Submission date: 10 January 2024

Abstract:

Let C_1,C_2 ⊆ 𝔾_m^N(ℂ) be irreducible closed algebraic curves, with N ≥ 3. Suppose C_1 is not contained in an algebraic subgroup of 𝔾_m^N(ℂ) of dimension 1 and C_1 ∪ C_2 is not contained in an algebraic subgroup of 𝔾_m^N(ℂ) of dimension 2. It is a conjecture that at most finitely many points x ∈ C_1 have the property that there is a positive integer n such that x^n ∈ C_2 and [n]C_1 ⊈ C_2, where [n]C_1={x^n:x ∈ C_1}. We prove this in the case where at least one of the two curves is not defined over \overline{ℚ}.

Mathematics Subject Classification: 11U09, 11U07, 03C64

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Full text arXiv 2401.05024: pdf, ps.