MODULAR FORMS

Problem sheet 1 KMB, Michaelmas 1997

1. For k an integer, let M_{k} denote the set of modular forms of weight k and let S_{k} denote the subset of cusp forms.

(a) Show that M_{k} is naturally a complex vector space, and that the map M_{k} → ℂ sending ∑_{n≥0} a_{n}q^{n} to a_{0} is a linear map with kernel S_{k}.

(b) Show that if f ∈ M_{k} and g ∈ S_{l} then fg ∈ S_{k+l}. Show that the constant functions are in M_{0}. Show that ⊕_{k∈ℤ} M_{k} is a commutative ring and that ⊕_{k∈ℤ} S_{k} is an ideal.

(c) By using question 3 of example sheet 0, one can prove that ζ(4) = π^{4}/90 and that ζ(6) = π^{6}/945. Assuming these results, and by considering (the first couple of terms in the q-expansion of) (G_{4}(z))^{3} and (G_{6}(z))^{2}, show that S_{12} is non-zero. Deduce that S_{k} is non-zero for all even k ≥ 16 as well.

2. Recall that the Riemann zeta function ζ(s) is defined for Re(s) > 1 by ζ(s) = ∑_{n≥1} n^{−s}. It is a basic fact from complex analysis that ζ(s) has a meromorphic continuation to all of ℂ and satisfies the functional equation

ζ(s) = 2^{s}π^{s−1} sin(\frac{1}{2}sπ)Γ(1 − s)ζ(1 − s),

where Γ(z) is the Gamma function (The Gamma function satisfies Γ(n) = (n − 1)! for n ≥ 1 an integer, and has a pole for n ≤ 0 an integer.) For the proof of the analytic continuation and the functional equation, see almost any book on complex analysis or analytic number theory. The first 27 pages of Titchmarsh’s book “The Theory Of The Riemann Zeta Function” contain no fewer than seven proofs of the functional equation.

(a) Using the functional equation, and question 3 of example sheet 0, show that ζ(n) is a non-zero rational number for every negative odd integer n.

(b) Show that if k ≥ 4 is an integer, then one can deduce from the calculation of the q-expansion of the Eisenstein series G_{k} that

\frac{ζ(1 − k)}{2} + ∑_{n≥1} σ_{k−1}(n)q^{n}

is a modular form of weight k.

Remark: The definition σ_{r}(n) = ∑_{0<d|n} d^{r} makes sense for n a negative integer, where we see σ_{r}(n) = σ_{r}(−n). If n = 0 then the definition gives σ_{r}(n) = 1^{r} + 2^{r} + 3^{r} + … which clearly doesn’t make sense, but one might be tempted, after looking at the definition ζ(s) = ∑_{n≥1} n^{−s}, to define σ_{r}(0) to be ζ(−r), which of course does make sense. Hence there is now an easy way of remembering the q-expansion of the modular form of weight k occurring in part (b), it’s just

\frac{1}{2} ∑_{n∈ℤ} σ_{k−1}(n)q^{n}.

This is of course not really maths as it stands, but nonetheless is a helpful reminder of the formula.

(3) Evaluating modular forms at elliptic points.

Show that if f is a modular form of weight not divisible by 4, then f(i) = 0 (Hint: use the fact that there are matrices in SL_{2}(ℤ) other than ±I which fix i.) Show similarly that if f is a modular form of weight not divisible by 6 then f(ω) = 0, where ω = e^{2πi/3}.

(4) Strange relations between σ_{k}(n).

(a) Use the fact (not yet proved) that the dimension of M_{10} is 1 to find a non-trivial formula for σ_{9}(n) involving σ_{3} and σ_{5} (see example sheet 0 for the relevant values of the zeta function.)

(b) If you have nothing better to do, then use the fact that the dimension of M_{14} is also 1 to find some non-trivial formulae for σ_{13}(n).

(c) Using part (c) of question 1, show (much to your relief) that the dimension of M_{k} is greater than 1 for even k ≥ 16.

(5) This is the first example of how modular forms and lattices are related.

(a) Let Λ′ be the lattice in ℝ^{8} given by {(a_{i})_{1≤i≤8} ∈ ℤ^{8} | ∑_{1≤i≤8} a_{i} is even }. Let Λ be the lattice generated by Λ′ and (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}). Show that if a = (a_{i}) ∈ Λ and b = (b_{i}) ∈ Λ then (a, b) = ∑_{i} a_{i}b_{i} is an integer. Show moreover that if a ∈ Λ then (a, a) is even.

(b) Let a_{n} denote the number of elements t of Λ such that (t, t) = 2n. Work out a_{0}, a_{1}, a_{2} and a_{3}. Observe that ∑ a_{n}q^{n} looks like it might be a modular form of weight 4, judging by the first few terms.

Remark: In fact, ∑ a_{n}q^{n} really is a modular form of weight 4. Moreover, this kind of technique (counting points in lattices with certain norms) gives modular forms in much greater generality; we may say something about this later, once we have defined modular forms of higher level.