Solving the Quispel-Roberts-Thompson maps using Kajiwara-Noumi-Yamada’s representation of elliptic curves

The paper proves that any (generic) QRT map ϕ = r_y ∘ r_x preserving a nonsingular biquadratic curve Γ(K0) linearizes to a translation on the elliptic torus T, and gives the explicit solution xn = c1 F12(u0 + n(hx − hy)), yn+1 = c2 G12(u0 + n(hx − hy)) with F12,G12 built from Weierstrass’ sigma function and divisors tied to hx, hy, e1, e2; see the embedding (4a), the specialization (17a,b), and Theorem 3.1 . The candidate solution reproduces the same construction: (i) uniformize Γ via u = ∫Res_{y=y+} ω with ω = dx∧dy/P, establishing w1,w2 and T = C/(Zw1+Zw2) exactly as in the paper’s Section 3 on periods ; (ii) define F12,G12 with the same divisors and use uniqueness on a genus-one curve to identify x = c1F12, y = c2G12, matching (17b) ; (iii) show the QRT involutions act as reflections u ↦ hy − u (holding y fixed) and u ↦ hx − u (holding x fixed), so their composition is translation by hx − hy, recovering Theorem 3.1’s iterate formulas . The only discrepancy is a naming swap of rx, ry in the model’s narrative relative to the paper’s convention (the model calls rx the vertical switch), but the geometry and final translation step agree. Overall, both are correct and essentially the same proof.