The cohomology of motivic A(2)

TL;DR

This paper computes the motivic cohomology of the subalgebra A(2) of the motivic Steenrod algebra over an algebraically closed field of characteristic zero, using a motivic May spectral sequence, and explores implications for a hypothetical motivic modular forms spectrum.

Contribution

It introduces a motivic version of the May spectral sequence to compute the cohomology of A(2) and applies Adams-Novikov spectral sequence techniques to speculate on the homotopy of a motivic modular forms spectrum.

Findings

Computed the cohomology of A(2) in the motivic setting.

Developed a motivic May spectral sequence method.

Speculated on the homotopy of a motivic modular forms spectrum.

Abstract

Working over an algebraically closed field of characteristic zero, we compute the cohomology of the subalgebra A(2) of the motivic Steenrod algebra that is generated by Sq^1, Sq^2, and Sq^4. The method of calculation is a motivic version of the May spectral sequence. Speculatively assuming that there is a "motivic modular forms" spectrum with certain properties, we use an Adams-Novikov spectral sequence to compute the homotopy of such a spectrum at the prime 2.