Average CM-values of Higher Green's Function and Factorization

TL;DR

This paper proves an averaged algebraicity conjecture for higher Green's function values at CM points and provides their ideal factorizations, extending classical results on singular moduli to a broader context.

Contribution

It introduces an averaged algebraicity result for higher Green's functions at CM points and derives their ideal factorizations, generalizing Gross-Zagier's work.

Findings

Proved an averaged algebraicity conjecture for higher Green's functions at CM points.

Derived the factorization of ideals generated by these algebraic values.

Extended classical singular moduli results to higher Green's functions.

Abstract

In this paper, we prove an averaged version of an algebraicity conjecture in \cite{GKZ87} concerning the values of higher Green's function at CM points. Furthermore, we give the factorization of the ideal generated by such algebraic value in the spirit of the famous work of Gross and Zagier on singular moduli.