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The arithmetic of hyperelliptic curves. (English) Zbl 0874.14012

González-Vega, Laureano (ed.) et al., Algorithms in algebraic geometry and applications. Proceedings of the MEGA-94 conference, Santander, Spain, April 5-9, 1994. Basel: Birkhäuser. Prog. Math. 143, 165-175 (1996).
A fundamental problem with respect to the arithmetic of elliptic curves \(E\) over number fields \(K\) is that of determining the group of rational points or Mordell-Weil group \(E(K)\) of \(E\) over \(K\). The task is, more precisely, to determine the torsion group \(E_{\text{tors}} (K)\), the rank \(r\) and a set of independent generators of infinite order \(P_1,\dots,P_r\) of \(E(K)\). Exactly the same problem arises for the Jacobians \({\mathcal J}\) of hyperelliptic curves \(C\) over number fields \(K\). The method for solving this problem for elliptic curves carries over to hyperelliptic curves, but there are some obstacles in the hyperelliptic case. The author sketches the current level of progress concerning the arithmetic of hyperelliptic curves under the above constructive point of view. The report restricts to the field of rationals \(K=\mathbb{Q}\). Following Cassels, Flynn and Gordon-Grant, complete 2-descent or 2-descent via isogeny can be employed for special classes of hyperelliptic curves \(C\) to determine the finite factor group \({\mathcal J}(\mathbb{Q})/2{\mathcal J}(\mathbb{Q})\). As for elliptic curves, on tries to compute the Selmer group \(S\) and takes into account that the extent to which the group \(S\) yields \({\mathcal J}(\mathbb{Q})/2{\mathcal J}(\mathbb{Q})\) is measured by the Tate-Shafarevich group. Once \({\mathcal J}(\mathbb{Q})/2{\mathcal J}(\mathbb{Q})\) is known, a set of generators of \({\mathcal J}(\mathbb{Q})\) arises from a complete set of representatives in \({\mathcal J}(\mathbb{Q})\) for that factor group and a set of points of bounded height in \({\mathcal J}(\mathbb{Q})\) – both sets being finite. Heights can be defined on \({\mathcal J}(\mathbb{Q})\) via the embedding of \({\mathcal J}(\mathbb{Q})\) into a suitable projective space over \(\mathbb{Q}\) whose dimension depends on the genus \(g\) of \(C\).
By a method of Chabauty worked out by Coleman, one obtains rational points on the hyperelliptic curve \(C\) of genus \(g\) over \(\mathbb{Q}\) from its Jacobian \({\mathcal J}\) over \(\mathbb{Q}\) or at least derives a bound for the number of rational points on \(C\) over \(\mathbb{Q}\). This is possible provided that the Mordell-Weil group \({\mathcal J}(\mathbb{Q})\) has rank \(\leq g-1\).
Many interesting examples illustrate this short but well-written survey. It remains the task of extending the methods described to a larger class of hyperelliptic curves and to improve the efficiency of the corresponding algorithms, especially those for computing \({\mathcal J}(\mathbb{Q})/2{\mathcal J}(\mathbb{Q})\) and the ones for determining \(C(\mathbb{Q})\).
For the entire collection see [Zbl 0841.00016].

MSC:

14G05 Rational points
14H52 Elliptic curves
14Q05 Computational aspects of algebraic curves
14G25 Global ground fields in algebraic geometry
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