2022/10/10 by Banerjee, Debargha, Merel, Loïc
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2210.04439
For N integer ≥1, 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[μN,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 XN and the classical modular curves X(n), for n even integer, both dominate X(2), which produces a morphism between jacobians JN→ 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.