Rational products of singular moduli

Abstract

International audienc

Fields generated by sums and products of singular moduli

“…The following simple lemma can be found in [3], where it is credited to Lenstra. Since the article [3] did not appear yet, we include a short proof for the reader's convenience. Proof Note first of all G does contain elements of order > 2 because it is not abelian.…”

“…In the sequel we denote ∆ y = ∆ and ∆ x = 4∆. Note that ∆ ≡ 1 mod 8, see [4,Subsection 3.2.2]. This implies that there are no subdominant singular moduli of discriminant 4∆, see [4,Proposition 2.6].…”

“…In particular, [Q(x) : Q] = h(∆ x ). For all details, see, for instance, [6, §7 and §11] Starting from the ground-breaking article of André [2] equations involving singular moduli were studied by many authors, see [1,4,8] for a historical account and further references. In particular, Kühne [7] proved that equation x + y = 1 has no solutions in singular moduli x and y, and Bilu et al [5] proved the same for the equation xy = 1.…”

“…In particular, Kühne [7] proved that equation x + y = 1 has no solutions in singular moduli x and y, and Bilu et al [5] proved the same for the equation xy = 1. These results where generalized in [1] and [4]. Here the statement about x + y is (a special case of) Theorem 1.2 from [1], and the statement about xy is Theorem 1.1 from [4].…”

“…These results where generalized in [1] and [4]. Here the statement about x + y is (a special case of) Theorem 1.2 from [1], and the statement about xy is Theorem 1.1 from [4].…”