Suppose $\rho_1, \rho_2$ are two $\ell$-adic Galois representations of the absolute Galois group of a number field, such that the algebraic monodromy group of one of the representations is connected and the representations are locally potentially equivalent at a set of places of positive upper density. We classify such pairs of representations and show that up to twisting by some representation, it is given by a pair of representations one of which is trivial and the other abelian. Consequently, assuming that the first representation has connected algebraic monodromy group, we obtain that the representations are potentially equivalent, provided one of the following conditions hold: (a) the first representation is absolutely irreducible; (b) the ranks of the algebraic monodromy groups are equal; (c) the algebraic monodromy group of the second representation is also connected and (d) the commutant of the image of the second representation remains the same upon restriction to subgroups of finite index of the Galois group.

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关于局部潜在等价伽罗瓦表示的结构

假设 $\rho_1, \rho_2$ 是一个数域的绝对伽罗瓦群的两个 $\ell$-adic Galois 表示，使得其中一个表示的代数单项群是连通的，并且这些表示在正高密度位置的集合。我们对这样的表示对进行分类，并表明在通过某种表示扭曲之前，它是由一对表示给出的，其中一个是平凡的，另一个是阿贝尔的。因此，假设第一个表示已经连接代数单一群，我们得到这些表示可能是等价的，前提是以下条件之一成立：（a）第一个表示是绝对不可约的；(b) 代数单峰群的秩相等；