Chap. 2: Analytic continuation

Like Galois Theory (in algebra), analytic continuation (in complex variables) is a first year graduate subject. Experience, however, shows that few get much from it. Yet, both subjects are powerful modern tools. With them you can glean information from the action of fundamental groups. This books presents the nonabelian theory of Riemann surfaces. That includes two key topics:

  1. How to picture compact Riemann surface covers of the sphere with nonabelian monodromy groups (not even necessarily solvable).
  2. How to work with the entire collection of covers created by taking one such and "dragging it by its branch points."

In "What Gauss Told Riemann About Abel's Theorem" I declare Riemann was set in motion by exactly these topics. Yet, 150 years later barriers remain to the understanding he sought in generalizing the two main theorems of Abel for complex torii. In this book I focus on two problems.

  1. How to introduce just enough group theory to show – at least to those sympathetic to Riemann (and me) – you can get to exciting modern applications that will solve serious problems.
  2. How to avoid algebraic equations for specific Riemann surfaces and parameter spaces attached to them, without losing a sense of "seeing" them.

To accomplish these goals we need very firm ground. That means a classical starting point. So, all complex variable starting points must go to classical texts. I've chosen the first half of Ahlfors (or Conway) graduate texts in complex variables. My experiences teaching complex variables show that Ahlfors is an excellent text in many ways. Yet, its clever combination of algebra and complex variables often fails to comfort students who usually have already chosen one of the two modes as "theirs." This is a mini-version of the problems posed in #3 and #4.

The main theorem of this analytic continuation chapter follows from statements from Cauchy's theorems (like the residue theorem), some knowledge of linear fractional transformations, and how analytic continuation works with branches of log. I review all these in the opening sections of Chap. 2, quoting assiduously particular pages in Alfhors. A student should not mistake the main theorem as a mere algebraic consequence of Kummer theory. My goal is to reveal, peeping out from under this abelian theory, the difficulties in Riemann's full existence theorem. From this we will know what to track in the nonabelian theory so we can distinguish one cover from other inequivalent covers, though they look very similar.

That chapter also reviews basic Galois theory. We know many first year algebra courses give only a little Galois theory. They also leave the feeling that it is an historical subject, without serious modern use. So, Chap. 2 motivates a different view by starting some simple applications that appear later in the book. The best examples in the book use group theory that is but one step beyond those groups – dihedral and alternating – that appear in opening discussions in 1st year graduate algebra.

A group theorist – to whom those examples appear so easy – may not realize the barriers they raise to understanding the nonabelian theory of Riemann surface covers. Fortunately, once over that barrier, you often find you need only learn a little more group theory. The point of this book is to pass you through that barrier using easy groups. Should a researcher do so successfully, we'll show by example precisely what you might ask a group theorist to go further.

An historical point shows why we need the profinite theory. Abel introduced the most basic modular curves, those parametrizing elliptic curves with a p division point. It was Galois who recognized, and used, that there was a series of modular curves corresponding to a series of powers of p . We anticipate the non-abelian version of this for Modular Towers in the abelian theory of this chapter. Please don't skip that part. Chap. 1 of the Fried-Jarden book is very elementary and it has all the profinite basics we need until later chapters.