Hyperelliptic curves on (1, 4)-polarised abelian surfaces

Borówka, Paweł; Ortega, Ángela

doi:10.1007/s00209-018-2174-2

Cited by 5 publications

(36 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Assume that the group G = η, ξ is non-isotropic. The following theorem summarises the results from [4] in the non-isotropic case.…”

Section: Klein Coverings: Non-isotropic Casementioning

confidence: 89%

“…Remark 3.4. As pointed out in [4], the conditions of Theorem 3.1 carry natural 'dualisations'. In the first condition one can take the dual surface.…”

Section: Klein Coverings: Non-isotropic Casementioning

confidence: 95%

“…Putting the two theorems together we get the main result of the paper as follows. [4,7] that every 2-torsion point on JH can be written uniquely as a sum of an even number of at most g Weierstrass points, so the Weil paring of η 1 , η 2 ∈ JH [2] can be computed as the parity of the number of common Weierstrass points in the presentations of η 1 , η 2 .…”

Section: Introductionmentioning

confidence: 99%

“…A Klein covering of a curve C is a 4:1étale covering f : C → C with monodromy group isomorphic to the Klein four-group V 4 = Z 2 × Z 2 . In particular, it is defined by a Klein four-subgroup G of the set of 2-torsion points on the Jacobian JC, denoted by JC[2].In [4] we have shown that there are two disjoint cases of Klein coverings that depends on the values of the Weil pairing restricted to G.…”

mentioning

confidence: 99%

“…In [4] we have shown that there are two disjoint cases of Klein coverings that depends on the values of the Weil pairing restricted to G.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Klein coverings of genus 2 curves

Borówka

Ortega

2019

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

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 ...

show abstract

“…Assume that the group G = η, ξ is non-isotropic. The following theorem summarises the results from [4] in the non-isotropic case.…”

Section: Klein Coverings: Non-isotropic Casementioning

confidence: 89%

“…Remark 3.4. As pointed out in [4], the conditions of Theorem 3.1 carry natural 'dualisations'. In the first condition one can take the dual surface.…”

Section: Klein Coverings: Non-isotropic Casementioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…In [4] we have shown that there are two disjoint cases of Klein coverings that depends on the values of the Weil pairing restricted to G.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Klein coverings of genus 2 curves

Borówka

Ortega

2019

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Hyperelliptic genus 3 curves with involutions and a Prym map

Borówka,

Shatsila

2024

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

We characterize genus 3 complex smooth hyperelliptic curves that admit two additional involutions as curves that can be built from five points in with a distinguished triple. We are able to write down explicit equations for the curves and all their quotient curves. We show that, fixing one of the elliptic quotient curve, the Prym map becomes a 2:1 map and therefore the hyperelliptic Klein Prym map, constructed recently by the first author with A. Ortega, is also 2:1 in this case. As a by‐product we show an explicit family of polarized abelian surfaces (for ), such that any surface in the family satisfying a certain explicit condition is abstractly non‐isomorphic to its dual abelian surface.

show abstract

Non-simple polarised abelian surfaces and genus 3 curves with completely decomposable Jacobians

Auffarth,

Borówka

2024

Annali di Matematica

View full text Add to dashboard Cite

Hyperelliptic curves on (1, 4)-polarised abelian surfaces

Cited by 5 publications

References 14 publications

Klein coverings of genus 2 curves

Klein coverings of genus 2 curves

Hyperelliptic genus 3 curves with involutions and a Prym map

Non-simple polarised abelian surfaces and genus 3 curves with completely decomposable Jacobians

Contact Info

Product

Resources

About