Denominators of Igusa class polynomials

Abstract

In [22], the authors proved an explicit formula for the arithmetic intersection number (CM(K).G1) on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field K. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. One of the main tools in the proof was a previous result of the authors [21] generalizing the singular moduli formula of Gross and Zagier. The current paper combines the arguments of [21, 22] and presents a direct proof of the main arithmetic intersection formula. We focus on providing a stream-lined account of the proof such that the algorithm for implementation is clear, and we give applications and examples of the formula in corner cases.