论文标题
费马特曲线的海森伯格覆盖
The Heisenberg covering of the Fermat curve
论文作者
论文摘要
对于$ n $ integer $ \ ge1 $，K。Murty和D. Ramakrishnan定义了$ n $ th Heisenberg曲线，是上半平面的压缩商的$ X'_n $，由模块化组的某个非共同子组。他们询问Manin-Drinfeld原则是否成立，即那些曲线的尖端支撑的除数是否是雅各布式的扭力。我们给出了$ n $ -th Heisenberg曲线的$ {\ bf z} [μ_n，1/n] $的型号，作为$ n $ th $ th fermat曲线的覆盖。我们表明，Manin-Drinfeld原则以$ n = 3 $持有，但不以$ n = 5 $。我们表明，生成器的描述和由于heisenberg的覆盖物以及更高的类似性质的覆盖率来解释了Fermat曲线的cuspidal子组引起的关系。曲线$ x_n $和经典的模块化曲线$ x（n）$，对于$ n $甚至整数，均主导$ x（2）$，它在jacobians $ j_n \ rightarrow j（n）$之间产生形态。我们证明，后者的图像$ 0 $或椭圆曲线为$ j $ -invariant $ 0 $。顺便说一句，我们对$ x'_ {n} $的同源性进行了描述。
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 $X'_{N}$.
