  In 2002 Polterovich has notably established that on closed aspherical
symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which we
call Hamiltonian torsion, must in fact be trivial. In this paper we prove the
first higher-dimensional Hamiltonian no-torsion theorems beyond the
symplectically aspherical case. We start by showing that closed symplectic
Calabi-Yau and negative monotone symplectic manifolds do not admit Hamiltonian
torsion. Going still beyond topological constraints, we prove that every closed
positive monotone symplectic manifold $(M,\omega)$ admitting Hamiltonian
torsion is geometrically uniruled by holomorphic spheres for every
$\omega$-compatible almost complex structure, partially answering a question of
McDuff-Salamon. This provides many additional no-torsion results, and as a
corollary yields the geometric uniruledness of monotone Hamiltonian
$S^1$-manifolds, a fact closely related to a celebrated result of McDuff from
2009. Moreover, the non-existence of Hamiltonian torsion implies the triviality
of Hamiltonian actions of lattices like $SL(k,\mathbb{Z})$ for $k \geq 2,$ as
well as those of compact Lie groups. Finally, for monotone symplectic manifolds
admitting Hamiltonian torsion, we prove an analogue of Newman's theorem on
finite transformation groups for several natural norms on the Hamiltonian
group: such subgroups cannot be contained in arbitrarily small neighborhoods of
the identity. Our arguments rely on generalized Morse-Bott methods, as well as
on quantum Steenrod powers and Smith theory in filtered Floer homology.
