MODULAR FORMS

Problem sheet 0 KMB, Michaelmas 1997

Example sheet 0 is just a collection of exercises, each of which proves a result which was used in some way in the course. The proofs of the results were either deemed “not interesting enough to include in the course” or “not number-theoretic enough to include in the course” by me, and this example sheet contains sketches of the proofs. Of course, if I assume a result in the course, you have every right to assume the result in the exam, so in particular doing these questions will probably not increase your mark in the exam (although it will increase your depth of knowledge!)

1. In this question we shall fill in some of the details concerning the calculation of the q-expansion of the Eisenstein series G_{k}.

(a) Prove that for k ≥ 4 and z ∈ ℌ the series

G_{k}(z) := \begin{matrix}∑ & ′ \\ m,n \end{matrix} \frac{1}{(mz + n)^{k}}

converges absolutely (where Σ′ denotes the sum over all elements of ℤ^{2} other than (0, 0).)

(b) Let F denote the set \{z ∈ ℌ | −\frac{1}{2} ≤ Re(z) ≤ \frac{1}{2} and |z| ≥ 1 \}. Prove that if z ∈ F and (m, n) ∈ ℤ^{2} then |mz + n|^{2} ≥ m^{2} − |mn| + n^{2} ≥ min{|mω + n|^{2}, |mσ + n|^{2} } where ω = e^{2πi/3} and σ = e^{πi/3}. Deduce that G_{k}(z) converges uniformly in F.

(c) Justify the interchange of the sum and the limit in the argument I gave which proved that G_{k}(z) was bounded at infinity.

2. In the computation of the q-expansion of G_{k}(z) I used the fact that ∑_{r∈ℤ} 1/(z +r)^{2} = π^{2}/ sin^{2}(πz) for z ∈ ℌ. This question sketches a proof of this fact.

(a) Consider the function

F (z) = −\frac{π^{2}}{sin^{2}(πz)} + ∑_{r∈ℤ} \frac{1}{(z + r)^{2}}

defined for z ∈ ℂ\ℤ. Show that this function can be extended to a holomorphic function on all of ℂ.

(b) Show that F (z/2) + F ((z + 1)/2) = 4F (z), and by considering sup_{x∈[0,1]} |F (x)| show that F is identically zero.

3. In this question we shall prove that ζ(k) is a rational multiple of π^{k} for k ≥ 2 an even integer, and we shall also give a method for computing these rational numbers.

(a) Let

F (z) = \frac{1}{z} + ∑_{m=1}^{∞} (\frac{1}{z + m} + \frac{1}{z − m}) + \frac{2iπ}{1 − e^{2iπz}}

for z ∈ ℂ\ℤ. Show that F is differentiable and that F′ = 0. (Hint: use the lemma proved in the lectures which evaluated ∑_{m∈ℤ} 1/(z + m)^{2}.)

(b) Show that π cot(πz) = iπ − \frac{2iπ}{1−q} where q = e^{2iπz}. Deduce that for |w| < π

w cot w = cw + 1 − 2 ∑_{n≥1,m≥1} \frac{w^{2n}}{π^{2n}m^{2n}}

for some c ∈ ℂ. Deduce that c = 0 and evaluate F.

(c) Define the Bernoulli numbers B_{m} by the formal power series identity

\frac{x}{e^{x} − 1} = 1 − \frac{x}{2} + ∑_{m≥1}(−1)^{m+1}B_{m} \frac{x^{2m}}{(2m)!}.

Show that the B_{n} are rational, and that the sum converges for |x| < 2π. Moreover, by setting x = 2iw, show that w cot w = 1 − ∑_{m≥1} B_{m}2^{2m}w^{2m}/(2m)! for |w| < 1.

(d) Deduce that if k ≥ 2 is even, then

ζ(k) = \frac{2^{k−1}}{k!} π^{k}B_{k/2}.

(e) Verify a couple of the following statements:

ζ(2) = π^{2}/6, ζ(4) = π^{4}/90, ζ(6) = π^{6}/945,
ζ(8) = π^{8}/9450, ζ(10) = π^{10}/93555, ζ(12) = 691π^{12}/638512875,
ζ(14) = 2π^{14}/18243225.

(I used the computer algebra package PARI to generate these, no doubt many other packages would do just as well.)

4. We say that a subgroup Γ of a topological group G is discrete if there exists an open subset U of G such that U ∩ Γ = {1}. This question is a proof that if Γ is a discrete subgroup of the topological group G, and X is a compact subset of G, then Γ ∩ X is finite. We used this result when computing properties of Γ\ℌ and we shall possibly use it again.

(a) Show that there are neighbourhoods V_{1} and V_{2} of 1 ∈ G such that V_{1} · V_{2} ⊆ U (Hint: Show W = {(g, h) ∈ G × G | gh ∈ U } is open.) Show furthermore that there is a neighbourhood V of 1 ∈ G such that V = V^{−1} and V · V ⊆ U.

(b) By considering the open cover

X = ⋃_{g∈G}(gV ∩ X)

show that Γ ∩ X is finite.