Weighted, GIT and Néron--Tate heights on genus-two moduli and Jacobians
Abstract
We compare three heights attached to a curve of genus two: the weighted height of its Igusa--Clebsch invariants, the GIT height of a binary sextic defining it, and the Néron--Tate height on its Jacobian. For a morphism from an abelian variety to a weighted projective space we define a weighted Néron--Tate height and show that the weighted height differs from it by a linear form and a bounded function; the linear form vanishes when inversion is induced by an automorphism of the weighted projective space. For binary sextics we prove the explicit inequality hinv≤ hGIT + 1/2 log 302, with nonnegative local differences at the finite places that vanish at places of good reduction. We then construct a family of semistable binary sextics over E× E, where E:y2=x3−2, whose relative Jacobian carries a section of infinite order, and we show that the Jacobian over the base point is absolutely simple. Along every curve in the parameter space through the base point, the Néron--Tate height of the section is a positive multiple of the weighted moduli height up to an error of square-root size, and up to a bounded error when the curve is rational. As applications, we obtain explicit families of genus-two Jacobians of positive rank over Q, an infinite congruence lattice of fibres of positive height, and canonical-limit formulas along elliptic curves in E× E.
Related
- Factorization patterns and fields of definition of ℓ-torsion points on the Jacobians of genus 3 hyperelliptic curves
- Rational torsion on simple genus two Jacobians
- A complete curve of genus 5 in the moduli space of curves of genus 3 and related questions
- The moduli space of curves of genus 16 is uniruled
- The Weil-Petersson diameter of moduli space for large genus
BibTeX
@misc{salami2026weighted,
title = {{Weighted, GIT and Néron--Tate heights on genus-two moduli and Jacobians}},
author = {Sajad Salami},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13850},
eprint = {2610.13850},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13850},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weighted, GIT and Néron--Tate heights on genus-two moduli and Jacobians
AU - Sajad Salami
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13850
UR - https://arxiv.org/abs/2610.13850
AB - We compare three heights attached to a curve of genus two: the weighted height of its Igusa--Clebsch invariants, the GIT height of a binary sextic defining it, and the Néron--Tate height on its Jacobian. For a morphism from an abelian variety to a weighted projective space we define a weighted Néron--Tate height and show that the weighted height differs from it by a linear form and a bounded function; the linear form vanishes when inversion is induced by an automorphism of the weighted projective space. For binary sextics we prove the explicit inequality h_inv≤ h_GIT + 1/2 log 302, with nonnegative local differences at the finite places that vanish at places of good reduction. We then construct a family of semistable binary sextics over E× E, where E:y^2=x^3−2, whose relative Jacobian carries a section of infinite order, and we show that the Jacobian over the base point is absolutely simple. Along every curve in the parameter space through the base point, the Néron--Tate height of the section is a positive multiple of the weighted moduli height up to an error of square-root size, and up to a bounded error when the curve is rational. As applications, we obtain explicit families of genus-two Jacobians of positive rank over Q, an infinite congruence lattice of fibres of positive height, and canonical-limit formulas along elliptic curves in E× E.
ER -