ±1, b ∈ (1/2d)ℤ if a = 1, b ∈ (1/2d)ℤ(τ) for a = −1, Re τ = 0, b ∈ (1/2d)ℤ(2τ + 1) for a = −1, Re τ = −1/2.

Assume further that d = 1, i.e. that σ is an involution. Then, obviously, we may get b = 0 if and only if F ix(σ) ≠ ∅.

F ix(σ) = ∅ if and only if Re(τ) = 0 and we may choose the origin in such a way that b = 1/2 for a = 1, or b = τ/2 for a = −1.

Moreover σ normalizes a finite group of translations T, if and only if, identifying T with a subgroup of C, aT̄ = T.

Proof. We look first for a vector w such that aw̄ − w + b := β be either a real vector, or an imaginary vector.

We look therefore at the image of the linear map w → aw̄ − w. Its complexification ℂ ⊗ ℂ → ℂ ⊗ ℂ has a matrix

( \begin{matrix}−1 & a \\ ā & −1 \end{matrix} ) whose determinant is zero, whence, the image of the

above linear map equals ℝ(a − 1) for a ≠ 1, and ℝi (Im τ) for a = 1.

We can therefore achieve that β be a real vector, unless a is real, a ≠ 1, i.e., a = −1, in which case we can achieve that β be an imaginary vector.

Since σ^{2}(z) = z + aβ̄ + β, if β is real we get 2β ∈ (1/d)ℤ, if a = 1. If a ≠ ±1 and β is real, then we have (1 ± τ)β ∈ (1/d)Γ, so β ∈

(1/d)ℤ. If a = −1 and β is imaginary, then 2β ∈ (1/d)Γ and the first assertion follows.

If d = 1, we observe that if Re(τ) = −1/2, the involutions z ↦ z̄+ 1/2, z ↦ −z̄+ 1/2 + τ = −z̄+ iIm(τ) have respectively i/2Im(τ), 1/4 as fixed points, thus in both cases we can assume β = 0.

It is immediate to verify that if Re(τ) = 0 and a = 1, b = 1/2, or a = −1, b = τ/2 there are no fixed points.

The third assertion follows from the fact that σ^{−1}(z) = az̄ − ab̄, whence, for a translation z → z +c, conjugation by σ yields z → z +ac̄. Q.E.D.

Remark 4.5. Assume now that σ is an antiholomorphic involution of an elliptic curve C as in the previous lemma. Then there are only three possible topological types for the action of σ on C;

• F ix(σ) = ∅

• F ix(σ) is homeomorphic to S^{1}:

this occurs if |τ| = 1, a = ±τ, or if Re(τ) = −1/2, a = ±1, b = 0.

• F ix(σ) has two connected components homeomorphic to S^{1}: this occurs if Re(τ) = 0, a = ±1, b = 0.

Proof. F ix(σ) = ∅: there are two cases which are obviously topological equivalent (exchange the two basis vectors of Γ, 1, and τ).

If F ix(σ) ≠ ∅, then we may assume b = 0, and the topological type is completely determined by the integral conjugacy class of the matrix

A = ( \begin{matrix}m & m′ \\ n & n′ \end{matrix} ), whose square is the identity and which has 1 and −1

as eigenvalues.

If A is diagonalizable then F ix(σ) has two connected components

homeomorphic to S^{1}, otherwise A is conjugated to the matrix ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

and F ix(σ) is homeomorphic to S^{1}. Q.E.D.

Remark 4.6. Let G be a Bagnera- de Franchis group: then G has a first incarnation as a group of translations of E, and a second one as a direct product G = G′ × T, where T is a group of translation and G′ is cyclic and a subgroup of the multiplicative group. If Ĝ is an extended Bagnera- de Franchis group, we let σ be an element in Ĝ−G: it conjugates G, sending T to itself. In the first incarnation, group conjugation is given, as we saw in the previous lemma, by complex conjugation followed by multiplication by a (if σ acts by σ(z) = az̄+b).

It follows that the extension 0 → G → Ĝ → ℤ/2 → 0 splits if and only if, c′ being the translation vector of σ^{2}, then c′ lies in the image of the endomorphism of G given by s + Id, s being the action of σ. In the second incarnation, let z → ξz be a generator g′ of G′, and let σ(z) = az̄ + b: then σ conjugates g′ to the transformation z → ξ̄z + a(ξ̄− 1)b̄.

Whence, if the action is trivial then G′ ≅ ℤ/2ℤ and 2b ∈ Γ.

Proof. We need only to remark that ac̄ ∈ Γ holds if and only if c ∈ Γ.

Q.E.D.

We can now give the list of all the possible groups Ĝ.

Lemma 4.7. Let us consider the extension

(∗) 0 → G → Ĝ → ℤ/2 =< σ >→ 0.

We have the following possibilities for the action of σ on G:

1. If G = ℤ/2, then Ĝ = ℤ/2 × ℤ/2.

2. If G = ℤ/2 × ℤ/2, then either σ acts as the identity on G and Ĝ = ℤ/2 × ℤ/2 × ℤ/2, if (∗) splits, Ĝ = ℤ/4 × ℤ/2 if (∗) does not split, and in this latter case the square of σ is the generator of T.

Or σ acts as ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ), the sequence splits, Ĝ = D_{4}, the dihe-

dral group, and again the square of the generator of ℤ/4ℤ is the generator of T.

3. If G = ℤ/4, then σ acts as −Id on G and Ĝ = D_{4}.

4. If G = ℤ/4 × ℤ/2, then either Ĝ = T × D_{4} ≅ ℤ/2 × D_{4}, or Ĝ is isomorphic to the group G_{1} :=< σ, g, t, |σ^{2} = 1, g^{4} = 1, t^{2} = 1, tσ = σt, tg = gt, σg = g^{−1}tσ >, and its action on the second elliptic curve F is generated by the following transformations: σ(z) = z̄ + 1/2, g(z) = iz, t(z) = z + 1/2(1 + i). The group G_{1} is classically denoted by c_{1} (cf. [H-S] p. 39).

In particular, in both cases (∗) splits.

5. If G = ℤ/3, then σ acts as −Id on G and Ĝ = S_{3}.

6. If G = ℤ/3 × ℤ/3, then we may choose G′ so that σ acts as −Id × Id on G = G′ × T and Ĝ = S_{3} × ℤ/3.

7. If G = ℤ/6, then σ acts as −Id on G and Ĝ = D_{6}.

Proof. Thanks to 4.3 we know that in the cases 1,3,5,7 the sequence (∗) splits. Case 1: if G = ℤ/2, then σ acts as the identity on G whence Ĝ = ℤ/2 × ℤ/2.

Cases 3,5,7: from the previous remark we can immediately determine Ĝ in these cases in which G is respectively equal to ℤ/4, ℤ/3, ℤ/6. In fact |ξ| = 1, whence ξ̄ = ξ^{−1}. Thus σ acts as −Id on G = G′, the extension splits and Ĝ is a dihedral group: respectively D_{4}, D_{3} = S_{3}, D_{6}.

Case 6: G = ℤ/3 × ℤ/3. Any element σ of order 2 makes (∗) split. Since T = ℤ/3 is invariant and σ acts as −1 on G/T, we can take eigenspaces T and G′ such that σ acts on G′ as −Id and as +Id or −Id on T. The second case is not possible, since, looking at the first incarnation, we obtain that the two eigenvalues of the action of σ on the lattice Γ of E (either = 1 or −1) reduce to −1 modulo 3, thus they equal −1 and σ acts holomorphically, a contradiction.

Case 2: G = ℤ/2 × ℤ/2.

σ acts on G and trivially on T = ℤ/2, whence either the action is

the identity, or is given by ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ). In the latter case, since as we

noticed H^{2}(ℤ/2, G) = \frac{ker(σ−1)}{(1+σ)G}, then H^{2}(ℤ/2, G) = 0, so the extension splits and Ĝ = D_{4} (and an element of order 4 is given by σe_{1} whose square is indeed e_{2}, the generator of T).

If σ acts as the identity on G, Ĝ is abelian, and the exact sequence splits if and only if there is no element of order 4. Hence the only possibilities for Ĝ are the following:

• Ĝ = ℤ/2 × ℤ/2 × ℤ/2, and the sequence splits,

• Ĝ = ℤ/4 × ℤ/2.

In the latter case, we recall that by lemma 4.2 the square of σ is the generator of T.

Case 4: G is, non canonically, isomorphic to ℤ/4×T, with T ≅ ℤ/2. σ^{2} belongs to T, moreover σ acts as Id on T and as −Id on G/T. Case 4.I: there is an element g such that σgσ^{−1} = g^{−1}. In this case,

for each g ∈ G we get (σg)^{2} = σgσ^{−1}g(σ)^{2} = (σ)^{2} because Id + s = 0. Whence, we distinguish two cases: Case 4.I.a): (σ)^{2} = 1. Here, it follows rightaway that Ĝ ≅ T ⊕ D_{4}. Case 4.I.b): (σ)^{2} = t, t being the generator of T. This case can be excluded as follows: first of all, notice that for the

elliptic curve F we have τ = i, and t equal to the translation by the semiperiod 1/2(1 + i).

We have σgσ^{−1} = g^{−1} for each element of order 4, in particular for the multiplication by i. Furthermore, up to multiplying σ by a power of the above element, we may assume (cf. the discussion preceding lemma 4.4) that a = 1, i.e., that σ(z) = z̄ + b.

Then, our conjugation relation reads out as follows: a(−1 − i)b̄ ≡ 0 (mod Γ), or, more simply, as (1 + i)b̄ ≡ 0 (mod Γ). Whence, b is a semiperiod b = 1/2(x + iy) (x, y ∈ ℤ) and the above

condition means that x + y ≡ 0 (mod 2).

But a previous calculation shows that the square of σ is a translation by ab̄ + b = x ≡ 0 (mod Γ), contradicting (σ)^{2} = t.

There remains the following Case 4.II: σgσ^{−1} = g^{−1}t. In this case we can always choose σ such that (σ)^{2} = 1. In fact, if

(σ)^{2} = t, then (σg)^{2} = σgσ^{−1}g(σ)^{2} = g^{−1}tg(σ)^{2} = g^{−1}tgt = 1.

Again, here σgσ^{−1} = g^{−1}t for each element of order 4 in G. As in case 4.I.b, we choose g as the element given (on F) by multiplication by i, and we have that t is the translation by the semiperiod (1 + i)/2.

In this case, however, we can only multiply σ by the square of g, whence we may only assume a = 1 or a = i.

Since we are assuming (σ)^{2} = 1, we get ab̄ + b ≡ 0 (mod Γ), furthermore σgσ^{−1} = g^{−1}t reads out, if σ(z) = az̄ + b, as follows:

(**) −a(1 + i)b̄ ≡ 1/2(1 + i) (mod Γ), i.e., (1+i)[ab̄+1/2] ∈ Γ whence, as before, b is a semiperiod b = 1/2(x+

iy) (x, y ∈ ℤ), and (**) simply means x + y ≡ 1 (mod 2). Up to exchanging σ with σt, we may assume x = 1, y = 0, and either

• σ(z) = z̄ + 1/2

• σ(z) = iz̄ + 1/2 However, the condition ab̄ + b ≡ 0 (mod Γ) is only verified in the

former case.

Our group Ĝ is generated by elements t, σ of order 2, g of order 4, t, g^{2} are in the centre, moreover the relation σgσ^{−1} = g^{−1}t holds: Ĝ is thus isomorphic to the group G_{1}.

