Gross-zagier on Singular Moduli: the Analytic Proof

The famous results of Gross and Zagier compare the heights of Heegner points on modular curves with special values of the derivatives of related L-functions. When specialized to the level 1 case (i.e., the full modular curve H/Γ, where Γ = SL2(Z)), we recover an astounding formula for the differences of singular moduli (the Heegner points on the full modular curve) in terms of an explicit prime factorization. The goal of this talk is to sketch an analytic proof of this formula, following Gross and Zagier’s paper in Crelle’s Journal. First, the formula. Let τ lie in an imaginary quadratic extension K of Q. To each τ , we associate a discriminant d in the usual way: if aτ + bτ + c = 0 and (a, b, c) = 1, then d = b − 4ac. By the theory of complex multiplication, j(τ) is an algebraic integer in an abelian extension of K, of degree h over Q, where h is the class number of primitive binary quadratic forms of discriminant d or alternatively the class number of the imaginary quadratic field of discriminant d. Its Galois conjugates are the numbers j(τ ′), where τ ′ runs over the roots of primitive quadratic polynomials of discriminant d. Let d1 and d2 be two relatively prime negative fundamental discriminants (i.e., integers that are either 1 or the discriminant of a quadratic number field). Let D = d1d2 and let w1 and w2 be the number of roots of unity in the quadratic orders of discriminant d1 and d2, respectively. Define J(d1, d2) =  ∏