３．IUTT-IV P35 Corollary 2.2. (Construction of Suitable Initial Θ-Data) Suppose that X = P1Q is the projective line over Q, and that D ⊆ X is the divisor consisting of the three points “0”, “1”, and “∞”. We shall regard X as the “λ-line” ? i.e., we shall regard the standard coordinate on X = P1Q as the “λ” in the Legendre form “y2 = x(x?1)(x?λ)” of the Weierstrass equation defining an elliptic curve ? and hence as being equipped with a natural classifying morphism UX → (Mell)Q [cf. the discussion preceding Proposition 1.8]. Let KV ⊆ UX(Q) be a compactly bounded subset [i.e., regarded as a subset of X(Q) ? cf. [GenEll], Example 1.3, (ii)] whose support contains the nonarchimedean prime “2”. Then: 略
４．IUTT-IV P39 Corollary 2.3. (Diophantine Inequalities) Let X be a smooth, proper, geometrically connected curve over a number field; D ⊆ X a reduced divisor; UX def = X\D; d a positive integer; ε ∈ R>0 a positive real number. Write ωX for the canonical sheaf on X. Suppose that UX is a hyperbolic curve, i.e., that the degree of the line bundle ωX(D) is positive. Then, relative to the notation reviewed above, one has an inequality of “bounded discrepancy classes” htωX(D) ≦ (1 + ε)(log-diffX + log-condD) of functions on UX(Q)?d ? i.e., the function (1 + ε)(log-diffX + log-condD) ? htωX(D) is bounded below by a constant on UX(Q)?d [cf. [GenEll], Definition 1.2, (ii), as well as Remark 2.3.1, (ii), below].