Q.E.D.

5. The split case

In this section we want to treat the case in which the extension

(∗) 0 → G →^{i} Ĝ →^{π} ℤ/2 → 1,

that we studied in the last section, splits.

Let σ be an antiholomorphic involution on an elliptic curve C. We recall the analysis of the antiholomorphic maps σ with σ^{2} equal to a translation given in the last section, it can be summarized as follows

\begin{matrix}Case & a \\ Re(τ) = 0, |τ| > 1 & ±1 \\ τ = i & 1 ≡ −1 \\ τ = i & i ≡ −i \\ |τ| = 1, −1/2 < Re(τ) < 0 & ±τ \\ τ = ρ & 1 ≡ ρ ≡ ρ^{2} \\ τ = ρ & −1 ≡ −ρ ≡ −ρ^{2} \\ Re(τ) = −1/2, |τ| > 1 & ±1 \end{matrix}

In the above table we have used the fact that if τ = i, by conjugating with the automorphism of C given by multiplication by i we can reduce the case a = −1 to the case a = 1, the case a = i to the case a = −i.

Analogously, if τ = ρ, by conjugating with the automorphisms of C given by multiplication by ρ, or by ρ^{2}, we can reduce the cases a = ρ, ρ^{2} to the case a = 1, the cases a = −ρ, −ρ^{2} to the case a = −1.

We consider first of all the action of Ĝ on the curve E. Let σ(z) = az̄+ b be any antiholomorphic involution of Ĝ acting on E. By lemma 4.4 we know that we have the following possibilities for b: if Re(τ) = 0, a = 1, we have b = 0, 1/2; if Re(τ) = 0, a = −1, we have b = 0, τ/2; if |τ| = 1, or Re(τ) = −1/2, we can assume b = 0.

Definition 5.1. We call cases 1, 3, 5, 7 of 4.7 the simple dihedral cases: they are characterized by the property that G = ℤ/q, Ĝ = D_{q} (where we have respectively q = 2, 4, 3, 6).

Remark 5.2. In a dihedral case with G = ℤ/q, Ĝ = D_{q}, Ĝ is generated by elements g and σ, where g acts on E by g(z) = z + c, c being a torsion element in P ic^{0}(E) of order precisely q and such that ac̄ ≡ −c (mod Γ).

Proof. We choose generators σ and g such that σ ∘ g ≡ g^{−1} ∘ σ, equivalently ac̄ ≡ −c (mod Γ). Q.E.D.

Lemma 5.3. The subgroup H_{q} of P ic^{0}(E)_{q} on which σ acts as −Id is isomorphic to ℤ/q, if and only if either |τ| = 1, a = ±τ, q = 2, 4, 3, 6 or Re(τ) = −1/2, a = ±1, q = 2, 4, 3, 6, or Re(τ) = 0, a = ±1 and q = 3.

If Re(τ) = 0, a = ±1, q = 2, 4, 6, then H_{q} is isomorphic to ℤ/q×ℤ/2.

Proof. We know that the map z ↦ az̄ is represented by the matrix

( \begin{matrix}m & m′ \\ n & n′ \end{matrix} ) on ℤ ⊕ ℤτ, hence the action on P ic^{0}(E) with basis 1/q,

τ/q is given by the reduction modulo q of the above integral matrix.

We have to solve the equation ac̄ + c ≡ 0 (mod Γ), with c ∈ P ic^{0}(E)_{q}, so we consider the kernel of the reduction of the matrix

M = ( \begin{matrix}m + 1 & m′ \\ n & n′ + 1 \end{matrix} ) modulo q.

For |τ| = 1, a = ±τ, we have: m = n′ = 0, n = m′ = ±1, while for Re(τ) = −1/2, a = ±1, we have: n = 0, m = ±1, n′ = m′ = −m, therefore the kernel of the linear map az̄+ z on P ic^{0}(E)_{q} is isomorphic to ℤ/q.

It remains to consider the case Re(τ) = 0, a = ±1. We have:

n = m′ = 0, m = ±1, n′ = −m. So the matrix M = ( \begin{matrix}2 & 0 \\ 0 & 0 \end{matrix} ) if

a = 1, M = ( \begin{matrix}0 & 0 \\ 0 & 2 \end{matrix} ) if a = −1. This tells us that the kernel of M

is isomorphic to ℤ/3 if q = 3, while it is isomorphic to ℤ/2 × ℤ/q, if q = 2, 4, 6. Q.E.D.

Lemma 5.4. If G = ℤ/2 × ℤ/2 and Ĝ = ℤ/2 × ℤ/2 × ℤ/2, we must have Re(τ) = 0, a = ±1, and b = 0, 1/2 in case a = 1, b = 0, τ/2, in case a = −1.

If G = ℤ/2 × ℤ/2 and Ĝ = D_{4}, we have either |τ| = 1, a = ±τ, or Re(τ) = −1/2, a = ±1.

Proof. If G = ℤ/2 × ℤ/2 and Ĝ = ℤ/2 × ℤ/2 × ℤ/2, since Ĝ is abelian, the map d ↦ ad̄ is the identity on the points of two torsion and one can easily see (cf. the proof of the previous lemma) that this can happen if and only if Re(τ) = 0 and a = ±1.

If G = ℤ/2×ℤ/2, Ĝ = D_{4}, we know from 4.7 that σ acts as ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} )

on the points of two torsion of E, then we have either |τ| = 1, a = ±τ, or Re(τ) = −1/2, a = ±1.

Q.E.D.

Lemma 5.5. If G = ℤ/4 × ℤ/2, then again we must have Re(τ) = 0, a = ±1, and b = 0, 1/2 in case a = 1, b = 0, τ/2, in case a = −1.

In the case Ĝ ≅ T × D_{4} ≅ ℤ/2 × D_{4}, the subgroup G ⊂ P ic^{0}(E)_{4} consists of the points c of four torsion satisfyng the equation ac̄ ≡ −c.

In the case Ĝ = G_{1}, we can choose generators of Ĝ, σ(z) = az̄ + b, g(z) = z + η, t(z) = z + ϵ, with η of order 4, ϵ of order 2 such that:

• if a = 1, ϵ = 1/2 and η = 1/4 + τ/4, b = 0, 1/2;

• if a = −1, ϵ = τ/2, η = 1/4 + τ/4, b = 0, τ/2.

Proof. In both cases the action of σ on P ic^{0}(E)_{2} ⊂ G is trivial, whence we get the same conditions upon a, b, τ as in the previous lemma.

Let g be any element in G of order 4: then the following relations hold

σ ∘ g = g^{−1} ∘ σ, in the case Ĝ = ℤ/2 × D_{4},

σ ∘ g = g^{−1}t ∘ σ, in the case Ĝ = G_{1} (where t is the generator of T). If Ĝ ≅ ℤ/2 × D_{4}, then the above equation shows that G is contained

in the subgroup of P ic^{0}(E)_{4} where σ acts as −Id. But we have seen in the proof of 5.3 that this subgroup is isomorphic to ℤ/2 × ℤ/4 if Re(τ) = 0, a = ±1, whence it equals G.

If Ĝ = G_{1}, then the relation aη̄ + η ≡ ϵ (mod Γ), gives 2Re(η) ≡ ϵ (mod Γ) if a = 1, 2iIm(η) ≡ ϵ (mod Γ) if a = −1, therefore ϵ ≡ 1/2 if a = 1, ϵ ≡ τ/2 if a = −1.

If a = 1, it follows that Re(η) = ±1/4, and since 2η is of two torsion but distinct from ϵ, we get that η = ±1/4 + ±τ/4. Up to replacing g with its inverse, or up to composing with t, we achieve η = 1/4 + τ/4, and an entirely similar argument works in the case a = −1.

Q.E.D.

Lemma 5.6. If G = ℤ/3 × ℤ/3, Ĝ = S_{3} × ℤ/3, then G is isomorphic to the group P ic^{0}(E)_{3} and the action of any antiholomorphic involution σ on P ic^{0}(E)_{3} has 1 and −1 as eigenvalues.

Therefore the datum of a Ĝ- action is equivalent to the datum of an isomorphism class of an antiholomorphic involution.

We have thus the usual following possibilities for σ:

• Re(τ) = 0, a = 1, b = 0, 1/2,

• Re(τ) = 0, a = −1, b = 0, τ/2,

• |τ| = 1, a = ±τ,

• Re(τ) = −1/2, a = ±1.

Proof. Since σ is an antiholomorphic involution, the eigenvalues of the action of σ on the lattice Γ of E are 1 and −1, thus the same holds for their reduction modulo 3, therefore we have all the possible values of a and b (see lemma 4.4). Q.E.D.

Let us now consider the action of Ĝ on F.

Lemma 5.7. Consider a cyclic group G′ of automorphisms on an elliptic curve F, generated by a transformation g having a fixed point 0.

Then the order d of g equals 2, 4, 3, 6 and F ix(g) is as follows:

• If d = 2, F ix(g) is the subgroup F_{2} of the 2-torsion points of F.

• If d = 4, F ix(g) is the subgroup of F_{2} isomorphic to ℤ/2 generated by (1 + i)/2.

• If d = 3, F ix(g) is the subgroup of F_{3} isomorphic to ℤ/3 generated by (1 − ρ)/3.

• If d = 6, F ix(g) is only the origin.

Proof. The proof is a simple computation, we notice that if d = 3, or d = 6, Γ ≅ ℤ[ρ]/(ρ^{2} + ρ + 1). Q.E.D.

Remark 5.8. Let σ be an antiholomorphic involution in Ĝ acting on F.

If σ ∘ g = g^{−1} ∘ σ, then σ acts on the sets F ix(g), F ix(g^{2}) − F ix(g), F ix(g^{3}) − F ix(g).

Moreover, F ix(g) is a subgroup of P ic^{0}(F) and σ acts on F ix(g) by an affine transformation.

Notice that g acts trivially on F ix(g), whence the action on F ix(g) = F ix(g^{−1}) is independent on the choice of σ in the simple dihedral case, when we take for g a generator of the subgroup G: we thus get in this case a topological invariant of the real hyperelliptic surface.

Corollary 5.9. In the cases 5, 7 of 4.7 we may pick as the origin in F a common fixed point of g and σ, therefore if G = ℤ/3 we may assume σ(z) = ±z̄, if G = ℤ/6 we may assume σ(z) = z̄.

Proof. We have already observed that we may choose a = ±1. If G = ℤ/6, by exchanging σ with g^{n}∘σ we see that we can assume a = 1.

If G = ℤ/3, g(z) = ρz, F ix(g^{2}) − F ix(g) is only the origin, and σ must act on F ix(g^{2}) − F ix(g) by the previous remark.

If G = ℤ/6, g(z) = −ρz, F ix(g) is only the origin and σ must act on F ix(g) by the previous remark. Q.E.D.

Lemma 5.10. If G = ℤ/4, Ĝ = D_{4} we may assume a = 1, and either σ(z) = z̄, or σ(z) = z̄ + (1 + i)/2.

Proof. We know that a = ±1, ±i, but by exchanging σ with g^{n} ∘ σ we can assume a = 1.

By 5.7 and 5.8 we know that σ acts on the subgroup of F_{2} generated by (1 + i)/2, that concludes the proof. Q.E.D.

