Papers › Some uniform effective results on André--Oort for sums of powers in ℂⁿ
Some uniform effective results on André--Oort for sums of powers in ℂⁿ
Guy Fowler
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We prove an Andr\'e--Oort-type result for a family of hypersurfaces in ℂⁿ that is both uniform and effective. Let K_* denote the single exceptional imaginary quadratic field which occurs in the Siegel--Tatuzawa lower bound for the class number. We prove that, for m, n ∈ℤ_(>0), there exists an effective constant c(m, n)>0 with the following property: if pairwise distinct singular moduli x₁, …, xₙ with respective discriminants Δ₁, …, Δₙ are such that a₁ x₁ᵐ + …+ aₙ xₙᵐ ∈ℚ for some a₁, …, aₙ ∈ℚ ∖{0} and # { Δᵢ : ℚ(√(Δᵢ)) = K_*} ≤1, then maxᵢ |Δᵢ |≤c(m, n). In addition, we prove an unconditional and completely explicit version of this result when (m, n) = (1, 3) and thereby determine all the triples (x₁, x₂, x₃) of singular moduli such that a₁ x₁ + a₂ x₂ + a₃ x₃ ∈ℚ for some a₁, a₂, a₃ ∈ℚ ∖{0}.
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