Elliptic subfields and automorphisms of genus 2 function fields

Tony Shaska; Helmut Voelklein

Home/Archive/arXiv

arXiv · math/0107142

Elliptic subfields and automorphisms of genus 2 function fields

Tony Shaska·Helmut Voelklein

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study genus 2 function fields with elliptic subfields of degree 2. The locus $Ł_2$ of these fields is a 2-dimensional subvariety of the moduli space $\mathcal M_2$ of genus 2 fields. An equation for $Ł_2$ is already in the work of Clebsch and Bolza. We use a birational parameterization of $Ł_2$ by affine 2-space to study the relation between the j-invariants of the degree 2 elliptic subfields. This extends work of Geyer, Gaudry, Stichtenoth and others. We find a 1-dimensional family of genus 2 curves having exactly two isomorphic elliptic subfields of degree 2; this family is parameterized by the j-invariant of these subfields.

Explore related subjects

math.AG math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Tony Shaska, Helmut Voelklein. 2001-07-19. Elliptic subfields and automorphisms of genus 2 function fields. https://arxiv.org/abs/math/0107142

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Good minimal models with nowhere vanishing holomorphic $1$-forms

Popa and Schnell show that any holomorphic 1-form on a smooth projective variety of general type has zeros. In this article, we show that a smooth good minimal model has a holomorphic 1-form without zero if and only if it admits an analytic fiber bundle structure over a positive dimensional abelian variety.

math.AG↗

arXiv2026

The automorphism groups of Kummer surfaces in characteristic two and their complex analogues

We calculate the automorphism group of the Kummer surface associated with a curve of genus 2 or the product of two elliptic curves in characteristic two under the assumption that the Kummer surface is a $K3$ surface. Moreover we discuss the complex $K3$ surfaces with the same Picard lattice as these Kummer surfaces. The paper has two appendices.

math.AG↗

arXiv2026

Algebraic Zhou valuations

In this paper, we generalize Zhou valuations, originally defined on complex domains, to the framework of general schemes. We demonstrate that an algebraic version of the Jonsson--Mustaţă conjecture is equivalent to the statement that every Zhou valuation is quasi-monomial. By introducing a mixed version of jumping numbers and Tian functions associated with valuations, we obtain characterizations of a valuation being a Zhou valuation or computing some jumping number using the Tian functions. Furthermore, we establish the correspondence between Zhou valuations in algebraic settings and their counterparts in analytic settings.

math.AG↗