### August 2009

Last time, we talked about twisted endomorphisms.  Now, we’re moving on to the second paragraph of the paper: generalized theta divisors.  In the meantime, we’re going to have to talk a bit about vector bundles and their moduli.

Alright, I’m back, and newly married (thus the long hiatus from posting).  And now it’s time to get back to math.  I’m currently attempting to read a paper by Beauville, Narasimhan and Ramanan titled “Spectral Curves and the Generalized Theta Divisor.”  This paper is one of the early uses of the Hitchin System to prove results about the moduli space of vector bundles over a curve.  The main result can be roughly stated as “a generic vector bundle is the pushforward of a line bundle.” This is a very nice result, and I’m going to work towards it in small steps, largely because I’m still trying to understand it myself.

Theorem(Shih,1974): If 2,3 or 7 is not a square in $\mathbf{Z}/p\mathbf{Z}$ then $PSL_2(\mathbf{Z}/p\mathbf{Z})$ is the Galois group of a polynomial with integral coefficients.