November 10, 2011
Consider a compact Riemann surface (or smooth projective algebraic curve in characteristic zero) . One of the first facts one observes in their theory is that the group of automorphisms of is quite large when has genus zero or one. When has genus zero, it is the projective line, and its automorphism group is , a fact which generalizes naturally to higher projective spaces. When , the curve acquires the structure of an elliptic curve from any distinguished point. Thus, translations by any element of act on . So has points corresponding to each element of (and a few more, such as inversion in the group law).
But when the genus of is at least two, things change dramatically. It is a famous theorem that the number of automorphisms is bounded:
Theorem 1 (Hurwitz) If is a compact Riemann surface of genus , then there are at most automorphisms of .
I won’t write out a proof here; a discussion is in lecture 9 of the notes I’m taking in an algebraic curves class. In fact, the hard part is to show that there are finitely many automorphisms, after which it is a combinatorial argument.
This bound is often sharp. There are infinitely many genera together with Riemann surfaces with exactly automorphisms; there are explicit constructions that I’m not very familiar with. It is false in characteristic , because the Riemann-Hurwitz formula is no longer necessarily true (because of the existence of non-separable morphisms), and counterexamples are given in the notes.
What I want to describe today is that this bound is often not sharp.
Theorem 2 If is a compact Riemann surface of genus for a prime, then has order strictly less than . (more…)
December 12, 2009
I realize I’ve been slow as of late.
Though the decomposition of the square-integrable 1-forms was a Hilbert space, -decomposition, it reflects many smoothness properties in which we are interested. The goal of this post is twofold: first, show that the smooth differentials in the first two components are respectively the exact and coexact ones; and second, show that the closedness of a square-integrable 1-form in some neighborhood implies the smoothness of the corresponding term. The ultimate goal is the existence theorems for harmonic functions on Riemann surfaces.
Smooth differentials in
I now claim that any smooth differential is exact and any smooth differential in is co-exact. This is nontrivial because of the way we took completions. I will only prove the claim for .
We already have closedness of by the previous post. To show exactness, it will be sufficient to show that for any closed smooth curve ,
because we could use a path integral to define the antiderivative. We can approximate by a simple closed curve homotopic to , so we can assume at the outset that is a homeomorphism .
Proposition 1 Given as above, there is a closed differential such that
for any closed .
To prove this, find an annular neighborhood of the compact set , which is divided into as in the figure.
December 6, 2009
Yesterday I defined the Hilbert space of square-integrable 1-forms on a Riemann surface . Today I will discuss the decomposition of it. Here are the three components:
1) is the closure of 1-forms where is a smooth function with compact support.
2) is the closure of 1-forms where is a smooth function with compact support.
3) is the space of square-integrable harmonic forms.
Today’s goal is:
Theorem 1 As Hilbert spaces,
The proof will be divided into several steps. (more…)