On singular moduli for arbitrary discriminants

TL;DR

This paper generalizes Gross and Zagier's formula for singular moduli differences to arbitrary discriminants, including non-maximal orders, with applications to genus 2 curves and explicit formulas for prime p>2.

Contribution

It extends the existing formulas for singular moduli differences to non-maximal orders and arbitrary discriminants, providing explicit and conjectural closed-form expressions.

Findings

Derived a computable formula for v_p(J(d1,d2)) for any discriminants and prime p>2.

Provided a simple closed form when d1 is squarefree and d2 is any quadratic order discriminant.

Proposed a conjectural formula for cases with relatively prime conductors.

Abstract

Let d1 and d2 be discriminants of distinct quadratic imaginary orders O_d1 and O_d2 and let J(d1,d2) denote the product of differences of CM j-invariants with discriminants d1 and d2. In 1985, Gross and Zagier gave an elegant formula for the factorization of the integer J(d1,d2) in the case that d1 and d2 are relatively prime and discriminants of maximal orders. To compute this formula, they first reduce the problem to counting the number of simultaneous embeddings of O_d1 and O_d2 into endomorphism rings of supersingular curves, and then solve this counting problem. Interestingly, this counting problem also appears when computing class polynomials for invariants of genus 2 curves. However, in this application, one must consider orders O_d1 and O_d2 that are non-maximal. Motivated by the application to genus 2 curves, we generalize the methods of Gross and Zagier and give a computable formula for v_p(J(d1,d2)) for any distinct pair of discriminants d1,d2 and any prime p>2. In the case that d1 is squarefree and d2 is the discriminant of any quadratic imaginary order, our formula can be stated in a simple closed form. We also give a conjectural closed formula when the conductors of d1 and d2 are relatively prime.