Rational points and the elliptic Chabauty method. (Q1424572)

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 \textit{E. V. Flynn} and \textit{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.