The torsion and B-free conjectures for cyclotomic points on hyperelliptic Jacobians

Let $\ell$ be a prime, $k\geq 0$, and $d$ a fixed dimension. A cyclotomic point means a rational point defined over a cyclotomic extension, and $D_{\ell^{k+1}}$ denotes the dihedral group with the indicated rotation order. Torsion and B-free conjectures. There is a negative conclusion to the stated assertion concerning the existence, for each $k$, of a $\mathbb Q$-cyclotomic point of order $\ell^{k+1}$ among all hyperelliptic Jacobians of any fixed dimension $d$. Without any bound $B$, for each $\ell^{k+1}$ there is a $\mathbb Q$-cyclotomic point on some hyperelliptic Jacobian corresponding to a $(D_{\ell^{k+1}},\mathbf C_{2^r})$ regular inverse Galois problem involution realization, with $r$ dependent on $\ell^{k+1}$. These conjectures relate torsion on hyperelliptic Jacobians to the regular inverse Galois problem, while the source records that the relevant involution realizations are known only in limited cases.