Retract rationality and algebraic tori

TL;DR

This paper characterizes the $p$-retract rationality of algebraic $k$-tori using their character lattices, establishing equivalences with retract rationality and providing examples based on prime sets.

Contribution

It introduces a characterization of $p$-retract rationality for algebraic tori and links it to the broader concept of retract rationality, including the Noether problem.

Findings

A $k$-torus is retract rational iff it is $p$-retract rational for all primes $p$.

The Noether problem for retract rationality is affirmative iff it is for all $p$-retract rational cases.

Examples of tori are given that are $p$-retract rational only for primes outside a specified set.

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 $p\notin S$.