Retract rationality and algebraic tori

Abstract

For any prime number $p$ and field $k$, we characterize the $p$-retract rationality of an algebraic $k$-torus in terms of its character lattice. We show that a $k$-torus is retract rational if and only if it is $p$-retract rational for every prime $p$, and that the Noether problem for retract rationality for a group of multiplicative type $G$ has an affirmative answer for $G$ if and only if the Noether problem for $p$-retract rationality for $G$ have a positive answer for all $p$. For every finite set of primes $S$ we give examples of tori that are $p$-retract rational if and only if $p\notin S$.

References in corpus (2)

• Unirational threefolds with no universal codimension 2 cycle
• The rationality problem for forms of $\overline{M_{0,n}}$

Cited by in corpus (1)

• On the Noether Problem for torsion subgroups of tori