Number Theory Seminar

Date: November 13, 2019

Time: 1:45PM - 2:45PM

Location: BLOC 220

Speaker: Mike Fried, University of California, Irvine

Title: Spaces of sphere covers and Riemann's two types of Θ functions

Abstract: Riemann surface covers X → P¹_z of the sphere, uniformized by a complex variable z, arise by giving the branch points and generators g₁ …g_r of a finite group G where the g_is have product-one.

By taking any one such cover, and dragging it by its branch points you create a space of such covers.

A Fundamental Problem: For a given G and the conjugacy classes of the g_is, describe the connected components of the space.

This talk will explain the following case/result: Spaces of r-branch point 3-cycle covers, of degree n, or their Galois closures
of degree n!/2, have one (resp. two) component(s) if r=n-1 (resp. r ≥ n).

Each space is determined by the type of natural θ functions they support. This improves a Fried-Serre formula on when sphere covers with odd-order branching lift to unramified Spin covers of the sphere. We will use the case n=4, to see these Θs and differentiate between their even and odd versions. Riemann used both for different purposes.

This is a special case of a general result about components of spaces of sphere covers. Hyperelliptic jacobians then appear as one case of a general problem entwining The Torsion Conjecture and the Regular Inverse Galois problem. A recent series of Ellenberg-Venkatesh-Westerland used these results, but only got to the hyperelliptic jacobian case.

URL: Link