TY - UNPB

T1 - Reduction modulo p of the Noether problem

AU - Ambrosi, Emiliano

AU - Valloni, Domenico

PY - 2023/2/8

Y1 - 2023/2/8

N2 - Let \(k\) be an algebraically closed field of characteristic \(p \geq 0\) and \(V\) be a faithful \(k\)-rational representation of a finite \(\ell\)-group \(G\), where \(\ell\) is a prime number. The Noether problem asks whether \(V/G\) is a stably rational variety. While if \(\ell=p\) it is well-known that \(V/G\) is always rational, when \(\ell\neq p\), Saltman and then Bogomolov constructed \(\ell\)-groups for which \(V/G\) is not stably rational. Hence, the geometry of \(V/G\) depends heavily on the characteristic of the field. We show that for all the groups \(G\) constructed by Saltman and Bogomolov, one cannot interpolate between the Noether problem in characteristic \(0\) and \(p\). More precisely, we show that it does not exist a complete valuation ring \(R\) of mixed characteristic \((0,p)\) and a smooth proper \(R\)-scheme \(X\rightarrow \mathrm{Spec}(R)\) whose special fiber and generic fiber are both stably birational to \(V/G\). The proof combines the integral \(p\)-adic Hodge theoretic results of Bhatt-Morrow-Scholze with the study of indefinitely closed differential forms in positive characteristic.

AB - Let \(k\) be an algebraically closed field of characteristic \(p \geq 0\) and \(V\) be a faithful \(k\)-rational representation of a finite \(\ell\)-group \(G\), where \(\ell\) is a prime number. The Noether problem asks whether \(V/G\) is a stably rational variety. While if \(\ell=p\) it is well-known that \(V/G\) is always rational, when \(\ell\neq p\), Saltman and then Bogomolov constructed \(\ell\)-groups for which \(V/G\) is not stably rational. Hence, the geometry of \(V/G\) depends heavily on the characteristic of the field. We show that for all the groups \(G\) constructed by Saltman and Bogomolov, one cannot interpolate between the Noether problem in characteristic \(0\) and \(p\). More precisely, we show that it does not exist a complete valuation ring \(R\) of mixed characteristic \((0,p)\) and a smooth proper \(R\)-scheme \(X\rightarrow \mathrm{Spec}(R)\) whose special fiber and generic fiber are both stably birational to \(V/G\). The proof combines the integral \(p\)-adic Hodge theoretic results of Bhatt-Morrow-Scholze with the study of indefinitely closed differential forms in positive characteristic.

KW - math.AG

KW - math.NT

M3 - Preprint

BT - Reduction modulo p of the Noether problem

ER -