In this talk, we will discuss the Tate conjecture for Shimura varieties of type (G(U(1,n-1) x U(n-1,1))). We will first describe the supersingular locus of these Shimura varieties, which provides Gysin maps induced by the closed immersions of the irreducible components of the supersingular locus.

Using these maps, we reduce the Tate conjecture to showing that a certain matrix has nonzero determinant, where the entries are given by Hecke actions and incidence numbers. The overall strategy is similar to the proof of the Tate conjecture for more general Shimura varieties, but expressed in a slightly more elaborate language.