On Certain Torsion Subgroups of Jacobians of Decomposable Curves of Genus 2 and Its
Applications to Symbolic Integration and Primality Proving
Date: July 17th (Wednesday)
Time: 15:30-16:00
Abstract
Given two elliptic curves $E_1$ and $E_2$ defined over a number field $K$, we
construct under some assumptions a curve $C$ of genus 2 defined over $K$ such that its
jacobian is $K$-isogenous to the product $E_1 \times E_2$ (following Serre, one says that
the jacobian of a curve of genus $g$ is decomposable if its jacobian is isogenous to a product
of $g$ elliptic curves). If $E_1$ and $E_2$ have a non-trivial rational torsion subgroup, this
method allows to produce some curves of genus 2 with a ``big'' rational torsion group. We
achieve this in the case $K=Q$ (the field of rational numbers), and break the previous
record for the order of a rational torsion subgroup of the jacobian of a curve of genus 2, and
produce by the way a minoration of the conjectural bound in the dimension 2 case. The
conjecture we refer to is the uniform boundness conjecture for the order of the torsion groups
of abelian varieties. Thanks to a theorem of Risch, by-products take place in the problem of
the integration of algebraic functions. Another more cryptographical application is
connected with ``primality proving''.
