Elliptic subcovers of a curve of genus 2. II: The refined Humbert invariant
type: Item
instance of: scholarly article
title: Elliptic subcovers of a curve of genus 2. II: The refined Humbert invariant
zbMATH Open document ID: 1451.14065
author: Ernst Kani
published in: Journal of Number Theory
publication date: 2018-08-29T00:00:00Z
review text: Let \(C/K\) be a curve of genus 2 over an arbitrary field \(K\). The first part of this two-part paper [\textit{E. Kani}, Ann. Math. Qué. 43, No. 2, 281--303 (2019; Zbl 1435.14029)] established a bijection between the set of equivalence classes of the elliptic subcovers of \(C/K\) and the set of certain primitive representations of an intrinsic quadratic form \(q_C\) called the \textit{refined Humbert invariant}. This second part explains how to compute the refined Humbert invariant explicitly from an ``\(N\)-presentation'' of the Jacobian variety \(J\) of \(C\) as \(J\cong (E\times E')/\mathrm{Graph}(-\psi)\), where \((E,E',\psi)\) is a certain triple with \(E, E'\) isogenious elliptic curves and \(\psi:E[N]\overset{\sim}{\rightarrow} E'[N]\) (\(1<N\in\mathbb{Z}\)). The domain lattice of \(q_C\) is the saturation \(NS'(J)\) of the Neron-Severi group \(NS(J)\) in \(NS(J_{\bar{K}})\) modulo the theta line whose rank is \(1+\mathrm{rank}_\mathbb{Z}(\mathrm{Hom}_K(E,E'))\). The main ingredient (Theorem 25) explicitly computes the coefficients \(a,b,c\) of a binary quadratic form \(q_C(X\bar D_1+Y\bar D_2)=aX^2+bXY+cY^2\) for the images \(\bar D_1\), \(\bar D_2\) of certain divisors \(D_1,D_2\in NS'(J)\) given with relevant numerical invariants. In the final paragraph of the paper is illustrated a very impressive and representative example for a Legendre curve \(C_K:y^2=x(x^2-3)(x^2-\frac13)\) over \(K\): If \(K\not\supset\mathbb{Q}(\sqrt{-1})\) then \(q_{C_K}\sim 4X^2\); if \(\mathbb{Q}(\sqrt{-1})\subset K \not\supset\mathbb{Q}(\sqrt{-3})\) then \(q_{C_K}\sim 4X^2+4Y^2\); and if \(\mathbb{Q}(\sqrt{-1},\sqrt{-3})\subset K\) then \(q_{C_K}\sim 4X^2+4Y^2+4YZ+4Z^2\).
reviewed by: Hiroaki Nakamura
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 6929040
zbMATH Keywords: curve covers
zbMATH Keywords: elliptic curves
zbMATH Keywords: Jacobian varieties
zbMATH Keywords: abelian surfaces
zbMATH Keywords: Neron-Severi group
zbMATH Keywords: binary quadratic forms
zbMATH Keywords: Humbert invariant
zbMATH Keywords: Legendre curves
MaRDI profile type: Publication
full work available at URL: j.jnt.2018.05.011
OpenAlex ID: W2809392040
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DOI: 10.1016/J.JNT.2018.05.011
description: scientific article; zbMATH DE number 6929040