Quadratic twists of abelian varieties with real multiplication

Let $F$ be a totally real number field and $A\mathrm{/}F$ a principally polarized abelian variety with real multiplication by the ring of integers $\mathcal{O}$ of a totally real field. Assuming $A$ admits an $\mathcal{O}$-linear 3-isogeny over $F$, we prove that a positive proportion of the quadratic twists $A_d$ have rank 0. We also prove that a positive proportion of $A_d$ have rank $\dim A$,assumingtheTate-Shafarevichgroupsarefinite\mathrm{.}If$A$ is the Jacobian of a hyperelliptic curve $C$, we deduce that a positive proportion of twists $C_d$ have no rational points other than those fixed by the hyperelliptic involution.