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The arithmetic of certain quartic curves. (English) Zbl 0589.14029
Let \(F(X,Y,Z)\) be a non-singular quadratic form with rational coefficients, and let the curves \(\mathcal E\) and \(\mathcal D_ j\) \((1\leq j\leq 3)\) be defined by \(F(x^ 2,y^ 2,z^ 2)=0\), \(F(X,y^ 2,z^ 2)=0\) etc. then the \(\mathcal D_ j\) are of genus 1, whereas \(\mathcal E\) has genus 3, and the Jacobian of \(\mathcal E\) is isogenous to the product of the Jacobians of the \(\mathcal D_ j\). Using this correspondence, a procedure is described for deciding whether there is a point on \(\mathcal E\) defined over some algebraic number field of odd degree. In sections 3–5, some very interesting examples are discussed in detail. We only mention theorem 4.1 which presents a certain \(F\) that satisfies: (i) \(\mathcal E\) has rational points everywhere locally; (ii) \(\mathcal E\) has no point defined over an algebraic number field of odd degree; (iii) for \(1\leq j\leq 3\), \(\mathcal D_ j\) has infinitely many rational points; i.e. \(\mathcal E\) is far from satisfying the Hasse principle. In order to work out this example, numerous details only briefly mentioned in the article had to be checked.

MSC:
14H45 Special algebraic curves and curves of low genus
11D25 Cubic and quartic Diophantine equations
14G05 Rational points
14G25 Global ground fields in algebraic geometry
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References:
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