Consider a compact Riemann surface (or smooth projective algebraic curve in characteristic zero) {X}. One of the first facts one observes in their theory is that the group {\mathrm{Aut}(X)} of automorphisms of {X} is quite large when {X} has genus zero or one. When {X} has genus zero, it is the projective line, and its automorphism group is {\mathrm{PGL}_2(\mathbb{C})}, a fact which generalizes naturally to higher projective spaces. When {g = 1}, the curve {X} acquires the structure of an elliptic curve from any distinguished point. Thus, translations by any element of {X} act on {X}. So {\mathrm{Aut}(X)} has points corresponding to each element of {X} (and a few more, such as inversion in the group law).

But when the genus of {X} is at least two, things change dramatically. It is a famous theorem that the number of automorphisms is bounded:

Theorem 1 (Hurwitz) If {X} is a compact Riemann surface of genus {g \geq 2}, then there are at most {84(g-1)} automorphisms of {X}.

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 {g} together with Riemann surfaces with exactly {84(g-1)} automorphisms; there are explicit constructions that I’m not very familiar with. It is false in characteristic p, 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 {X} is a compact Riemann surface of genus {g = p+1} for {p>84} a prime, then {\mathrm{Aut}(X)} has order strictly less than {84(g-1)}. (more…)

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Now we’re going to use the machinery already developed to prove the existence of harmonic functions.

Fix a Riemann surface {M} and a coordinate neighborhood {(U,z)} isomorphic to the unit disk {D_1} in {\mathbb{C}} (in fact, I will abuse notation and identify the two for simplicity), with {P \in M} corresponding to {0}.

First, one starts with a function {h: D_1 - \{0 \} \rightarrow \mathbb{C}} such that:

1. {h} is the restriction of a harmonic function on some {D_{1+\epsilon} - 0} 2. {d {}^* h = 0} on the boundary {\partial D_1} (this is a slight abuse of notation, but ok in view of 1).

The basic example is {z^{-n} + \bar{z}^{n}}.

Theorem 1 There is a harmonic function {f: M - P \rightarrow \mathbb{C}} such that {f-h} is continuous at {P}, and {\phi df \in L^2(M)} if {\phi} is a bounded smooth function that vanishes in a neighborhood of {P}.

 

In other words, we are going to get harmonic functions that are not globally defined, but whose singularities are localized. (more…)

I realize I’ve been slow as of late.

Though the decomposition of the square-integrable 1-forms was a Hilbert space, L^2-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 E term.  The ultimate goal is the existence theorems for harmonic functions on Riemann surfaces.

Smooth differentials in {E, E^*}

I now claim that any smooth differential {\omega \in E} is exact and any smooth differential in {E^*} is co-exact. This is nontrivial because of the way we took completions. I will only prove the claim for {E}.

We already have closedness of {\omega} by the previous post. To show exactness, it will be sufficient to show that for any closed smooth curve {c},

\displaystyle \int_c \omega = 0

because we could use a path integral to define the antiderivative. We can approximate {c} by a simple closed curve homotopic to {c}, so we can assume at the outset that {c} is a homeomorphism {S^1 \rightarrow M}

Proposition 1 Given {c} as above, there is a closed differential {\eta_c} such that\displaystyle ( \omega, {}^* \eta_c) = \int_c \omega   

for any closed {\omega}

To prove this, find an annular neighborhood of the compact set {c(S^1) \subset M}, which is divided into {R_1, R_2} as in the figure.

(more…)

Yesterday I defined the Hilbert space of square-integrable 1-forms {L^2(X)} on a Riemann surface {X}. Today I will discuss the decomposition of it. Here are the three components:

1) {E} is the closure of 1-forms {df} where {f} is a smooth function with compact support.

2) {E^*} is the closure of 1-forms {{}^* df} where {f} is a smooth function with compact support.

3) {H} is the space of square-integrable harmonic forms.

Today’s goal is:

Theorem 1 As Hilbert spaces,

\displaystyle L^2(X) = E \oplus E^* \oplus H.   

The proof will be divided into several steps. (more…)

It’s now time to do some more manipulations with differential forms on a Riemann surface. This will establish several notions we will need in the future.

The Hodge star

Given the 1-form {\omega} in local coordinates as {u dz + v d\bar{z}}, define

\displaystyle ^*{\omega} := -iu dz + iv d\bar{z} .

In other words, given the decomposition {T^*(X) = T^{*(1,0)}(X) \oplus T^{*(0,1)}(X)}, we act by {-i} on the first sumamand and by {i} on the second. This shows that the operation is well-defined. Note that {^*{}} is conjugate-linear and {^*{}^2 = -1}. Also, we see that {^*{} dx = dy, ^*{dy} = -dx} if {z = x + iy}.  This operation is called the Hodge star.

From the latter description of the Hodge star we see that for any smooth {f},

\displaystyle d ^*{} df = d( -if_z dz + if_{\bar{z}} d\bar{z}) = 2i f_{z \bar{z}} dz \wedge d\bar{z}.

From the definitions of {f_{z}, f_{\bar{z}}}, this can be written as {-2i \Delta f dz \wedge d\bar{z}} if {\Delta} is the usual Laplacian with respect to the local coordinates {x,y}.

The Hodge star allows us to define co things. A form {\omega} is co-closed if {d ^*{} \omega = 0}; it is co-exact if {\omega = ^* df} for {f} smooth. (more…)

Finally, it’s time to get to the definition of a Riemann surface. A Riemann surface is a connected one-dimensional complex manifold. In other words, it’s a Hausdorff space {M} which is locally homeomorphic to {\mathbb{C}} via charts (i.e., homeomorphisms) {\phi_i:U_i \rightarrow V_i} for {U_i \subset M, V_i \subset \mathbb{C}} open and such that {\phi_j \circ \phi_i^{-1}: V_i \cap V_j \rightarrow V_i \cap V_j} is holomorphic.

Examples

Evidently an open subset of a Riemann surface is a Riemann surface. In particular, an open subset of {\mathbb{C}} is a Riemann surface in a natural manner.

The Riemann sphere {P^1(\mathbb{C}) := \mathbb{C} \cup \{ \infty \}} or {S^2} is a Riemann sphere with the open sets {U_1 = \mathbb{C}, U_2 = \mathbb{C} - \{0\} \cup \{\infty\}} and the charts

\displaystyle \phi_1 =z, \ \phi_2 = \frac{1}{z}.

The transition map is {\frac{1}{z}} and thus holomorphic on {U_1 \cap U_2 = \mathbb{C}^*}.

An important example comes from analytic continuation, which I will briefly sketch below. A function element is a pair {(f,V)} where {f: V \rightarrow \mathbb{C}} is holomorphic and {V \subset \mathbb{C}} is an open disk. Two function elements {(f,V), (g,W)} are said to be direct analytic continuations of each other if {V \cap W \neq \emptyset} and {f \equiv g } on {V \cap W}. By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).

Starting with a given function element {\gamma = (f,V)}, we can consider the totality {X} of all equivalence classes of function elements that can be obtained by continuing {\gamma} along curves in {\mathbb{C}}. Then {X} is actually a Riemann surface. (more…)