# Motivic Steenrod operations in characteristic $p$

## TL;DR

This paper defines motivic Steenrod operations in characteristic p, proves their properties, and applies them to quadratic forms over fields of characteristic 2, extending classical cohomological tools.

## Contribution

It introduces Steenrod operations in characteristic p motivic cohomology and establishes their algebraic properties, including Adem relations and applications to quadratic forms.

## Key findings

- Steenrod operations coincide with pth powers on certain motivic cohomology groups.
- Operations satisfy Adem relations and Cartan formula in mod p Chow groups.
- New results on quadratic forms over fields of characteristic 2 using Steenrod squares.

## Abstract

Using the recent work of Frankland and Spitzweck, we define Steenrod operations $P^{n}$ on the mod $p$ motivic cohomology of smooth varieties defined over a base field of characteristic $p$. We show that $P^{n}$ is the $p$th power on $H^{2n,n}(-,\mathbb{F}_{p})\cong CH^{n}(-)/p$ and prove an instability result for the operations. Restricted to mod $p$ Chow groups, we show that the operations satisfy the expected Adem relations and Cartan formula. For $p=2$, we use the new Steenrod squares to obtain new results on quadratic forms over a base field of characteristic $2$.