On Serre's injectivity question and norm principle

TL;DR

This paper affirms Serre's injectivity question for certain reductive groups over fields of characteristic not 2, by linking it to norm principles and analyzing obstructions related to Dynkin diagram types.

Contribution

It establishes Serre's injectivity for groups with Dynkin diagrams of types A, B, C, and some D, using norm principles and scalar obstructions, extending previous results.

Findings

Positive answer for groups with Dynkin components of type A, B, C.

Obstruction related to spinor norms affects D_n cases.

Connection between Serre's question and norm principles clarified.

Abstract

Let $k$ be a field of characteristic not $2$. We give a positive answer to Serre's injectivity question for any smooth connected reductive $k$-group whose Dynkin diagram contains connected components only of type $A_n$, $B_n$ or $C_n$. We do this by relating Serre's question to the norm principles proved by Barquero and Merkurjev. We give a scalar obstruction defined up to spinor norms whose vanishing will imply the norm principle for the non-trialitarian $D_{n}$ case and yield a positive answer to Serre's question for connected reductive $k$-groups whose Dynkin diagrams contain components of non-trialitarian type $D_n$ also. We also investigate Serre's question for reductive $k$-groups whose derived subgroups admit quasi-split simply connected covers.