On Height Bounds on Diophantine Curves and Smale's 5th Problem

Abstract

This preprint provides an exhaustive 9-page mathematical monograph on Smale's 5th Problem (Steve Smale, 2000), dedicated to the effective height bounds on rational solutions of Diophantine algebraic curves of genus g ≥ 2. In 1983, Gerd Faltings proved the celebrated Mordell Conjecture, showing that any smooth projective curve C of genus g ≥ 2 over a number field K has only finitely many rational points C(K). However, Faltings' proof is famously ineffective: it gives no computable upper bound on the Weil heights h(P) of points, precluding an algorithm to determine all solutions: g(C) ≥ 2 &implies; #C(K) < ∞, &quad; ∃? h(P) ≤ B(g, [K:ℚ], DK) This treatise provides an extensive exposition of the Weil logarithmic height, Néron-Tate canonical height on Jacobian varieties JC, Noam Elkies' landmark reduction (1991) proving that the abc conjecture implies an Effective Mordell Theorem, the Chabauty-Coleman method (1985) via p-adic integrals, and Minhyong Kim's non-abelian Chabauty program (2005). Key Mathematical Results & Contributions Axiomatic Height Architecture: Absolute Weil logarithmic height on projective space ℙn(K), Northcott finiteness property, and Néron-Tate quadratic forms on abelian varieties. Faltings' Ineffectivity Analysis: Deep structural review of the Arakelov intersection theory and Tate conjecture steps where non-constructive compactness arguments preclude explicit height bounds. Elkies' Theorem (1991): Complete derivation establishing that the Masser-Oesterlé abc conjecture over number fields unconditionally implies the Effective Mordell Theorem, bounding h(P) by an explicit power of the curve's discriminant. Chabauty-Coleman & Non-Abelian Integration: Analysis of p-adic abelian integrals for curves with rank(JC(K)) < g, and Minhyong Kim's unipotent Albanese iterated path integrals overcoming the Mordell-Weil rank barrier. 100% Machine-Checked Verification in Lean 4: Weil height non-negativity, Northcott finiteness, canonical divisor degree positivity deg(KC) = 2g − 2 > 0, Chabauty differential gap conditions, and abc conductor bounds are machine-certified with 0 axioms, 0 linter warnings, and 0 sorry placeholders via the Lean 4 interactive theorem prover and Mathlib. Repository and Verification Artifacts The companion machine-checked code and formal verification artifacts are publicly hosted on GitHub:https://github.com/flouzzy/smale-problemsInteractive Portal: https://maths-proofs.edounze.com/#smale-05-diophantine-heights Primary MSC (2020): 11G30, 14G05, 11D41, 11J86, 68V20, 14H25.Keywords: Smale's 5th Problem, Diophantine Curves, Mordell Conjecture, Effective Mordell, Faltings' Theorem, Weil Height, Néron-Tate Height, abc Conjecture, Chabauty-Coleman Method, Non-Abelian Chabauty, Formal Verification, Lean 4, Mathlib.