[Paper Review] An arithmetic intersection formula for denominators of Igusa class polynomials

Kristin Lauter, Bianca Viray|Project Muse (Johns Hopkins University)|Oct 29, 2012

Coding theory and cryptography4 citations

TL;DR

This paper establishes an explicit arithmetic intersection formula for the ℓ-part of the intersection number $(\operatorname{CM}(K).\textup{G}_{1})_{\ell}$ on the Siegel moduli space of abelian surfaces, generalizing prior work by Bruinier-Yang and Yang. Using a novel approach based on the embedding problem for orders in quaternion algebras and generalized Gross-Zagier theory, the authors derive a computable formula for the ℓ-adic valuation of denominators in Igusa class polynomials for any primitive quartic CM field $K$, providing a tight bound even when $\mathcal{O}_K$ is not monogenic.

ABSTRACT

In this paper we prove an explicit formula for the arithmetic intersection number (CM(K).G1)_{\ell} on the Siegel moduli space of abelian surfaces, generalizing the work of Bruinier-Yang and Yang. 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. Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for the intersection number (CM(K).G1)_{\ell} under strong assumptions on the ramification of the primitive quartic CM field K. Yang later proved this conjecture assuming that O_K is freely generated by one element over the ring of integers of the real quadratic subfield. In this paper, we prove a formula for (CM(K).G1)_{\ell} for more general primitive quartic CM fields, and we use a different method of proof than Yang. We prove a tight bound on this intersection number which holds for all primitive quartic CM fields. As a consequence, we obtain a formula for a multiple of the denominators of the Igusa class polynomials for an arbitrary primitive quartic CM field. Our proof entails studying the Embedding Problem posed by Goren and Lauter and counting solutions using our previous article that generalized work of Gross-Zagier and Dorman to arbitrary discriminants.

Motivation & Objective

• To compute the ℓ-adic valuation of the denominators in Igusa class polynomials for genus 2 curves with complex multiplication.
• To generalize prior formulas by Bruinier-Yang and Yang, which required restrictive assumptions on the CM field $K$.
• To provide a tight upper bound on $(\operatorname{CM}(K).\textup{G}_{1})_{\ell}$ for all primitive quartic CM fields $K$, even when $\mathcal{O}_K$ is not freely generated.
• To establish a method for computing these denominators that improves the efficiency of algorithms used in cryptographic construction of genus 2 curves.

Proposed method

• The authors study the embedding problem of the ring of integers $\mathcal{O}_K$ into the endomorphism algebra of a product of two $\ell$-isogenous elliptic curves over $\overline{\mathbb{F}}_\ell$, with respect to the product polarization.
• They use a generalization of Gross-Zagier and Dorman's theory on Heegner points to count solutions to the embedding problem over $\mathbb{Z}_\ell$-orders in quaternion algebras.
• The key technical tool is a criterion for when an element $w$ is in the ring of multipliers of an ideal $I$ in a maximal order, based on trace and norm conditions modulo powers of $\ell$.
• They derive a formula for the number of such ideals $I_p$ of norm $p^{v_\ell(\delta)}$ satisfying certain integrality and trace conditions, using $p$-adic valuation and norm constraints.
• The proof relies on analyzing the structure of orders $R_p u + R_p \delta$ in the $\ell$-adic quaternion algebra, and showing that they are $p$-power multiples of ideals of bounded norm.
• The final count is expressed as a product over primes dividing $\delta$, with each local factor given by a sum over $j \equiv v_\ell(\delta) \pmod{2}$ of a class number-like function $\mathfrak{I}^{(p)}_{j - r_p}(\operatorname{Tr}(w), \operatorname{N}(w))$.

Experimental results

Research questions

• RQ1 What is the exact $\ell$-adic valuation of the denominators in Igusa class polynomials for a primitive quartic CM field $K$?
• RQ2 Can the arithmetic intersection number $(\operatorname{CM}(K).\textup{G}_{1})_{\ell}$ be computed explicitly for all primitive quartic CM fields, without assuming $\mathcal{O}_K$ is monogenic?
• RQ3 How does the number of solutions to the embedding problem in $\ell$-adic quaternion algebras relate to the denominators of Igusa class polynomials?
• RQ4 What is a tight, uniform upper bound for $(\operatorname{CM}(K).\textup{G}_{1})_{\ell}$ across all primitive quartic CM fields $K$?
• RQ5 Can the formula be simplified in special cases, such as when $\mathcal{O}_K$ is freely generated over the ring of integers of the real quadratic subfield?

Key findings

• The paper provides an explicit formula for $(\operatorname{CM}(K).\textup{G}_{1})_{\ell}$ under the assumption that $\mathcal{O}_K$ is freely generated over the ring of integers of the real quadratic subfield, valid for all $\ell$ outside a finite set.
• A tight upper bound on $(\operatorname{CM}(K).\textup{G}_{1})_{\ell}$ is established for all primitive quartic CM fields $K$, regardless of the structure of $\mathcal{O}_K$.
• The number of solutions to the embedding problem is shown to be equal to a product over primes $p \mid \delta$, $p \neq \ell$, of local counts involving $\mathfrak{I}^{(p)}_{j - r_p}(\operatorname{Tr}(w), \operatorname{N}(w))$ for $j \equiv v_\ell(\delta) \pmod{2}$.
• The norm of the ideal $R_p u + R_p \delta$ divides $\delta^2 p^{-r_p}$, and the valuation $v_p(\operatorname{N}(u)) \geq 2c_p + r_p$ is proven to be necessary for $w$ to be in the ring of multipliers.
• In the case $c_p = 0$, the number of such ideals is exactly 1, corresponding to a unique ideal of norm $\delta$ containing $u$, and the formula simplifies significantly.
• The final result yields a formula for a multiple of the denominators of Igusa class polynomials for any primitive quartic CM field $K$, enabling improved bounds in algorithmic computation.