论文标题
密度分支的素数
A Density of Ramified Primes
论文作者
论文摘要
令$ k $为$ \ mathbb {q} $带有奇数类号码的奇数奇数字段的完全真实数字字段，因此每个完全正面的单位都是一个单位的平方，因此$ 2 $ in $ k/\ mathbb {q} $ intert inter。我们定义一个数字字段的家庭$ \ {k（p）\} _ p $，具体取决于$ k $，并由理性的素数$ p $索引，这些$ p $完全分为$ k/\ kathbb {q} $，因此$ p $始终以$ k（p）$ k（p）$ $ 2 $ $ 2 $ 2 $。以短字符总和为标准猜想的条件，这种理性的总理$ p $的密度在$ k（p）/\ mathbb {q} $中表现出两个可能的分支因素化之一，严格介于$ 0 $和$ 1 $之间，并且在$ [k：\ mathbb bbb的公式中都明确地给出了。在立方体情况下，我们的结果是无条件的。我们的证明依赖于对主要理想旋转联合分布的详细研究。
Let $K$ be a cyclic totally real number field of odd degree over $\mathbb{Q}$ with odd class number, such that every totally positive unit is the square of a unit, and such that $2$ is inert in $K/\mathbb{Q}$. We define a family of number fields $\{K(p)\}_p$, depending on $K$ and indexed by the rational primes $p$ that split completely in $K/\mathbb{Q}$, such that $p$ is always ramified in $K(p)$ of degree $2$. Conditional on a standard conjecture on short character sums, the density of such rational primes $p$ that exhibit one of two possible ramified factorizations in $K(p)/\mathbb{Q}$ is strictly between $0$ and $1$ and is given explicitly as a formula in terms of $[K:\mathbb{Q}]$. Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.
