Vojta’s Height Inequality for curves (Mochizuki’s formulation)
Establish that for any number field L, any geometrically connected smooth projective curve X/L with a reduced divisor D such that U=X−D is a hyperbolic curve, and any integer d≥1, the inequality h_{ω_X(D)}(P) ≤ (1+ε)(log-diff_X(P)+log-con_D(P)) holds for all points P in U(\overline{L}) of degree at most d and every ε>0, where log-diff_X(P) denotes the normalized arithmetic degree of the different of L(P)/L and log-con_D(P) denotes the normalized arithmetic degree of the reduced pullback of D along the morphism associated to P.
References
In conjectured a general inequality, which is now known as Vojta's Height Inequality for Curves (\Cref{con:vojta-inequality}), and showed that this inequality implies the $abc$-conjecture (\Cref{con:abc}). Let $L$ be a number field. Let $(X,D)/L$ be a pair consisting of a geometrically connected, smooth, projective curve over $L$ and $D\subset X$ a reduced divisor such that $U=X-D$ is a (punctured) hyperbolic curve over $L$. Let $d\geq 1$ be an integer. Then for every $\varepsilon>0$, the following inequality $$_{\omega_X(D)}\lesssim (1+\varepsilon)\left(log-diff_X+log-con_D\right)$$ holds on $U(\overline{L}){\leq d}=(X-D)(\overline{L}){\leq d}.