We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli j(τ ), j(τ ′ ) such that the numbers 1, j(τ ) m and j(τ ′ ) n are linearly dependent over Q for some positive integers m, n, must be of degree at most 2. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in C 2 defined over Q. On the other hand, we show that, with "obvious" exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to an hyperbola xy = A, where A ∈ Q.
We show that the field Q(x, y), generated by two singular moduli x and y, is generated by their sum x + y, unless x and y are conjugate over Q, in which case x + y generates a subfield of degree at most 2. We obtain a similar result for the product of two singular moduli. CDC, the IRN GandA (CNRS) and the ALGANT Program. Our calculations were performed using the PARI/GP package [9]. The sources are available from the second author.
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.
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