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2009 Jacobians of Genus-2 Curves with a Rational Point of Order 11
Nicolas Bernard,, Franck Leprévost, Michael Pohst
Experiment. Math. 18(1): 65-70 (2009).

Abstract

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 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 $\Q$. 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 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 $\Q$ 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 $\Qb$-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.

Citation

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Nicolas Bernard,. Franck Leprévost. Michael Pohst. "Jacobians of Genus-2 Curves with a Rational Point of Order 11." Experiment. Math. 18 (1) 65 - 70, 2009.

Information

Published: 2009
First available in Project Euclid: 27 May 2009

zbMATH: 1244.11064
MathSciNet: MR2548987

Subjects:
Primary: 11G30 , 11Y40 , 14H40 , 14Q05

Keywords: genus-2 curves , Jacobians , modular curves , rational point of order 11 , torsion

Rights: Copyright © 2009 A K Peters, Ltd.

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Vol.18 • No. 1 • 2009
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