Igusa class polynomials, embeddings of quartic CM fields, and arithmetic intersection theory

TL;DR

This paper investigates a conjectured intersection number on arithmetic Hilbert modular surfaces related to CM points, providing numerical evidence, exploring anomalies, and comparing with Igusa class polynomials and embedding problem solutions.

Contribution

It extends the analysis of Bruinier and Yang's conjecture to new fields, offering numerical support and insights into related arithmetic phenomena.

Findings

Numerical evidence supports the conjecture in new cases.

Identifies interesting anomalies in the data.

Draws comparisons with Igusa class polynomial denominators and embedding solutions.

Abstract

Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, CM(K).T_m, where CM(K) is the zero-cycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field K, and T_m is the Hirzebruch-Zagier divisors parameterizing products of elliptic curves with an m-isogeny between them. In this paper, we examine fields not covered by Yang's proof of the conjecture. We give numerical evidence to support the conjecture and point to some interesting anomalies. We compare the conjecture to both the denominators of Igusa class polynomials and the number of solutions to the embedding problem stated by Goren and Lauter.