The silver gene is dominant, only needing one copy for the black of the coat to be diluted (Zz or ZZ).
Since it only dilutes black, chestnut horses don't express this gene, looking like normal chestnuts and only carrying it.
There is, however, a gene or multiple genes that will dilute chestnut's mane and tail called flaxen. Meanwhile this gene won't affect black pigment, not being expressed in bay and black horses.