A Habiro formalism for CM points

In this talk, we will present a Habiro-ring-like object encoding the information of CM points and singular moduli, which may contribute to a Habiro formalism for CM points.

 

We will start with the original Habiro ring defined by Habiro, explain how to view elements in the Habiro ring as power series expansions at roots of unity as well as functions on roots of unity, and sketch Habiro's proof of the injectivity of these associations via congruences of cyclotomic polynomials.

 

We will then recall the theory of singular moduli and Hilbert class polynomials, draw the comparison to roots of unity and cyclotomic polynomials, and build a Habiro-ring-like object via the comparison. We will also discuss our expectations on the 'correct' Habiro formalism for CM points.