complex analysis

Let {\phi: \Omega_{SO} \rightarrow \Lambda} be a genus. We might ask when {\phi} satisfies the following multiplicative property:

Property: For any appropriate fiber bundle {F \rightarrow E \rightarrow B} of manifolds, we have

\displaystyle \phi(E) = \phi(B) \phi(F). \ \ \ \ \ (1)

When {B} is simply connected, this is true for the signature by an old theorem of Chern, Hirzebruch, and Serre.

A special case of the property (1) is that whenever {E \rightarrow B} is an even-dimensional complex vector bundle, then we have

\displaystyle \phi(\mathbb{P}(E)) = 0,

for {\mathbb{P}(E)} the projectivization: this is because {\mathbb{P}(E) \rightarrow B} is a fiber bundle whose fibers are odd-dimensional complex projective spaces, which vanish in the cobordism ring.

Ochanine has given a complete characterization of the genera which satisfy this property.

Theorem 1 (Ochanine) A genus {\phi} annihilates the projectivizations {\mathbb{P}(E)} of even-dimensional complex vector bundles if and only if the associated log series {g(x) = \sum \frac{\phi(\mathbb{CP}^{2i})}{2i+1} x^{2i+1}} is given by an elliptic integral

\displaystyle g(x) = \int_0^x Q(u)^{-1/2} du,

for {Q(u) = 1 - 2\delta u^2 + \epsilon u^4} for constants {\delta, \epsilon}.

Such genera are called elliptic genera. Observe for instance that in the case {\epsilon = 1, \delta = 1}, then

\displaystyle g(x) = \int_0^x \frac{du}{1 - u^2} = \tanh^{-1}(u),

so that we get the signature as an example of an elliptic genus (the signature has {\tanh^{-1}} as logarithm, as we saw in the previous post).

I’d like to try to understand the proof of Ochanine’s theorem in the next couple of posts. In this one, I’ll describe the proof that an elliptic genus in fact annihilates projectivizations {\mathbb{P}(E)} of even-dimensional bundles {E}. (more…)

Apologies for the lack of posts lately; it’s been a busy semester. This post is essentially my notes for a talk I gave in my analytic number theory class.

Our goal is to obtain bounds on the distribution of prime numbers, that is, on functions of the form {\pi(x)}. The closely related function

\displaystyle \psi(x) = \sum_{n \leq x} \Lambda(n)

turns out to be amenable to study by analytic means; here {\Lambda(n)} is the von Mangolt function,

\displaystyle \Lambda(n) = \begin{cases} \log p & \text{if } n = p^m, p \ \text{prime} \\ 0 & \text{otherwise} \end{cases}.

Bounds on {\psi(x)} will imply corresponding bounds on {\pi(x)} by fairly straightforward arguments. For instance, the prime number theorem is equivalent to {\psi(x) = x + o(x)}.

The function {\psi(x)} is naturally connected to the {\zeta}-function in view of the formula

\displaystyle - \frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^\infty \Lambda(n) n^{-s}.

In other words, {- \frac{\zeta'}{\zeta}} is the Dirichlet series associated to the function {\Lambda}. Using the theory of Mellin inversion, we can recover partial sums {\psi(x) = \sum_{n \leq x} \Lambda(x)} by integration of {-\frac{\zeta'}{\zeta}} along a vertical line. That is, we have

\displaystyle \psi(x) = \frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} -\frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds ,

at least for {\sigma > 1}, in which case the integral converges. Under hypotheses on the poles of {-\frac{\zeta'}{\zeta}} (equivalently, on the zeros of {\zeta}), we can shift the contour appropriately, and estimate the integral to derive the prime number theorem. (more…)

First, thanks to all who kindly contributed advice on the previous post.

A long on-and-off project of mine has been to learn several complex variables. My latest attempt started a few days back, though the commencement of the fall semester may derail it. As is now customary for myself, I have started writing a set of notes that I intend to make grow reasonably large, though right now it is less than thirty pages (even with parts of my AG notes on coherence copied and pasted in). I shall post here an excerpt on the Weierstrass preparation theorem, which will assume only the definition of a holomorphic function in several variables). (more…)

So.  First off, surely the five remaining readers of this increasingly erratic blog have noticed the change of theme.

