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On the Computation of Class Polynomials with "Thetanullwerte" and Its Applications to the Unit Group Computation Leprévost, Franck ; ; in Experimental Mathematics (2011), 20(3), 271-281 Detailed reference viewed: 122 (0 UL)Jacobians of genus 2 curves with a rational point of order 11 Leprévost, Franck ; Bernard, Nicolas ; in Experimental Mathematics (2009), 18(1), 65-70 On the one hand, it is well-known that Jacobians of (hyper)elliptic curves defined over $\Q$ having a rational point of order $l$ can be used in many applications, for instance in the construction of ... [more ▼] On the one hand, it is well-known that Jacobians of (hyper)elliptic curves defined over $\Q$ having a rational point of order $l$ can be used in many applications, for instance in the construction of class groups of quadratic fields with a non-trivial $l$-rank. On the other hand, it is also well-known that $11$ is the least prime number which is not the order of a rational point of an elliptic curve defined over $\Q$. It is therefore interesting to look for curves of higher genus, whose Jacobians have a rational point of order $11$. This problem has already been addressed, and Flynn found such a family $\Fl_t$ of genus $2$ curves. Now, it turns out, that the Jacobian $J_0(23)$ of the modular genus $2$ curve $X_0(23)$ has the required property, but does not belong to $\Fl_t$. The study of $X_0(23)$ leads to a method to partially solving the considered problem. Our approach allows us to recover $X_0(23)$, and to construct another $18$ distinct explicit curves of genus $2$ defined over $\Q$ and whose Jacobians have a rational point of order $11$. Of these $19$ curves, $10$ do not have any rational Weierstrass point, and $9$ have a rational Weierstrass point. None of these curves are $\Qb$-isomorphic to each other, nor $\Qb$-isomorphic to an element of Flynn's family $\Fl_t$. Finally, the Jacobians of these new curves are absolutely simple. [less ▲] Detailed reference viewed: 206 (13 UL)Units in some parametric families of quartic fields Leprévost, Franck ; ; in Acta Arithmetica (2007), 127(3), 205-216 Detailed reference viewed: 117 (0 UL)Rational torsion of J_0(N) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5, 7 or 10 Leprévost, Franck ; ; in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (2004), 74 Detailed reference viewed: 138 (0 UL)Familles de polynômes liées aux courbes modulaires X_1(l) unicursales et points rationnels non-triviaux de courbes elliptiques quotient Leprévost, Franck ; ; in Acta Arithmetica (2003), 110(4), 401-410 Detailed reference viewed: 96 (0 UL) |
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