Jacobians of Genus-2 Curves with a Rational Point of Order 11

Abstract

On the one hand, it is well known that Jacobians of (hyper)elliptic curves defined over ℚ having a rational point of orderI can be used in many applications, for instance in the constructionof class groups of quadratic fields with a nontrivial l-rank.On the other hand, it is also well known that 11 is the leastprime number that is not the order of a rational point of an ellipticcurve defined over ℚ. It is therefore interesting to look forcurves of higher genus whose Jacobians have a rational point oforder 11. This problem has already been addressed, and Flynnfound such a family 𝔉 t of genus-2 curves. Now it turns out thatthe Jacobian J 0(23) of the modular genus-2 curve X 0(23) hasthe required property, but does not belong to 𝔉 t . The study ofX 0(23) leads to a method giving a partial solution of the consideredproblem. Our approach allows us to recover X 0(23) and toconstruct another 18 distinct explicit curves of genus 2 definedover ℚ whose Jacobians have a rational point of order 11. Ofthese 19 curves, 10 do not have any rational Weierstrass point,and 9 have a rational Weierstrass point. None of these curvesare ℚ̄-isomorphic to each other, nor ℚ̄-isomorphic to an elementof Flynn's family 𝔉 t . Finally, the Jacobians of these new curvesare absolutely simple.