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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 *X* → **P**^{1}_{z} of the sphere, uniformized by a complex variable *z*, arise by giving the branch points and generators *g*_{1} …*g*_{r} of a finite group *G* where the *g*_{i}s 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*_{i}s, 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.