Klein coverings of genus 2 curves

Abstract

We investigate the geometry ofétale 4 : 1 coverings of smooth complex genus 2 curves with the monodromy group isomorphic to the Klein four-group. There are two cases, isotropic and non-isotropic depending on the values of the Weil pairing restricted to the group defining the covering. We recall from our previous work [4] the results concerning the non-isotropic case and fully describe the isotropic case. We show that the necessary information to construct the Klein coverings is encoded in the 6 points on P 1 defining the genus 2 curve. The main result of the paper is the fact that, in both cases the Prym map associated to these coverings is injective. Additionally, we provide a concrete description of the closure of the image of the Prym map inside the corresponding moduli space of polarised abelian varieties.are both injective onto their images.The idea of the proof is that the Klein covering provides a tower of curves in which one can find elliptic curves that generate the Prym variety. Then we show that all the information is actually contained in the division of 6 points on P 1 into 3 pairs in the isotropic case and 2 triples in the non-isotropic case. This in particular shows that R iso 2 and R ni 2 are irreducible. The paper is structured as follows. In the Preliminaries we collect the necessary results on curves with involutions. An interesting result on its own is the characterisation of non-hyperelliptic curves of genus 3 admitting 2 commuting involutions (see Lemma 2.15) and their Jacobians, (see Proposition 2.16). Such curves will play crucial role in the isotropic Klein coverings.Section 3 is devoted to non-isotropic Klein coverings. It consists of a summary of results from [4] and Theorem 3.8 that states that the Prym map in this case is injective. Section 4 explains the construction of isotropic Klein coverings. Here, we characterise the curves that appear as quotient curves and the covering maps between them. The conclusion is Theorem 4.11 that states that the Prym map is injective also in this case. In Section 5 we give a precise description, in terms of period matrices, of the image of the Prym map in both cases.Acknowledgements. We would thank Jennifer Paulhus for showing us the LMF database which contains examples of families of curves with prescribed automorphism groups. We are grateful to Klaus Hulek for suggesting related questions to investigate, Section 5 is the outcome of one of these.In the paper we deal with elliptic curves, i.e. genus 1 curve with a chosen point. In this way, an elliptic curve is a one dimensional abelian variety. Thus, the involution −1 becomes a (hyper)elliptic involution and we define the Weierstrass points on an elliptic curve as the 2-torsion points, i.e. the fixed points of the involution.A double covering f : C → C can be defined from upstairs as the quotient of C by an involution σ so that C = C /σ, or from downstairs by fixing a branching divisor B on C and a line bundle L such that L ⊗2 = O C (B). In the paper we use both perspectives. If σ is the involution ...

Hyperelliptic genus 3 curves with involutions and a Prym map

“…In the paper, we consider hyperelliptic curves of genus 3 that admit two additional involutions. The motivation to study them came from the Klein covering construction that has been described in [7,8,10]. A covering is called Klein if it is Galois 4:1 covering with the deck group being isomorphic to the Klein four-group.…”

“…Such curves are classically interesting examples, since the decomposition allows us to understand their periods in terms of periods of curves of smaller genera, especially in terms of elliptic curves, see [13,16,17,21,22]. In particular, genus 3 curves with completely decomposable Jacobians have recently gained a lot of attention [8,14,15,20]. Therefore, in Proposition 4.3 we show explicit equations of all the curves appearing in the construction.…”