Modular Forms

The modular group SL(2, ℤ) acts on the upper half-plane by Möbius transformations. The fundamental domain — points not equivalent to each other under the group — is the strip

F = {τ ∈ ℍ : |Re(τ)| ≤ 1/2 and |τ| ≥ 1}

This is the "ice cream cone" shape with a cusp going to i∞. Two generators: S: τ → −1/τ (the unit-disk flip) and T: τ → τ + 1 (horizontal shift). Every element of SL(2, ℤ) is a word in S and T.

The j-invariant:
j(τ) = q⁻¹ + 744 + 196884 q + 21493760 q² + …
where q = e^(2πiτ). j is uniquely characterized by being meromorphic, modular (j(γτ) = j(τ) for γ ∈ SL(2, ℤ)), with the cusp asymptotics q⁻¹ + 744 + …

Monstrous moonshine: the coefficient 196884 = 196883 + 1, where 196883 is the dimension of the smallest non-trivial irreducible representation of the Monster Group (the largest sporadic finite simple group, 8 × 10⁵³ elements). McKay noticed this in 1978. Conway and Norton conjectured a vast generalization. Borcherds proved it in 1992 (Fields Medal 1998). Vertex operator algebras and string theory turned out to be involved. No one fully understands why.

The Heegner points: special points τ = (−1 + √(−d))/2 for d = 1, 2, 3, 7, 11, 19, 43, 67, 163. At each, j(τ) is an integer. The case d = 163 gives j(τ) = 640320³ and explains why e^(π√163) is almost an integer to 12 decimals — one of the most famous "coincidences" in mathematics, which isn't a coincidence at all.