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:
- How to picture compact Riemann surface covers of the
sphere with nonabelian monodromy groups (not even necessarily solvable).
- 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.
- 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.
- 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.