Simultaneous supersingular reductions of abelian varieties

For a point $x_0$ in a Shimura variety attached to a Shimura datum of Hodge type $(G,X)$, we have an associated abelian scheme $A_0$. Fixing a non-empty finite set $\mathcal{S}$ of primes, we consider the simultaneous supersingular reduction modulo $\mathrm{\ell }\in \mathcal{S}$ of (several copies of) $p$-adic Hecke orbits of $A_0$. We give a precise description of the image of this map. As an application, we give a more conceptual proof of Mazur’s conjecture on non-torsionness of higher Heegner points on an abelian variety which is a quotient of the Jacobian of a Shimura curve. Our arguments simplify those of C.Cornut and V.Vatsal in two important aspects: (1) we do not need to assume the $p$-adic group $G^1({\mathbb{Q}}_p)\mathrm{/}{Z}_{G^1({\mathbb{Q}}_p)}$ to be simple; (2) we do not need to consider separately the geometric" part and chaotic" part in the Hecke orbits.