# Number Theory Seminar

Speaker: Peter Koymans.

Title: The spin of prime ideals and applications.

Abstract: Let $K$ be a cyclic, totally real extension of $\mathbb{Q}$ of degree at least $3$, and let $\sigma$ be a generator of $Gal(K/\mathbb{Q})$. We further assume that the totally positive units are exactly the squares of units. In this case, Friedlander, Iwaniec, Mazur and Rubin define the spin of an odd principal ideal $a$ to be
$spin(\sigma, a) = (\alpha/\sigma(a))_K$,
where $\alpha$ is a totally positive generator of $a$ and $(*/*)_K$ is the quadratic residue symbol in $K$. Friedlander, Iwaniec, Mazur and Rubin prove equidistribution of $spin(\sigma, p)$ as $p$ varies over the odd principal prime ideals of $K$. In this talk I will show how to extend their work to more general fields and give various arithmetic applications.