Comparing arithmetic intersection formulas for denominators of Igusa class polynomials

TL;DR

This paper proves the equivalence of two formulas for intersection numbers on arithmetic Hilbert modular surfaces for certain quartic CM fields, enabling broader cryptographic applications involving Igusa class polynomials.

Contribution

It demonstrates that the Bruinier-Yang and Lauter-Viray formulas agree for a larger class of quartic CM fields without relying on previous assumptions.

Findings

The two formulas are shown to be equivalent for a broader class of fields.

The proof does not depend on prior results by Yang or Lauter-Viray.

This equivalence extends the applicability of the Bruinier-Yang formula.

Abstract

Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for CM(K).G_1 under strong assumptions on the ramification in K. Yang later proved this conjecture under slightly stronger assumptions on the ramification. In recent work, Lauter and Viray proved a different formula for CM(K).G_1 for primitive quartic CM fields with a mild assumption, using a method of proof independent from that of Yang. In this paper we show that these two formulas agree, for a class of primitive quartic CM fields which is slightly larger than the intersection of the fields considered by Yang and Lauter and Viray. Furthermore, the proof that these formulas agree does not rely on the results of Yang or Lauter and Viray. As a consequence of our proof, we conclude that the Bruinier-Yang formula holds for a slightly largely class of quartic CM fields K than what was proved by Yang, since it agrees with the Lauter-Viray formula, which is proved in those cases. The factorization of these intersection numbers has applications to cryptography: precise formulas for them 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.