The Heisenberg covering of the Fermat curve

Abstract

For N integer $\ge 1$, K. Murty and D. Ramakrishnan defined the Nth Heisenberg curve, as the compactified quotient $X^{\prime }_N$ 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 $\mathbf {Z}[\mu _N,1/N]$ of the Nth Heisenberg curve as covering of the Nth 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 $X^{\prime }_{N}$.