Lemma 5.11. If G = ℤ/3 × ℤ/3, Ĝ = D_{3} × ℤ/3, we may choose generators of Ĝ acting on F given by σ(z) = −z̄, g(z) = ρz, t(z) = z + (1 − ρ)/3.

Proof. By 4.7 we know that we may assume that Ĝ is generated by σ(z) = az̄+ b, g(z) = ρz, t(z) = z + (1 − ρ)/3 such that σ ∘ g = g^{−1} ∘ σ, σ ∘ t = t ∘ σ. We have already remarked that we may choose a = ±1, and one can easily see that the relation σ ∘ t = t ∘ σ holds only for a = −1.

Since σ ∘ g = g^{−1} ∘ σ, we can assume b = 0, by 5.8 as in the case G = ℤ/3.

Q.E.D.

Lemma 5.12. If G = ℤ/4 × ℤ/2 we have the following cases:

• If Ĝ = ℤ/2 × D_{4}, we can choose the following set of generators of Ĝ acting on F: g(z) = iz, t(z) = z + (1 + i)/2, σ(z) = z̄.

• If Ĝ = G_{1}, we can choose the following set of generators of Ĝ acting on F: g(z) = iz, t(z) = z + (1 + i)/2, σ(z) = z̄ + 1/2.

Proof. For the case Ĝ = G_{1} see lemma 4.7.

Assume Ĝ = ℤ/2 × D_{4}, then we have already observed in 4.7 that we may assume that Ĝ is generated by the transformations g(z) = iz, t(z) = z +(1+i)/2, σ(z) = az̄+b, such that σ ∘g = g^{−1} ∘σ, σ ∘t = t∘σ.

Therefore by composing σ with a power of g we may assume a = 1, and the relation σ∘g = g^{−1}∘σ gives by 5.8 either b = 0, or b = (1+i)/2. The second case can be excluded by exchanging σ with t ∘ σ. Q.E.D.

Lemma 5.13. If G = ℤ/2, Ĝ = ℤ/2×ℤ/2 generated by σ(z) = az̄+b, g(z) = −z, we have the following possibilities for F, a, b:

• Re(τ) = 0, a = 1, b an element in F_{2}.

• |τ| = 1, a = τ, b = 0.

• Re(τ) = −1/2, a = 1, b = 0. Moreover there are three different topological types of the action of the affine transformation σ on F ix(g), where g is a generator of G (see 5.8), namely:

• If Re(τ) = 0, a = 1, b = 0, then σ acts as the identity on F ix(g).

• If Re(τ) = 0, a = 1, b ∈ P ic^{0}(E)_{2} of order 2, then σ acts as a translation on F ix(g).

• If |τ| = 1, a = τ, or Re(τ) = −1/2, a = 1, then σ acts as a linear

map with matrix ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) on F ix(g).

Proof. Since Ĝ is abelian, we have the condition 2b ≡ 0 (mod Γ), while the condition σ^{2} = 1 reads ab̄ + b ≡ 0 (mod Γ).

By exchanging σ with −σ, a gets multiplied by −1, so we obtain the statement on the possible values of a.

Assume now that Re(τ) = 0, a = 1.

The conditions 2b ≡ 0 (mod Γ), ab̄+ b = b̄+ b = 2Re(b) ≡ 0 (mod Γ) give us the following possibilities for b: b = 0, 1/2, τ/2 (1 + τ)/2.

Assume now that |τ| = 1, a = τ. The condition 2b ≡ 0 (mod Γ), allows us to write b = x/2 + τy/2, with x, y ∈ {0, 1}.

The condition ab̄+ b = τb̄+ b ≡ 0 (mod Γ) implies x ≡ y (mod 2), so either b = 0, or b = (1 + τ)/2, but by conjugation with the translation ϕ(z) = z + 1/2, we can assume b = 0, σ(z) = τz̄.

Assume now that Re(τ) = −1/2.

We may assume a = 1, and the condition 2b ≡ 0 (mod Γ) allows us to write b = x/2 + τy/2, with x, y ∈ {0, 1}, while the condition ab̄+b ≡ 0 (mod Γ) reads 2Re(b) = x−y/2 = n ∈ ℤ. Thus y ≡ 0 (mod 2) and either b = 0, or b = 1/2, but by translating by ϕ(z) = z + τ/2, we can assume b = 0 and σ(z) = z̄.

Finally we know from 5.8 that σ acts as an affine transformation on F ix(g) = P ic^{0}(E)_{2}, that this action is a topological invariant, and we observe that there are the following 4 different types of affine transformations acting on P ic^{0}(E)_{2}:

the identity, a translation, the map A = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ), the map A com-

posed with a translation.

Then the statement follows by an easy computation. Q.E.D.

There remains to treat the case G = ℤ/2 × ℤ/2.

Lemma 5.14. Let G = ℤ/2 × ℤ/2, then either Ĝ = ℤ/2 × ℤ/2 × ℤ/2, or Ĝ = D_{4} and we can choose generators of Ĝ as follows: g(z) = −z, t(z) = z + ϵ, ϵ of order 2, σ(z) = az̄ + b, where

• if Ĝ = ℤ/2 × ℤ/2 × ℤ/2 we have the following cases:

Re(τ) = 0, a = 1: if ϵ = 1/2, b = 0, τ/2; if ϵ = τ/2, or ϵ = 1/2 + τ/2, we can choose b = 0, 1/2. |τ| = 1, a = τ, ϵ = (1 + τ)/2, b = 0. Re(τ) = −1/2, a = 1, ϵ = 1/2, b = 0.

• if Ĝ = D_{4} we have the following cases:

Re(τ) = 0, if a = 1, b = τ/4, τ/4 + 1/2, ϵ = τ/2; if a = −1, b = −1/4, −1/4 + τ/2, ϵ = 1/2;

|τ| = 1, if a = τ, b = 1/4 − τ/4, ϵ = 1/2 + τ/2; if a = −τ, b = 1/4 + τ/4, ϵ = 1/2 + τ/2;

Re(τ) = −1/2, if a = 1, b = 1/4 + τ/2, ϵ = 1/2; if a = −1, b = −1/4, ϵ = 1/2;

Proof. Assume first of all G = ℤ/2 × ℤ/2, Ĝ = ℤ/2 × ℤ/2 × ℤ/2. By exchanging σ with −σ, a gets multiplied by −1, thus we have the

statement for a.

Since Ĝ is abelian, we have the condition 2b ≡ 0 (mod Γ), while the condition σ^{2} = 1 reads ab̄ + b ≡ 0 (mod Γ).

Assume first of all Re(τ) = 0, a = 1, then as in the case G = ℤ/2, the conditions 2b ≡ 0 (mod Γ), and ab̄ + b ≡ 0 (mod Γ), give us the following possibilities for b: b = 0, 1/2, τ/2, (1 + τ)/2.

Now, by composing σ with t, we obtain the statement.

If |τ| = 1, a = τ, as in the case G = ℤ/2, we find σ(z) = τz̄. Since σ must commute with t, we must have aϵ̄ = τϵ̄ = ϵ, which yields ϵ = (1 + τ)/2.

If Re(τ) = −1/2, we can assume a = 1, and as in the case G = ℤ/2, we find σ(z) = z̄. The condition aϵ̄ = ϵ̄ = ϵ yields ϵ = 1/2.

If G = ℤ/2 × ℤ/2, Ĝ = D_{4}, we know by 4.7 that we may choose generators g, t, σ such that σ(−z) = −σ(z) + ϵ or equivalently 2b ≡ ϵ ∈ 1/2Γ−Γ, σ(z +ϵ) = σ(z)+ϵ, or equivalently aϵ̄ ≡ ϵ (mod Γ), while σ^{2} = 1 yields ab̄ + b ∈ Γ. An element of order 4 in Ĝ is σ ∘ g, whose square is t.

Assume Re(τ) = 0, a = 1, then the conditions 2b ∈ 1/2Γ − Γ and 2Re(b) ≡ 0 (mod Γ) imply 2b ≡ τ/2 (mod Γ), therefore we can choose b = ±τ/4, or b = ±τ/4 + 1/2, ϵ = τ/2. But by composing σ with t, we can assume b = τ/4, or b = τ/4 + 1/2.

A similar computation gives the result for a = −1.

If |τ| = 1, a = τ, then the conditions 2b ∈ 1/2Γ − Γ and τb̄ + b ≡ 0 (mod Γ) tell us that we can choose b = 1/4 − τ/4, ϵ = (1 + τ)/2.

In fact by the first condition we can write 2b = n/2 + m/2τ, with n, m ∈ {0, 1}. 2(ab̄ + b) = 2(τb̄ + b) = (n + m)/2 + τ(n + m)/2 ∈ 2Γ, thus n + m ≡ 0 (mod 4), and we obtain b ≡ 1/4 − τ/4 (mod Γ). A similar computation gives the result for a = −τ.

If Re(τ) = −1/2, a = 1, the conditions 2b ∈ 1/2Γ − Γ and b̄ + b = 2Re(b) ≡ 0 (mod Γ) tell us that we can choose b ≡ 1/4 + τ/2, ϵ = 1/2.

In fact we can write 2b = n/2 + m/2τ, with n, m ∈ {0, 1}, then 2Re(b) = n/2 − m/4 ≡ 0 (mod Γ) and we have m ≡ 2n (mod 4), whence we may take b ≡ 1/4 + τ/2, and therefore ϵ = 1/2.

A similar computation gives the result for a = −1.

Q.E.D.

6. The non split case

In this section we want to treat the case where the exact sequence (∗) does not split. We have

G = T × ℤ/2 ≅ ℤ/2 × ℤ/2,

0 → T × ℤ/2 →^{j} ℤ/4 × ℤ/2 → ℤ/2 → 1

and σ^{2} = t, where t is the generator of T, a translation.

We get as generators σ, and g, where g acts on F by multiplication by −1.

We consider first the action of Ĝ on F.

Let σ(z) = az̄ + b, then b is a half-period for the following reasons: σ, g commute, a condition which is equivalent to 2b ≡ 0 (mod Γ), and moreover b ≠ 0, else σ^{2} is the identity.

If b is a half-period, then necessarily σ^{2} has order at most two, and it has order exactly two iff (ab̄ + b) ∉ Γ.

Lemma 6.1. The case Re(τ) = 0, |τ| > 1 is impossible for the curve F. In the case τ = i for the curve F we can assume a = i.

Proof. In fact, assume a = ±1: then, since (ab̄ + b) ∈ 1/2Γ − Γ, either 2Re(b) ∈ 1/2Γ − Γ or 2iIm(b) ∈ 1/2Γ − Γ, contradicting 2b ∈ Γ which in this case is equivalent to 2Re(b) ∈ Γ and 2iIm(b) ∈ Γ.

Finally, if τ = i the case a = −i can be reduced to the case a = i modulo changing coordinates via the automorphism of F given by multiplication by i.

Q.E.D.

Proposition 6.2. In the non split case, let |τ| = 1 for F. Then we can choose an element σ of Ĝ of order 4 such that σ(z) = τz̄+ 1/2. In this case σ^{2} is the translation by the vector t = (1 + τ)/2.

If Re(τ) = −1/2, we can choose an element σ of Ĝ of order 4 such that σ(z) = z̄ + τ/2. In this case σ^{2} is the translation by the vector t = 1/2.

If τ = ρ, the case a = ρ can be reduced to the case a = 1, by conjugating with the automorphism of F given by multiplication by ρ. Thus we can always assume σ(z) = z̄ + τ/2, σ^{2}(z) = z + 1/2.

