Singular units and isogenies between CM elliptic curves

Abstract

In this note, we will apply the results of Gross–Zagier, Gross–Kohnen–Zagier and their generalizations to give a short proof that the differences of singular moduli are not units. As a consequence, we obtain a result on isogenies between reductions of CM elliptic curves.

Deformations of Theta Integrals and A Conjecture of Gross-Zagier

“…In addition, there are other applications of these expected results. For cases (2) and (3), by combining the analogue of Theorem 1.2 and the idea in [Li21b], we hope to obtain non-existence result of genus 2 curves with CM Jacobian and having everywhere good reduction in certain families, generalizing the main result in [HP17]. In the last case, we expect a variation of our construction to lead to a proof of the factorization conjecture of CM-values of twisted Borcherds product in [BY07].…”

On a Conjecture of Yui and Zagier

“…If d 1 d 2 is not a perfect square, this subgroup can be identified with Gal(H/E) with H the composite of the ring class fields H d j associated to the order of discriminant d j (see Prop. 3.2 in [13]). This observation and the above lemma give the following corollary.…”

Around the support problem for Hilbert class polynomials

“…This follows from the fact that differences of singular moduli are never units in the ring of algebraic integers, see [33,Corollary 1.3].…”