Linear Independence of Powers of Singular moduli of Degree Three

Abstract

We show that two distinct singular moduli j(τ ), j(τ ′ ), such that for some positive integers m, n the numbers 1, j(τ ) m and j(τ ′ ) n are linearly dependent over Q generate the same number field of degree at most 2. This completes a result of Riffaut, who proved the above theorem except for two explicit pair of exceptions consisting of numbers of degree 3. The purpose of this article is to treat these two remaining cases.

Equations in three singular moduli: the equal exponent case

“…Alongside this work, there have also been a number of results which classify explicitly all the CM-points lying on particular families of algebraic curves (contained in C 2 ), e.g. [1,6,19,16,7]. In this article, we prove the following two results, which consider instead two families of algebraic surfaces in C 3 .…”

“…We recall here those explicit André-Oort results in two dimensions which we will make use of in the sequel. These are due to Riffaut [19] and also joint with Luca [16].…”

Trinomials with given roots

“…As we have seen, they cannot be dismissed using the method of Proposition 6.4. Instead, we use a version of the argument from [15].…”

Equations with powers of singular moduli

“…There are only two such pairs, as indicated in the statement of the theorem. However, in a more recent work, Luca and Riffaut [9] eliminated these two remaining pairs. Consequently, Theorem 1.5 together with [9, Theorem 1.2] completely solve the above equation for distinct singular moduli, and we deduce the following Theorem.…”