Quadratic Chabauty: \(p\)-adic heights and integral points on hyperelliptic curves (Q334476)

Let \(X/\mathbb{Q}\) be a hyperelliptic curve of genus \(g\) with Jacobian \(J\) of Mordell-Weil rank \(r\). If \(r< g\), the method of Coleman-Chabauty gives an effective method to determine \(X(\mathbb{Q})\).NEWLINENEWLINEThe authors consider the case \(r = g\). They prove that there is a Coleman function \(\rho: X(\mathbb{Q}_p) \to \mathbb{Q}_p\) on \(X \times_\mathbb{Q} \mathbb{Q}_p\) and a finite set of values \(T\) such that \(\rho(\mathcal{U}(\mathbb{Z}[1/p])) \subset T\) with \(\mathcal{U}(\mathbb{Z}[1/p])\) the set of \(p\)-integral solutions of an affine integral equation of \(X\). Further, they prove that \(T\) is effectively computable if \(X\) has good reduction at \(p\), and that \(\rho\) is effectively computable from a basis of \(J(\mathbb{Q}) \otimes_\mathbb{Z} \mathbb{Q}\), see Theorem 3.1.NEWLINENEWLINELet NEWLINE\[NEWLINEf_i(z) = \int_{\infty}^{z}\omega_i.NEWLINE\]NEWLINE Then there exists a non-trivial linear combination of the \(f_i\) vanishing on \(\mathcal{U}(\mathbb{Z}[1/p])\), or there exist constants \(\alpha_{ij} \in \mathbb{Q}_p\) such that the Coleman function \(\rho\) is defined using products of Coleman integrals, NEWLINE\[NEWLINE\rho(z) = \tau(z) - \sum_{0 \leq i \leq j< g}\alpha_{ij}f_i(z)f_j(z),NEWLINE\]NEWLINE with \(\tau\) defined in Theorem 2.2 as an iterated Coleman integral. For the computation of \(\rho\), one needs to compute iterated Coleman integrals, \(p\)-adic height pairings and a basis of \(J(\mathbb{Q}) \otimes_\mathbb{Z} \mathbb{Q}\). The computation of \(\tau\) is described in section 4, and the computation of \(T\) in section 5.NEWLINENEWLINEThe authors conclude with some examples illustrating the methods of their article.