(G,F)-points on ℚ-algebraic varieties

Let ℚ be the field of rational numbers, and let C be an algebraically closed field containing ℚ. Let G ∈ ℚ[x,y,z] be a polynomial, then the zero set of G is Z(G) = {P ∈ Cⁿ | G(P) = 0}. A set V ⊂ Cⁿ is called a ℚ-algebraic variety if V = Z(G) for some polynomial G in ℚ[x,y,z]. The set V(G) = {P ∈ ℚ³ | G(P) = 0} is called the set of ℚ-rational points of V. Let F : ℚ³ → ℚ³, (x,y,z) ⟼ (f(x),f(y),f(z)) be a vector function, where f ∈ ℚ[x]. It is easy to show that the function obtained by the composition of G and F, denoted as G ○ F, is still in ℚ[x,y,z]. Moreover, let V(G ○ F) be the set of ℚ-rational points of the ℚ-algebraic variety corresponding to G ○ F, i.e., V(G ○ F) = {P ∈ ℚ³ | G ○ F(P) = 0}. A rational point P is called a (G,F)-point on V(G) if P belongs to the intersection of V(G) and V(G ○ F), that is P ∈ V(G) ∩ V(G ○ F). Denote ⟨G,F⟩ as the set consisting of all (G,F)-points on V(G). Obviously, ⟨G,F⟩ is the set of ℚ-rational points of a ℚ-algebraic variety, that is, ⟨G,F⟩ = {P ∈ ℚ³ | G(P) = 0 and G ○ F(P) = 0}. In this paper, we consider the algebraic variety ⟨G,F⟩ for some specific functions G and F. For these specific functions G and F, we prove that ⟨G,F⟩ will be isomorphic to a certain elliptic curve. We also analyze some properties of these elliptic curves.