On the Chow ring of very general abelian varieties and a question of Pirola

We prove that for a very general abelian variety of dimension $\ge 4$, a divisor $D\in {\mathrm{C}\mathrm{H}}^1(A)$ that satisfies $D^2=0$ in ${\mathrm{C}\mathrm{H}}^2(A)$ is of torsion. The same result is also established for a very general Jacobian in genus $4$. We use then the second statement in order to prove a conjecture of Pirola, which states that any rational section of the Kummer fibration $K=J\mathrm{/}\pm \mathrm{I}\mathrm{d}\to {\mathcal{M}}_4$, where $J\to {\mathcal{M}}_4$ is the Jacobian fibration, must be a multiple of the Griffiths-Pirola section given by the difference of the two trigonal divisors.