I want to next discuss the second inequality in class field theory, which is an upper bound on the norm index of the idele group.  There are two ways I know of to prove this: one analytic, one algebraic.  I will first sketch the analytic one. I say sketch because to do a full proof would get into the details of Dirichlet series, lattice points in homogeneously expanding domains, the construction of a certain fundamental domain for the action of the units, etc., etc., and I’d rather outline those ideas rather than do all the details because this is a series on class field theory. What I do plan on doing properly, however, is the algebraic (due to Chevalley in 1940) proof of the second inequality, which heavily uses results of field theory (e.g. Kummer theory) and local fields (e.g. power index computations).  I still thought it worthwhile to sketch the analytic approach, though. Rather than jumping right into it (I have to first say something about how the ideal and idele groups are connected), I decided to give an expository post on L-functions and Dirichlet’s theorem—in the case of the rational numbers.

1. Ramblings on the Riemann-zeta function

Recall that the Riemann-zeta function is defined by {\zeta(s) =  \sum n^{-s}}, and that it is intimately connected with the distribution of the prime numbers because of the product formula

\displaystyle  \zeta(s) = \prod_p (1 -  p^{-s})^{-1}

valid for {Re(s)>1}, and which is a simple example of unique factorization. In particular, we have

\displaystyle  \log \zeta(s) = \sum_p p^{-s} +  O(1) , \ s \rightarrow 1^+.

It is known that {\zeta(s)} has an analytic continuation to the whole plane with a simple pole with residue one at {1}. The easiest way to see this is to construct the analytic continuation for {Re (s)>0}. For instance, {\zeta(s) - \frac{1}{s-1}} can be represented as a certain integral for {Re(s)>1} that actually converges for {Re(s)>0} though. (The functional equation is then used for the rest of the analytic continuation.) The details are here for instance. As a corollary, it follows that

\displaystyle  \sum_p p^{-s} = \log  \frac{1}{s-1} + O(1) , \ s \rightarrow 1^+.

This fact can be used in deducing properties about the prime numbers. (Maybe sometime I’ll discuss the proof of the prime number theorem on this blog.) Much simpler than that, however, is the proof of Dirichlet’s theorem on the infinitude of primes in arithmetic progressions. I will briefly outline the proof of this theorem, since it will motivate the idea of L-functions.

Theorem 1 (Dirichlet) Let {\{an+b\}_{n  \in \mathbb{Z}}} be an arithmetic progression with {a,b} relatively prime. Then it contains infinitely many primes.

The idea of this proof is to note that the elements of the arithmetic progression {\{an+b\}} can be characterized by so-called “Dirichlet characters.” This is actually a general and very useful (though technically trivial) fact about abelian groups, which I will describe now.


I fell a bit behind on the continuation of the class field theory series because I was setting up a new laptop. Before I resume that, I want to talk about something very weird that I learned today.

Let {U \subset \mathbb{C}} be a set that omits at least two points. If {f: U \rightarrow U} holomorphic and is such that f(w)=w, {f'(w)=1} at one {w \in U}, then {f} is the identity.

This is a striking rigidity phenomenon!

But how do we prove it? The idea is to consider the sequence of iterates {f, f \circ f, \dots}. Suppose for simplicity {P=0}. Then in a neighborhood of {0}, we can write {f = z + cz^m + \dots }, where the {\dots} are omitted higher terms. If {f} is not identically the identity, then {c \neq 0}.

So, similarly, by direct computation, in some neighborhood of {P}, we have {f \circ f = z + 2c z^m + \dots}. Similarly, if we define {g_1 = f, g_2 = f \circ f, } for notational convenience, we have

\displaystyle g_k = z + kc z^m + \dots.

But the {g_k} are all holomorphic maps into {U}. Since {U} omits at least two points, the family {g_k} is normal by Montel’s theorem and consequently has a subsequence {g_{k_i}} that converges uniformly on compact sets.

