I will discuss a natural appearance of unramified mixed Tate realizations in the Hecke-trivial part of
the cohomology of the moduli space $A_g$ of principally polarized abelian varieties and show how
this can be used to understand the primitive stable cohomology of minimal compactifications.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/6370