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On Certain Torsion Subgroups of Jacobians of
   Decomposable Curves of Genus 2 and Its Applications to
         Symbolic Integration and Primality Proving 
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Franck Leprevost
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Date:  July 17th (Wednesday)\\
Time:  15:30-16:00\\

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Abstract
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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''. 