Thus the derivatives {g^{(m)}_{k_i}(0) = m! k_i c} converge, which is impossible unless {c=0}.

Huh? I didn’t exactly see that coming. If {U} is the unit disk, then at least it looks familiar. A holomorphic map {f} of the unit disk into itself sending zero to zero must satisfy {|f'(0)| \leq 1}, and if equality holds {f} is a rotation. So perhaps this result should be thought of as a generalization of Schwarz’s lemma? (Nevertheless, the use of Montel’s theorem is quite a sledgehammer to prove something as elementary as Schwarz.)

I should say where I got this from: Krant’z Complex Analysis: The Geometric Viewpoint. Krantz didn’t prove exactly this, but the argument is the same.  Either this is standard fare that I missed when learning basic complex analysis, or I’m turning Climbing Mount Bourbaki into a comedy routine.

Apologies for the embarrassingly bad pun in the title.

Distributions in general

First, it’s necessary to talk about distributions on an arbitrary open set {\Omega \subset \mathbb{R}^n}, which are not necessarily tempered. In particular, they may “grow arbitrarily” as one approaches the boundary. So, instead of requiring a functional on a Schwarz space, we consider functionals on {C_0^{\infty}(\Omega),} the space of smooth functions compactly supported in {\Omega}. However, we need some notion of continuity, which would require a topology on {C_0^{\infty}(\Omega)}. There is now the tricky question of how we would require completeness of the topological vector space {C_0^{\infty}(\Omega)}, which we of course desire. We can get such a topology by talking about “strict inductive limits” and whatnot, but since I don’t really find that particularly fun, I’ll sidestep it (but not really—most of the ideas will still remain).

Anyway, the idea here will be to consider auxiliary spaces {C^{\infty}(K)} for {K \subset \Omega} compact. This is the space of smooth functions {f: \Omega \rightarrow \mathbb{R}} which are supported in {K}. We give the space a Frechet topology by the family of seminorms

\displaystyle ||f||_a := \sup_K |D^a f|. (more…)

A friend of mine is taking a course on analytic number theory in the spring and needs to learn basic complex analysis in a couple of weeks.  I decided to do a post (self-contained, except for Stokes’ formula) on deducing the Cauchy theorems and their applications from Stokes’ theorem now instead of later–when I’ll talk about several complex variables.  It might be objected that Stokes’ theorem is just Green’s theorem for n=2, commonly used in undergraduate treatments, but my goal was to take an expository challenge: write something rigorous on complex variables in as short a space as possible without sacrificing readability.  So Stokes’ theorem for manifolds is preferable to Green’s theorem as stated in a vague way about “insides of a curve” (before, say, the Jordan curve theorem is proved) and the traditional proof of Green’s theorem via rectangular decompositions.

So, let’s consider an open set {O \subset \mathbb{C}}, and a {C^2} function {f: O \rightarrow \mathbb{C}}. We can consider the differential

\displaystyle df := f_x dx + f_y dy

which is a complex-valued 1-form on {O}. It is also convenient to write the differential using the {z} and {\bar{z}}-derivatives I talked about earlier, i.e.

\displaystyle f_z := \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) f, \quad f_{\bar{z}} := \frac{1}{2}\left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) f.

The reason these are important is that if {w_0 \in O}, we can choose {A,B \in \mathbb{C}} with

\displaystyle f(w_0+h) = f(w_0) + Ah + B \bar{h} + o(|h|), \ h \in \mathbb{C}

by differentiability, and it is easy to check that {A=f_z(w_0), B=f_{\bar{z}}(w_0)}. So we can define a function {f} to be holomorphic if it satisfies the differential equation

\displaystyle f_{\bar{z}} = 0,

which is equivalent to being able to write

\displaystyle f(w_0 + h) =f(w_0) + Ah + o(|h|)

for each {w_0 \in O} and a suitable {A \in \mathbb{C}}. In particular, it is equivalent to a difference quotient definition. The derivative {f_z} of a holomorphic function thus satisfies all the usual algebraic rules, under which holomorphic functions are closed. (more…)

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…)

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…)

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