费马曲线的海森堡覆盖
The Heisenberg covering of the Fermat curve
AI总结：
本文给出第N个海森堡曲线作为费马曲线覆盖的模型，证明N=3时Manin-Drinfeld原理成立、N=5时不成立，解释了费马曲线尖点子群描述，并研究了雅可比簇态射像及X'_N同调。
摘要：
对于整数N≥1，K. Murty和D. Ramakrishnan定义了第N个海森堡曲线，即上半平面被模群的某个非同余子群商的紧化X'_N。他们询问Manin-Drinfeld原理是否成立，即这些曲线尖点上支撑的除子在雅可比簇中是否为挠元。我们给出了第N个海森堡曲线作为第N个费马曲线覆盖在Z[μ_N,1/N]上的模型。我们证明当N=3时Manin-Drinfeld原理成立，但N=5时不成立。我们表明Rohrlich关于费马曲线尖点子群的生成元与关系的描述可由海森堡覆盖及类似性质的更高覆盖解释。曲线X_N与偶整数n对应的经典模曲线X(n)均支配X(2)，这产生了雅可比簇间的态射J_N→J(n)。我们证明后者的像为0或j不变量为0的椭圆曲线。顺便，我们给出了X'_N同调的描述。
英文摘要：
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 ${\bf Z}[μ_N,1/N]$ 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 $X'_{N}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
