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Some special curves of genus 5. (English) Zbl 0870.11017

Let \(E\) be the irreducible curve in the projective space \(\mathbb{P}^4\) given by the equations: \[ Q_1(\ell,m)=r^2, \qquad Q_2(\ell,m)=s^2, \qquad Q_3(\ell,m)=t^2, \] where \(Q_i(\ell,m)\) \((i=1,2,3)\) are three non-proportional non-singular diagonal quadratic forms with rational coefficients. If the curve \(E\) does not possess an effective rational divisor of degree 5, then \(E\) has no points defined over an algebraic number field of odd degree. In this paper the author studies the problem of how to determine whether or not \(E\) can possess an effective rational divisor of degree 5. Applying his results on the curve \(C\) defined by the equations \[ 4\ell^2-11m^2=r^2, \qquad 17\ell^2+m^2=s^2, \qquad \ell^2+m^2=t^2, \] it follows that \(C\) has no points in any algebraic number field of odd degree.

MSC:

11D25 Cubic and quartic Diophantine equations
14G05 Rational points
11G05 Elliptic curves over global fields
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14G25 Global ground fields in algebraic geometry
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