  Bernoulli-Carlitz numbers were introduced by L. Carlitz in 1935, they are the
analogues in positive characteristic of Bernoulli numbers. We prove a
conjecture formulated by F. Pellarin and the first author on the non-vanishing
modulo a given prime of families of Bernoulli-Carlitz numbers. We then show
that the "exceptional zeros" of certain $L$-series are intimately connected to
the Bernoulli-Carlitz numbers.
