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Rational points and the elliptic Chabauty method. (Points rationnels et méthode de Chabauty elliptique.) (French) Zbl 1097.11014

The main result of this paper is the calculation of the rational points on a specific hyperelliptic curve \(Y^2 = F(X)\) over \(\mathbb{Q}\) of genus 4 and Mordell-Weil rank 4. The interest in this result is the extension of the method of Chabauty that it employs. The author factors \(F(X) = F_1(X)F_2(X)\) over a field \(k\) of degree 3, and reduces the problem of finding rational points the problem of finding \(k\)-rational points with rational \(X\)-coordinate on the elliptic curve \(Y^2 = F_1(X)\). She then applies the elliptic Chabauty method as developed by E. V. Flynn and J. L. Wetherell in [ Manuscr. Math. 100, No. 4, 519–533 (1999; Zbl 1029.11024)]. A key step in the method requires finding the common zeros of two \(p\)-adic power series in two variables, for which he author gives an explicit version of Sugatani’s Weierstrass Preparation Theorem. The paper also includes a survey of the elliptic Chabauty method and other strategies.

MSC:

11D41 Higher degree equations; Fermat’s equation
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11D45 Counting solutions of Diophantine equations
13J05 Power series rings
14G05 Rational points

Citations:

Zbl 1029.11024

Software:

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References:

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