Title: Hecke-towers and modular products
Abstract:
There are three important product formulas for expressions of the type $\prod (j(\sigma) − j(z) ) . Here j is the modular j-invariant which distinguishes isomorphism classes of elliptic curves and generates the field of modular functions for SL_2(\mathbb{Z}), and z and $\sigma$ are each either complex variables or run over a complete set of representatives of classes of imaginary quadratic numbers of fixed discriminant. These three identities are:
1) The denominator formula for the Monster Lie-algebra,
2) Borcherds' product formulas for Hilbert class functions,
3) The Gross–Zagier formula for norms of singular moduli.
Borcherds notes that, despite the similarity of the “left hand sides” of these identities, their proofs are wildly different, and “there does not seem to be any obvious way to deduce any of these 3 formulas from the others.” Motivated by this statement, we will show how these three identities can be derived from a cohesive theory, incorporating both algebraic and analytic generating functions.