On height functions on Jacobian surfaces
type: Item
instance of: scholarly article
title: On height functions on Jacobian surfaces
zbMATH Open document ID: 0926.14013
author: Kentaro Yoshitomi
published in: Manuscripta Mathematica
publication date: 1999-04-29T00:00:00Z
full work available at URL: http://hdl.handle.net/10466/14492
review text: Let \(K\) be an algebraic number field, \(C\) a hyperelliptic curve of genus \(2\) defined by \(y^{2}=f(x)\), where \[ f(x)=x^{5}+a_{1}x^{4}+ a_{2}x^{3}+a_{3}x^{2}+a_{4}x+a_{5}\in K[x], \] \(\mathcal B\) the set of finite Weierstrass points, \(J\) the Jacobian variety of \(C\), \(\Theta\) the theta divisor of \(J\). For a divisor \(D\) of degree \(0\) on \(C\), denote by \(\overline D\) its image in \(J=\text{ Pic}^{0}(C)\). For \(P\in C(\mathbb C)\), let \(u^{(P)}\in\mathbb C^2\) (which is defined modulo the period lattice \(\Lambda\)) be the hyperelliptic integral \[ \left(\int_{\infty}^{P}{dx\over 2y},\;\int_{\infty}^{P}{xdx\over 2y}\right). \] For \(z\in\mathbb C^{2}\), let \(\tilde z\) denote the image of \(z\) in \(J=\mathbb C^{2}/\Lambda\). The map \[ (P_{1},P_{2})\mapsto u^{(P_{1})}+u^{(P_{2})} \] gives a surjection from the symmetric \(2\)-product \(S^{2}(C)\) of \(C\) to \(J\). For two points \(P_{1}\) and \(P_{2}\) on \(C\) with coordinates \((x_{1},y_{1})\) and \((x_{2},y_{2})\), respectively, and for \(u=u^{(P_{1})}+u^{(P_{2})}\), define \[ \Phi(u)={\mathfrak p}_{111}(u)- {\mathfrak p}_{12}(u){\mathfrak p}_{122}(u)+ {\mathfrak p}_{22}(u){\mathfrak p}_{112}(u), \] where \[ {\mathfrak p}_{11}(u)={F(x_{1},x_{2})-2y_{1}y_{2}\over (x_{1}-x_{2})^{2}}, \qquad {\mathfrak p}_{12}(u)=-x_{1}x_{2}, \quad {\mathfrak p}_{22}(u)=x_{1}+x_{2}, \] \[ {\mathfrak p}_{111}(u)= 2{y_{2}\psi(x_{1},x_{2})- y_{1}\psi(x_{2},x_{1})\over (x_{1}-x_{2})^{3}}, \qquad {\mathfrak p}_{112}(u)= 2{x_{2}^{2}y_{1}- x_{1}^{2}y_{2}\over x_{1}-x_{2}}, \] \[ {\mathfrak p}_{122}(u)= -2{x_{2}y_{1}- x_{1}y_{2}\over x_{1}-x_{2}}, \qquad {\mathfrak p}_{222}(u)= 2{y_{1}- y_{2}\over x_{1}-x_{2}}, \] where \[ \psi(x_{1},x_{2})=x_{1}^{3}x_{2}( 3x_{1}+x_{2})+4a_{1}x_{1}^{3}x_{2} +a_{2}x_{1}^{2}(x_{1}+3x_{2})+2a_{3}x_{1} (x_{1}+x_{2})+ a_{4} (3x_{1}+x_{2})+ 4a_{5} \] and \[ F(x_{1},x_{2})=x_{1}^{2}x_{2}^{2}( x_{1}+x_{2})+2a_{1}x_{1}^{2}x_{2}^{2} +a_{2}x_{1}x_{2}(x_{1}+x_{2})+2a_{3} x_{1} x_{2}+ a_{4} (x_{1…
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 1160799
zbMATH Keywords: height functions
zbMATH Keywords: Jacobian surfaces
zbMATH Keywords: Jacobian variety of a hyperelliptic curve
zbMATH Keywords: Néron's local pairing
zbMATH Keywords: canonical local height at archimedean places
zbMATH Keywords: Birch and Swinnerton-Dyer conjecture
reviewed by: Michel Waldschmidt
MaRDI profile type: Publication
OpenAlex ID: W2057446284
DOI: 10.1007/S002290050053
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description: scientific article; zbMATH DE number 1160799