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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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