We study the automorphisms of modular curves associated to Cartan subgroups of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_2$ 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 nonsplit curves of prime level $p\ge 13$: the curve $X_{\text{ ns}}^{+}(p)$ has no nontrivial automorphisms, whereas the curve $X_{\text{ ns}}(p)$ has exactly one nontrivial automorphism. Moreover, as an immediate consequence of our results we compute the automorphism group of $X_0^{\ast }(n):=X_0(n)∕W$, where $W$ is the group generated by the Atkin–Lehner involutions of $X_0(n)$ and $n$ is a large enough square.

我们研究与 Cartan 子群相关的模曲线的自同构${\mathrm{总帐}<!--FUNCTION\; APPLICATION-->}_2$以及它们的归一化器的某些子组。我们证明如果$n$足够大，所有的自同构都是由复上半平面的分枝覆盖引起的。我们得到素数水平的非分裂曲线的新结果$p\ge 13$: 曲线$X_{\text{ ns}}^{+}(p)$没有非平凡的自同构，而曲线$X_{\text{ ns}}(p)$恰好有一个非平凡的自同构。此外，作为我们结果的直接结果，我们计算了自同构群$X_0^{*}(n)：=X_0(n)∕W$， 在哪里$W$是由 Atkin-Lehner 对合生成的群$X_0(n)$和$n$是一个足够大的正方形。