Proof. We observe first of all that by exchanging σ with σg, a gets multiplied by −1, while conjugating σ with an automorphism of F given by multiplication by λ a gets multiplied by λ^{2}.

Therefore, we may assume a = τ if |τ| = 1, and a = 1 if Re(τ) = −1/2.

Furthermore, by exchanging σ with σ^{−1}, we can substitute b with −ab̄.

Case |τ| = 1.

Since 2b ∈ Γ, we may write b = x/2 + yτ/2, with x, y ∈ {0, 1}.

The condition ab̄ + b = τb̄ + b ∈ 1/2Γ − Γ yields x + y ≡ 1 (mod 2), so either b = 1/2, or b = τ/2. But by exchanging b with −τb̄, we can assume b = 1/2.

Whence we have: σ(z) = τz̄ + 1/2, and therefore σ^{2}(z) = z + 1/2 + τ/2.

Case Re(τ) = −1/2.

The condition ab̄+b = b̄+b ∈ 1/2Γ−Γ is equivalent to the condition 2Re(b) ∈ 1/2Γ − Γ, that implies Re(b) ≡ ±1/4 (modΓ).

Since 2b ∈ Γ, we easily see that b is either congruent to τ/2, or to 1/2 + τ/2. But by exchanging b with −ab̄ = −b̄, we can assume b = τ/2.

Thus we may assume σ(z) = z̄ + τ/2, and σ^{2}(z) = z − 1/2 ≡ z + 1/2 (mod Γ).

In the case τ = ρ, we observe that the two maps z ↦ τz̄ + 1/2 and z ↦ z̄ + τ/2 are conjugated by the automorphism given by multiplication by ρ. Q.E.D.

Proposition 6.3. In the non split case for the curve E, only the case Re(τ) = 0, |τ| ≥ 1, a = ±1 occurs.

If a = 1, σ(z) = z̄ + b, we have σ^{2}(z) = z + 1/2, b ≡ 1/4.

If a = −1, we have σ(z) = −z̄ + b, σ^{2}(z) = τ/2, b ≡ 1/4τ.

Proof. The fact that Ĝ is abelian implies that for every d ∈ 1/2Γ−Γ, we must have ad̄≡ d (mod Γ), whence the map d ↦ ad̄ is the identity on the points of two torsion of E.

If |τ| = 1, we know that a = ±τ, but then for d = 1/2 we have ad̄= ±τ/2 ≢ 1/2 (mod Γ), a contradiction.

If Re(τ) = −1/2, then a = ±1 and if we take d = τ/2, ad̄= ±τ̄/2 = ∓(τ + 1)/2 ≢ τ/2 (mod Γ), again a contradiction.

Therefore we can assume Re(τ) = 0, a = ±1. In this case the map d ↦ ad̄ is the identity on the points of two torsion of E.

If a = 1, the condition σ^{2}(z) = ab̄ + b = b̄ + b = 2Re(b) ∈ 1/2Γ − Γ implies Re(b) ≡ ±1/4, and σ^{2}(z) = z + 1/2. By 4.4, we can assume b ≡ ±1/4 and by exchanging σ with σ^{−1} we can assume b = 1/4.

If a = −1, the condition σ^{2}(z) = ab̄+b = −b̄+b = 2iIm(b) ∈ 1/2Γ−Γ implies Im(b) ≡ ±1/4Im(τ) and σ^{2}(z) = z + τ/2, again we conclude by 4.4 and by substituting σ with σ^{−1}.

Q.E.D.

Remark 6.4. Observe that, by remark 2.7, 1, in the non split case we must have F ix(σ) = S(ℝ) = ∅.

7. Topology of the real part of S

In this section we want to describe the topology of the fixed point locus of the involution σ acting on a real hyperelliptic surface S.

Remark 7.1. The fixed point locus of σ can only be a disjoint union of tori and Klein bottles. In fact we have the Albanese map

[figure: S; α; σ; S; α; A = ℂ/Λ; σ̄; A = ℂ/Λ]

and each component of the real locus is a S^{1} bundle on S^{1}.

We have the following

Lemma 7.2. Let σ be as above with F ix(σ) ≠ ∅ and assume that the group G is of odd order, i.e., either ℤ/3 or ℤ/3 × ℤ/3. Then the connected components of F ix(σ) are homeomorphic to S^{1} × S^{1}.

Proof.

Let π : E × F → (E × F)/G be the projection. We notice (see 2.7) that if z ∈ F ix(σ), then ∀z̃ ∈ π^{−1}(z), ∃ σ̃ : E × F → E × F such that σ̃(z̃) = z̃. We have the following commutative diagram,

[figure: E × F; π; σ̃; E × F; π; S; σ; S]

where σ̃ is an antiholomorphic involution on E × F.

Let C be the connected component of F ix(σ) that contains z and let C̃ be the connected component of π^{−1}(C) that contains z̃.

Clearly σ̃(C̃) = (C̃), moreover σ̃ is a lift of the identity of C and with σ̃(z̃) = (z̃), whence σ̃ is the identity on C̃.

We know that the components of F ix(σ) can be either tori or Klein bottles.

Since C̃ ≅ S^{1} ×S^{1} in the latter case C̃ would be an oriented covering of odd degree (equal to |G|) of a Klein bottle. But this is impossible since every oriented covering of a Klein bottle factors through the orientation covering which has degree 2. Q.E.D.

Using the Harnack - Thom - Krasnov inequality we are able to bound the number of connected components of the real part of S in all the cases given by the list of Bagnera - de Franchis. We will show later that for all G we can find real hyperelliptic surfaces S that reach these bounds (i.e., we can find M - surfaces).

Remark 7.3. 1. If G = ℤ/2, then h^{0}(S(ℝ), ℤ/2) ≤ 4; 2. If G = ℤ/2 × ℤ/2, then h^{0}(S(ℝ), ℤ/2) ≤ 3;

3. If G = ℤ/4, then h^{0}(S(ℝ), ℤ/2) ≤ 3;

4. If G = ℤ/4 × ℤ/2, then h^{0}(S(ℝ), ℤ/2) ≤ 2;

5. If G = ℤ/3, then h^{0}(S(ℝ), ℤ/2) ≤ 2;

6. If G = ℤ/3 × ℤ/3, then h^{0}(S(ℝ), ℤ/2) ≤ 2;

7. If G = ℤ/6, then h^{0}(S(ℝ), ℤ/2) ≤ 2.

Proof. We recall the Harnack-Thom-Krasnov inequality:

∑_{m}(dim_{ℤ/2}H^{m}(S(ℝ), ℤ/2)) ≤ ∑_{m}(dim_{ℤ/2}H^{m}(S, ℤ/2) − 2λ_{m}),

where

λ_{m} := dim_{ℤ/2}(1 + σ)H^{m}(S, ℤ/2).

Now we have

∑_{m}(dim_{ℤ/2}H^{m}(S(ℝ), ℤ/2)) = 2h^{0}(S(ℝ), ℤ/2) + h^{1}(S(ℝ), ℤ/2),

and since the components of S(ℝ) are either Tori or Klein bottles, we have h^{1}(S(ℝ), ℤ/2) = 2h^{0}(S(ℝ), ℤ/2). Thus we find

4h^{0}(S(ℝ), ℤ/2) ≤ ∑_{m}(dim_{ℤ/2}H^{m}(S, ℤ/2)).  (9)

We notice that b_{2}(S) = 2, and we have the following list for H_{1}(S, ℤ) (see e.g. [Su]):

• G = ℤ/2, H_{1}(S, ℤ) = ℤ^{2} ⊕ (ℤ/2)^{2};

• G = ℤ/2 ⊕ ℤ/2, H_{1}(S, ℤ) = ℤ^{2} ⊕ ℤ/2; • G = ℤ/4, H_{1}(S, ℤ) = ℤ^{2} ⊕ ℤ/2;

• G = ℤ/4 ⊕ ℤ/2, H_{1}(S, ℤ) = ℤ^{2};

• G = ℤ/3, H_{1}(S, ℤ) = ℤ^{2} ⊕ ℤ/3;

• G = ℤ/3 ⊕ ℤ/3, H_{1}(S, ℤ) = ℤ^{2};

• G = ℤ/6, H_{1}(S, ℤ) = ℤ^{2}. The Universal Coefficients Theorem and Poincaré duality allow us to compute all the Betti numbers of S with coefficients in ℤ/2. Then by 9 we obtain the given bounds on the number of connected components of S(ℝ). Q.E.D.

We want now to show how one can compute the number k of Klein bottles, respectively the number t of 2-dimensional tori in the real part S(ℝ) of a real hyperelliptic surface.

For every connected component V of F ix(σ) = S(ℝ) the inverse image π^{−1}(V) splits as the G-orbit of any of its connected components, let W be one such.

We have already observed that there is a lift σ̃ of σ such that W is in the fixed locus of σ̃.

If σ̃_{1}, σ̃_{2} are two distinct antiholomorphic involutions, then π^{−1}(V)∩ F ix(σ̃_{1})∩F ix(σ̃_{2}) = ∅, in fact otherwise there would exist a component W_{h} such that σ̃_{1}_{|W_{h}} = σ̃_{2}_{|W_{h}} = Id, thus σ̃_{1}σ̃_{2} = Id, a contradiction.

We conclude from the above argument that the connected components of F ix(σ) correspond bijectively to the set C obtained as follows: consider all the lifts σ̃ of σ which are involutions and pick one representative σ̃_{i} for each conjugacy class.

Then we let C be the set of equivalence classes of connected components of ∪F ix(σ̃_{i}), where two components A, A′ of F ix(σ̃_{i}) are equivalent if and only if there exists an element g ∈ G, such that g(A) = A′.

Let σ̃ be an antiholomorphic involution which is a lift of σ and such that F ix(σ̃) ≠ ∅.

Since σ̃ is of product type, F ix(σ̃) is a disjoint union of 2^{a_{1}+a_{2}} copies of S^{1} × S^{1}, where a_{i} ∈ {0, 1}.

In fact if an antiholomorphic involution σ̂ on an elliptic curve C has fixed points, then F ix(σ̂) is a disjoint union of 2^{a} copies of S^{1}, where a = 1 if the matrix of the action of σ̂ on H_{1}(C, ℤ) is diagonalizable, else a = 0.

What said insofar describes the number of such components V: in order to determine their nature, observe that V = W/H, where

W is as before and H ⊂ G is the subgroup such that HW = W.

Since the action of G on the first curve E is by translations, the action on the first S^{1} is always orientation preserving, whence V is a Klein bottle if and only if H acts on the second S^{1} by some orientation reversing map, or equivalently H has some fixed point on the second S^{1}.

Let h ∈ H be a transformation having a fixed point on the second S^{1}: since the direction of this S^{1} is an eigenvector for the tangent action of h, it follows that the tangent action is given by multiplication by −1.

The existence of such an element h for the second S^{1} is obvious if a_{1} = a_{2} = 0 and σ̃ commutes with h (this always occurs except in the last case of 2 of 4.7) and in this case we get 1 Klein bottle.

In the other cases one needs a more delicate analysis whose details we omit here.

In the next section we shall give a precise description of all the isomorphism classes of real hyperelliptic surfaces and we shall also determine the real part S(ℝ) for each isomorphism class.

8. Moduli space of real hyperelliptic surfaces

In this section we describe all the isomorphism classes of of real hyperelliptic surfaces. Each slot in the following tables corresponds to a fixed topological type.

