  For $N$ integer $\ge1$, K. Murty and D. Ramakrishnan defined the $N$-th
Heisenberg curve, as the compactified quotient $X'_N$ of the upper half-plane
by a certain non-congruence subgroup of the modular group. They ask whether the
Manin-Drinfeld principle holds, namely if the divisors supported on the cusps
of those curves are torsion in the Jacobian. We give a model over ${\bf
Z}[\mu_N,1/N]$ of the $N$-th Heisenberg curve as covering of the $N$-th Fermat
curve. We show that the Manin-Drinfeld principle holds for $N=3$, but not for
$N=5$. We show that the description by generator and relations due to Rohrlich
of the cuspidal subgroup of the Fermat curve is explained by the Heisenberg
covering, together with a higher covering of a similar nature. The curves $X_N$
and the classical modular curves $X(n)$, for $n$ even integer, both dominate
$X(2)$, which produces a morphism between jacobians $J_N\rightarrow J(n)$. We
prove that the latter has image $0$ or an elliptic curve of $j$-invariant $0$.
In passing, we give a description of the homology of $X'_{N}$.
