论文标题
Galois对紫红色表面及其螺线管的作用
Galois action on Fuchsian surface groups and their solenoids
论文作者
论文摘要
让$ c $成为一条复杂的代数曲线，该曲线由Fuchsian $γ$统一。在本文的第一部分中，我们确定了与$γ$相关的自动晶体组，与Belyaev完成其相称的$ \ Mathrm {comm}（comm}（γ）$代数曲线。反过来，这一事实得出了$γ$算术的算术不变性的证明。在第二部分中，我们关注$γ$算术的情况。我们发现的进一步的Galois不变性列表包括：i）$ \ mathrm {comm}（γ）$，ii）等式的可溶性$ x^2+\ sin^2 \ sin^2 \ frac {2π} {2π} {2k+1} $在不可或缺的quaternion algebra of $ g $γ$和$γ$ $ up $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a的属性中。
Let $C$ be a complex algebraic curve uniformised by a Fuchsian group $Γ$. In the first part of this paper we identify the automorphism group of the solenoid associated with $Γ$ with the Belyaev completion of its commensurator $\mathrm{Comm}(Γ)$ and we use this identification to show that the isomorphism class of this completion is an invariant of the natural Galois action of $\mathrm{Gal}(\mathbb C/\mathbb Q)$ on algebraic curves. In turn this fact yields a proof of the Galois invariance of the arithmeticity of $Γ$ independent of Kazhhdan's. In the second part we focus on the case in which $Γ$ is arithmetic. The list of further Galois invariants we find includes: i) the periods of $\mathrm{Comm}(Γ)$, ii) the solvability of the equations $X^2+\sin^2 \frac{2π}{2k+1}$ in the invariant quaternion algebra of $Γ$ and iii) the property of $Γ$ being a congruence subgroup.
