  We consider the problem of testing hypotheses on the copula density from $n$
bi-dimensional observations. We wish to test the null hypothesis characterized
by a parametric class against a composite nonparametric alternative. Each
density under the alternative is separated in the $L_2$-norm from any density
lying in the null hypothesis. The copula densities under consideration are
supposed to belong to a range of Besov balls. According to the minimax
approach, the testing problem is solved in an adaptive framework: it leads to a
$\log\log$ term loss in the minimax rate of testing in comparison with the
non-adaptive case. A smoothness-free test statistic that achieves the minimax
rate is proposed. The lower bound is also proved. Besides, the empirical
performance of the test procedure is demonstrated with both simulated and real
data.
