On the probabilities of local behaviors in abelian field extensions
On the probabilities of local behaviors in abelian field extensions
复制标题
关于阿贝尔域扩张中局部行为的概率
作者：
M. Wood
对于数域K和有限交换群G，我们确定了K的随机G-扩张按导体序时的各种局部完备化的概率。特别地，对于K的一个固定素数，我们确定了K的一个随机G-扩张在K处非分歧时，K分裂成r个素数的概率。我们发现，这些概率是很好的表现，大部分是独立的。这类似于切波塔列夫的密度定理，它给出了在一个固定的扩展中K的随机素数在扩展中分裂成r个素数的概率。我们也给出了有界导体G-扩张的个数的渐近性。事实上，我们给出了一类扩展不变量，包括导体，我们得到相同的计数和概率结果。与此相反，我们证明，无论是与Chebotarev概率的类比，也不独立的概率举行时，扩展排序判别。
Abstract For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime ℘ of K, we determine the probability that ℘ splits into r primes in a random G-extension of K that is unramified at ℘. We find that these probabilities are nicely behaved and mostly independent. This is in analogy to Chebotarev’s density theorem, which gives the probability that in a fixed extension a random prime of K splits into r primes in the extension. We also give the asymptotics for the number of G-extensions with bounded conductor. In fact, we give a class of extension invariants, including conductor, for which we obtain the same counting and probabilistic results. In contrast, we prove that neither the analogy with the Chebotarev probabilities nor the independence of probabilities holds when extensions are ordered by discriminant.