College of Liberal Arts & Sciences

# Algebra Seminar

**Abstract: **

A basic problem in algebraic number theory is to determine when a given number field $L$ has an unramified Galois extension $M$ for which $\textbf{Gal}(M/L)$ is isomorphic to a given group $G$. A folklore conjecture is that for all finite groups $G$ and number fields $K$, there are infinitely many such $M/L$ for which $L$ is a quadratic extension of $K$. In this talk I will discuss work with F. Bleher and J. Gillibert on the case in which $G$ is the three by three Heisenberg group over $Z/n$ for some odd integer $n > 1$. We show that the folklore conjecture is true when $K$ contains a primitive $n$-th root of unity. The main part of the proof is showing there exists an unramified $G$-cover of a hyperelliptic curve which can be specialized to give the result.