### September 2009

Today we’re back to some material from the first post in this series, and going to prove an actual theorem about vector bundles.  Next time, we’ll be getting into the heart of the paper, and that may be my last post on B-N-R.

I’ve been a lot more active recently, now that my life has quieted down a bit into reading papers, running seminars, taking classes, and teaching a bit, instead of the craziness of a wedding.  So now, something I’ve been meaning to do, but which hasn’t been done yet: updating the blogroll.  So I ask that anyone who has a blog that discusses stuff like what we discuss, post here to be added.  Now, in the broadest sense, that means math, but most especially algebra, algebraic geometry, differential geometry, complex geometry, number theory, and representation theory.  But I’m most assuredly missing things, so if you think that we authors (or our readers) would be interested in your blog, post on this thread.

The last post was on the generalities of Abelian varieties, and constructing a map.  This time, we’re going to do it for a specific one, and the maps involved will all be useful later.  We start out with $\pi:Y\to X$ a finite morphism of curves.

Now, we continue our tour through Beauville, Narasimhan, Ramanan.  We’ve talked about Twisted Endomorphisms and we’ve talked about the Generalized Theta Divisor on the Moduli Space of Vector Bundles.  So today we’ll talk a bit about Abelian Varieties, Principal Polarizations, and we’ll prove some lemmas.  This one’s going to be a bit longer than the last few.

Let’s examine the role of topology in the study of fields and arithmetic. A topology on a field compatible with the field operations is given by an absolute value, which in turn defines a metric. Outside of number theory, people usually mean the standard real absolute  value $|x| = \pm x$ when they talk about an absolute value. Note that on $\mathbf{Q}$ this absolute value carries the Archimedian property that $\mathbf{Z}$ is an unbounded set. What about the other case?