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.

Abstract