i will first describe certain conjectural tate classes in the middle dimensional etale cohomology of many shimura varieties over finite fields (e.g. hilbert and picard modular surfaces at inert primes). according to the tate conjecture, there should exist corresponding algebraic cycles. surprisingly, we find that these cycles are provided by the supersingular (or more precisely basic) loci of these shimura varieties. this is based on a joint work with liang xiao.