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Strong abc-conjecture (Mochizuki’s formulation)

Establish that for the pair (X,D)=(P^1, {0,1,∞}) over ℚ (the fundamental hyperbolic tripod), and any integer d≥1, the inequality h_{ω_{P^1}(D)}(P) ≤ (1+ε)(log-diff_{P^1}(P)+log-con_D(P)) holds for all points P∈(P^1−{0,1,∞})(\overline{ℚ}) of degree at most d and every ε>0.

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Background

The strong abc-conjecture is a specialized form of the abc-conjecture formulated on the hyperbolic tripod P1−{0,1,∞}, relating heights to arithmetic discriminants and conductors of points of bounded degree. It plays a crucial role in reduction arguments that connect local height inequalities on compactly bounded subsets to the general Vojta framework.

In the paper, this strong form is positioned within Mochizuki’s methodology: proving it on compactly bounded subsets and then invoking equivalences and refinements (including Mochizuki’s general position theorem) to deduce Vojta’s Height Inequality and the classical abc-conjecture.

References

In Conjecture 14.4.12, the following conjecture is called the strong $abc$-conjecture: Let $(X,D)=(\mathbb{P}1,{0,1,\infty})$ be the fundamental hyperbolic tripod over $\mathbb{Q}$. Let $d\geq 1$ be an integer. Then for every $\varepsilon>0$ the following inequality $${\omega{\mathbb{P}1}(D)}\lesssim (1+\varepsilon)\left(log-diff_{\mathbb{P}1}+log-con_D\right)$$ holds on $U_{tpd}(\overline{\mathbb{Q}}){\leq d}=(\mathbb{P}1-D)(\overline{\mathbb{Q}}){\leq d}=\left(\mathbb{P}1-{0,1,\infty}\right)(\overline{\mathbb{Q}}){\leq d}.

— Construction of Arithmetic Teichmuller Spaces IV: Proof of the abc-conjecture (2403.10430 - Joshi, 15 Mar 2024) in Section 2.6 (The strong $abc$-conjecture)