Torsion points of a section of an abelian scheme over a p-adic field have rigid-analytic neighborhoods isolating each torsion order; a nodal-fiber example shows that bad reduction destroys this separation.
Torsion points of sections of Lagrangian torus fibrations and the Chow ring of hyper-K\"ahler manifolds
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abstract
Let $\phi:X\rightarrow B$ be a Lagrangian fibration on a projective irreducible hyper-K\"ahler manifold of dimension $\leq8$. Let $M\in {\rm Pic}\,X$ be a line bundle whose restriction to the general fiber $X_b$ 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.
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On the $p$-adic distribution of torsion values for a section of an abelian scheme
Torsion points of a section of an abelian scheme over a p-adic field have rigid-analytic neighborhoods isolating each torsion order; a nodal-fiber example shows that bad reduction destroys this separation.