Computing integral points on hyperelliptic curves using quadratic Chabauty. (English) Zbl 1376.11053
The paper under review deals with the problem of computing all integer solutions \((x, y)\) to the hyperelliptic equation \(y^2 = f(x)\), where \(f(x)\) is a separable polynomial of degree at least 3 with integer coefficients. R. F. Coleman’s interpretation [Duke Math. J. 52, 765–770 (1985; Zbl 0588.14015)] of the method of Chabauty, allowing one to determine the rational points on a curve whose Jacobian has Mordell-Weil rank less than its genus. Over the last decade, M. Kim [Invent. Math. 161, No. 3, 629–656 (2005; Zbl 1090.14006)] has initiated a program aimed at removing this restricting on rank, allowing the study of rational points on hyperbolic curves through the use of nonabelian geometric objects generalizing the role of the Jacobian in the Chabauty-Coleman method. In this frame, the authors gave a method which they call quadratic Chabauty [J. Reine Angew. Math. 720, 51–79 (2016; Zbl 1350.11067)] based on \(p\)-adic height pairings to \(p\)-adically approximate the set of integer solutions of equation \(y^2 = f(x)\) in the case when the Jacobian of the corresponding algebraic curve has Mordell-Weil rank equal to its genus.
In this paper, a full algorithm is presented for the computation in practice of all integer solutions of hyperelliptic equation \(y^2 = f(x)\). It combines the quadratic Chabauty method with the Mordell-Weil sieve. All the necessary computations are described and an analysis of the \(p\)-adic precision which must be maintained throughout the computation is provided. A few examples are given.
In this paper, a full algorithm is presented for the computation in practice of all integer solutions of hyperelliptic equation \(y^2 = f(x)\). It combines the quadratic Chabauty method with the Mordell-Weil sieve. All the necessary computations are described and an analysis of the \(p\)-adic precision which must be maintained throughout the computation is provided. A few examples are given.
Reviewer: Dimitros Poulakis (Thessaloniki)
MSC:
| 11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |
| 11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |
| 11Y50 | Computer solution of Diophantine equations |
| 14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |
Keywords:
hyperelliptic equations; integer solution; Chabauty-Coleman method; quadratic Chabauty method; Mordell-Weil sieveReferences:
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