AN (Formula presented)-MOTIVIC v₁-SELF-MAP OF PERIODICITY 1

We consider a nontrivial action of C2 on the type 1 spectrum Y: = M₂(1) ^ C(n), which is well-known for admitting a 1-periodic v₁-self-map. The resultant finite C2-equivariant spectrum y^C2 can also be viewed as the complex points of a finite (Formula presented)-motivic spectrum (Formula presented). In this paper, we show that one of the 1-periodic v₁-self-maps of y can be lifted to a self-map of y^C2 as well as (Formula presented). Further, the cofiber of the self-map of (Formula presented) is a realization of the subalgebra (Formula presented) (1) of the R-motivic Steenrod algebra. We also show that the C2-equivariant self-map is nilpotent on the geometric fixed-points of y^C2.