Semisimple pointed isogeny graphs for abelian varieties

In this paper we show that if ${\varphi }_i:A_i\to A$ is a semisimple pointed $K$-rational $\mathrm{\ell }$-isogeny graph of order $n$ for a prime $\mathrm{\ell }$, then the group of $\mathrm{\ell }$-torsion points $A[\mathrm{\ell }](\bar{K})$ contains a subspace of dimension $n$ generated by $K$-rational points. We also show that the same result is true for elliptic curves without the semisimplicity condition. Furthermore, we give an explicit counterexample for abelian varieties of higher dimension to show that the semisimplicity condition is indeed necessary.