In the column of the values of σ_{1}, we always have two possible values, that correspond to two antiholomorphic maps which are topologically equivalent, but not analytically equivalent. Nevertheless they are in the same connected component of the moduli space of real elliptic curves, since the curve with τ_{1} = i has an isomorphism given by the multiplication by i that conjugates the two different antiholomorphic maps.

Assume first of all that we are in the split case for the extension (∗). We set E = ℂ/ℤ + τ_{1}ℤ, F = ℂ/ℤ + τ_{2}ℤ, τ_{j} = x_{j} + iy_{j}, σ̃ = (σ_{1}, σ_{2}) :

E × F → E × F. The action of G is given as in the Bagnera - de Franchis list.

The first case that we will consider is the one with G = ℤ/2, Ĝ = ℤ/2 × ℤ/2. Let g be a generator of G: its action on E is a translation by an element η ∈ P ic^{0}(E)_{2}, thus g(z_{1}, z_{2}) = (z_{1} + η, −z_{2}), ∀(z_{1}, z_{2}) ∈ E × F.

Let us now give a list of the topological invariants that distinguish the different cases presented in the forthcoming table.

Recall that for an antiholomorphic involution σ on an elliptic curve C, the number ν(σ) of connected components of F ix(σ), which is either 0, or 1 or 2, is a topological invariant of the involution.

1) In our case, given an antiholomorphic involution on a hyperelliptic surface S, since we have exactly two lifts of σ on E × F, σ̃ = (σ_{1}, σ_{2}), σ̃ ∘ g the sets

1)(a): {ν(σ_{1}), ν(σ_{1} ∘ g)},

1)(b): {ν(σ_{2}), ν(σ_{2} ∘ g)}, are topological invariants of σ. Since G acts on E by translation and on F by multiplication by

−1, also the parity of ν(σ_{i}) (determined by the alternative: the action on the first homology group is diagonalizable or not) is a topological invariant.

Other topological invariants are:

2) the homology of S(ℝ);

3) the action of the involution σ_{2} on the fixed point locus of the action of g on F (see 5.8);

4) the action of Ĝ′ on Λ′′ ⊗ ℝ, where Λ′′ = π_{1}(E) ≅ ℤ^{2}, given by the orbifold fundamental group exact sequence:

1 → Λ′′ ⊕ Γ → Π̂ → Ĝ → 1, where Γ = π_{1}(F) (see section 2). In particular, given a transformation in G, we have the condition

whether its first action is representable by a translation in Λ′′ ⊗ ℝ which is an eigenvector for σ_{1}.

G, Ĝ  τ_{1}  τ_{2}  σ_{1}(z_{1}), η  σ_{2}(z_{2}) S(ℝ)
ℤ/2  x_{1} = 0  x_{2} = 0  z̄_{1}, η = \frac{1}{2}  z̄_{2}  4K
(ℤ/2)^{2}  or −z̄_{1}, η = \frac{τ_{1}}{2}
ℤ/2 |τ_{1}| = 1, x_{2} = 0 ±τ_{1}z̄_{1}, η = \frac{1+τ_{1}}{2} , z̄_{2} 4K
(ℤ/2)^{2}  if |τ_{1}| = 1,
∪ x_{1} = −\frac{1}{2},  ±z̄_{1}, η = \frac{1}{2}

if x_{1} = −\frac{1}{2}

ℤ/2 x_{1} = 0, x_{2} = 0, z̄_{1}, η = \frac{1}{2} z̄_{2} + \frac{τ_{2}}{2} 2T
(ℤ/2)^{2}  or −z̄_{1}, η = \frac{τ_{1}}{2}
ℤ/2 x_{1} = 0, x_{2} = 0, z̄_{1}, η = \frac{τ_{1}}{2} z̄_{2} + \frac{τ_{2}}{2} 2T
(ℤ/2)^{2}  or −z̄_{1}, η = \frac{1}{2}
ℤ/2  x_{1} = 0, |τ_{2}| = 1,  z̄_{1}, η = \frac{τ_{1}}{2}  τ_{2}z̄_{2} if 2T
(ℤ/2)^{2}  or  or −z̄_{1}, η = \frac{1}{2} |τ_{2}| = 1
x_{2} = −\frac{1}{2}  z̄_{2} if

x_{2} = −\frac{1}{2}

ℤ/2  x_{1} = 0,  x_{2} = 0,  z̄_{1}, η = \frac{1+τ_{1}}{2}  z̄_{2}  2T
(ℤ/2)^{2}  or −z̄_{1}, η = \frac{1+τ_{1}}{2}
ℤ/2 x_{1} = 0, x_{2} = 0, z̄_{1}, η = \frac{1+τ_{1}}{2} z̄_{2} + \frac{τ_{2}}{2} 2T
(ℤ/2)^{2}  or −z̄_{1}, η = \frac{1+τ_{1}}{2}
ℤ/2 x_{1} = 0, x_{2} = 0, z̄_{1}, η = \frac{1}{2} z̄_{2} + \frac{1}{2} ∅
(ℤ/2)^{2}  or −z̄_{1}, η = \frac{τ_{1}}{2}
ℤ/2 x_{1} = 0, x_{2} = 0, z̄_{1}, η = \frac{1}{2} z̄_{2} + \frac{1+τ_{2}}{2} ∅
(ℤ/2)^{2}  or −z̄_{1}, η = \frac{τ_{1}}{2}
ℤ/2 x_{1} = 0, x_{2} = 0, z̄_{1}, η = \frac{τ_{1}}{2} z̄_{2} + \frac{1+τ_{2}}{2} ∅
(ℤ/2)^{2}  or −z̄_{1}, η = \frac{1}{2}
ℤ/2 x_{1} = 0, x_{2} = 0, z̄_{1}, η = \frac{1+τ_{1}}{2} z̄_{2} + \frac{1}{2} ∅
(ℤ/2)^{2}  or −z̄_{1}, η = \frac{1+τ_{1}}{2}
ℤ/2 x_{1} = 0, x_{2} = 0, z̄_{1}, η = \frac{1+τ_{1}}{2} z̄_{2} + \frac{1+τ_{2}}{2} ∅
(ℤ/2)^{2}  or −z̄_{1}, η = \frac{1+τ_{1}}{2}
ℤ/2  |τ_{1}| = 1, x_{2} = 0, ±τ_{1}z̄_{1}, η = \frac{1+τ_{1}}{2} z̄_{2} + \frac{1+τ_{2}}{2} ∅
(ℤ/2)^{2} ∪ x_{1} = −\frac{1}{2}  if |τ_{1}| = 1
|τ_{1}| ≥ 1  ±z̄_{1}, η = \frac{1}{2}

if x_{1} = −\frac{1}{2}

ℤ/2  x_{1} = 0,  x_{2} = 0,  z̄_{1} + \frac{1}{2}, η = \frac{τ_{1}}{2}  z̄_{2}  ∅
(ℤ/2)^{2}  or −z̄_{1} + \frac{τ_{1}}{2} , η = \frac{1}{2}
ℤ/2 x_{1} = 0, x_{2} = 0, z̄_{1} + \frac{1}{2}, η = \frac{τ_{1}}{2} z̄_{2} + \frac{τ_{2}}{2} ∅
(ℤ/2)^{2}  or −z̄_{1} + \frac{τ_{1}}{2} , η = \frac{1}{2}
ℤ/2 x_{1} = 0, x_{2} = 0, z̄_{1} + \frac{1}{2}, η = \frac{τ_{1}}{2} z̄_{2} + \frac{1+τ_{2}}{2} ∅
(ℤ/2)^{2}  or −z̄_{1} + \frac{τ_{1}}{2} , η = \frac{1}{2}
ℤ/2 x_{1} = 0, |τ_{2}| = 1, z̄_{1} + \frac{1}{2}, η = \frac{τ_{1}}{2} τ_{2}z̄_{2} ∅
(ℤ/2)^{2}  ∪ x_{2} = −\frac{1}{2}, or −z̄_{1} + \frac{τ_{1}}{2} , η = \frac{1}{2} if |τ_{2}| = 1,

z̄_{2}

if x_{2} = −\frac{1}{2}

ℤ/2 x_{1} = 0, |τ_{2}| = 1 z̄_{1}, η = \frac{1}{2} τ_{2}z̄_{2}, 2K
(ℤ/2)^{2}  ∪ x_{2} = −\frac{1}{2} or −z̄_{1}, η = \frac{τ_{1}}{2} if |τ_{2}| = 1,

z̄_{2},

if x_{2} = −\frac{1}{2}
ℤ/2 |τ_{1}| = 1,  |τ_{2}| = 1,  ±τ_{1}z̄_{1}, η = \frac{1+τ_{1}}{2}  τ_{2}z̄_{2},  2K
(ℤ/2)^{2} ∪ x_{1} = −\frac{1}{2} ∪ x_{2} = −\frac{1}{2}  if |τ_{1}| = 1  if |τ_{2}| = 1,
±z̄_{1}, η = \frac{1}{2}  z̄_{2},
if x_{1} = −\frac{1}{2}  if x_{2} = −\frac{1}{2}
ℤ/2 x_{1} = 0, x_{2} = 0, z̄_{1}, η = \frac{τ_{1}}{2} z̄_{2}, 4T
(ℤ/2)^{2}  or −z̄_{1}, η = \frac{1}{2}
ℤ/2 x_{1} = 0, |τ_{2}| = 1, z̄_{1}, η = \frac{1+τ_{1}}{2} τ_{2}z̄_{2}, T
(ℤ/2)^{2}  ∪ x_{2} = −\frac{1}{2}, or −z̄_{1}, η = \frac{1+τ_{1}}{2} if |τ_{2}| = 1,

z̄_{2},

if x_{2} = −\frac{1}{2}
ℤ/2 |τ_{1}| = 1, x_{2} = 0 ±τ_{1}z̄_{1}, η = \frac{1+τ_{1}}{2} z̄_{2} + \frac{τ_{2}}{2} , T
(ℤ/2)^{2} ∪ x_{1} = −\frac{1}{2}  if |τ_{1}| = 1
±z̄_{1}, η = \frac{1}{2}
if x_{1} = −\frac{1}{2}

Let us now explain how to distinguish the different cases listed in the above table:

S(ℝ) = 4K: the first two cases of the list are distinguished by the parity of ν(σ_{1}) (ν = 2, respectively ν = 1).

S(ℝ) = 2T: the third case listed with S(ℝ) = 2T is distinguished from all the others by the parity of ν(σ_{2}) = 1.

The fourth case is distinguished from all the others by the invariant 3), or by 1)(b) (the set of values of ν(σ_{2}) equals {2, 2}).

The first is distinguished from the second by the invariant 1)(a) (the respective sets are {2, 0}, {2, 2})) and from the fifth by the invariant 4): in fact the action of g on Λ′′ ⊗ ℝ is given by z_{1} ↦ z_{1} + 1/2 in the first case, by z_{1} ↦ z_{1} + (1 + τ_{1})/2 in the fifth case, σ_{1}(z_{1}) = z̄_{1} in both cases, but 1/2 is an eigenvector of the action of σ_{1} on Λ′′, while (1 + τ_{1})/2 is not.

The second is distinguished from the fifth by the invariant 1)(a) (the respective sets are {2, 2}, {2, 0})).

S(ℝ) = ∅: the sixth is distinguished from all others by the parity of ν(σ_{1}) = 1.

