arXiv subjects
Publications and source records attributed to Christine McMeekin.
For K a cyclic cubic number field with odd class number containing a unit w such that Norm(w)=Norm(1-w)=-1, we prove that the density of rational primes p that satisfy the given spin relation is equal to 1/2. Furthermore, we prove that this density restricted to rational primes that are 1 mod 4 is 1/4 and this density restricted to rational primes that are -1 mod 4 is 3/4.
For cyclic totally real number fields $K$ with odd prime degree $n$, odd class number, $2$ inert, and the property that every totally positive unit is a square, the density of rational primes $p$ that satisfy the spin relation spin$(\mathfrak{p},σ)$spin$(\mathfrak{p},σ^{-1})=1$ for all $σ\neq 1 \in$ Gal$(K/\mathbb{Q})$ where $\mathfrak{p}$ is a prime of $K$ above $p$ is given by the formula \[ D_K=\frac{m_Kn+1}{n2^n} \] where $m_K$ is a computable and bounded invariant of the number field $K$. This formula is modified in the erratum from the original version due to an error in the inert case. As the inert case is insubstantial, the strength of the results is not significantly changed.
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.