Remarks on the Milnor conjecture over schemes

TL;DR

This paper surveys recent generalizations of the Milnor conjecture to schemes, discusses counterexamples over certain p-adic curves, and presents new results for curves over local fields and surfaces over finite fields.

Contribution

It introduces a new perspective on the unramified Milnor question and provides novel results in the context of schemes, especially for curves and surfaces.

Findings

Counterexamples for smooth complete p-adic curves without rational theta characteristic

New approaches to the unramified Milnor question

Results on curves over local fields and surfaces over finite fields

Abstract

The Milnor conjecture has been a driving force in the theory of quadratic forms over fields, guiding the development of the theory of cohomological invariants, ushering in the theory of motivic cohomology, and touching on questions ranging from sums of squares to the structure of absolute Galois groups. Here, we survey some recent work on generalizations of the Milnor conjecture to the context of schemes (mostly smooth varieties over fields of characteristic not 2). Surprisingly, a version of the Milnor conjecture fails to hold for certain smooth complete p-adic curves with no rational theta characteristic (this is the work of Parimala, Scharlau, and Sridharan). We explain how these examples fit into the larger context of an unramified Milnor question, offer a new approach to the question, and discuss new results in the case of curves over local fields and surfaces over finite fields.