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Texas A&M Number Theory Seminar

Department of Mathematics
Blocker 220
Wednesdays, 1:45–2:45 PM

Mike Fried

Emeritus University of California, Irvine

Wednesday, November 13, 2019

Blocker 220, 1:45PM

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

Abstract: Riemann surface covers XP1z of the sphere, uniformized by a complex variable z, arise by giving the branch points and generators g1gr of a finite group G where the gis 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 gis, 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. rn).

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.