Torsion points of sections of Lagrangian torus fibrations and the Chow ring of hyper-K\"ahler manifolds

Let $\varphi :X\to B$ be a Lagrangian fibration on a projective irreducible hyper-K"ahler manifold of dimension $\le 8$. Let $M\in {\rm Pic},X$bealinebundlewhoserestrictiontothegeneralfiber$X_b$of$\phi$ is topologically trivial. We prove that if the fibration has maximal variation or is isotrivial, the set of points $b$ such that the restriction $M_{\mid X_b}$ is torsion is dense in $B$. We give an application to the Chow ring of $X$. We prove a similar result for elliptic fibrations which gives a toy model for the argument.