§ 2.1. The issue of bounding heights: the ABC and Szpiro Conjectures A brief exposition of various conjectures related to this issue of bounding heights of rational points may be found in [Fsk], §1.3. In this context, the case where the algebraic curve under consideration is the projective line minus three points corresponds most directly to the so-called ABC and − by thinking of this projective line as the “λ-line” that appears in discussions of the Legendre form of the Weierstrass equation for an elliptic curve − Szpiro Conjectures. In this case, the height of a rational point may be thought of as a suitable weighted sum of the valuations of the q-parameters of the elliptic curve determined by the rational point at the nonarchimedean primes of potentially multiplicative reduction [cf. the discussion at the end of [Fsk], §2.2; [GenEll], Proposition 3.4]. Here, it is also useful to recall [cf. [GenEll], Theorem 2.1] that, in the situation of the ABC or Szpiro Conjectures, one may assume, without loss of generality, that, for any given finite set Σ of [archimedean and nonarchimedean] valuations of the rational number field Q, the rational points under consideration lie, at each valuation of Σ, inside some compact subset [i.e., of the set of rational points of the projective line minus three points over some finite extension of the completion of Q at this valuation] satisfying certain properties.
