The Gross-Zagier formula: a brief introduction

act on the upper half-plane by linear fractional transformations. We may form the quotient Y0(N) = Γ0(N)\H, a Riemann surface with finitely many cusps. Compactifying gives a curve X0(N) which is in fact defined over Q; the map z → (j(z), j(Nz)) ∈ AC realizes Y0(N) as a (highly singular) plane curve with Q-coefficients. Over a general field k of characteristic zero, the k-points of the curve X0(N) (away from the cusps) parametrize diagrams (φ : E → E′) where E/k, E′/k are elliptic curves and φ : E → E′ is a k-rational isogeny with kerφ % Z/NZ over k. There is a canonical Q-rational involution wN : X0(N) → X0(N) which sends the diagram (φ : E → E′) to the diagram (φ̂ : E′ → E). Over C, elliptic curves are simply quotients C/Λ for lattices Λ = ω1Z + ω2Z ⊂ C, ω1/ω2 / ∈ R; the Weierstrass ℘-function