The Heisenberg covering of the Fermat curve

TL;DR

This paper constructs models of Heisenberg curves covering Fermat curves, investigates the Manin-Drinfeld principle's validity for specific cases, and explores the relationships between these curves and classical modular curves, revealing their Jacobian structures.

Contribution

It provides explicit models of Heisenberg coverings of Fermat curves over integers, verifies the Manin-Drinfeld principle for N=3 but not for N=5, and links these coverings to classical modular curves and their Jacobians.

Findings

Manin-Drinfeld principle holds for N=3

Manin-Drinfeld principle fails for N=5

Morphisms between Jacobians are either trivial or elliptic with j-invariant 0

Abstract

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 $X'_{N}$.