Generators of the algebraic symplectic bordism ring

In this paper, we study the $\eta$-completed part of the motivic spectrum $\text{MSp}$ constructed by Panin and Walter, representing the universal $\text{Sp}$-oriented cohomology theory. In particular, we investigate the inclusion $({\text{MSp}}_{\eta }^{\wedge }{)}^{\ast }\hookrightarrow {\text{MGL}}^{\ast }$ of the cofficient rings, by studying the motivic Adams spectral sequence associated to $\text{MSp}$, mimiking a strategy used by Levine,Yang, Zhao for ${\text{MSL}}^{\ast }$. In order to give a description of $({\text{MSp}}_{\eta }^{\wedge }{)}^{\ast }$, we refine the Pontryagin-Thom construction in a way that allows one to obtain symplectic bordism classes from a large family of varieties that carry a certain “symplectic twist”, and we prove a criterion to select generators among these classes.