The last one is distinguished from all others by the parity of ν(σ_{2}) = 1.

The seventh is distinguished from all others by the invariant 3), or by 1)(b) (the set of values of ν(σ_{2}) equals {2, 2}, while in all the other cases it equals either {0, 2}, or {0, 0}, or {1, 1} in the last case).

The first case is distinguished from the second, the third, the fifth, the ninth case by the invariant 1)(b): in the first case the set of values of ν(σ_{2}) equals {0, 2}, in the others {0, 0}.

The first case is distinguished from the the eighth case by the invariant 1)(a): in the first case the set of values of ν(σ_{1}) equals {2, 0}, in the eight case it equals {0, 0}.

The first case is distinguished from the fourth by the invariant 4): in fact the action of g on Λ′′ ⊗ℝ is given by z_{1} ↦ z_{1} +1/2 in the first case, by z_{1} ↦ z_{1} + (1 + τ_{1})/2 in the fourth case, σ_{1}(z_{1}) = z̄_{1} in both cases, but 1/2 is an eigenvector of the action of σ_{1} on Λ′′, while (1 + τ_{1})/2 is not.

The second case is distinguished from the third by the invariant 4): in fact the action of g on Λ′′ ⊗ℝ is given by z_{1} ↦ z_{1} +1/2 in the second case, by z_{1} ↦ z_{1} + τ_{1}/2 in the third case, σ_{1}(z_{1}) = z̄_{1} in both cases, but 1/2 is the +1 eigenvector of the action of σ_{1} on Λ′′, while τ_{1}/2 is the −1 eigenvector.

The second case is distinguished from the fourth and the eighth case by the invariant 1)(b): in the second case the set of values of ν(σ_{2}) equals {0, 0}, in the other cases {0, 2}.

The second case is distinguished from the fifth by the invariant 4): in fact the action of g on Λ′′ ⊗ℝ is given by z_{1} ↦ z_{1} +1/2 in the second case, by z_{1} ↦ z_{1} +(1+τ_{1})/2 in the fifth case, σ_{1}(z_{1}) = z̄_{1} in both cases, but 1/2 is an eigenvector of the action of σ_{1} on Λ′′, while (1 + τ_{1})/2 is not.

The second case is distinguished from the the ninth case by the invariant 1)(a): in the second case the set of values of ν(σ_{1}) equals {2, 0}, in the ninth case it equals {0, 0}.

The third case is distinguished from the fourth by the invariant 1)(b): in the third case the set of values of ν(σ_{2}) equals {0, 0}, in the fourth case {0, 2}.

The third case is distinguished from the fifth by the invariant 4): in fact the action of g on Λ′′ ⊗ ℝ is given by z_{1} ↦ z_{1} + τ_{1}/2 in the third case, by z_{1} ↦ z_{1} +(1+τ_{1})/2 in the fifth case, σ_{1}(z_{1}) = z̄_{1} in both cases, but τ_{1}/2 is an eigenvector of the action of σ_{1} on Λ′′, while (1 + τ_{1})/2 is not.

The third case is distinguished from the eighth and the ninth case by the invariant 1)(a): in the third case the set of values of ν(σ_{1}) equals {2, 2}, in the eighth and the ninth case it equals {0, 0}.

The fourth case is distinguished from the fifth and the ninth case by the invariant 1)(b): in the fourth case the set of values of ν(σ_{2}) equals {0, 2}, in the other cases {0, 0}.

The fourth case is distinguished from the eighth case by the invariant 1)(a): in the fourth case the set of values of ν(σ_{1}) equals {0, 2}, in the eighth case it equals {0, 0}.

The fifth case is distinguished from the eighth case by the invariant 1)(b): in the fifth case the set of values of ν(σ_{2}) equals {0, 0}, in the eighth case {2, 0}.

The fifth case is distinguished from the ninth case by the invariant 1)(a): in the fifth case the set of values of ν(σ_{1}) equals {2, 0}, in the ninth case it equals {0, 0}.

The eighth case is distinguished from the ninth case by the invariant 1)(b): in the eighth case the set of values of ν(σ_{2}) equals {2, 0}, in the ninth case {0, 0}.

S(ℝ) = 2K: the parity of ν(σ_{1}) distinguishes the two cases.

S(ℝ) = T: the parity of ν(σ_{2}) distinguishes the two cases.

Let us now consider the case G = ℤ/4. If g is a generator of G, we denote by g(z_{1}, z_{2}) = (z_{1} + η, iz_{2}) the action of g on E × F.

The topological invariants that distinguish the different cases here are the above mentioned invariant 2), and the analogous of the invariants 1)(a) and 1)(b) that we had in the case G = ℤ/2:

1′)(a) the set {ν(σ_{1} ∘ g^{n}), for n = 0, 1, 2, 3}.

1′)(b) the set {ν(σ_{2} ∘ g^{n}), for n = 0, 1, 2, 3}.

ℤ/4 x_{1} = 0, i  z̄_{1}, η = \frac{1}{2} + \frac{τ_{1}}{4}  z̄_{2},  2T
D_{4}  or −z̄_{1}, η = \frac{1}{4} + \frac{τ_{1}}{2}
ℤ/4 x_{1} = 0, i z̄_{1} + \frac{1}{2}, η = \frac{1}{2} + \frac{τ_{1}}{4} z̄_{2}, T
D_{4}  or −z̄_{1} + \frac{τ_{1}}{2} , η = \frac{1}{4} + \frac{τ_{1}}{2}
ℤ/4 x_{1} = 0, i z̄_{1}, η = \frac{τ_{1}}{4} z̄_{2} + \frac{1+i}{2}, T
D_{4}  or −z̄_{1}, η = \frac{1}{4}
ℤ/4 x_{1} = 0, i z̄_{1} + \frac{1}{2}, η = \frac{1}{2} + \frac{τ_{1}}{4} z̄_{2} + \frac{1+i}{2}, T
D_{4}  or −z̄_{1} + \frac{τ_{1}}{2} , η = \frac{1}{4} + \frac{τ_{1}}{2}
ℤ/4 x_{1} = 0, i z̄_{1}, η = \frac{1}{2} + \frac{τ_{1}}{4} z̄_{2} + \frac{1+i}{2}, ∅
D_{4}  or −z̄_{1}, η = \frac{1}{4} + \frac{τ_{1}}{2}
ℤ/4 x_{1} = 0, i z̄_{1} + \frac{1}{2}, η = \frac{τ_{1}}{4} z̄_{2}, ∅
D_{4}  or −z̄_{1} + \frac{τ_{1}}{2} , η = \frac{1}{4}
ℤ/4 x_{1} = 0, i z̄_{1} + \frac{1}{2}, η = \frac{τ_{1}}{4} z̄_{2} + \frac{1+i}{2}, ∅
D_{4}  or −z̄_{1} + \frac{τ_{1}}{2} , η = \frac{1}{4}
ℤ/4 x_{1} = 0, i z̄_{1}, η = \frac{τ_{1}}{4} z̄_{2}, 3T
D_{4}  or −z̄_{1}, η = \frac{1}{4}
ℤ/4 |τ_{1}| = 1 i τ_{1}z̄_{1}, η = \frac{1−τ_{1}}{4} z̄_{2}, 3K
D_{4} ∪ x_{1} = −\frac{1}{2}  or −τ_{1}z̄_{1}, η = \frac{1+τ_{1}}{4}
if |τ_{1}| = 1,
z̄_{1}, η = \frac{1}{4} + \frac{τ_{1}}{2}
−z̄_{1}, η = \frac{1}{4}
if x_{1} = −\frac{1}{2}
ℤ/4 |τ_{1}| = 1 i τ_{1}z̄_{1}, η = \frac{1−τ_{1}}{4} z̄_{2} + \frac{1+i}{2} K
D_{4} ∪ x_{1} = −\frac{1}{2}  or −τ_{1}z̄_{1}, η = \frac{1+τ_{1}}{4}
if |τ_{1}| = 1,
z̄_{1}, η = \frac{1}{4} + \frac{τ_{1}}{2}
−z̄_{1}, η = \frac{1}{4}
if x_{1} = −\frac{1}{2}

In the list, if S(ℝ) = T, the second and the third cases are distinguished by invariant 1′)(a): in the second case the set of values of ν(σ_{1}) equals {2, 2, 2, 2}, in the third it equals {0, 2, 0, 2}.

The first case is distinguished from the others by the invariant 1′)(b): in the first case {ν(σ_{2} ∘ g^{n}), for n = 0, 1, 2, 3} = {2, 1, 2, 1}, while in the other cases {ν(σ_{2} ∘ g^{n}), for n = 0, 1, 2, 3} = {0, 1, 0, 1}.

If S(ℝ) = ∅, the first case is distinguished from the third case by the invariant 1′)(a): in the first case the set of values of ν(σ_{1}) equals {2, 0, 2, 0}, in the third it equals {0, 0, 0, 0}.

The second case is distinguished from the other cases by the invariant 1′)(b): in the second case the set of values of ν(σ_{2}) equals {2, 1, 2, 1}, in the other cases it equals {0, 1, 0, 1}.

If G = ℤ/4×ℤ/2, we take generators g, t, such that on F, g(z_{2}) = iz_{2}, t(z_{2}) = z_{2} + (1 + i)/2.

If Ĝ ≅ G_{1}, we see that the topological type of S(ℝ) distinguishes the cases. The same holds if Ĝ ≅ D_{4} × ℤ/2.

ℤ/4 × ℤ/2 x_{1} = 0 i  z̄_{1},  z̄_{2} + \frac{1}{2}, ∅
G_{1}  G =< \frac{1+τ_{1}}{4} > × < \frac{1}{2} > ϵ = \frac{1+i}{2}
or −z̄_{1}
G =< \frac{−1+τ_{1}}{4} > × < \frac{τ_{1}}{2} >
ℤ/4 × ℤ/2 x_{1} = 0 i z̄_{1} + \frac{1}{2}, z̄_{2} + \frac{1}{2}, T
G_{1}  G =< \frac{1+τ_{1}}{4} > × < \frac{1}{2} > ϵ = \frac{1+i}{2}
or −z̄_{1} + \frac{τ_{1}}{2}
G =< \frac{−1+τ_{1}}{4} > × < \frac{τ_{1}}{2} >

ℤ/4 × ℤ/2 x_{1} = 0 i  z̄_{1},  z̄_{2},  2T
D_{4} × ℤ/2  G =< \frac{τ_{1}}{4} > × < \frac{1}{2} > ϵ = \frac{1+i}{2}
or −z̄_{1}
G =< \frac{1}{4} > × < \frac{τ_{1}}{2} >
ℤ/4 × ℤ/2 x_{1} = 0 i z̄_{1}, z̄_{2}, K ⊔ T
D_{4} × ℤ/2  G =< \frac{τ_{1}}{4} > × < \frac{1+τ_{1}}{2} > ϵ = \frac{1+i}{2}
or −z̄_{1}
G =< \frac{1}{4} > × < \frac{1+τ_{1}}{2} >
ℤ/4 × ℤ/2 x_{1} = 0, i z̄_{1} + \frac{1}{2}, z̄_{2}, T
D_{4} × ℤ/2  G =< \frac{τ_{1}}{4} > × < \frac{1}{2} > ϵ = \frac{1+i}{2}
or −z̄_{1} + \frac{τ_{1}}{2}
G =< \frac{1}{4} > × < \frac{τ_{1}}{2} >
ℤ/4 × ℤ/2 x_{1} = 0, i z̄_{1} + \frac{1}{2}, z̄_{2}, K
D_{4} × ℤ/2  G =< \frac{τ_{1}}{4} > × < \frac{1+τ_{1}}{2} > ϵ = \frac{1+i}{2}
or −z̄_{1} + \frac{τ_{1}}{2}
G =< \frac{1}{4} > × < \frac{1+τ_{1}}{2} >

If G = ℤ/3, we denote by g(z_{1}, z_{2}) = (z_{1} + η, ρz_{2}) the action of a generator g of G on E × F.

