MA342J: Introduction to Modular Forms
Tutorial 2, February 9
j-invariant
In class we proved that
Γ\H ∼
= P 1 (C) .
In particular there must be a holomorphic function
j:H→C
such that
az + b = j(z)
cz + d
for every integer matrix with determinant 1 and a simple pole at infinity, i.e.
j
j(z) =
1
+ a0 + a1 q + . . .
q
Moreover, since we already put its residue at ∞ to be equal 1, it is unique
up to adding constants. Therefore a0 can be anything but all an , n ≥ 1 are
uniquely defined by our requirements. By geometric reasons (which we will see
soon in this course) it is convenient to put a0 = 744. This function j is called
j-invariant.
Question 1: Construct
j(z) =
1
+ 744 + a1 q + a2 q 2 + . . .
q
in terms of Eisenstein series, then compute a couple of its first Fourier coefficients, e.g. a1 and a2 .
Question 2: Prove that j takes different values on different Γ-orbits, i.e.
j(z1 ) = j(z2 ) implies that z2 = gz1 for some g ∈ Γ. Hint: use “zero counting
formula”.
Question 3: Compute the values j(i) and j(ρ).
1