Article (Scientific journals)
Jacobians of genus 2 curves with a rational point of order 11
Leprévost, Franck; Bernard, Nicolas; Pohst, Michael
2009In Experimental Mathematics, 18 (1), p. 65-70
Peer Reviewed verified by ORBi
 

Files


Full Text
Bernard_Leprevost_Pohst_20080528.pdf
Author preprint (140.97 kB)
Request a copy

All documents in ORBilu are protected by a user license.

Send to



Details



Abstract :
[en] 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.
Disciplines :
Mathematics
Identifiers :
UNILU:UL-ARTICLE-2010-112
Author, co-author :
Leprévost, Franck ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Computer Science and Communications Research Unit (CSC)
Bernard, Nicolas ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Computer Science and Communications Research Unit (CSC)
Pohst, Michael
Language :
English
Title :
Jacobians of genus 2 curves with a rational point of order 11
Publication date :
2009
Journal title :
Experimental Mathematics
ISSN :
1944-950X
Publisher :
A K Peters, Natick, United States - Massachusetts
Volume :
18
Issue :
1
Pages :
65-70
Peer reviewed :
Peer Reviewed verified by ORBi
Available on ORBilu :
since 23 April 2013

Statistics


Number of views
138 (14 by Unilu)
Number of downloads
1 (0 by Unilu)

Scopus citations®
 
4
Scopus citations®
without self-citations
4
OpenCitations
 
5
WoS citations
 
5

Bibliography


Similar publications



Contact ORBilu