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Finding rational points on bielliptic genus 2 curves. (English) Zbl 1029.11024
Summary: The authors discuss a technique for trying to find all rational points on curves of the form $$Y^2= f_3X^6+ f_2X^4+ f_1X^2+ f_0$$, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty’s theorem may be applied. However, they concentrate on the situation when the rank is at least 2. In this case, they derive an associated family of elliptic curves, defined over a number field $$\mathbb{Q}(\alpha)$$. If each of these elliptic curves has rank less than the degree of $$\mathbb{Q}(\alpha): \mathbb{Q}$$, then they describe a Chabauty-like technique which may be applied to try to find all the points $$(x,y)$$ defined over $$\mathbb{Q}(\alpha)$$ on the elliptic curves, for which $$x\in \mathbb{Q}$$. This in turn allows them to find all $$\mathbb{Q}$$-rational points on the original genus 2 curve. They apply this to give a solution to a problem of Diophantus (where the sextic in $$X$$ is irreducible over $$\mathbb{Q}$$), which simplifies the recent solution given in the second author’s Ph.D. thesis, Univ. California, Berkeley (1997). The authors also present two examples where the sextic in $$X$$ is reducible over $$\mathbb{Q}$$.

##### MSC:
 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 14G05 Rational points
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