Elements of prime order in Tate-Shafarevich groups of abelian varieties over $\mathbb{Q}$

For each prime $p$, we show that there exist geometrically simple abelian varieties $A\mathrm{/}\mathbb{Q}$ with non-trivial $p$-torsion in their Tate-Shafarevich groups. Specifically, for any prime $N\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}p)$, let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point $P$ of order $p$, and let $B = A_f/\langle P \rangle$\mathrm{.}Thenthenumberofpositiveintegers$d \leq X$, such that the Tate-Shafarevich group of ${\widehat{B}}_d$ has non-trivial $p$-torsion, is $\gg X\mathrm{/}\log X$, where ${\widehat{B}}_d$ is the dual of the $d$-th quadratic twist of $B$. We prove this more generally for abelian varieties of ${\mathrm{G}\mathrm{L}}_2$-type with a $p$-isogeny satisfying a mild technical condition. In the special case of elliptic curves, we give stronger results, including many examples where $\mathrm{S}\mathrm{h}\mathrm{a}(E_d)[p]\mathrm{\ne }0$ for an explicit positive proportion of integers $d$.