### 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.

Dear Readers,

In spite of orals closing in a little more every day, I clearly haven’t been updating so much recently. I’d started a post about using Minkowski’s geometry of numbers to think about class numbers and unit groups and such things… but honestly that stuff is quite well-worn and at this point it wouldn’t be a good use of time to think carefully about how best to choose my words and explain this to the world when better expositors like William Stein or James Milne have already done so. Instead, I will talk about the expository part of my exam where I will talk about a particular case of the inverse Galois Problem.

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.