[Paper Review] The strong $ABC$ conjecture over function fields (after McQuillan and Yamanoi)

Carlo Gasbarri|ArXiv.org|Nov 19, 2008

Meromorphic and Entire Functions16 references3 citations

TL;DR

This paper presents a proof of the strong $ABC$ conjecture over function fields using two distinct approaches: McQuillan's global method based on logarithmic pluricanonical forms and Yamanoi's local method using Ahlfors theory and Nevanlinna theory. The key result establishes a sharp inequality relating the height of a section to the radical of the divisor, confirming the conjecture in the function field setting and providing a foundational framework for potential generalizations to higher-dimensional varieties.

ABSTRACT

The $abc$ conjecture predicts a highly non trivial upper bound for the height of an algebraic point in terms of its discriminant and its intersection with a fixed divisor of the projective line counted without multiplicity. We describe the two independent proofs of the strong $abc$ conjecture over function fields given by McQuillan and Yamanoi. The first proof relies on tools from differential and algebraic geometry; the second relies on analytic and topological methods. They correspond respectively to the Nevanlinna and the Ahlfors approach to the Nevanlinna Second Main Theorem.

Motivation & Objective

• To establish the strong $ABC$ conjecture over function fields, generalizing the classical $abc$ conjecture to geometric and arithmetic settings.
• To provide a rigorous proof of the conjecture using two complementary approaches: McQuillan's global analytic method and Yamanoi's local Nevanlinna-theoretic method.
• To demonstrate that the function field case, while simpler than the number field case, still requires deep analytic and geometric tools due to the non-isotrivial nature of key families.
• To lay the groundwork for extending the $ABC$ conjecture to higher-dimensional varieties and families of general type by analyzing the interplay between height, ramification, and divisor theory.
• To show that the conjecture implies effective finiteness results for integral points on hyperbolic curves and solutions to Diophantine equations, mirroring the implications in number fields.

Proposed method

• McQuillan's method uses logarithmic pluricanonical forms and the $p$-adic and complex geometry of semistable curves to derive a global inequality involving the height and the radical of the divisor.
• Yamanoi's approach applies Ahlfors theory and Nevanlinna theory to analytic maps from bordered Riemann surfaces to moduli spaces, deriving a local inequality involving the area of preimages and boundary lengths.
• The proof involves constructing a metric on the canonical bundle $K_n$ over the moduli space ${\cal M}_{0,n}$ and using the first Chern class $c_1(K_n)$ to define the height functional.
• A key technical step is the use of the Cauchy–Schwarz inequality to control the integral of the boundary length, linking it to the growth of the area functional.
• The argument proceeds by contradiction: assuming the conjecture fails leads to a contradiction with a non-integrability result from Nevanlinna theory, as formalized in Theorem \ref{nonintyama}.
• The two methods are shown to be complementary: McQuillan's global method and Yamanoi's local method can be combined to suggest a path toward proving the conjecture in higher dimensions.

Experimental results

Research questions

• RQ1 Can the strong $ABC$ conjecture over function fields be proven using analytic methods from Nevanlinna theory and Ahlfors theory?
• RQ2 How do the global and local approaches to the $ABC$ conjecture—McQuillan’s pluricanonical method and Yamanoi’s boundary-length control—compare in terms of applicability and strength?
• RQ3 To what extent can the function field $ABC$ conjecture be generalized to families of higher-dimensional varieties or surfaces of general type?
• RQ4 What is the role of the moduli space ${\cal M}_{0,n}$ and its tautological line bundle in formulating the conjecture in geometric terms?
• RQ5 Can the effective finiteness of integral points on hyperbolic curves be derived from the conjecture, and if so, how does the height bound depend on the discriminant and degree of the number field?

Key findings

• The strong $ABC$ conjecture over function fields is proven using two independent methods: McQuillan’s global analytic approach and Yamanoi’s local Nevanlinna-theoretic method.
• The conjecture is shown to imply effective finiteness of integral points on hyperbolic curves and effective bounds on solutions to Diophantine equations in two variables over number fields.
• A key inequality is established: $A(F,f^*c_1(K_n)) \leq n({\cal D}_n,f) + R_g + \epsilon A(F,f^*c_1(K_n)) + C\deg(g)(A(R,g^*\eta) + n({\cal D}_{n-1},\iota) + \chi^+(R) + \ell(\partial F,f^*c_1(K_n)))$, which implies the $ABC$ conjecture.
• The proof relies on a contradiction argument: assuming the conjecture fails leads to a non-integrable function, contradicting a known result in Nevanlinna theory.
• The geometric analogue of the $abc$ conjecture for polynomials—Mason’s theorem—is recovered as a special case of the main result.
• The authors suggest that a hybrid approach combining McQuillan’s global methods and Yamanoi’s local techniques may be the most promising path toward proving the conjecture in higher dimensions.