论文标题
在$ \ m athrm {gal}的表示（\ overline {\ mathbb {q}}}/\ mathbb {q}）$，$ \ wideHat {gt} $和$ \ mathrm {aut}
On representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, $\widehat{GT}$ and $\mathrm{Aut}(\hat{F}_2)$
论文作者
论文摘要
通过Belyi的工作，绝对Galois组$ g _ {\ Mathbb {Q}} = \ Mathrm {gal}（\ overline {\ Mathbb {Q}}/\ Mathbb {Q} $ a = \ mathrm {aut}（\ wideHat {f_2}）$，两个生成器上免费的profinite组$ \ widehat {f_2} $的自动形态组。 $ g _ {\ mathbb {q}} $的图像位于$ \ widehat {gt} $，grothendieck-teichmüllergroup中。众所周知，$ g _ {\ mathbb {q}} $的每种Abelian表示都可以扩展到$ \ wideHat {gt} $，但Lochak和Schneps提出了构建$ \ wideHat {gt} $的不可约性非阿贝尔表示的挑战。我们实际上是通过证明$ g _ {\ mathbb {q}} $的丰富算术定义的表示形式可以扩展到$ \ wideHat {gt} $的有限索引子组。实际上，通过将这些表示形式一直扩展到$ a = \ mathrm {aut}（\ wideHat {f_2}）$的有限索引子组来实现。我们通过开发Grunewald和Lubotzky的作品的涂鸦版本来做到这一点，该版本为离散组$ \ mathrm {aut}（aut}（f_d）$提供了丰富的表示形式。
By work of Belyi, the absolute Galois group $G_{\mathbb{Q}}=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be embedded into $A=\mathrm{Aut}(\widehat{F_2})$, the automorphism group of the free profinite group $\widehat{F_2}$ on two generators. The image of $G_{\mathbb{Q}}$ lies inside $\widehat{GT}$, the Grothendieck-Teichmüller group. While it is known that every abelian representation of $G_{\mathbb{Q}}$ can be extended to $\widehat{GT}$, Lochak and Schneps put forward the challenge of constructing irreducible non-abelian representations of $\widehat{GT}$. We do this virtually, namely by showing that a rich class of arithmetically defined representations of $G_{\mathbb{Q}}$ can be extended to finite index subgroups of $\widehat{GT}$. This is achieved, in fact, by extending these representations all the way to finite index subgroups of $A=\mathrm{Aut}(\widehat{F_2})$. We do this by developing a profinite version of the work of Grunewald and Lubotzky, which provided a rich collection of representations for the discrete group $\mathrm{Aut}(F_d)$.
