On height functions on Jacobian surfaces (Q1387918)

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 \[ F(x_{1},x_{2})=x_{1}^{2}x_{2}^{2}( x_{1}+x_{2})+2a_{1}x_{1}^{2}x_{2}^{2} +a_{2}x_{1}x_{2}(x_{1}+x_{2})+2a_{3} x_{1} x_{2}+ a_{4} (x_{1}+x_{2})+ 2a_{5}. \] Let \(v\) be an archimedean place of \(K\), \(\widehat{\lambda}_{v}\) the canonical local height on \(J-\Theta\) normalized by \(\widehat{\lambda}_{v}(2z)=4\widehat{\lambda}_{v}(z)+ v\bigl(\Phi(z)\bigr)\). Denote by \(\langle a,b\rangle_{v}\) the Néron local pairing explicitly given by \(\langle a,b\rangle_{v}=g_{a}(b)\) where \(g_{a}\) is Green's function attached to \(a\). Assume \(P_{1}\) and \(P_{2}\) on \(C(K)\) are such that \(b=P_{1}-P_{2}\) satisfies \(\overline b\not\in\Theta\). Let \(z_{b}=u^{(P_{1})}-u^{(P_{2})}\in\mathbb C^{2}\). A basis of the tangent space at \(P_{i}\) is \[ 2y_{i}{\partial\over\partial x}= f'(x_{i}){\partial\over\partial y}. \] A uniformizer at \(P_{i}\) is \[ {x-x_{i}\over 2y_{i}} \quad \text{if }P_{i}\not\in{\mathcal B} \quad\text{and } {y-y_{i}\over f'(x_{i})} \quad \text{if }P_{i}\in{\mathcal B}. \] The main result of this paper is that Néron's local pairing and the canonical height are related by \[ \langle b,b \rangle_{v}= 2\widehat{\lambda}_{v}(\widetilde z_{b}). \] This enables the author to compute numerically the canonical local height at archimedean places. In particular he checks numerically the Birch and Swinnerton-Dyer conjecture in some cases.