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Jacobians of genus-2 curves with a rational point of order 11. (English) Zbl 1244.11064

Summary: On the one hand, it is well known that Jacobians of (hyper)elliptic curves defined over \(\mathbb Q\) 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 nontrivial \(l\)-rank. On the other hand, it is also well known that 11 is the least prime number that is not the order of a rational point of an elliptic curve defined over \(\mathbb Q\). 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 E. V. Flynn [J. Number Theory 36, No. 3, 257–265 (1990; Zbl 0757.14025)] found such a family \(\mathfrak F_t\) 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 \(\mathfrak F_t\). The study of \(X_0(23)\) leads to a method giving a partial solution of 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 \(\mathbb Q\) 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 \(\overline {\mathbb Q}\)-isomorphic to each other, nor \(\overline {\mathbb Q}\)-isomorphic to an element of Flynn’s family \(\mathfrak F_t\). Finally, the Jacobians of these new curves are absolutely simple.

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G10 Abelian varieties of dimension \(> 1\)
14G25 Global ground fields in algebraic geometry
14H40 Jacobians, Prym varieties

Citations:

Zbl 0757.14025
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