Finite $p$-Groups of Birational Automorphisms and Characterizations of Rational Varieties

We prove that there exists a constant $R(n)$ such that a rationally connected variety of dimension $n$ over an algebraically closed field is rational if its birational automorphism group contains a $p$-subgroup of maximal rank for some prime number $p>R(n)$. This answers a question of Prokhorov and Shramov. Some applications related to the Jordan property are discussed.