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Familles de courbes de genre 2 munies d’une classe de diviseurs rationnels d’ordre 15, 17, 19 ou 21. (Families of curves of genus 2 with a rational divisor class of order 15, 17, 19 or 21). (French) Zbl 0758.14017

Summary: We construct here, for \(\ell=15\), 17, 19 or 21 a family with one parameter \(t\) of curves of genus 2 defined over \(\mathbb{Q}\), such that its jacobian has a point of order \(\ell\) rational over \(\mathbb{Q}(t)\). The same method allows to construct, for \(\ell=2g^ 2+2g+1\) or \(2g^ 2+3g+1\), a one parameter family of hyperelliptic curves of genus \(g\) over \(\mathbb{Q}\), such that its jacobian has a point of order \(\ell\) rational over \(\mathbb{Q}(t)\).

MSC:

14H40 Jacobians, Prym varieties
14H10 Families, moduli of curves (algebraic)
14G05 Rational points
14H52 Elliptic curves
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