On representations of Gal(Q‾/Q) , GTˆ and Aut(Fˆ2)
On representations of Gal(Q‾/Q) , GTˆ and Aut(Fˆ2)
复制标题
关于 Gal(Q−/Q) 、GT− 和 Aut(F−2) 的表示
DOI：
10.1016/j.jalgebra.2021.06.005
复制
发表时间：
2022
影响因子：
0.9
通讯作者：
Lubotzky, Alexander
中科院分区：
文献类型：
--
作者：
Bleher, Frauke M.;Chinburg, Ted;Lubotzky, Alexander
利用Belyant [2]的工作，有理数域Q的绝对Galois群G Q= Gal（Q/Q）可以嵌入到两个生成元上的自由profinite群F <$2的自同构群A= Aut（F <$2）中. G Q的像位于G T，Grothendieck-Teichmüller群中。虽然已知G Q的每一个阿贝尔表示都可以推广到G T，Lochak和Schneps [13]提出了构造G T的不可约非阿贝尔表示的挑战。我们这样做实际上，即通过证明，一类丰富的算术定义的表示G Q可以扩展到有限指数子群G T。事实上，这是通过将这些表示一直扩展到A= Aut（F <$2）的有限指数子群来实现的。我们通过发展格鲁内瓦尔德和卢博茨基[7]的工作的profinite版本来做到这一点，这些工作为离散群Aut（F d）提供了丰富的表示集合。
By work of Belyĭ [2], the absolute Galois group G Q= Gal (Q‾/Q) of the field Q of rational numbers can be embedded into A= Aut (F ˆ 2), the automorphism group of the free profinite group F ˆ 2 on two generators. The image of G Q lies inside G T ˆ, the Grothendieck-Teichmüller group. While it is known that every abelian representation of G Q can be extended to G T ˆ, Lochak and Schneps [13] put forward the challenge of constructing irreducible non-abelian representations of G T ˆ. We do this virtually, namely by showing that a rich class of arithmetically defined representations of G Q can be extended to finite index subgroups of G T ˆ. This is achieved, in fact, by extending these representations all the way to finite index subgroups of A= Aut (F ˆ 2). We do this by developing a profinite version of the work of Grunewald and Lubotzky [7], which provided a rich collection of representations for the discrete group Aut (F d).