Math 678, Lecture 1

Notes by Bryden Cais

Sept. 4, 2002

1 Modular Functions

Observe that we have a discrete action of ℤ on ℝ by translations:

z ↦ z + a.

We can ask for the ℝ vector space of smooth functions invariant under this action, that is, the space of smooth fnctions on ℝ/ℤ. Fourier analysis tells us that this is spanned by the functions

{e(nz) : n ∈ ℤ} ,

where e(z) = exp(2πiz). The situation becomes much more interesting when we consider the following generalization: Let ℍ = {z ∈ ℂ : ℑ(z) > 0} be the usual upper half plane. Then we have an action of SL_{2}(ℤ) on ℍ via fractional linear transformation:

(\begin{matrix}a & b \\ c & d \end{matrix}) z = \frac{az + b}{cz + d}.

(The right way to think about this action is as the action inherited from the natural action of SL_{2}(()ℤ) on ℂ^{2} \ {0}/ℂ^{∗}). We then want to study the holomorphic functions on ℍ invariant under the action of SL_{2}(ℤ). This turns out to be the wrong generalization: it isn’t hard to show that ℍ/SL_{2}(ℤ) has the structure of a compact Riemann surface (as long as we add the point at infinity). It is a fact that any holomorphic function on a compact Riemann surface must be constant. So we relax the conditions on the functions under consideration to holomorphic functions f on ℍ such that

f(γz) = j(γ, z)f(z),

for some simple function j with no poles in ℍ. Clearly, if we are to have a genuine action of SL_{2}(ℤ) on these functions, we must have

f(γ_{1}γ_{2}z) = f(γ_{1}(γ_{2}z)),

which implies that we must have

j(γ_{1}γ_{2}, z) = j(γ_{1}, γ_{2}z)j(γ_{2}, z).  (1)

The simplest function j that one can imagine is the function j(γ, z) = (cz + d)^{k}, where

γ = (\begin{matrix}a & b \\ c & d \end{matrix}) .

It is not difficult to verify that this function satisfies (1). (Notice that it is sufficient to check this for k = 1 since z ↦ z^{k} is a homomorphism.)

We will therefore define (at least for now) a modular function to be a holomorphic function f on ℍ satisfying

f(γz) = (cz + d)^{k}f(z)

for all γ = \begin{matrix}a & b \\ c & d \end{matrix} ∈ SL_{2}(ℤ) and z ∈ ℍ. Observe that since

(\begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix}) , (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) ∈ SL_{2}(ℤ),

any such f must have k even, and is periodic with period 1, so it has a fourier series expansion.

One of the main threads running throughout the course will be the connection between L series and modular functions. To motivate this connection, consider the Riemann zeta function defined by

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} .

It is not hard to see that this series defines a holomorphic function of s for ℜ(s) > 1. Moreover, ζ(s) satisfies a functional equation:

π^{−s/2}Γ (\frac{s}{2}) ζ(s) = π^{s−1/2}Γ (\frac{1 − s}{2}) ζ(1 − s).  (2)

The standard way to prove this is to take mellin transforms of both sides of the identity

∑_{n∈ℤ}e(−n^{2}/z) = \sqrt{\frac{z}{i}} ∑_{n∈ℤ}e(n^{2}z),  (3)

which in turn is proved using the Poisson summation formula. This states that for any smooth function f with compact support we have

∑_{n∈ℤ}f(n) = ∑_{n∈ℤ} f̂(z)

Exercise 1. Prove the Poisson summation formula.

2 Divisor Sums and Sums of Squares

Modular functions are ubiquitous in modern number theory. Define

σ_{k−1}(n) = \begin{cases}\frac{(k−1)!ζ(k)}{(2πi)^{k}} & if n = 0 \\ ∑_{d|n} d^{k−1} & otherwise \end{cases} .

In 1916, Ramanujan stated the remarkable result

∑_{a+b=m} σ_{r−1}(a)σ_{s−1}(b) = \frac{(r − 1)!(s − 1)!ζ(r)ζ(s)}{(r + s − 1)!ζ(r + s)}σ(r + s − 1) + ε,

where ε is an error term. It turns out that for m < 12, we have ε = 0 (which will turn out to be because the space of cusp forms of dimension m < 12 is just {0}). This identity is an analytic way of expressing very important algebraic information about the structure of modular forms of a given weight.

Another number theoretic context in which modular forms naturally arise is that of the problem of the number of ways of representing a positive integer as the sum of k squares. Let r_{k}(n) denote the number of ways that n can be represented as the sum of k squares. The formulae

r_{2}(n) = ∑_{d|n} (\frac{−4}{d})
r_{4}(n) = 8(∑_{d|n} d − ∑_{4d|n} 4d)

are classical. It turns out that r_{k}(n) san similarly be expressed as such a divisor sum, though as k increases things tend to get much more complicated. This result comes straight out of the theory of modular functions.

3 The τ function

A beautiful example of the interplay between modular functions and number theory arises when one considers the function

Δ(z) = q ∏_{n=1}^{∞}(1 − q^{n})^{24} = ∑_{n=1}^{∞} τ(n)q^{n},

where q = e(z). One can show that Δ is a modular function (of weight 12). The numbers τ(n) are obviously integers. Moreover, they enjoy remarkable number theoretic properties. Namely, we have

τ(m)τ(n) = τ(mn)

whenever (m, n) = 1, and

|τ(p)| ≤ 2p^{11/2}

for all primes p. Both of these results are deep (especially the latter, which was only proved recently by Deligne) and come from the theory of modular forms.