论文标题
关于减少代数数的订单和索引的分布
On the distribution of the order and index for the reductions of algebraic numbers
论文作者
论文摘要
令$α_1，\ ldots，α_r$为数字字段中的代数$ k $，生成$ r $ in $ k^\ times $的子组。我们在GRH下调查了$ k $的Primes $ \ Mathfrak P $的数量，以使每个订单（α_i\ bmod \ Mathfrak p）$的每个订单都在于与$α_i$相关的给定算术进程。我们还研究了Primes $ \ Mathfrak P $，其中$（α_i\ bmod \ Mathfrak P）$的索引是固定的整数，或者位于给定的整数中，每个$ i $。可以考虑$ \ Mathfrak P $的Frobenius共轭类别的其他条件。这些结果是从2006年开始的Ziegler定理的概括，这涉及此问题的$ r = 1 $。
Let $α_1,\ldots,α_r$ be algebraic numbers in a number field $K$ generating a subgroup of rank $r$ in $K^\times$. We investigate under GRH the number of primes $\mathfrak p$ of $K$ such that each of the orders of $(α_i\bmod\mathfrak p)$ lies in a given arithmetic progression associated to $α_i$. We also study the primes $\mathfrak p$ for which the index of $(α_i\bmod\mathfrak p)$ is a fixed integer or lies in a given set of integers for each $i$. An additional condition on the Frobenius conjugacy class of $\mathfrak p$ may be considered. Such results are generalizations of a theorem of Ziegler from 2006, which concerns the case $r=1$ of this problem.
