Singular Moduli and the Ideal Class Group

Abstract

Let $d_1$ and $d_2$ be discriminants of distinct quadratic imaginary orders $\cO_{d_1}$ and $\cO_{d_2}$ and let $J(d_1,d_2)$ denote the product of differences of CM $j$-invariants with discriminants $d_1$ and $d_2$. In 2012, Lauter and Viray generalized the methods of Gross, Zagier, and Dorman to give a computable formula for $v_p(J(d_1,d_2))$ for any distinct pair of discriminants $d_1,d_2$ and any prime $p>2$. Further, in the case that $d_1$ is square-free and $d_2$ is the discriminant of any quadratic imaginary order, they gave a simple closed form. To do this, they related the question to a particular counting problem. We recap the developments of Gross-Zagier, Dorman, and Lauter-Viray before making progress on the remaining cases by considering a related counting problem.