论文标题
Prime和复合水平的Cartan模块化曲线的自态
Automorphisms of Cartan modular curves of prime and composite level
论文作者
论文摘要
我们研究了与$ \ mathrm {gl} _2（\ mathbb z/n \ mathbb z）$及其正常化的某些子组相关的模块化曲线的自动形态曲线。我们证明，如果$ n $足够大，所有的自动形态都是由复杂上半平面的受损覆盖率引起的。我们将获得针对质量级的非分类曲线$ p \ ge 13 $的新结果：曲线$ x _ {\ text {ns}}}}^+（p）$没有非平凡的自动形态，而曲线curve $ x _ {\ text {\ text {ns}}}（p）$ auty not-nor-nim-triv autim autiv altriv autim triv autim triv autim triv autiv autiv autiv autiv。此外，作为我们结果的直接结果，我们计算$ x_0^*（n）的自动形态组：= x_0（n）/w $，其中$ w $是由$ x_0（n）$和$ n $的Atkin-Lehner所产生的组，而$ n $是足够大的广场。
We study the automorphisms of modular curves associated to Cartan subgroups of $\mathrm{GL}_2(\mathbb Z/n\mathbb Z)$ and certain subgroups of their normalizers. We prove that if $n$ is large enough, all the automorphisms are induced by the ramified covering of the complex upper half-plane. We get new results for non-split curves of prime level $p\ge 13$: the curve $X_{\text{ns}}^+(p)$ has no non-trivial automorphisms, whereas the curve $X_{\text{ns}}(p)$ has exactly one non-trivial automorphism. Moreover, as an immediate consequence of our results we compute the automorphism group of $X_0^*(n):=X_0(n)/W$, where $W$ is the group generated by the Atkin-Lehner involutions of $X_0(n)$ and $n$ is a large enough square.
