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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