Jacobians isomorphic to a product of two elliptic curves and ternary quadratic forms
type: Item
instance of: scholarly article
title: Jacobians isomorphic to a product of two elliptic curves and ternary quadratic forms
zbMATH Open document ID: 1356.11030
author: Ernst Kani
published in: Journal of Number Theory
publication date: 2014-06-25T00:00:00Z
review text: The author addresses the problem raised in [\textit{T. Hayshida} and \textit{M. Nishi}, J. Math. Soc. Japan 17, 1--16 (1965; Zbl 0132.41701)]: Whether or not the product \(E_1\times E_2\) of two elliptic curves over an algebraically closed field can be the Jacobian of a smooth irreducible curve \(C\) of genus 2 (this is equivalent to the question of whether or not there is such a curve \(C\) on \(E_1\times E_2\)). The author translates this problem into a classification problem about quadratic and ternary forms and answers the above question in the case that \(E_1\) and \(E_2\) are isogenous elliptic curves with complex multiplication. Namely, let \(q_{E_1, E_2}\) be the quadratic form on \(\text{Hom}(E_1, E_2)\) defined by \(q_{E_1, E_2}(f):=\text{deg}(f)\). It is proved in the paper under review that there is no genus 2 curve on \(E_1\times E_1\) if and only if \(q_{E_1, E_2}\) is equivalent to one of the 15 forms \(ax^2+bxy+cy^2\) whose coefficients \((a, b, c)\) are in the following list: \[ \{k(1,1,1): k=1,2,4,6,10\}\cup\{k(1,0,1) : k=1,2, 6\}\cup \{(1, 1, 2), (1, 1, 4)\} \] \[ \cup\{2(1,1,c): c=3, 9\}\cup \{2(1,0,c): c=2,5\}\cup \{2(2,0,3)\}. \]
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 6308466
zbMATH Keywords: elliptic curves with complex multiplication
zbMATH Keywords: curves of genus 2
zbMATH Keywords: Jacobians
zbMATH Keywords: product surfaces
zbMATH Keywords: abelian variety
zbMATH Keywords: Humbert invariant
zbMATH Keywords: binary and ternary quadratic forms
zbMATH Keywords: idoneal numbers
zbMATH Keywords: mass formula
reviewed by: Vladimir L. Popov
MaRDI profile type: Publication
full work available at URL: j.jnt.2013.12.006
OpenAlex ID: W1977001616
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DOI: 10.1016/J.JNT.2013.12.006
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description: scientific article; zbMATH DE number 6308466