We notice that F ix(g) = ℤ(1 − ρ)/3ℤ(1 − ρ) and the action of σ_{2}(z_{2}) = z̄_{2} on F ix(g) is −Id, while the action of σ_{2}(z_{2}) = −z̄_{2} on F ix(g) is Id, so these two actions are topologically different.

This invariant and S(ℝ) are sufficient to distinguish all the cases.

ℤ/3 x_{1} = 0, ρ  z̄_{1}, η = \frac{τ_{1}}{3}  z̄_{2}, 2T
S_{3}  or −z̄_{1}, η = \frac{1}{3}
ℤ/3 x_{1} = 0, ρ z̄_{1}, η = \frac{τ_{1}}{3} −z̄_{2}, 2T
S_{3}  or −z̄_{1}, η = \frac{1}{3}
ℤ/3 x_{1} = 0, ρ z̄_{1} + \frac{1}{2}, η = \frac{τ_{1}}{3} z̄_{2}, ∅
S_{3}  or −z̄_{1} + \frac{τ_{1}}{2} , η = \frac{1}{3}
ℤ/3 x_{1} = 0, ρ z̄_{1} + \frac{1}{2}, η = \frac{τ_{1}}{3} −z̄_{2}, ∅
S_{3}  or −z̄_{1} + \frac{τ_{1}}{2} , η = \frac{1}{3}
ℤ/3 |τ_{1}| = 1, ρ τ_{1}z̄_{1}, η = \frac{1−τ_{1}}{3} z̄_{2} T
S_{3} ∪ x_{1} = −\frac{1}{2}  or −τ_{1}z̄_{1}, η = \frac{1+τ_{1}}{3}
if |τ_{1}| = 1
z̄_{1}, η = \frac{1−τ_{1}}{3}
or −z̄_{1}, η = \frac{1}{3}
if x_{1} = −\frac{1}{2}
ℤ/3 |τ_{1}| = 1, ρ τ_{1}z̄_{1}, η = \frac{1−τ_{1}}{3} −z̄_{2}, T
S_{3} ∪ x_{1} = −\frac{1}{2}  or −τ_{1}z̄_{1}, η = \frac{1+τ_{1}}{3}
if |τ_{1}| = 1
z̄_{1}, η = \frac{1−τ_{1}}{3}
or −z̄_{1}, η = \frac{1}{3}
if x_{1} = −\frac{1}{2}

If G = ℤ/3 × ℤ/3, with generators g, t acting on F by g(z_{2}) = ρz_{2}, t(z_{2}) = z_{2} + (1 − ρ)/3, the homology of S(ℝ) is the only topological invariant needed in order to distinguish the three cases.

ℤ/3 × ℤ/3 x_{1} = 0, ρ  z̄_{1},  −z̄_{2}, 2T
S_{3} × ℤ/3  G =< \frac{τ_{1}}{3} > × < \frac{1}{3} > ϵ = \frac{1−ρ}{3}

or −z̄_{1}

G =< \frac{1}{3} > × < \frac{τ_{1}}{3} >
ℤ/3 × ℤ/3 x_{1} = 0, ρ z̄_{1} + \frac{1}{2}, −z̄_{2}, ∅
S_{3} × ℤ/3  G =< \frac{τ_{1}}{3} > × < \frac{1}{3} > ϵ = \frac{1−ρ}{3}
or −z̄_{1} + \frac{τ_{1}}{2}
G =< \frac{1}{3} > × < \frac{τ_{1}}{3} >
ℤ/3 × ℤ/3 |τ_{1}| = 1, ρ τ_{1}z̄_{1}, −z̄_{2}, T
S_{3} × ℤ/3 ∪ x_{1} = −\frac{1}{2}  G =< \frac{1−τ_{1}}{3} > × < \frac{1+τ_{1}}{3} > ϵ = \frac{1−ρ}{3}

or −τ_{1}z̄_{1}

G =< \frac{1+τ_{1}}{3} > × < \frac{1−τ_{1}}{3} >
if |τ_{1}| = 1

z̄_{1}

G =< \frac{1−τ_{1}}{3} > × < \frac{1}{3} > or −z̄_{1}

G =< \frac{1}{3} > × < \frac{1−τ_{1}}{3} >
if x_{1} = −\frac{1}{2}

If G = ℤ/6, we take a generator g of G such that g(z_{1}, z_{2}) = (z_{1} + η, −ρz_{2}) and also here the homology of S(ℝ) is the only topological invariant needed to distinguish the four cases.

ℤ/6 x_{1} = 0, ρ  z̄_{1}, η = \frac{τ_{1}}{6}  z̄_{2}, 2T
D_{6}  or −z̄_{1}, η = \frac{1}{6}
ℤ/6 x_{1} = 0, ρ z̄_{1}, η = \frac{1}{2} + \frac{τ_{1}}{6}  z̄_{2}, T
D_{6}  or −z̄_{1}, η = \frac{τ_{1}}{2} + \frac{1}{6}
ℤ/6 x_{1} = 0, ρ z̄_{1} + \frac{1}{2}, η = \frac{τ_{1}}{6}  z̄_{2}, ∅
D_{6}  or −z̄_{1} + \frac{τ_{1}}{2} , η = \frac{1}{6}
ℤ/6 |τ_{1}| = 1, ρ τ_{1}z̄_{1}, η = \frac{1−τ_{1}}{6}  z̄_{2}, 2K
D_{6} ∪ x_{1} = −\frac{1}{2}  or −τ_{1}z̄_{1}, η = \frac{1+τ_{1}}{6}
if |τ_{1}| = 1,
z̄_{1}, η = \frac{1}{6} + \frac{τ_{1}}{3}
or −z̄_{1}, η = \frac{1}{6}
if x_{1} = −\frac{1}{2}

Let us now consider the case where G = ℤ/2 × ℤ/2. Here, we may choose as generators of G the generator t of T and

another element g, which is not canonically defined.

Let η_{1}, ϵ_{1}, ϵ_{2} be such that g(z_{1}, z_{2}) = (z_{1} + η_{1}, −z_{2}), t(z_{1}, z_{2}) = (z_{1} + ϵ_{1}, z_{2} + ϵ_{2}).

Here we have the same topological invariants 1), 2), as in the case G = ℤ/2.

Furthermore the normal subgroup T of G is of order 2, let as usual t be a generator. We may consider all the possible liftings of t to a vector t′ in the lattice Ω′ = Λ′ ⊕ Γ.

The condition whether there exists such a t′ whose two components are eigenvectors for the action of σ̃ on Ω′ is a topological invariant of the real hyperelliptic surface (notice that the two possible choices for σ_{2} differ just up to multiplication by −1).

G, Ĝ  τ_{1}  τ_{2}  σ_{1}(z_{1}), η_{1}, ϵ_{1}  σ_{2}(z_{2}), ϵ_{2} S(ℝ)
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0  z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{τ_{1}}{2}  z̄_{2}, ϵ_{2} = \frac{1+τ_{2}}{2} 2K
(ℤ/2)^{3}  or −z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{τ_{1}}{2} z̄_{2}, ϵ_{2} = \frac{1}{2} 2K
(ℤ/2)^{3}  or −z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2}
(ℤ/2)^{2} x_{1} = 0, |τ_{2}| = 1  z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{τ_{1}}{2}  τ_{2}z̄_{2}, ϵ_{2} = \frac{1+τ_{2}}{2} 2K
(ℤ/2)^{3}  or  or −z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2}  if
x_{2} = −\frac{1}{2}  |τ_{2}| = 1
z̄_{2}, ϵ_{2} = \frac{1}{2}

if x_{2} = −\frac{1}{2},

(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{1+τ_{1}}{2} z̄_{2}, ϵ_{2} = \frac{τ_{2}}{2} 2K
(ℤ/2)^{3}  or −z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1+τ_{1}}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{1+τ_{1}}{2} z̄_{2}, ϵ_{2} = \frac{1+τ_{2}}{2} 2K
(ℤ/2)^{3}  or −z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1+τ_{1}}{2}
(ℤ/2)^{2} x_{1} = 0, |τ_{2}| = 1,  z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{1+τ_{1}}{2}  τ_{2}z̄_{2}, ϵ_{2} = \frac{1+τ_{2}}{2} 2K
(ℤ/2)^{3}  or  or −z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1+τ_{1}}{2}  if
x_{2} = −\frac{1}{2}  |τ_{2}| = 1

z̄_{2}, ϵ_{2} = \frac{1}{2} if x_{2} = −\frac{1}{2},

(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{τ_{1}}{2}  z̄_{2} + \frac{τ_{2}}{2} , ϵ_{2} = \frac{1}{2} T
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{1+τ_{1}}{2}  z̄_{2} + \frac{τ_{2}}{2} , ϵ_{2} = \frac{1}{2} T
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1+τ_{1}}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{1+τ_{1}}{2} , ϵ_{1} = \frac{1}{2} z̄_{2}, ϵ_{2} = \frac{τ_{2}}{2} T
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{1+τ_{1}}{2} , ϵ_{1} = \frac{τ_{1}}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{1+τ_{1}}{2} , ϵ_{1} = \frac{1}{2}  z̄_{2}, ϵ_{2} = \frac{1+τ_{2}}{2} T
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{1+τ_{1}}{2} , ϵ_{1} = \frac{τ_{1}}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{1+τ_{1}}{2} , ϵ_{1} = \frac{τ_{1}}{2}  z̄_{2}, ϵ_{2} = \frac{1+τ_{2}}{2} T
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{1+τ_{1}}{2} , ϵ_{1} = \frac{1}{2}

(ℤ/2)^{2} x_{1} = 0 x_{2} = 0  z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{τ_{1}}{2}  z̄_{2} + \frac{1}{2}, ϵ_{2} = \frac{τ_{2}}{2} ∅
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{1+τ_{1}}{2} z̄_{2} + \frac{1}{2}, ϵ_{2} = \frac{τ_{2}}{2} ∅
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1+τ_{1}}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{1+τ_{1}}{2} , ϵ_{1} = \frac{1}{2} z̄_{2} + \frac{1}{2}, ϵ_{2} = \frac{τ_{2}}{2} ∅
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{1+τ_{1}}{2} , ϵ_{1} = \frac{τ_{1}}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1} + \frac{1}{2}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1+τ_{1}}{2}  z̄_{2}, ϵ_{2} = \frac{1+τ_{2}}{2} ∅
(ℤ/2)^{3}  or
−z̄_{1} + \frac{τ_{1}}{2} , η_{1} = \frac{1}{2}, ϵ_{1} = \frac{1+τ_{1}}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1} + \frac{1}{2}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2} z̄_{2}, ϵ_{2} = \frac{1+τ_{2}}{2} ∅
(ℤ/2)^{3}  or
−z̄_{1} + \frac{τ_{1}}{2} , η_{1} = \frac{1}{2}, ϵ_{1} = \frac{τ_{1}}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2} z̄_{2}, ϵ_{2} = \frac{τ_{2}}{2} 3T
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{τ_{1}}{2}

