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 Sq1, Sq2, and Sq4. 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.
Homology, Homotopy and Applications, Vol. 11 (2009), No. 2, pp.251-274.
Available as: ps ps.gz pdf