  To find interesting items in genome-wide association studies or next
generation sequencing data, a crucial point is to design powerful false
discovery rate (FDR) controlling procedures that suitably combine discrete
tests (typically binomial or Fisher tests). In particular, recent research has
been striving for appropriate modifications of the classical Benjamini-Hochberg
(BH) step-up procedure that accommodate discreteness. However, despite an
important number of attempts, these procedures did not come with theoretical
guarantees. The present paper contributes to fill the gap: it presents new
modifications of the BH procedure that incorporate the discrete structure of
the data and provably control the FDR for any fixed number of null hypotheses
(under independence). Markedly, our FDR controlling methodology allows to
incorporate simultaneously the discreteness and the quantity of signal of the
data (corresponding therefore to a so-called $\pi\_0$-adaptive procedure). The
power advantage of the new methods is demonstrated in a numerical experiment
and for some appropriate real data sets.
