分野別セミナー

[講演概要]
Let $K/k$ be a Galois extension of number fields with Galois group $G$. For a conjugacy class $C$ of $G$, the least unramified prime with Frobenius element in $C$ is known to be at most a fixed absolute power $\alpha$ of the discriminant of $K$ due to the celebrated work of Lagarias, Montgomery, and Odlyzko (1979). This exponent $\alpha$ has been extensively studied with the primary method exploiting statistics of zeros of L-functions. We generalize an alternative method based on detecting sign changes. For $G=S_n$, our method improves this exponent $\alpha$ to decay exponentially with $n$ as $n \to \infty$. The ideas also apply to general groups $G$ and certain conjugacy invariant subsets $C$.

This talk is based on joint work with Peter J. Cho and Robert Lemke Oliver.