No Singular Modulus Is a Unit

Abstract

A result of the second-named author states that there are only finitely many CM-elliptic curves over C whose j-invariant is an algebraic unit. His proof depends on Duke's Equidistribution Theorem and is hence noneffective. In this article, we give a completely effective proof of this result. To be precise, we show that every singular modulus that is an algebraic unit is associated with a CM-elliptic curve whose endomorphism ring has discriminant less than 10 15 . Through further refinements and computerassisted arguments, we eventually rule out all remaining cases, showing that no singular modulus is an algebraic unit. This allows us to exhibit classes of subvarieties in C n not containing any special points.

Singular units and isogenies between CM elliptic curves

“…This was an improvement over an earlier result for all (ineffectively) large discriminants in [Hab15]. By specializing Theorem 1.1 to m = 1 and d 1 = −3, we recover the main result of [BHK20]. By allowing both CM points to vary, we actually have the following more general result.…”

“…Remark 1.4. When one of the discriminants is fixed, the proof in [BHK20] can be adapted to prove the result above. However, this involves eliminating finitely many cases by computer calculation, and it is not clear if the same strategy works with both discriminants varying.…”

“…The results in [Hab15,BHK20] originated from a question of Masser, which was motivated by effective results of André-Oort type. As a generalization of Corollary 1.2 in [BHK20], we can deduce the following result from Theorem 1.1.…”

“…In [BHK20], it was shown that j(z) is not a unit for any CM point z ∈ Y (1). This was an improvement over an earlier result for all (ineffectively) large discriminants in [Hab15].…”

Trinomials with given roots

“…Indeed, assume the contrary: k 1 ≥ 3 and k 1 + k 2 ≥ 5. Writing ∆ = −p 2 1 m, Proposition 3.3 implies that min{p 2 1 , (m + (p 1 − 2) 2 )/4} is suitable for ∆. However, since k 1 + k 2 ≥ 5, we have m > 4p 2 1 , which implies that the minimum is, actually, p 2 1 .…”