Speaker: Dr. Michael Griffin, Kennesaw State University
Title: “Hecke-towers and modular product formulas”
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 SL2(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) Borcherd’s product formulas for the 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, each identity building from the previous, and rooted in the algebraic structure of spaces of modular objects.

The Discrete Math Seminar (DMS) is intended for Kennesaw State faculty working in the various areas of algebra, number theory, and discrete mathematics to get together to discuss their current work or related questions. Seminars often involve advanced mathematical knowledge. However, the seminars are open to anyone interested in attending.

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