(ℤ/2)^{2} x_{1} = 0 x_{2} = 0  z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2}  z̄_{2}, ϵ_{2} = \frac{1+τ_{2}}{2} 2T
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{τ_{1}}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2} z̄_{2} + \frac{τ_{2}}{2} , ϵ_{2} = \frac{1}{2} 2T
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{τ_{1}}{2}
(ℤ/2)^{2} x_{1} = 0, |τ_{2}| = 1, z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2} τ_{2}z̄_{2}, ϵ_{2} = \frac{1+τ_{2}}{2} 2T
(ℤ/2)^{3} or or if
x_{2} = −\frac{1}{2} −z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{τ_{1}}{2}  |τ_{2}| = 1

z̄_{2}, ϵ_{2} = \frac{1}{2} if x_{2} = −\frac{1}{2},

(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1+τ_{1}}{2}  z̄_{2}, ϵ_{2} = \frac{1+τ_{2}}{2} 2T
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{1+τ_{1}}{2}

(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{τ_{1}}{2}  z̄_{2}, ϵ_{2} = \frac{τ_{2}}{2} 2K ⊔ T
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2}
(ℤ/2)^{2} x_{1} = 0 x_{2} = 0 z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{1+τ_{1}}{2}  z̄_{2}, ϵ_{2} = \frac{1}{2} 2K ⊔ T
(ℤ/2)^{3}  or
−z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1+τ_{1}}{2}

G, Ĝ  τ_{1}  τ_{2}  σ_{1}(z_{1}), η_{1}, ϵ_{1}  σ_{2}(z_{2}), ϵ_{2} S(ℝ)
(ℤ/2)^{2} |τ_{1}| = 1, x_{2} = 0 τ_{1}z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{1+τ_{1}}{2}  z̄_{2} + \frac{τ_{2}}{4} ,  2T
D_{4}  or  or −τ_{1}z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1+τ_{1}}{2} ϵ_{2} = \frac{τ_{2}}{2} or
x_{1} = −\frac{1}{2}  if |τ_{1}| = 1,  −z̄_{2} − \frac{1}{4}
z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2}  ϵ_{2} = \frac{1}{2}
or −z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2}
if x_{1} = −\frac{1}{2}
(ℤ/2)^{2} |τ_{1}| = 1, x_{2} = 0 τ_{1}z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{1+τ_{1}}{2}  τ_{2}z̄_{2} + \frac{1−τ_{2}}{4} , 2T
D_{4}  or  or −τ_{1}z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1+τ_{1}}{2} ϵ_{2} = \frac{1+τ_{2}}{2} or
x_{1} = −\frac{1}{2}  if |τ_{1}| = 1,  −τ_{2}z̄_{2} + \frac{1+τ_{2}}{4}
z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2}  ϵ_{2} = \frac{1+τ_{2}}{2}
or −z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2} or z̄_{2} + \frac{1}{4} + \frac{τ_{2}}{2}
if x_{1} = −\frac{1}{2}  ϵ_{2} = \frac{1}{2}
or −z̄_{2} − \frac{1}{4}
or ϵ_{2} = \frac{1}{2}
if x_{2} = −\frac{1}{2}
(ℤ/2)^{2} |τ_{1}| = 1, x_{2} = 0 τ_{1}z̄_{1}, η_{1} = \frac{1}{2}, ϵ_{1} = \frac{1+τ_{1}}{2}  z̄_{2} + \frac{1}{2} + \frac{τ_{2}}{4} ,  ∅
D_{4}  or  or −τ_{1}z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1+τ_{1}}{2} ϵ_{2} = \frac{τ_{2}}{2} or
x_{1} = −\frac{1}{2}  if |τ_{1}| = 1,  −z̄_{2} − \frac{1}{4} + \frac{τ_{2}}{2}
z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2}  ϵ_{2} = \frac{1}{2}
or −z̄_{1}, η_{1} = \frac{τ_{1}}{2} , ϵ_{1} = \frac{1}{2}
if x_{1} = −\frac{1}{2}

Let us now explain how to distinguish the different cases listed in the tables above. Assume first of all Ĝ = ℤ/2 × ℤ/2 × ℤ/2.

S(ℝ) = 2K: the first case is distinguished from the second and the fourth since in the first case ϵ_{2} = \frac{1+τ_{2}}{2} , therefore any lifting of t to a vector t′ ∈ Ω′ cannot be an eigenvector for the action of σ_{2}, while in the second and the fourth case ϵ_{2} = \frac{1}{2}, respectively \frac{τ_{2}}{2} , which are eigenvectors for the action of σ_{2}.

The first case is distinguished from the third and the sixth by the parity of ν(σ_{2}). In fact ν(σ_{2}) = 2 in the first case, while in the third and the sixth case we have ν(σ_{2}) = 1.

The first case is distinguished from the fifth since in the first case ϵ_{1} is an eigenvector for the action of σ_{1} on Λ′, while in the fifth case ϵ_{1} = \frac{1+τ_{1}}{2} , which is not an eigenvector for the action of σ_{1}.

The second case is distinguished from the third and the sixth by the parity of ν(σ_{2}). In fact ν(σ_{2}) = 2 in the second case, while in the third and the sixth case we have ν(σ_{2}) = 1.

The second case is distinguished from the fourth since in the second case ϵ_{1} is an eigenvector for the action of σ_{1} on Λ′, while in the fourth case ϵ_{1} = \frac{1+τ_{1}}{2} , which is not an eigenvector for the action of σ_{1}.

The second case is distinguished from the fifth since in the second case ϵ_{2} is an eigenvector for the action of σ_{2} on Γ, while in the fifth case ϵ_{2} = \frac{1+τ_{2}}{2} , which is not an eigenvector for the action of σ_{2}.

The third case is distinguished from the fourth and the fifth by the parity of ν(σ_{2}). In fact ν(σ_{2}) = 1 in the third case, while in the other two cases we have ν(σ_{2}) = 2.

The third case is distinguished from the sixth since in the third case ϵ_{1} is an eigenvector for the action of σ_{1} on Λ′, while in the sixth case ϵ_{1} = \frac{1+τ_{1}}{2} , which is not an eigenvector for the action of σ_{1}

The fourth and the fifth cases are distinguished from the sixth by the parity of ν(σ_{2}). In fact ν(σ_{2}) = 1 in the sixth case, while in the other two cases we have ν(σ_{2}) = 2.

The fourth case is distinguished from the fifth since in the fourth case ϵ_{2} is an eigenvector for the action of σ_{2} on Γ, while in the fifth case ϵ_{2} = \frac{1+τ_{2}}{2} , which is not an eigenvector for the action of σ_{2}.

S(ℝ) = T: the first two cases are distinguished from the the last three cases by the invariant 1)(b): in the first two cases the set of values of ν(σ_{2}) equals {2, 0}, in the last three cases it equals {2, 2}.

The first case is distinguished from the second since in the first case ϵ_{1} is an eigenvector for the action of σ_{1} on Λ′, while in the second case ϵ_{1} = \frac{1+τ_{1}}{2} , which is not an eigenvector for the action of σ_{1}.

The third case is distinguished from the fourth and the fifth since in the third case ϵ_{2} is an eigenvector for the action of σ_{2} on Γ, while in the other two cases ϵ_{2} = \frac{1+τ_{2}}{2} , which is not an eigenvector for the action of σ_{2}.

The fourth case is distinguished from the fifth since in the fourth case ϵ_{1} is the +1 eigenvector for the action of σ_{1} on Λ′, while in the fifth case ϵ_{1} is the −1 eigenvector for the action of σ_{1}.

S(ℝ) = ∅: the first three cases are distinguished from the the last two cases by the invariant 1)(b): in the first three cases the set of values of ν(σ_{2}) equals {0, 2}, in the last two cases it equals {2, 2}.

The second case is distinguished from the first and the third since in the first and the third case ϵ_{1} is an eigenvector for the action of σ_{1} on Λ′, while in the second case ϵ_{1} = \frac{1+τ_{1}}{2} , which is not an eigenvector for the action of σ_{1}.

The first case is distinguished from the third since in the first case ϵ_{1} is the −1 eigenvector for the action of σ_{1} on Λ′, while in the third case ϵ_{1} is the +1 eigenvector for the action of σ_{1}.

The fourth case is distinguished from the fifth since in the fifth case ϵ_{1} is an eigenvector for the action of σ_{1} on Λ′, while in the fourth case ϵ_{1} = \frac{1+τ_{1}}{2} , which is not an eigenvector for the action of σ_{1}.

S(ℝ) = 2T: the third case is distinguished from all the others by the parity of ν(σ_{2}) = 1.

The second case is distinguished from all the others by the invariant 1)(b). In fact the set of values of ν(σ_{2}) is {2, 0} in the second case, while in the other cases it is either {2, 2}, or {1, 1} (only in the third case).

The first case is distinguished from the last one since in the first case ϵ_{1} is an eigenvector for the action of σ_{1} on Λ′, while in the last case ϵ_{1} = \frac{1+τ_{1}}{2} , which is not an eigenvector for the action of σ_{1}.

S(ℝ) = 2K ⊔ T: the first case is distinguished from the second one since in the first case ϵ_{1} is an eigenvector for the action of σ_{1} on Λ′, while in the second case ϵ_{1} = \frac{1+τ_{1}}{2} , which is not an eigenvector for the action of σ_{1}.

If Ĝ = D_{4}, the cases with S(ℝ) = 2T are distinguished by the parity of ν(σ_{2}), which is equal to 2 in the first case, and equal to 1 in the second case.

We finally give the table for the non split case, where G = ℤ/2 × ℤ/2 is generated by elements g, t such that g(z_{1}, z_{2}) = (z_{1} + η_{1}, −z_{2}), g_{2}(z_{1}, z_{2}) = (z_{1} + ϵ_{1}, z_{2} + ϵ_{2}). Recall the by now standard notation τ_{j} = x_{j} + iy_{j}.

G, Ĝ  τ_{1}  τ_{2}  σ̃_{1}, η_{1}, ϵ_{1}  σ̃_{2}, ϵ_{2} S(ℝ)
(ℤ/2)^{2} x_{1} = 0, |τ_{2}| = 1 z̄_{1} + \frac{1}{4},  τ_{2}z̄_{2} + \frac{1}{2},  ∅
ℤ/4 × ℤ/2  or  ϵ_{1} = \frac{1}{2}  ϵ_{2} = \frac{1+τ_{2}}{2}
x_{2} = −\frac{1}{2} η_{1} = \frac{1+τ_{1}}{2} if |τ_{2}| = 1
or −z̄_{1} + \frac{τ}{4}  z̄_{2} + \frac{τ}{2}
ϵ_{1} = \frac{τ_{1}}{2}  ϵ_{2} = \frac{1}{2}
η_{1} = \frac{1+τ_{1}}{2} if x_{2} = −\frac{1}{2}

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Address of the authors:

Fabrizio Catanese: Mathematisches Institut der Georg - August - Universität Göttingen Bunsentraße 3-5 D-37073 Göttingen e-mail: catanese@uni-math.gwdg.de

Paola Frediani: Dipartimento di Matematica dell’Università di Pisa via Buonarroti 2 I-56127 Pisa e-mail: frediani@dm.unipi.it