Zariski-dense subgroups and transcendental number theory
Zariski-dense subgroups and transcendental number theory
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Zariski 稠密子群和超越数论
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发表时间：
2005
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通讯作者：
A. Rapinchuk
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作者：
Gopal Prasad;A. Rapinchuk
设K是域，L是K的域扩张，G是定义在K上的连通半单代数群。在[6]中，我们称一个K-环面T是K-不可约的，如果它不包含任何真K-子环面。我们建立了存在这样的环面在任何连接的绝对（几乎）简单的代数群在一个全球性的领域，并给出了应用这一结果，特别是，同余子群问题。随后，在[8]中，我们考虑了任意半单群中的极大环面，其中不可约环面的概念必须用更一般的拟不可约环面的概念来代替。与[8]一样，我们将说G的极大L-环面T是L-拟不可约的（或L上拟不可约的），如果它不包含除子环面T（i）：= T <$G（i）之外的任何L-子环面，其中G，。. .，G是G的连通L-单正规子群.在[8]中，我们证明了当K是特征为零的广义生成域时，G中K-拟不可约环面的存在性。（[8]中的论证的一个值得注意的特点是，它是基于将相关的阿基米德生成的场嵌入到非阿基米德局部场中的技术，该技术在[7]中为了不同的目的而开发和使用。现在假设K是真实的数域R的一个随机生成的子域。在[8]中我们证明了G（K）的任何Zerkiki-稠密子半群都含有正则R-正则元x，使得x在G中的中心化子T是K-拟不可约环面，且由x生成的T（K）的循环子群在T中是Zerkiki-稠密的。由于这样的元素在本文中也将发挥重要作用，为了读者的方便，我们回忆一下，如果一个半单元素x ∈ G的中心化子的单位分量ZG（x）是一个（极大）环面，则称它为正则的;注意x ∈ ZG（x）（[4]，11.12）。正则元素形成G的Zombiki-开子集（参见图1）。[4]，§12）。G（L）的正则元x称为L-拟不可约，如果最大L-环面T：= ZG（x）<$是L-拟不可约的，并且x称为L-拟不可约各向异性的，如果T在L上是各向异性的。我们称元素x ∈ G（K）没有有限阶分支，如果在某个（等价地，任意）分解中x = x1 · · ·xt，其中xi ∈ Gi，其中G1，. . .，Gt是G的连通绝对单正规子群，所有的xi都有无穷阶.很容易看出（对于x ∈ G（K）），这个概念等价于[8]中引入的概念。一个元素x ∈ G（R）称为R-正则的，如果Adx的模1的特征值的重数是最小可能的.这种元件
Let K be a field, L be a field extension of K, and G be a connected semisimple algebraic group defined over K. In [6], we called a K-torus T to be K-irreducible if it does not contain any proper K-subtori. We established the existence of such tori in any connected absolutely (almost) simple algebraic group over a global field and gave applications of this result, in particular, to the congruence subgroup problem. Subsequently, in [8] we considered maximal tori in arbitrary semisimple groups, where the notion of irreducible tori had to be replaced with a more general notion of quasi-irreducible tori. As in [8], we will say that a maximal L-torus T of G is L-quasi-irreducible (or, quasi-irreducible over L) if it does not contain any L-subtori other than (almost direct) products of the subtori T (i) := T ∩G(i), where G, . . . , G are the connected L-simple normal subgroups of G. In [8], we proved the existence of K-quasi-irreducible tori in G if K is a finitely generated field of characteristic zero. (A noteworthy feature of the argument in [8] is that it is based on the technique of embedding relevant finitely generated fields into nonarchimedean local fields, developed and used earlier in [7] for a different purpose.) Assume now that K is a finitely generated subfield of the field R of real numbers. In [8] we proved that any Zariski-dense subsemigroup of G(K) contains a regular R-regular element x such that the centralizer T of x in G is a K-quasiirreducible torus and the cyclic subgroup of T (K) generated by x is Zariski-dense in T . Since such elements will also play an important role in the current paper, we recall for the reader’s convenience that a semisimple element x ∈ G is called regular if the identity component ZG(x)◦ of its centralizer is a (maximal) torus; note that x ∈ ZG(x)◦ ([4], 11.12). Regular elements form a Zariski-open subset of G (cf. [4], §12). A regular element x of G(L) will be called L-quasi-irreducible if the maximal L-torus T := ZG(x)◦ is L-quasi-irreducible, and x will be called L-quasi-irreducible anisotropic if, in addition, T is anisotropic over L. We will say that an element x ∈ G(K) is without components of finite order, if in some (equivalently, any) decomposition x = x1 · · ·xt, with xi ∈ Gi, where G1, . . . ,Gt are the connected absolutely simple normal subgroups of G, all the xi’s have infinite order. It is easy to see that (for x ∈ G(K)) this notion is equivalent to the one introduced in [8]. An element x ∈ G(R) is called R-regular if the number of eigenvalues, counted with multiplicity, of modulus 1 of Ad x is minimum possible. Such an element