论文标题
$ e_8 $ - 单一，不变理论和模块化形式
$E_8$-singularity, invariant theory and modular forms
论文作者
论文摘要
作为代数表面，$ e_8 $ -singularity $ x^5+y^3+z^2 = 0 $可以从商$ c_y/\ text {sl}（2，2，13）$上获得模块化曲线$ x（13）$，其中$ y \ y \ y \ y \ subset \ subbb {cp cp {cp cp cut culte cpect $ \ text {sl}（2，13）$ - 不变的多项式和$ c_y $是$ y $的锥体。它与kleinian奇异性$ \ mathbb {c}^2/γ$不同，其中$γ$是二进制二十面体群。这给了Arnol'd和Brieskorn关于Icosahedron和$ e_8 $之间神秘关系的问题，即$ e_8 $ - singularity不一定是克莱琳·伊克萨希德里亚奇点。特别是，$ e_8 $ singularity的方程式具有无限种类的不同模块化参数化，并且$ e_8 $ singularity的多种不同类型的不同构造。它们形成了模块化曲线$ x（13）$的$ e_8 $ singularity结构的变体，我们为此提供了代数版本，几何版本，$ j $ - 功能版本和Poincarényology$ 3 $ -SPHERE以及更高的尺寸提升，即Milnor的Exototototototototototototothotic $ 7 $ -Sphere。此外，$ q_ {18} $和$ e_ {20} $ - $ x（13）$的奇异结构的变化。因此，三个不同的代数表面，$ e_8 $，$ q_ {18} $和$ e_ {20} $的方程式可以从同一标签$ c_y/\ c_y/\ text {sl}（2，2，13）$上实现奇异性，模块化曲线$ x（13）$，并且具有相同的模块化参数。
As an algebraic surface, the equation of $E_8$-singularity $x^5+y^3+z^2=0$ can be obtained from a quotient $C_Y/\text{SL}(2, 13)$ over the modular curve $X(13)$, where $Y \subset \mathbb{CP}^5$ is a complete intersection curve given by a system of $\text{SL}(2, 13)$-invariant polynomials and $C_Y$ is a cone over $Y$. It is different from the Kleinian singularity $\mathbb{C}^2/Γ$, where $Γ$ is the binary icosahedral group. This gives a negative answer to Arnol'd and Brieskorn's questions about the mysterious relation between the icosahedron and $E_8$, i.e., the $E_8$-singularity is not necessarily the Kleinian icosahedral singularity. In particular, the equation of $E_8$-singularity possesses infinitely many kinds of distinct modular parametrizations, and there are infinitely many kinds of distinct constructions of the $E_8$-singularity. They form a variation of the $E_8$-singularity structure over the modular curve $X(13)$, for which we give its algebraic version, geometric version, $j$-function version and the version of Poincaré homology $3$-sphere as well as its higher dimensional lifting, i.e., Milnor's exotic $7$-sphere. Moreover, there are variations of $Q_{18}$ and $E_{20}$-singularity structures over $X(13)$. Thus, three different algebraic surfaces, the equations of $E_8$, $Q_{18}$ and $E_{20}$-singularities can be realized from the same quotients $C_Y/\text{SL}(2, 13)$ over the modular curve $X(13)$ and have the same modular parametrizations.
