Algebra Seminar

Ted Chinburg (University of Pennsylvania)
Constructing unramified Heisenberg group extensions of number fields

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.

Event Date: 
March 22, 2021 - 3:30pm to 4:30pm
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